Journal articles on the topic 'Generalized associated Legendre function of first kind'

We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences as the partial sums of the Hurwitz zeta function. These transformation coefficients satisfy many properties which are analogous to known identities and expansions of the Stirling numbers of the first kind and to the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove in the article include new series expansions of the Dirichlet beta function, the Legendre chi function, BBP-type series identities for special constants, alternating and exotic Euler sum variants, alternating zeta functions with powers of quadratic denominators, and particular series defining special cases of the Riemann zeta function constants at the positive integers s ≥ 3.

ABSTRACT Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind, which is an elementary transform of the hypergeometric function and other related functions belong to subclasses of starlike and convex functions defined in the unit disk associated with the exponential mapping. Several differential subordination implications are derived for analytic functions involving Bessel function and the operator introduced by Baricz et al. [Differential subordinations involving generalized Bessel functions, Bull. Malays. Math. Sci. Soc. 38(3) (2015), 1255–1280]. These results are obtained by constructing suitable class of admissible functions. Examples involving trigonometric and hyperbolic functions are provided to illustrate the obtained results.

In the present paper, we obtain some characterizations for a certain generalized Bessel function of the first kind to be in the subclasses SpT(?,?), UCT(?,?), PT(?) and CPT(?) of normalized analytic functions in the open unit disk U. Furthermore, we consider an integral operator related to the generalized Bessel Function which we have characterized here.

Abstract Sufficient conditions on associated parameters p, b and c are obtained so that the generalized and “normalized” Bessel function up(z) = up,b,c(z) satisfies the inequalities ∣(1 + (zu″p(z)/u′p(z)))2 − 1∣ < 1 or ∣((zup(z))′/up(z))2 − 1∣ < 1. We also determine the condition on these parameters so that . Relations between the parameters μ and p are obtained such that the normalized Lommel function of first kind hμ,p(z) satisfies the subordination . Moreover, the properties of Alexander transform of the function hμ,p(z) are discussed.

<abstract><p>A useful family of integral operators and special functions plays a crucial role on the study of mathematical and applied sciences. The purpose of the present paper is to give sufficient conditions for the families of integral operators, which involve the normalized forms of the generalized Lommel functions of the first kind to be univalent in the open unit disk. Furthermore, we determine the order of the convexity of the families of integral operators. In order to prove main results, we use differential inequalities for the Lommel functions of the first kind together with some known properties in connection with the integral operators which we have considered in this paper. We also indicate the connections of the results presented here with those in several earlier works on the subject of our investigation. Moreover, some graphical illustrations are provided in support of the results proved in this paper.</p></abstract>

The main object of this paper is to find necessary and sufficient conditions for generalized Bessel functions of first kind zup(z) to be in the classes SPp(α,β) and UCSP(α,β) of uniformly spiral-like functions and also give necessary and sufficient conditions for z(2-up(z)) to be in the above classes. Furthermore, we give necessary and sufficient conditions for I(κ,c)f to be in UCSPT(α,β) provided that the function f is in the class Rτ(A,B). Finally, we give conditions for the integral operator G(κ,c,z)=∫0z(2-up(t))dt to be in the class UCSPT(α,β). Several corollaries and consequences of the main results are also considered.

The 2-D sextic oscillator is studied as a family of axial symmetric parametric integrable Hamiltonian systems, presenting a bifurcation analysis of the different flows. It includes the "elliptic core" model in 1-D nonlinear oscillators, recently proposed in the literature. We make use of the energy-momentum mapping, which will give us the fundamental fibration of the four-dimensional phase space. Special attention is given to the singular values of the energy-momentum mapping connected with rectilinear and circular orbits. They are related to the saddle-center and pitchfork scenarios with the associated homoclinic and heteroclinic trajectories. We also study how the geometry of the phase space evolves during the transition from the one-dimensional to the two-dimensional model. Within an elliptic function approach, the solutions are given using Legendre elliptic integrals of the first and third kind and the corresponding Jacobi elliptic functions.

Abstract In the present article we introduce two new combinatorial interpretations of the r-Whitney numbers of the second kind obtained from the combinatorics of the differential operators associated to the grammar G := {y → yxm, x → x}. By specializing m = 1 we obtain also a new combinatorial interpretation of the r-Stirling numbers of the second kind. Again, by specializing to the case r = 0 we introduce a new generalization of the Stirling number of the second kind and through them a binomial type family of polynomials that generalizes Touchard’s polynomials. Moreover, we recover several known identities involving the r-Dowling polynomials and the r-Whitney numbers using the combinatorial differential calculus. We construct a family of posets that generalize the classical Dowling lattices. The r-Withney numbers of the first kind are obtained as the sum of the Möbius function over elements of a given rank. Finally, we prove that the r-Dowling polynomials are a Sheffer family relative to the generalized Touchard binomial family, study their umbral inverses, and introduce [m]-Stirling numbers of the first kind. From the relation between umbral calculus and the Riordan matrices we give several new combinatorial identities

We study the properties of the (noncommutative) bm-independence of algebras, indexed by partially ordered sets. The index sets are given by positive cones, in particular the symmetric cones, which include the positive-definite Hermitian matrices with complex or quaternionic entries. We formulate and prove the general versions of the bm-Central Limit Theorems for bm-independent random variables, indexed by lattices in such positive cones. The limit measures we obtain are symmetric and compactly supported on the real line. Their (even) moment sequences (gn)n≥0 satisfy the generalized recurrence for the Catalan numbers: [Formula: see text], where the coefficients γ(r) are computed by the Euler's beta-function of the first kind, related to the given positive cone. Example of a nonsymmetric cone, the Vinberg's cone, is also studied in this context.

In this article, the Lagrange expansion of the second kind is used to generate a novel zero-truncated Katz distribution; we refer to it as the Lagrangian zero-truncated Katz distribution (LZTKD). Notably, the zero-truncated Katz distribution is a special case of this distribution. Along with the closed form expression of all its statistical characteristics, the LZTKD is proven to provide an adequate model for both underdispersed and overdispersed zero-truncated count datasets. Specifically, we show that the associated hazard rate function has increasing, decreasing, bathtub, or upside-down bathtub shapes. Moreover, we demonstrate that the LZTKD belongs to the Lagrangian distribution of the first kind. Then, applications of the LZTKD in statistical scenarios are explored. The unknown parameters are estimated using the well-reputed method of the maximum likelihood. In addition, the generalized likelihood ratio test procedure is applied to test the significance of the additional parameter. In order to evaluate the performance of the maximum likelihood estimates, simulation studies are also conducted. The use of real-life datasets further highlights the relevance and applicability of the proposed model.

In this study, a general, analytical solution of a steady creeping or Stokes flow impinging on a stationary spherical cap-shaped bubble on a solid flat surface is provided. The phenomena usually take place in bubble∕pore formation in materials and manufacturing processing and MEMS, boiling heat transfer, and nucleation and growth of gas bubbles in tissues of animals and human, etc. In view of high capillary pressure and small liquid pressure, the shape of the bubble in a microscale can be considered as a spherical cap on a surface. In this model, shear stresses associated with the no-slip condition, interfacial mass transport such as condensation and evaporation are absent on the bubble surface. An analytical solution of the Stokes equations for zero Reynolds number flow in a toroidal coordinate system is found by decomposing the flow into a stagnation flow and a flow disturbed by the bubble and applying the separation-of-variables method. The stream function can be expressed in terms of a difference in Legendre functions of the first kind. The effects of impinging velocity and contact angle of the bubble on the flow pattern and pressure distribution are provided.

Abstract In this paper, we consider a polynomial generalization, denoted by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k), of the restricted Stirling numbers of the first and second kind, which reduces to these numbers when a = 1 and b = 0 or when a = 0 and b = 1, respectively. If a = b = 1, then $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) gives the cardinality of the set of Lah distributions on n distinct objects in which no block has cardinality exceeding m with k blocks altogether. We derive several combinatorial properties satisfied by $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) and some additional properties in the case when a = b = 1. Our results not only generalize previous formulas found for the restricted Stirling numbers of both kinds but also yield apparently new formulas for these numbers in several cases. Finally, an exponential generating function formula is derived for $\begin{array}{} u_m^{a,b} \end{array}$ (n, k) as well as for the associated Cauchy numbers.

Polydisperse suspensions with particles of a finite number N of size classes have been widely studied in laboratory experiments. However, in most real-world applications the particle sizes are distributed continuously. In this paper, a well-studied one-dimensional kinematic model for batch sedimentation of polydisperse suspensions of small equal-density spheres is extended to suspensions with a continuous particle size distribution. For this purpose, the phase density function Φ = Φ(t, x, ξ), where ξ ∈ [0, 1] is the normalized squared size of the particles, is introduced, whose integral with respect to ξ on an interval [ξ1, ξ2] is equivalent to the volume fraction at (t, x) occupied by particles of that size range. Combining the Masliyah–Lockett–Bassoon (MLB) model for the solid-fluid relative velocity for each solids species with the concept of phase density function yields a scalar, first-order equation for Φ, namely the equation of the generalized kinetic theory. Three numerical schemes for the solution of this equation are introduced, and a numerical example and an L1 error study show that one of these schemes introduces less numerical diffusion and less spurious oscillations near discontinuities than the others. Several numerical examples illustrate the simulated behavior of this kind of suspensions. Numerical results also illustrate the solution of an eigenvalue problem associated with the equation of the generalized kinetic theory.

AbstractIn this article we derive analytic and Fourier aspects of a Kronecker limit formula for second-order Eisenstein series. Let Γ be any Fuchsian group of the first kind which acts on the hyperbolic upper half-space H such that the quotient Γ\H has finite volume yet is non-compact. Associated to each cusp of Γ\H, there is a classically studied first-order non-holomorphic Eisenstein series E(s, z) which is defined by a generalized Dirichlet series that converges for Re(s) > 1. The Eisenstein series E(s, z) admits a meromorphic continuation with a simple pole at s = 1. Classically, Kronecker’s limit formula is the study of the constant term 1 (z) in the Laurent expansion of E(s, z) at s = 1. A number of authors recently have studied what is known as the second-order Eisenstein series E*(s, z), which is formed by twisting the Dirichlet series that defines the series E(s, z) by periods of a given cusp form f. In the work we present here, we study an analogue of Kronecker’s limit formula in the setting of the second-order Eisenstein series E* (s, z), meaning we determine the constant term 2(z) in the Laurent expansion of E*(s, z) at its first pole, which is also at s = 1. To begin our investigation, we prove a bound for the Fourier coefficients associated to the first-order Kronecker limit function 1. We then define two families of convolution Dirichlet series, denoted by and with m ∈ ℕ, which are formed by using the Fourier coefficients of 1 and the weight two cusp form f. We prove that for all m, and admit a meromorphic continuation and are holomorphic at s = 1. Turning our attention to the second-order Kronecker limit function 2, we first express 2 as a solution to various differential equations. Then we obtain its complete Fourier expansion in terms of the cusp form f, the Fourier coefficients of the first-order Kronecker limit function 1, and special values (1) and (1) of the convolution Dirichlet series. Finally, we prove a bound for the special values (1) and (1) which then implies a bound for the Fourier coefficients of 2. Our analysis leads to certain natural questions concerning the holomorphic projection operator, and we conclude this paper by examining certain numerical examples and posing questions for future study.

This research paper aims to investigate the commutativity properties of prime rings in relation to generalized derivations and a left multiplier that fulfil specific algebraic conditions involving involution. Additionally, we present various examples studies to illustrate that the constraints imposed in our theorems are indeed necessary and cannot be omitted without compromising the validity of our results.Introduction: Consider a ring S that satisfies the associative property, and let Z(S) represent its center. An involution denoted by *, which is an additive function mapping S to itself. This involution has specific properties: for any α and β in S, applying the involution twice returns the original element ((α^* )^*=α), it distributes over addition ((α+β)^*=α^*+β^*), and it reverses the order of multiplication ((αβ)^*=β^* α^*). We categorize elements as hermitian when they remain unchanged under the involution (α^*=α), and as skew-hermitian when they change sign (α^*=-α). We use H(S) to represent the collection of all hermitian elements in S, and S(S) for all skew-hermitian elements. The involution is classified as first kind if H(S) is a subset of the center of S, denoted as Z(S). If this is not the case, it’s considered second kind, and in this scenario, the intersection of H(S) and Z(S) contains more than just the zero element. We also define several types of mappings on ring S. A left multiplier, △, is an additive map where △(υω)=△(υ)ω, ∀υ,ω∈S. A derivation, ψ, is an additive mapping that satisfies ψ(υω)=ψ(υ)ω+υψ(ω), ∀υ,ω∈S. Extending this concept, we define a generalized derivation, Γ, which is linked to a derivation ψ. This function satisfies Γ(υω)=Γ(υ)ω+υψ(ω), ∀υ,ω∈S. It’s worth noting that any derivation can be considered a generalized derivation associated with itself.Objectives: In this study, we intend to examine the commutativity of a prime ring S by utilizing generalized derivations Γ_1, Γ_2, and a left multiplier △, while adhering to specific algebraic identities that involve involution. Specifically, we will delve into the commutativity of rings S that fulfill the following algebraic conditions: • [Γ_1 (υ),Γ_2 (υ^*)]+△([υ,υ^*])∈Z(S), for all υ∈S; • Γ_1 (υ)∘Γ_2 (υ^*)+△(υ∘υ^*)∈Z(S), for all υ∈S; • [Γ_1 (υ),Γ_2 (υ^*)]+△(υ∘υ^*)∈Z(S), for all υ∈S; • Γ_1 (υ)∘Γ_2 (υ^*)+△([υ,υ^*])∈Z(S), for all υ∈S; • Γ(υυ^*)±Γ(υ)Γ(υ^*)+△([υ,υ^*])∈Z(S), for all υ∈S; • Γ(υυ^*)±Γ(υ^*)Γ(υ)+△(υ∘υ^*)∈Z(S), for all υ∈S. Lastly, we offer examples to illustrate that the constraints applied to our hypotheses are necessary and not redundant.Results: Building on the work of Nejjar ([8], Theorems 3.5, 3.8), who demonstrated that a 2-torsion-free prime ring with involution and a derivation ψ satisfying certain conditions must be commutative, we explore broader generalizations of these conditions. Specifically, Nejjar showed that if the derivation ψ meets either of the following criteria: [ψ(υ),ψ(υ)]±υ∘υ∈Z(S), ∀υ∈S or ψ(υ)∘ψ(υ)±υ∘υ∈Z(S), ∀υ∈S, then S is necessarily commutative. Our work extends these findings by introducing new identities for pairs of generalized derivations that are connected to a left multiplier △. Finally, we offer examples to illustrate that the constraints applied to our hypotheses are necessary and not redundant.Conclusions: In this research, we investigate the commutativity of prime rings S admitting an involution and generalized derivations satisfying some algebraic identities. We can conclude our paper with an open question.Open question: are these results correct if we replace the generalized derivation by the generalized (α,β)-derivation, where α and β are automorphisms of ring S?

AbstractWith a certain graphic interpretation in mind, we say that a function whose value at every point in its domain is a nonempty set of real numbers is an Abacus. It is shown that to every collection C of abaci there corresponds a logic, called an abacus logic, i.e.. a certain set of propositions partially ordered by generalized implication. It is also shown that to every collection C of abaci there corresponds a theory Jc in a classical propositional calculus such that the abacus logic determined by C is isomorphic to the poset of Jc. Two examples are given. In both examples abacus logic is a lattice in which there happens to be an operation of orthocomplementation. In the first example abacus logic turns out to be the Lindenbaum algebra of Jc. In the second example abacus logic is a lattice isomorphic to the ortholattice of subspaces of a Hilbert space. Thus quantum logic can be regarded as an abacus logic. Without suggesting “hidden variables” it is finally shown that the Lindenbaum algebra of the theory in the second example is a subalgebra of the abacus logic B of the kind studied in example 1. It turns out that the “classical observables” associated with B and the “quantum observables” associated with quantum logic are not unrelated. The value of a classical observable contains, in coded form, information about the “uncertainty” of a quantum observable. This information is retrieved by decoding the value of the corresponding classical observable.

The theory of Bessel functions is a rich subject due to its essential role in providing solutions for differential equations associated with many applications. As fractional calculus has become an efficient and successful tool for analyzing various mathematical and physical problems, the so-called fractional Bessel functions were introduced and studied from different viewpoints. This paper is primarily devoted to the study of developing two aspects. The starting point is to present a fractional Laplace transform via conformable fractional-order Bessel functions (CFBFs). We establish several important formulas of the fractional Laplace Integral operator acting on the CFBFs of the first kind. With this in hand, we discuss the solutions of a generalized class of fractional kinetic equations associated with the CFBFs in view of our proposed fractional Laplace transform. Next, we derive an orthogonality relation of the CFBFs, which enables us to study an expansion of any analytic functions by means of CFBFs and to propose truncated CFBFs. A new approximate formula of conformable fractional derivative based on CFBFs is provided. Furthermore, we describe a useful scheme involving the collocation method to solve some conformable fractional linear (nonlinear) multiorder differential equations. Accordingly, several practical test problems are treated to illustrate the validity and utility of the proposed techniques and examine their approximate and exact solutions. The obtained solutions of some fractional differential equations improve the analog ones provided by various authors using different techniques. The provided algorithm may be beneficial to enrich the Bessel function theory via fractional calculus.

For pure Yang-Mills theory on Minkowski space-time, formulated in functional spaces where the covariant divergence is an elliptic operator without zero modes, and for a trivial principal bundle over the fixed time Euclidean space with a compact, semisimple, connected and simply connected structure Lie group, a Green function for the covariant divergence has been found. It allows one to solve the first class constraints associated with Gauss’ laws and to identify a connection-dependent coordinatization of the trivial principal bundle. In a neighborhood of the global identity section, by using canonical coordinates of the first kind on the fibers, one has a symplectic implementation of the Lie algebra of the small gauge transformations generated by Gauss’ laws and one can make a generalized Hodge decomposition of the gauge potential one-forms based on the BRST operator. This decomposition singles out a pure gauge background connection (the BRST ghost as Maurer-Cartan one-form on the group of gauge transformations) and a transverse gauge-covariant magnetic gauge potential. After an analogous decomposition of the electric field strength into the transverse and the longitudinal part, Dirac’s observables associated with the transverse electric and magnetic components are identified as their restriction to the global identity section of the trivial principal bundle. The longitudinal part of the electric field can be re-expressed in terms of these electric and magnetic transverse parts and of the constraints without Gribov ambiguity. The physical Lagrangian, Hamiltonian, non-Abelian and topological charges have been obtained in terms of transverse Dirac’s observables, also in the presence of fermion fields, after a symplectic decoupling of the gauge degrees of freedom; one has an explicit realization of the abstract “Riemannian metric” on the orbit space. Both the Lagrangian and the Hamiltonian are nonlocal and nonpolynomial; like in the Coulomb gauge they are not Lorentz-invariant, but the invariance can be enforced on them if one introduces Wigner covariance of the observables by analyzing the various kinds of Poincare orbits of the system and by reformulating the theory on suitable spacelike hypersurfaces, following Dirac. By extending to classical relativistic field theory the problems associated with the Lorentz noncovariance of the canonical (presymplectic) center of mass for extended relativistic systems, in the sector of the field theory with P2>0 and W2≠0 one identifies a classical invariant intrinsic unit of length, determined by the Poincare Casimirs, whose quantum counterpart is the ultraviolet cutoff looked for by Dirac and Yukawa: it is the Compton wavelength of the field configuration (in an irreducible Poincare representation) multiplied by the value of its spin.

Abstract The two subunits of core binding factor (RUNX1 and CBFβ) play critical roles in hematopoiesis and are frequent targets of chromosomal translocations found in leukemia. CBFβ functions to increase the DNA-binding of the RUNX1 subunit 20–40 fold and to protect the Runx1 subunit against ubiqitination and subsequent proteasome degradation, making this protein-protein interaction critical for CBF function. Two of the most common translocations involving the subunits of CBF are the inv(16) and the t(8;21) which produce the chimeric proteins CBFβ-SMMHC and AML1-ETO, respectively. We are characterizing both the structures and biochemical properties of these translocation products to gain insights into their function and provide novel avenues for therapeutic development. The inv(16) results in the fusion of the N-terminal 165 amino acids of CBFβ to the coiled-coil region of smooth muscle myosin protein. The CBFβ-SMMHC fusion protein causes dysregulation of CBF function by means of anomalously tight binding to RUNX1. As there is still one copy of the normal CBFβ present in those leukemic cells harboring the inv(16), inhibition of CBFβ-SMMHC may enable the wildtype CBFβ to restore normal CBF function. Since binding to RUNX1 is required for the dysfunction associated with this protein, this binding represents an excellent target for inhibition as a potential therapeutic strategy. A small molecule inhibitor of this kind has the potential to be a useful and highly specific therapeutic agent. As an initial step in this direction, we have developed small molecules which bind to CBFβ and inhibit RUNX1 binding. This represents the first step toward our overall goal of developing compounds which can specifically inhibit CBFβ-SMMHC while minimally perturbing the activity of CBFβ itself. We previously solved the 3D structure of CBFβ using solution NMR methods and mapped the binding interface with RUNX1 by both chemical shift perturbation as well as Ala mutagenesis of the binding interface. Using this data, we employed virtual screening of CBFβ and experimental screening with NMR spectroscopy and FRET to identify initial lead compounds that could bind to CBFβ and inhibit its interaction with the RUNX1 Runt domain. Using a traditional medicinal chemistry approach, we have elaborated these compounds to identify structure-activity relationships (SAR). Based on the SAR results and NMR-based docking of the compounds to CBFβ, we have optimized the initial leads to generate compounds with low micromolar affinity which effectively inhibit the binding of RUNX1 to CBFβ. These compounds represent the first small molecule inhibitors of this protein-protein interaction. These compounds demonstrate activity in a protein localization assay in mammalian cells with no generalized toxicity, indicating they are good leads for further development. NMR-based docking of these molecules to CBFβ shows these compounds bind at a site displaced from the binding interface for RUNX1 on CBFβ, i.e. these compounds function as allosteric inhibitors of this protein-protein interaction, an approach which may be generalizable to other protein-protein interactions. We are pursuing several approaches to modify these compounds to achieve selective inhibition of CBFβ-SMMHC with minimal perturbation of native CBFβ activity.

В статье даётся краткий очерк антиномической природы биоэтического дискурса и возможностей его геометрической визуализации. Рассматриваются два варианта визуализации. Первый связан с представлением той или иной ситуации как системы полярностей, которая в свою очередь моделируется в рамках векторной модели. В простейшем случае тезис и антитезис рассматриваются как два перпендикулярных вектора, а синтез – как их векторная сумма. В этом случае можно ввести и более количественную оценку «меры многомерности» полярной системы – как величины проекции её векторного представления на суммарный вектор. С использованием этих конструкций разбирается один пример из биоэтики, связанный со столкновением принципов милосердия и правдивости (проблема «лжи во спасение»). Деяние (действие или бездействие) интерпретируется как своеобразный оператор на событиях, который переводит одни события в другие. Предполагается, что субъект в своих деяниях рассматривает различные возможности и выбирает те из них, которые максимизируют ту или иную ценностную меру субъекта, в данном случае – меру векторной проекции полярного вектора ситуации на суммарный вектор – вектор синтеза базисных полярностей. Второй вариант визуализации связан с понятием антиномий в биоэтике – таких противоречий, которые не являются формально-логическими ошибками. В отличие от последних, в антиномиях как тезис, так и антитезис имеют свой момент оправдания в рамках тех или иных условий. Используется также понятие «антинома» – логического субъекта антиномии, который предицируется тезисом и антитезисом антиномии. Редукции антиномии соответствуют двум крайним аспектам антинома, которые называются его «редуктами» – по аналогии с редукцией волновой функции в квантовой механике. Приводятся различные примеры антиномов: биоэты, глоболоки, холомеры. В биоэтах один редукт выражает в большей мере биологические (биоредукт), второй – этические (эторедукт) определения антинома. В глоболоках выделяются глобальный (глоборедукт) и локальный (локоредукт) виды редуктов: первый выражает более глобальные (универсальные) этические определения, второй – более локальные, связанные с ценностями и нормами того или иного сообщества. Наконец, холомеры – вид антиномов, где антиномически соединяются определения целого (холоредукт) и части (мероредукт). Даётся их интерпретация как многомерных ментальных объектов в некотором обобщённом пространстве, так что крайние их аспекты (редукции антиномии) можно представить как проекции более многомерного состояния. В заключении делается предположение о связи биоэтических проблем с идеей ментальной многомерности, что составляет основу возможной визуализации как интерпретации ментальной многомерности на векторном её представлении. The article provides a brief outline of the antinomic nature of bioethical discourse and the possibilities of its geometric visualization. Two visualization options are considered. The first is associated with the representation of a particular situation as a system of polarities, which in turn is modeled in the framework of a vector model. In the simplest case, the thesis and the antithesis are considered as two orthogonal vectors P1 and P2, and the synthesis is considered as their vector sum S = P1+P2. In this case, we can also introduce a more quantitative estimate of the “measure of multidimensionality” M(P) of the polar system – as the magnitude of the projection of its vector representation P on the sum vector S, i.e. M(P) = (P,es), where es = S/|S| is the unit vector of the vector S, and (P,es) is the scalar product of the vectors P and es. Using these constructs, the author analyzes one example from bioethics related to the clash of the principles of mercy and truthfulness (the problem of “lying for salvation”). An act (action or omission) is interpreted as a kind of an operator on events that transforms some events into others. It is assumed that the subject considers various possibilities in their actions and chooses those that maximize a particular value measure of the subject, in our case, the measure M(P) of the vector projection of the polar vector P of the situation on the sum vector S – the vector of synthesis of basic polarities. The second version of visualization is related to the concept of antinomies – such contradictions that are not formal logical errors – in bioethics. In contrast to errors, in antinomies, both the thesis and the antithesis have their moment of justification within the framework of certain conditions. The concept “antinome” is also used; it is the logical subject of antinomy, which is predicated by the thesis and the antithesis of antinomy. Antinomy reductions correspond to two extreme aspects of the antinome, which are called its “reducts” – by analogy with the reduction of the wave function in quantum mechanics. Various examples of antinomes are given: bioets, globolocs, and holomers. In bioets, one reduct expresses the biological (bioreduct) definition of the antinome, another the ethical (ethoreduct) one. In globolocs, global (globoreduct) and local (locoreduct) types of reducts are distinguished: the former expresses more global (universal) ethical definitions, the latter more local ones, related to the values and norms of a particular local community. Finally, holomers are a kind of antinomes in which the definitions of the whole (holoreduct) and the part (meroreduct) are antinomically connected. They are interpreted as multidimensional mental objects in some generalized space, so that their extreme aspects (antinomy reductions) can be represented as generalized projections of a more multidimensional state within certain constricted conditions (reduction intervals). In this case, it is possible to geometrically visualize such states as, for example, three-dimensional objects in space, through which antinomes can be modeled, and their reducts as two-dimensional projections of a three-dimensional body on certain projection planes (intervals of reducts). In this case, one of the central tasks of bioethics is to determine the boundaries of the demarcation of some intervals from others. For example, in solving the problem of abortion and the status of the human embryo, such a demarcation is expressed in the search for a time point that would separate the phase of a more biological definition (bioreduct) of the embryo from its more ethical state (ethoreduct). In conclusion, the author suggests that bioethical problems are connected with the idea of mental multidimensionality, which forms the basis of a possible visualization as an interpretation of mental multidimensionality in its vector representation.

The generalized Laguerre polynomials form a complete set (orthogonal and normalized) in the space ** with respect to a certain weighting function because they are the Eigen functions of a second-order differential operator. Here we shall show how to expand some well-known classical polynomials such as the Legendre and the Hermite polynomials in series of generalized Laguerre polynomials. Since the generalized Laguerre orthogonal polynomials are set of polynomials that are mutually orthogonal to each other with respect to a measure of weighting function that is just the integrand of the gamma function, thus it grants us the guarantee of the ability to expand the first kind of Bessel functions and the gamma function in terms of the generalized Laguerre polynomials. The series expansion in terms of the generalized Laguerre polynomials can be achieved by following various approaches. For instance, the series expansion of the first kind Bessel functions is gained by the generalized hypergeometric function approach, whereas the series expansion of gamma function was obtained directly by the usual way that is by calling the orthogonality and orthonormality properties of the generalized Laguerre polynomials. Another powerful technique to gain the series expansion of the classical polynomials is the generating function approach as it has been followed here to obtain a series expansion of the Legendre and the Hermite polynomials in terms of the generalized Laguerre polynomials. The series expansion in series of the generalized Laguerre polynomials has a variety of applications in mathematics, physics, and engineering

We derive all eighteen Gauss hypergeometric representations for the Ferrers function of the second kind, each with a different argument. They are obtained from the eighteen hypergeometric representations of the associated Legendre function of the second kind by using a limit representation. For the 18 hypergeometric arguments which correspond to these representations, we give geometrical descriptions of the corresponding convergence regions in the complex plane. In addition, we consider a corresponding single sum Fourier expansion for the Ferrers function of the second kind. In four of the eighteen cases, the determination of the Ferrers function of the second kind requires the evaluation of the hypergeometric function separately above and below the branch cut at [1,infty). In order to complete these derivations, we use well-known results to derive expressions for the hypergeometric function above and below its branch cut. Finally we give a detailed review of the 1888 paper by Richard Olbricht who was the first to study hypergeometric representations of Legendre functions.

This study explores the geometric properties of normalized Gaussian hypergeometric functions in a certain subclass of analytic functions. This work investigates the inclusion properties of integral operators associated with generalized Bessel functions of the first kind. By finding the conditions on the parameters of the Gaussian hypergeometric function, we identify criteria for the normalized Gaussian hypergeometric function to be in a certain subclass of analytic functions. Using Taylor coefficients, we obtained sufficient conditions for the integral operators associated with generalized Bessel functions of the first kind to belong to different subclasses of univalent functions. The results obtained are analyzed and compared with the existing literature, providing new insights into the subclasses.

In this paper, we obtain the necessary and sufficient conditions for generalized Bessel functions of the first kind zup(z) to be in the classes S(b, λ, β) and R(b, λ, β) of analytic functions with complex order and also give the necessary and sufficient conditions for z(2−up(z)) to be in the classes TS(b, λ, β) and TR(b, λ, β). Furthermore, we give the necessary and sufficient conditions for J(k, c) to be in the class TR(b, λ, β) provided that the function f is in the class Rτ (A, B). Finally, we give conditions for the integral operator G(k, c, z) = ∫0z(2 − up(t))dt to be in the class TR(b, λ, β). Several corollaries and consequences of the main results are also obtained.

Abstract In this paper certain classes of infinite sums involving special functions are evaluated analytically by application of basic quantum mechanical principles to simple models of half harmonic oscillator and a particle trapped inside an infinite potential well. The infinite sums $\sum^{\infty}_{n=0}\frac{2^{2n}}{(2n+1)!}\Gamma^{2}\left(n+\frac{3}{2}\right)\left[\hspace{0.2mm}_2\hspace{-0.03cm}F_1\left(-n,\frac{\nu+2}{2};\frac{3}{2};\frac{1}{2}\right)\right]^{2}$, $\sum^{\infty}_{n=0}\frac{\left[L_{\nu}^{2n+1-\nu}\left(\frac{b^{2}}{2}\right)\right]^{2}b^{4n}}{2^{2n}(2n+1)!}$ and $\sum^{\infty}_{n=1}\frac{\big[J_{\nu+1}(n\pi)\big]^{2}}{n^{2\nu}}$, where $_2\hspace{-0.03cm}F_1\left(-n,\frac{\nu+2}{2};\frac{3}{2};\frac{1}{2}\right)$ is generalized hypergeometric function, $L_{\nu}^{2n+1-\nu}\left(\frac{b^{2}}{2}\right)$ associated Laguerre polynomial and $J_{\nu+1}(n\pi)$ Bessel function of the first kind, are calculated for integer $\nu$. It is also demonstrated that the same procedure can be generalized by application to some classes of functions which are not regular wave functions leading to additional infinite sums, as a consequence of which the series $\sum_{n=1}^{\infty}\frac{\left[\mathsf{H}_{\nu}(n\pi)\right]^{2}}{n^{2\nu}}$ containing Struve functions of the first kind $\mathsf{H}_{\nu}(n\pi)$ are evaluated. Convergence of the evaluated series, additionally verified by the application of different convergence tests, is secured by the properties of the corresponding Hilbert space.

This article proposes and demonstrates a calibration-based integral formulation for resolving the forcing function in a mass–spring–damper system, given either displacement or acceleration data. The proposed method is novel in the context of vibrations, being thoroughly studied in the field of heat transfer. The approach can be expanded and generalized further to multi-variable systems associated with machine parts, vehicle suspensions, translational and rotational systems, gear systems, etc. when mathematically described by a system of constant property, linear, time-invariant ordinary differential equations. The analytic approach and subsequent numerical reconstruction of the forcing function is based on resolving a parameter-free inverse formulation for the equation(s) of motion. The calibration approach is formulated in the frequency domain and takes advantage of several observations produced by the dimensionality reduction leading to an algebratized system involving an input–output relationship and a transfer function possessing all the system parameters. The transfer function is eliminated in lieu of experimental data, from a calibration effort, thus leading to a reduction of systematic errors. These parameter-free, reduced systematic error aspects are the distinct and novel advantages of the proposed method. A first-kind Volterra integral equation is formed containing only the unknown forcing function and experimental data. As with all ill-posed problems, regularization must be introduced for system stabilization. A future-time technique is instituted for forming a family of predictions based on the chosen regularization parameter. The optimal regularization parameter is estimated using a combination of phase–plane analysis and cross-correlation principles. Finally, a numerical simulation is performed verifying the proposed approach.

Landing in the Midwest after a lifetime in Los Angeles, I was shocked to learn how “famous” that great city really is. It used to seem perfectly reasonable that the freeways on CHiPs looked just like the ones I rode to school. When I was five, I remember being secretly bummed that my mom never took us to the disco-classical mural from Xanadu, which I was convinced had to be hidden somewhere in Venice Beach. In high school, it never seemed strange that the Peach Pit on Beverly Hills 90210 was the same as the Rose City Diner. From the L.A. River to the Griffith Park Observatory, from the Hollywood Sign to Venice Beach, the places I had been in, through, and around were inscribed with meanings in ways that I could never fully grasp. Even marginalized localities like Inglewood, Compton, and East L.A., which especially during the 1980s and early 1990s were being ravaged by urban warfare, got to be the stars of movies, songs, and many music videos. And on April 29, 1992, the corner of Florence and Normandie “blew up” into a full blown riot, sparked by the acquittal of the four white officers who beat black motorist, Rodney King. I could watch the city burn on T.V. or from the hill behind my house. All my life, I lived with a foot in each L.A., the one that’s outside my living room and the one that’s inside my living room, oblivious to the fact that I lived in a famous city. It was only after I moved away from L.A. that I realized my homesickness could often be softened by a click of the remote. I could look for a familiar stretch of road, a bit of the skyline, or a clean but otherwise familiar segment of sidewalk, and it didn’t even matter who, what, where, or why was taking place in the story on screen. It was as though fragments of my life had been archived for me in media space. Some memories were real and some just recollections of other representations – like seeing the observatory in Bowfinger and wondering if I was remembering Rebel Without a Cause or a second grade field trip. But when I arrived here, the question that greeted me most often at parties was, “Why are you in Bowling Green!?!” And the second was, “Did you meet any famous people?” And so I tell them about how I went to driver’s education class with Mayim Byalik, the star of Blossom. Or that I met Annette Funicello one New Year’s Eve at my Uncle Phil’s house. Aside from the occasional queer chuckle about my brush with Blossom, this record is unimpressive. People are hoping for something a little bit more like, “I spent the night in jail with Poison,” “I was an extra on Baywatch,” or “I was at the Viper Room the night River Phoenix passed away.” In spite of my lackluster record of interactions with the rich and the famous, I would still get introduced as being “from California.” I had become the recipient of a second-rate, secondhand fame, noted for being from a place where, if I were more ambitious, I could have really rubbed shoulders with famous people. To young people, many of whom were itching to travel to a place like LA or New York, I was a special kind of failure. But if you aren’t famous, if you are a loser like me, life in L.A. isn’t about the a-list at all. It is about living in a city that captures the imagination, even as you walk down the street. So earning notoriety in a city that speaks in spectacle is an exercise in creativity. It seems like everybody, even the most down-to-earth people, are invested in developing a character, an image, a persona that can bubble up and be noticed in spite of the overwhelming glow of Hollywood. Even at my suburban high school, during the late 1980s and early 1990s, I knew upper-middleclass boys who got nose jobs and manicures. I knew girls who would go trolling for rich men to buy them pretty things that their parents couldn’t afford. There were kids whose parents helped them cheat their way into college. There were wannabe junkies who drove their moms’ minivans into the ghetto to score. I saw people panic, pout, and scream over cars and allowances and shoes. I know that consumer culture is growing stronger just about everywhere, but back home it happened a lot sooner and a lot stronger. Because of our proximity to Hollywood, the crest of the cultural tidal wave looks much higher and its force is much stronger. And I guess I was just too fat to be in California, so I left. However, every once in a while, somebody does manage to make a scene in L.A. A little loser, or whatever you want to call one of the peasants who tend to the vast fiefdoms of L.A.’s elites, rises from banality to achieve celebrity, even if it is a minor celebrity, in the City of Angels. One such figure is the notorious Daniel Ramos, who in 1991 became a central figure in the city’s struggle over its own image. Daniel Ramos was not a star, a politician, or a leader of industry – but before he even appeared in the news, he had trafficked illegally in making a name for himself. A teenager from the projects, Ramos was more widely known as “Chaka,” a graffiti writer credited with over 10,000 tags from San Diego to San Francisco. I had seen Chaka’s tags just about everywhere, and had determined that he might be superhuman. His name, taken after a hairy little missing link from the popular fantasy show, The Land of the Lost, made me smirk as it conjured up images of a sub-humanoid with broken dialect creeping out from the darkness with cans of paint, marking the walls with his sign, calling out to the rest of us half-humans stranded in the land of the lost. Meanwhile, L.A.’s rich and famous whizzed by, casting resentful glances at Chaka’s do-it-yourself media blitz. I knew that Chaka was “bad,” but my imagination loved him. And when he allegedly left his mark in the courthouse elevator on the day of his release from a five-month stretch in prison (Costello), I couldn’t help but feel glad to know that Chaka was still alive, that legends don’t die (his name even made it, through the hand of Dave Grohl, into Nirvana’s “Smells Like Teen Spirit Video” in 1991). For me, and I imagine for many others, it was the beginning of a political awakening. I wondered what was so bad about graffiti, even though I had been taught all my life that it was wrong. More than ten years later, as I sit by the railroad tracks in my small, Midwestern town, eagerly waiting for messages from California painted on the sides of boxcars, I find myself asking a related question – what is good about advertising? I’m not the first to make the welcomed association between graffiti and advertising. In an interview with the vastly capable scholar, Joe Austin, New York graffiti legend IZ THE WIZ explained it thusly: OK, now you’re on a poorer economic level and what do you have? Years ago, and even today, a boxer makes a name for himself in the boxing ring. So when this art form starts developing, why would it be any different? It’s all in the name. When you’re poor, that’s all you got. (40) Austin elaborates on this insight, explaining: The proliferation of posters, advertisements, and signs bearing the images and names of products and proprietors in twentieth-century cities is one obvious place to begin. These are the directly visible extensions of individual/corporate identities into the new shared urban public spaces of the streets, a quantitatively and qualitatively new site in human history where hundreds of thousands of often spectacularly displayed names abound, each catching the eyes of potential consumers and imprinting itself on their memories. (39) So, on one level, the story of Chaka is the story of a poor man who went toe to toe with big media, in a town run by big media, and held his own. It is the story of someone who has managed to say in no insignificant way, “I am here.” Or has Ramos himself yelled as he was being shackled by police, “I am the famous ‘Chaka’” (Walker A4). In spite of everything else, Ramos had a name that was widely recognized, respected by some, reviled by others. Nancy Macdonald, in her important study the culture of writing, shifts the focus away from the more solidly class-based argument employed by Austin in his study of the origins of New York graffiti art to one which lends itself more readily to understanding the culture of writing in the 1990s, after hip hop had become more accessible to middleclass enthusiasts. Macdonald explains, “Writers use the respect and recognition of their peers to validate their masculine identities” (124). While I am reluctant to downplay the class struggle that certainly seems to have implicitly informed Chaka’s quest for recognition, his outlaw appeal lends itself such an interpretation. In a city like Los Angeles, where middle class agency and upward mobility for the service class are not simply functions of wealth, but also of scrupulously maintained images, feelings of powerlessness associated with the lack of a compelling image are to be expected. It is the engine that drives the exuberant extravagance of consumer culture, lifestyle choices, and ultimately biopolitics. In a society where culture and capital are the dual poles which determine one’s social standing, the pursuit of notoriety is not simply a measure of masculinity – hijacking images is a way to assert one’s agency in spite of the diminished value of unskilled labor and the collective fear of underclass masculinities. In her book Wallbangin’: Graffiti and Gangs in L.A., Susan A. Philips provides discussion of Chaka’s contribution to L.A. graffiti. Notably, Chaka was seen by those in the graffiti community as an everyman, who was responsible for two significant cultural achievements: he “open[ed] up the style of the New York-based tags and creat[ed] the phenomenon of the individual tagger” (Phillips 320). He also, as Phillips notes, “wrote tags that you could read…in blockish gang-type lettering” (320). Unlike his New York graffiti-writing peers, which are best known for their beautiful “wildstyle,” Chaka did not typically traffic in multicolor murals and displays of painterly virtuosity. His chief accomplishment was his cunning pervasiveness and daring criminality. As such, his body of work should be seen as incompatible with High Art attempts to bring collectible graffiti into gallery spaces through the 1980s and ‘90s. Chaka’s medium, in a sense, has less to do with paint, than it has to do with the city and its rules. For the majority of the public, Chaka was seen as an individual face for the graffiti pandemic that was strategically linked in the public mind with specter of gang violence. However, to those familiar with the writing scene in L.A., Chaka is more than a lone individual: THE OG’Z OF THE LEGION OF DOOM WERE THE ONE RESPONSIBLE FOR BRINGING THE EARLY LOS ANGELES GRAFFITI SCENE TO IT’S KNEES! AND GAVE US MOST OF THE LEGENDS WE KNOW TODAY! I REMEMBER I TIME WHEN EVERY LOS ANGELES INTERSTATE HEAVEN ROCKED BY EITHER LEST-CAB-STANS-SUB OR THE CHAKA!!! (god i miss those days!) remember the CAB undercover story on the news where he did those loks on dope throwies on the 110 pasadena? I think it was chuck henry channel 7 ??? does anyone still have that on vhs? i had it on vhs along with the CHAKA PUBLIC SERVICE ANOUNCEMENT (that was great!). (Poncho1DEcrew) Instead of being an individual tagger, Chaka is recognized as a member of a crew (LOD), who managed to get up in legendary ways. In reclaiming freeway overpasses (the “Heavens”), walls, trains, road signs, and just about everything else for his crew, vicariously for the many other people who respect his name, and also for himself, Chaka is more than simply selfish, as is often suggested by his detractors. In the heavens is the right place to begin. High up in the sky, over the freeways, for all to see, the writing in the heavens is visible, mysterious, and ultimately risky. The problem of climbing along the girders underneath the bridges, escaping detection, but leaving something bold points to what distinguishes writing from an ad-campaign. Sure, some of what the tagger does is about simply being a recognized image all over the place. But the other part is about finding the place, working within environmental constraints, battling against time, stretching one’s limits, and doing it with style. While the image may be everywhere, the act of writing itself is a singularity, shrouded by secrecy, and defined by the moment of its doing. The aftereffect is a puzzle. And in the case of Chaka, the question is, “How the hell did this guy get up over 10,000 times?” While I can’t see how he did it and I don’t know where, exactly, he got all that paint, I do know one thing: Chaka went everywhere. He mapped the city out as a series of landmarks, he put his name to the space, and he claimed Los Angeles for people other than the ones who claim to own the rights to beam their generalized and monolithic messages into our living rooms. Instead of archiving the city in the banalities of mass media, he has created an archive of an alternative L.A., filled with singularities, and famous in the way that only one’s hometown can be. Instead of being a celebrity, renowned by virtue of a moderately unique character, his ability to generate money, and an elite image, Chaka represents an alternative fame. As a modern day “everyman” and folk hero, he brings a message that the city belongs to all people. Far from the naïve and mean-spirited equations between graffiti writing and canine scent-marking as a primitive drive to mark territorial boundaries with undesirable substances (writers:paint::dogs:piss), Chaka’s all-city message is not so much a practice of creating exclusionary spaces as it is an assertion of one’s identity in a particular space. A postmodern pilgrim, Chaka has marked his progress through the city leaving a perceptible record of his everyday experience, and opening up that possibility for others. This is not to say that it is necessary for all people to paint in order to break loose from the semiotic order of the city, it is only to say that is hopeful to realize that this order is not fixed and that is not even necessarily our own. Reflecting back on my own experience as one who has grown up very much in love in the produced spaces of the scripted and archived fame of Los Angeles, the realization that such an overwhelming place is open even to my own inscriptions is an important one. This realization, which has been many years in the making, was set into place by the curious fame of Chaka. For a writer and scholar disturbed by the “death of the author,” it comes as a relief to see writing resurrected in the anti-authoritarian practice of a teenage boy from the projects. References Austin, Joe. Taking the Train: How Graffiti Art Became an Urban Crisis in New York City. New York: Columbia UP, 2001. Costello, D. “Writing Was on the Wall.” Courier-Mail 9 May 1991. Macdonald, Nancy. The Graffiti Subculture: Youth, Masculinity and Identity in London and New York. Hampshire: Palgrave, 2001. Phillips, Susan A. Wallbangin’: Graffiti and Gangs in L.A. Chicago: U of Chicago P, 1999. Poncho1DEcrew. 50mm Los Angeles Forum. 18 June 2004. 11 July 2004 http://www.50mmlosangeles.com/>. Walker, Jill. “Letter from the Streets; Handwriting on the Wall: 10,000 Chakas.” Washington Post 4 May 1991: A4. Citation reference for this article MLA Style Heckman, Davin. "Being in the Shadow of Hollywood: Celebrity, Banality, and the Infamous Chaka." M/C Journal 7.5 (2004). echo date('d M. Y'); ?> <http://journal.media-culture.org.au/0411/12-heckman.php>. APA Style Heckman, D. (Nov. 2004) "Being in the Shadow of Hollywood: Celebrity, Banality, and the Infamous Chaka," M/C Journal, 7(5). Retrieved echo date('d M. Y'); ?> from <http://journal.media-culture.org.au/0411/12-heckman.php>.

<strong>Gyeongsang University Turnitin Trash Files</strong> <strong>HUMAN CAPITAL NEXUS AND GROWTH OF NIGERIA ECONOMY</strong> <strong>CHAPTER ONE</strong> <strong>INTRODUCTION</strong> <strong>Background to the Study </strong> Government expenditure equally known as public spending simply refers to yearly expenditure by the public sector (government) in order to achieve some macroeconomic aims notably high literacy rate, skilled manpower, high standard of living, poverty alleviation, national productivity growth, and macro-economic stability. It is also expenditure by public authorities at various tiers of government to collectively cater for the social needs of the people. Generally, it has been revealed that public expenditure plays a key role in realizing economic growth. This is because providing good education to individuals is one of the principal avenues of improving human resource quality in any economy. From this perspective, advancing school enrolment may subsequently lead to economic growth. Therefore, education remains the effective way to subdue poverty, illiteracy, underfeeding and accelerate economic growth in the long-term. Much attention has been channeled towards clarifying the relationship between education and economic growth, and so, has led to series of studies by economists over the past 30 years. There is substantial literature to back the fact that correlation exists between the two. (Sylvie, 2018). In line with the views of Hadir and Lahrech (2015), the fact that humans are the most worthy assets remains undisputable in both developed and developing countries. Therefore efficiency in human resource management is pertinent if development must be realized. In this sense, the major gateway to development is adequate investment in human capital which may be described as an individual&rsquo;s potential economic value in terms of skills, knowledge, and other intangible assets. In order to realize the well-known macroeconomic objective of economic growth, Nigeria being a developing country embarked on some programs in the educational sector with the aim of boosting human capital development. However, these programs have only served as conduits for enriching the corrupt political elite. Given the high prospects of achieving economic growth in Nigeria and the place of human capital development in its actualization, education, therefore, remains a top priority for the Nigeria government as well as concerned researchers. Thus, this study is one among other concerned studies that will attempt to examine economic growth and human capital nexus in Nigeria through education variables. In particular, using education as a measure of human capital, it will attempt to explore the impact of education variables on the growth of Nigeria&rsquo;s economy. According to Wamboye, (2015), education makes way for vital knowledge, skills, techniques and information for individuals to function in family and society. Education can groom a set of educated leaders to take on jobs in government services, public and private firms, and domestic and foreign firms. The growth of education can provide all kinds of grooming that would foster literacy and basic skills. Though alternative investments in the economy could generate more growth, it must not deviate from the necessary contributions; economic as well as non-economic, that education can make and has made to expediting macroeconomic growth (Clark, 2015). Todaro and Smith in Clark (2015), likewise called attention to the fact that, extension of education lead to an increasingly gainful labor force and provide it with expanded information and abilities, and boost employment and income-earning avenues for educators, schools, and employees. Economic growth, proxied by Gross Domestic Product (GDP), gives numerous advantages which include increasing the general living standard of the masses as estimated by per capita pay (income), making the distribution of income simpler to accomplish, thus, shortening the time span needed to achieve the fundamental needs of man to a considerable majority of the masses. The main source of per capita yield (output) in any nation, regardless of whether it is advanced or developing, is really increment in &#39;human productivity&#39;. Per capita yield (output) growth is notwithstanding a significant aspect of economic prosperity (Abramowitz, 1981). For the most part, it has been uncovered that individuals are the most important source of productivity growth and economic prosperity. Technology and technological hardware are the results of human inventions and innovativeness. The suggestion of UNESCO, that 26% of yearly planned expenditure (budget) in developing nations should be dedicated to education has become intangible, particularly in Nigeria. Planned expenditure on education in Nigeria ranges from only 5%-7% of total planned expenditure. The impact of the above situation on the economic prosperity of the nation as it concerns human capital development, capacity building, infrastructural advancement, etc, is troubling. On this note, the necessity of a well-thought out plan for rectifying this unwanted situation can&#39;t be over stressed. &nbsp; <strong>1.2 Statement of the Problem </strong> Sikiru (2011) as cited in Ajibola (2016) rightly pointed out that the role of education in any economy is no longer business as usual because of the knowledge based globalized economy where productivity greatly depends on the quantity and quality of human resource, which itself largely depends on investment in education. Governments continue to increase spending on education with a view toward enhancing the standard of education, build human capacity and attainment of economic growth. Ironically, this effort by government is still a far cry of UNESCO&rsquo;s recommendation of 26% total annual budget to education, and so, has not yielded the expected results. Thus, researchers sought to understand the relationship between government expenditure on education and economic growth and how they influence each other. These researches on the above subject matter, have given rise to divergent school of thoughts. Over time, Nigeria has indicated willingness to develop&nbsp; education in order to curtail illiteracy and quicken national development. Anyway regardless of the irreproachable evidence that education is key to the improvement of the economy; there exists a wide loop-hole in accessibility, quality and fairness (equity) in education (Ayo, 2014). Empirically verifiable facts in recent years have indicated that the Nigeria&nbsp;education system has continuously turned-out graduates who overtime have defaulted in adapting to evolving techniques and methods of production; due to inadequate infrastructure, underfunding, poor learning aids, outmoded curriculum, dearth of research and development. This has resulted to drastic reduction in employment and the advent of capacity underutilization. This paper assesses growth of Nigeria economy in relation to government expenditure on education and school enrollment from 1981 to 2018. Frequent adjustments and changes in education system in Nigeria, points to the fact that, all is not well with the countries education system. Government have experimented 6-3-3-4, 9-3-4 systems of education, among others. Enrollment in schools forms the main part of investment in human capital in most of the world&rsquo;s societies (Schultz, 2002). There are several explanations concerning why improvement in scholastic quality is not forthcoming in Nigeria as regards the above subject matter. Researchers disagree on whether changes in education attainment levels alters economic growth rate in the long-term. &nbsp;&ldquo;In Nigeria, average public education expenditure to total government expenditure between 1981 and 2018 is 5.68 per cent. It ranged between 0.51 and 10.8 per cent during the period under review&rdquo; (CBN Statistical Bulletin, 2019). However, the major problem therefore, is that despite an increase in the numeric value of budgetary allocation to education in Nigeria over the years, they still fall short of 26 % UNESCO,S recommendation. For instance, 2014, 10.6%; 2015, 9.5%; 2016, 6.1%, 2017, 5.41%, 2018, 7.0% and 2019, 7.2% percent respectively of total annual budget to education. The statistics presented above indicates that investment in education has not produced the desired level of human capital and economic growth in Nigeria. These uncertainties as it relates to government expenditure on education, school enrollment and growth of Nigeria economy gave birth to this research work. Furthermore, most studies relating to the subject matter, conducted analysis on times series data without subjecting these data sets to structural breaks, thereby giving rise to spurious results and therefore, unreliable recommendations. For instance, unit root test with structural breaks were not employed in majority of these studies. <strong>1.3 Research Questions </strong> The issues raised above have provoked series of questions which this study attempts to provide answers. These questions include; i. To what extent does government expenditure on education affect growth of Nigeria economy? ii. To what extent does primary school enrollment affect growth of Nigeria economy? iii. To what extent does secondary school enrollment affect growth of Nigeria economy? iv. To what extent does tertiary school enrollment affect growth of Nigeria economy? <strong>1.4 Objectives of the Study </strong> The main objective of the study is to access the effect of government expenditure on education and growth of Nigeria economy. Specific objectives of the study are to; i. Access the effect of government expenditure on growth of Nigeria economy. ii. Access the effect of primary school enrollment on growth of Nigeria economy. iii. Access the effect of secondary school enrollment on growth of Nigeria economy. iv. Access the effect of tertiary school enrollment on growth of Nigeria economy. <strong>1.5 Hypotheses of the Study </strong> The following hypotheses were tested in this study. i. Government expenditure on education has no significant effect on growth of Nigeria economy. ii. Primary school enrollment has no significant effect on growth of Nigeria economy. iii. Secondary school enrollment has no significant effect on growth of Nigeria economy. iv. Tertiary school enrollment has no significant effect on growth of Nigeria economy. <strong>1.6 Scope of the Study </strong> The study covers the time series analysis of government expenditure on education, school enrolment; primary, secondary and tertiary, and growth of Nigeria economy from 1981 to 2018. Based on available data. Justification for this study is on the premise that, time series data used for the study is a current data on government expenditure on education, education enrolment and economic growth in Nigeria. This study used annual data for the period 1981-2018, collected from the CBN Statistical Bulletin (2019) and World Bank databank. Variables employed for the study include; Real GDP Per Capita, government expenditure on education, primary, secondary and tertiary school enrolment. <strong>1.7 Significance of the Study </strong> Models of economic growth provide useful predictions that inform decisions made by policy makers. Agreeing with policy options based on inaccurate research studies could undermine government intervention particularly in the education sector. A good perception of the interaction among investment in education, its outcome, school enrolment and economic growth is appropriate policy measure, guarantees human capital development. Thus, a representative model that take cognisance of inter-play among public education expenditure, school enrolment and growth of the economy will lead to adequate disbursement and utilization of government funds. The outcome of this research will serve as a tool for policy makers in the Ministries of Finance, Education and the National Planning Commission including regulatory agencies not mentioned here. It will also serve as a reference material for subsequent research work in this field. <strong>1.8 Limitation of the Study </strong> This research x-rays Government Expenditure on Education, school (primary, secondary and tertiary) enrolment as they relate to Growth of Nigeria Economy. Time series data covering the period 1981 to 2018 is used for this study. A study undertaken in 2020, but can not access 2019 data on the variables used, stand as one of the limitations, since lag periods are essential in policy implementation. Data availability, genuineness and accuracy of same, time and financial constraints, constitute limitations to this research work. Effect of corruption on government expenditure and outbreak of Corona virus, resulting to closure of tertiary institutions in Nigeria, also constitute limitation to this study. <strong>1.9 Organization of the Study </strong> This research work comprises of five (5) chapters, these includes; Chapter one: this consists of background to the study, Problem Statement, research questions, research hypothesis and scope of the study. Chapter two: consisting of conceptual framework, theoretical review, review of related literatures and theoretical framework. Chapter three: explained the methodology this research adopted. Chapter four: presentation of results and discussion of findings. Chapter five: consists of summary of findings, conclusion, policy recommendation, contribution to knowledge and suggestion for further studies.&nbsp; <strong>CHAPTER TWO</strong> <strong>LITERATURE REVIEW AND THEORITICAL FRAMEWORK</strong> <strong>2.1 Conceptual Review</strong> <strong>2.1.1 Government&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </strong> Government is the sector of the economy focusing on giving different public services. Its structure differs by nation, yet in many nations, government involves such services as infrastructure, military, police, public travel, government provided education, alongside medical services and those working for the public sector itself, like, elected authorities. The government may offer types of assistance that a non-taxpayer can&#39;t be barred from, (for example, street lighting), goods which aids all of society instead of benefiting only one person. Finances for government goods and services are generally obtained through various techniques, including taxes, charges, and through monetary transfers from different tiers of government (for example from federal to state government). Various governments from around the globe may utilize their own strategies for financing public goods and services. <strong>2.1.2 Government Expenditure</strong> Government Expenditure refers to government spending both capital and recurrent. For the purpose of this study we limited our scope to government educational expenditure in Nigeria. The theory of government expenditure is the theory of the costs of availing goods and services through planned spending (budget). There are two ways to deal with the subject of growth of government, precisely, the expansion in total size of government spending and the expansion of government in terms of economic magnitudes. Government expenditure is spending made by the public sector (government) of a nation on aggregate needs and wants, for example, pension and arrangement of infrastructure, among others. Until the nineteenth century, government speding was constrained, as free enterprise theorists believed that financial resources left in the private sector could lead to higher returns. In the twentieth century, John Maynard Keynes advocated the job of government spending in influencing levels of wages and income distribution in the economy. From that point forward government spending has demonstrated an expanding pattern. The public expenditure trend of the government of a nation is essentially the manner in which assets (resources) are distributed to the various segments of the economy where spending is required. It is exemplified in the government&rsquo;s ways of spending money. In analyzing the trend of government spending hence, it is critical to realize that under a federal system of administration, government job in dealing with the economy is the joint duty of the different tiers of government (Eze and Ikenna 2014). <strong>2.1.3 Human Capital </strong> By and large, human capital is characterized as all skills that are indistinguishably helpful to numerous organizations, including the training organization. Industry-specific skills, conversely, foster efficiency (productivity) just in the industry in which the skills were obtained. In a serious market setting, laborers consistently get a pay that approaches their minor profitability and in this manner, on account of general human capital, laborers win a similar compensation any place they work. <strong>2.1.4 Economic Growth</strong> As per Haller (2012), economic growth or economic expansion means the way toward expanding the size of a country&rsquo;s economy, its macro-economic indicators, particularly the per capita GDP, in an incremental yet not mandatorily linear course, with beneficial outcomes on the socio-economic sector. IMF (2012) perceives economic expansion as the expansion in the market worth of commodities created in a country over a period of time after discounting for inflation. The rate of increment in real Gross Domestic Product is often used as an estimate of economic expansion. In the perspectives of Kimberly (2012), economic expansion is an expansion in the creation of commodities. Any expansion in the worth of a nation&rsquo;s created commodities is likewise characterized as economic expansion. Economic expansion means an expansion in real GNP per unit of labor input. This relates to labor efficiency variation with time. Economic expansion is routinely estimated with the pace of increment in GDP. It is often estimated in real terms (deducting the impact of inflation on the cost of all commodities created). Growth improves the living standard of the individuals in that specific nation. As per Jhingan (2004), one of the greatest aims of money policy approach as of late has been quick macroeconomic expansion. He thus, characterized economic prosperity (growth) as the event whereby the real per capita earnings (income) of a nation increments over a significant stretch of time. Economic expansion is estimated by the expansion in the quantity of commodities created in a nation. An expanding economy creates more commodities in each subsequent timespan. In this manner, growth happens when an economy&#39;s capacity to produce increases which in turn, is utilized to create a greater quantity of commodities. In a more extensive perspective, economic expansion means increasing the living standard of individuals, and reducing disparities in earnings. &nbsp; <strong>2.1.5 </strong><strong>Gross Domestic Product</strong> - GDP Investopedia designates Gross Domestic Product (GDP) as the financial worth of marketable commodities created in a nation during any given duration of time. It is normally computed on a yearly or a quarterly premise. It comprises household and government consumption, government pay-outs, investments and net exports that exist in a sovereign territory. Set forth plainly, GDP is a broad estimation of a country&#39;s aggregate economic activity. &nbsp; <strong>2.1.6 Education</strong> There is no singular meaning of education and this is on the grounds that it indicates various things to various individuals, cultures and societies (Todaro and Stephen, 1982). Ukeje (2002), considers education to be a process, a product and a discipline. When viewed as a process, education is a group of activities which involves passing knowledge across age-groups (generations). When viewed as a product, education is estimated by the characteristics and attributes displayed by the educated individual. Here, the informed (educated) individual is customarily considered to be an informed and refined individual. While as a discipline, education is perceived in terms of the pros of well-structured knowledge which learners are acquainted with. Education is a discipline concerned with techniques of giving guidance and learning in institutions of learning in lieu of informal socialization avenues like rural development undertakings and education via parent-child interactions). It comprises both inherent (intrinsic) and instrumental worth. It is attractive for the person as well as for the general public. Education as private commodity directly aids the individuals who get it, which thusly influences the person&#39;s future pay (income) stream. At the macroeconomic level, a workforce that is superior in terms of education is thought to expand the supply of human capital in the economy and increment its efficiency (productivity). Considering the externalities pervasive in education, it is broadly acknowledged that the state has a key task to carry out in guaranteeing fair distribution of educational chances (opportunities) to the whole populace. This is especially critical in developing nations, for example, Nigeria that experiences the ill effects of elevated poverty levels, inequality and market imperfections. Enrolment might be viewed as the process of commencing participation in a school, which is the number of learners (students) adequately registered as well as participating in classes (Oxford dictionaries). 2.1.7 Primary Education Pupils usually commence learning at the elementary level when they are as old as 5 years or more. Pupils go through 18 terms equivalent to 6 years at the elementary level and may be awarded a first school leaving certification upon successful completion of learning. Subjects treated at the elementary stage comprise arithmetic, foreign and indigenous languages, culture, home economics, religious studies, and agric science. Privately owned institutions of learning may opt to treat computer science, and fine arts. It is mandatory to participate in a Common Entrance Examination in order to meet requirements for induction into secondary institutions of learning. <strong>2.1.8 Secondary Education In Nigeria</strong> Decades after the advancement of elementary education, government gave attention to secondary education, because of the requirement for pupils to advance their education in secondary schools. Secondary education is defined as the completion of fundamental education that started at the elementary level, and seeks to establish the frameworks for long-term learning and human development, by providing subject and skill-centred guidance. It is equally a link between elementary learning and tertiary learning. It is given in two phases, junior and senior levels of three years each and it is six-year duration. It was only in 1909 that the colonial administration began to supplement the endeavors of the Christian Missions in giving secondary education. This was when King&#39;s College was established in Lagos as the colonial government&#39;s secondary institution of learning. As per Adesina and Fafunwa , numerous laws were enacted to improve the condition of secondary education in Nigeria. For the duration of the time the nation was under Colonial Governments, there were scarcely any secondary schools to give secondary education to those that were then ready to gain it. 2.1.9 Tertiary Education Institutions of tertiary learning comprise universities, colleges of education, polytechnics and monotechnics. Government has dominant control of university education, and regulates them through National Universities Commission (NUC). At the university level, first year selection criteria include: At least 5 credits in not more than two sittings in WAEC/NECO; and a score above the 180 benchmark in the Joint Admission and Matriculation Board Entrance Examination (JAMB). Prospective entrants who hold satisfactory national certificates of education (NCE) or national diplomas (ND) having 5 or more ordinary level credits may gain direct entry into universities at the undergraduate level. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <strong>2.2 Theoretical Review</strong> <strong>2.2.1 Wagner&rsquo;s Law of Expending State Activity </strong> Public expenditure has one its oldest theories rooted in Adolph Wagner&rsquo;s (1883) work. A German economist that came up with a fascinating hypothesis of development in 1883 which held that as a country builds its public sector up, government spending will consequently become more significant. Wagner built up a &ldquo;law of increasing state activity&quot; after empirical investigation on Western Europe toward the conclusive part of the nineteenth century. Wagner&#39;s Law as treated to in Likita (1999) contended that government administration development is a product of advancement in industrialization and economic development. Wagner believed that during industrialization, the expansion of real earnings per capita will be accompanied by increments in the portion of government spending in total spending. He stated that the coming of industrial communities can bring about greater political impetus for social advancement and expanded earnings. Wagner (1893) stated three central reasons for the expansion in state spending. To start with, activities in the public sector will supplant non-private sector activities during industrialization. State duties like authoritative and defensive duties will increment. Furthermore, governments will be expected to give social services and government assistance like education, and public health for the elderly, subsidized food, natural hazards and disaster aids, protection programs for the environment and social services. Thirdly, industrial expansion will lead to novel&nbsp; technology and erode monopoly. Governments will need to balance these impacts by offering public goods through planned spending. Adolf Wagner in Finanzwissenschaft (1883) and Grundlegung der politischen Wissenschaft (1893) identified state spending as an &ldquo;internal&rdquo; factor, controlled by the development of aggregate earnings. Thus, aggregate earnings give rise to state spending. Wagner&#39;s may be viewed as a long-term phenomenon which is best observed with lengthy time-series for better economic interpretation and factual (statistical) derivations. This is because these patterns were expected to manifest after 5 or 10 decades of present day industrial community. The hypothesis of public spending is the hypothesis of the costs of availing commodities through planned government spending as well as the theory of policies and laws enacted to bring about private spending. Two ways to deal with the topic of the growth of the government sector are, namely, the expansion in volume of non-private spending and the expansion of non-private sector. Okafor and Eiya (2011) investigated the factors responsible for increment of government spending utilizing BLUE-OLS estimator. They discovered that population, government borrowing, government income, and inflation significantly affected government at the 5% level, while inflation most certainly did not. Further, Edame (2014) examined the predictive factors of state infrastructure spending in Nigeria, utilizing error correction modeling. In this study, it was found that growth-pace of urbanization, public income, density of population, system of government, and foreign reserves collectively or separately impact Nigeria&rsquo;s state infrastructure spending. &nbsp; <strong>2.2.3</strong><strong>The Classical Theory of Economic Growth</strong> This theory signifies the underlying structure of economic reasoning and Adam Smith&#39;s &quot;The Wealth of Nations&quot; (1776) typically paves the way for classical economics. Prominent and remarkable advocates of the classical school are: Adam Smith (1723-1790), David Ricardo (1772-1823), Thomas Malthus (1766-1834), Karl Marx (1818-1883), John Stuart Mill (1808-1873), Jean-Baptiste Say (1767-1832) and so on. Basically Smith&#39;s theory says that the endowment of countries was put together not with respect to gold, but with respect to commerce: As when two economic agents trade valuable commodities, in order to reap the benefits of trade, endowment grows. The classicalists see that markets are self-regulating, when liberated from compulsion. The classicalists termed this figuratively as the &quot;invisible hand&quot;, which establishes equilibrium, when consumers choose among various suppliers, and failure is allowed among firms that fail to compete successfully. The classicalists often warned against the risks of &ldquo;trust&rdquo;, and emphasized on free market economy (Smith, 1776). Adam Smith connected the expansion in endowment of individuals to the expansion of the yield of production factors, which manifests in the improvement of productivity of labor and an expansion in the quantity of working capital. Much scrutiny was given to population expansion, to the expansion in the portion of laborers in material production, to investment and geographical findings, which added to far-reaching prosperity. The perspectives of Thomas Malthus on economic expansion, portraying the expansion of populace and the expansion in production appeared pessimistic. As per Malthus, when the proportion between population expansion and subsistence methods&nbsp; remains, when the populace is expanding increasingly, and subsistence methods expand steadily, the aftermath will be inadequate earth resources (land), and consequently a severe battle for few resources, the prevalence of wars, plagues, hunger, mass illness, etc (Ojewumi and Oladimeji, 2016). As a solution to this issue, Malthus proposed to limit the growth of the populace by the &quot;call to prudence&quot;, particularly the impoverished, and the birth of children on the bases that they were to be provided with decent means of subsistence. One among the most compelling classicalists was David Ricardo (1772 &ndash; 1823). Apparently, the hypothesis of comparative advantage which recommends that a country should engage exclusively in internationally competitive businesses and trade with different nations to acquire commodities lacking domestically is his most notable contribution. He contended the possibility of the presence of a natural market wages and expected that new technologies will result to a fall in the demand for labor. John Stuart Mill (1808-1873) to a great extent summarized the past ideologies of the classicalists. Specifically, he finished the classicalists&rsquo; hypothesis of economic dynamics that considers long-term economic patterns. At the core of this idea is the unceasing amassing of capital. As indicated by the hypothesis, the expansion in capital prompts an increment in the need for labor, and zero population growth gives rise to increment in real earnings, and therefore gives rise to population expansion in the long-run. When the amassing of capital is quicker than the expansion in the workforce, both of these processes can, in principle, remain forever. Increment in the quantity of laborers means having more &quot;mouths&quot;, hence the expansion in the demand for consumption and particularly for food. Food created in agribusiness, which, as we know, characterized by diminishing returns to scale. Therefore, issues of diminishing marginal productivity of capital emerge and the fall of incentives to invest. <strong>2.2.4 The Keynesian Approach of Public Expenditure </strong> John M. Keynes (1936), a British Economist and the pioneer of macroeconomics contended that public spending is a crucial determinant of economic posperity. Keynes hypothesis clearly stated that fiscal policy instrument (for example government expenditure) is a significant apparatus for obtaining stability and better economic expansion rate in the long-term. To obtain stability in the economy, this hypothesis endorses government action in the economy through macroeconomic policy especially fiscal policy. From the Keynesian view, government spending will contribute incrementally to economic expansion. Keynes contended that it is necessary for government to mediate in the economy since government could change financial downturns by raising finances from private borrowing and afterward restoring the funds to the private sector through several spending programs. Likewise, government capital and recurrent spending in the structure provision of class rooms, research centers, acquisition of teaching and learning aids including PCs and payment of salary will have multiplier effects on the economy. Spending on education will boost productivity as well as advancement by improving the quality of labour. It will likewise help in developing a stream of educated administrators in both the private and public sectors of the economy. Keynes classified public spending as an exogenous variable that can create economic prosperity rather than an endogenous phenomenon. In summary, Keynes acknowledged the functioning of the government to be significant as it can prevent economic downturn by expanding aggregate demand and in this manner, switching on the economy again by the multiplier effect. It is an apparatus that proffers stability the short-term yet this should be done carefully as excessive government spending leads to inflation while lack of spending aggravates unemployment. &nbsp; <strong>2.2.5 </strong><strong>Human Capital Theory</strong> Human capital theory, at first developed by Becker (1962), contends that workers have a set of abilities which they can improve or acquire by learning and instruction (education). Be that as it may, human capital hypothesis often assume for the most part expect that experiences are converted into knowledge and skills. It helps us comprehend the training activities of organizations. It (re-)introduced the view that education and training add up to investment in future efficiency (productivity) and not only consumption of resources. From this viewpoint, both firms and labourers rely upon investment in human capital to foster competitiveness, profitability, and earnings. In spite of the fact that these advantages are self-evident, these investments are tied to some costs. From the firm&#39;s perspective, investment in human capital contrast from those in physical capital, because the firm doesn&#39;t gain a property right over its investment in skills, so it and its employees need to agree on the sharing of costs and benefits derived from these investments. While investment in physical capital are solely the organization&#39;s own choice, investment in the abilities (skills) of its workforce include interaction with the workers to be groomed. In the basic formulation, Becker, assuming that commodity and labour markets are perfectly competitive, introduced the distinction between firm-specific and general human capital to answer the question: who bears the expenses of training? &nbsp; <strong>2.2.6 Neoclassical Growth Theories </strong> The neoclassical development hypotheses arose in the 1950s and 1960s, when regard for the issues of dynamic equilibrium declined and the issue of actualizing growth potentials through the adoption of novel technology, boosting productivity and improving the organization of production gained popularity. The principle advocates of this school are Alfred Marshall (1842-1924), Leon Walras (1834-1910), William Stanley Jevons (1835-1882), Irving Fisher (1867-1947) and others. The American economist Robert Solow (1924-present) along with other economists opposed the state&#39;s participation and rather supported the notion of permitting firms to competitively grow by utilizing the majority of the assets accessible to them. They hinged on the production theory and marginal productivity theory from the classical school, according to which, the earnings obtained production factors depend on their marginal products. Neoclassical scholars disagreed with neo-Keynesian views on growth on three grounds (UN, 2011): Firstly, in light of the fact that they are centered around capital accumulation, overlooking land, labour, technology and so on; On the second note, owing to the fact that they are rooted in the unchanging nature of capital share in earnings (income); On the third note, while the neoclassicists recognized the self-restoring equilibrium of the market mechanism, the former overlooked it. On this premise, they identified inflationary government spending as a source of instability in the economy. <strong>2.2.7 The Endogenous Growth Theory</strong> This was created as a response to exclusions and inadequacies in the Solow-Swan model. This theory throws light on long-term economic expansion pace based on the pace of population expansion and the pace of technical advancement which is autonomous with regards to savings rate. Since long-term economic expansion rate depended on exogenous factors, Romer (1994) saw that the neoclassical hypothesis had hardly limited implications. As per Romer, in models with exogenous technical change and exogenous population expansion, it never truly made a difference what the public administration did. The new growth theory doesn&#39;t rebuff the neoclassical growth theory. Perhaps it broadens the neoclassical growth hypothesis by incorporating endogenous technical advancement in growth models. The endogenous development models have been improved by Kenneth J. Arrow, Paul M. Romer, and Robert E. Lucas. The endogenous development model highlights technical advancement arising from pace investment, quantity of capital, and human capital supply. Romer saw natural assets as a lower priority than ideas. He refers to case of Japan which has limited natural assets but welcomed novel ideas and technology from the West. These included improved plans for production of producer durable goods for final production. Accordingly, ideas are key in economic prosperity. With respect to endogenous growth theory, Chude and Chude (2013) submitted that the major improvement in the endogenous growth hypothesis in relation to the past models lies in the fact that it treats the determinants of technology. That is, it openly attempts to model technology instead of expecting it to be exogenous. Momentously, it is a statistical clarification of technological improvement that introduced a novel idea of human capital, knowledge and abilities (skills) that empower employees to be increasingly productive. More often than not, economic expansion is a product of progress in technology, arising from effective utilization of productive resources through the process of learning. This is because human capital development has high rate or increasing rate of return. Therefore, the rate of growth depends heavily on what (the type of capital) a country invests in. Thus to achieve economic expansion, public expenditure in human capital development especially education spending must be increased. At the same time, the theory predicts unexpected additional benefits from advancement of a substantial valued-added knowledge economy, that can develop and preserve a competitive advantage in expanding industries. <strong>2.3 Empirical Review</strong> Bearing in mind the sensitive nature of the field being studied, many investigations had been conducted with the aim of clarifying the divergent ideological schools. For example, Amadi, and Alolote, (2020) explored government infrastructural spending and Nigeria&rsquo;s economic advancement nexus. The investigation uncovered that public spending on transport, communication, education and medical infrastructure significantly affect economic expansion, while spending on agric and natural resources infrastructure recorded a major adverse impact on economic expansion. Despite the fact that the investigation is recent, the time series variables were not exposed to unit root tests with breaks, and thus will yield misleading outcomes. Shafuda and Utpal (2020) explored government spending on human capital and Namibia&rsquo;s economic prosperity (growth) nexus from 1980 to 2015. The examination utilized human development indicators like healthcare outcomes, educational accomplishments and increment in national earnings in Namibia. The investigation uncovered huge effects of government spending on medical care and education on GDP expansion over the long-term. Study conducted in 2020 that utilized data from 1980-1915, comprises a setback to this work. Ihugba, Ukwunna, and Obiukwu (2019) explored government education spending and Nigeria&rsquo;s elementary school enrolment nexus by applying the bounds testing (ARDL) method of cointegration during the time of 1970 to 2017. The model utilized for the investigation attempted to recognize the interaction between two variables and their relationship with control variables; per capita earning (income), remittances, investment and population expansion. The bounds tests indicated that the variables that were studied are bound together over the long-term, when elementary school enrolment is the endogenous variable. The investigation saw that an inconsequential relationship exists between government education spending on elementary school enrolment while a positive relationship exists among remittances and primary school enrolment. Sylvie (2018) explored education and India&rsquo;s economic expansion nexus. The investigation inspected the connection among education and economic prosperity in India from 1975 to 2016 by concentrating on elementary, secondary and tertiary levels of education. It used econometric estimations with the Granger Causality Method and the Cointegration Method. The study indicated that there is convincing proof demonstrating a positive association between education levels and economic expansion in India which may impact government activities and shape the future of India. Ayeni, and Osagie (2018), explored education spending and Nigeria&rsquo;s economic expansion nexus from 1987 to 2016. The investigation uncovered that education spending was inconsistent with education sectoral yield (output), while recurrent education spending had meaningful relationship with real gross national output (or GNP), conversely, capital spending on education was weak. Ogunleye, Owolabi, Sanyaolu, and Lawal, (2017), utilized BLUE-OLS estimator to study the effect of advancement in human capital on Nigeria&rsquo;s economic expansion from 1981 to 2015. The empirical outcomes indicated that human capital development has strong effects on economic expansion (growth). Likewise, human capital development variables; secondary school enrolment, tertiary enrolment, aggregated government spending on health and aggregated government spending on education displayed positive and strong effect on economic expansion of Nigeria. Glylych, Modupe and Semiha (2016) explored education and Nigeria&rsquo;s economic expansion nexus utilizing BLUE-OLS estimator to unveil the interaction between education as human capital and real Gross Domestic Product. The investigation found a strong connection between GDP and different indicators (capital spending on education, recurrent spending on education, elementary school enrolment and secondary school enrolment) utilized in the investigation except for elementary school enrolment (PRYE). Lingaraj, Pradeep and Kalandi (2016) explored education expenditure and economic expansion nexus in 14 major Asian nations by utilizing balanced panel data from 1973 to 2012. The co-integration result indicated the presence of long-run relationships between education spending and economic expansion in all the nations. The findings additionally uncovered a positive and significant effect of education training on economic advancement of all the 14 Asian nations. Further, the panel vector error correction showed unidirectional Granger causality running from economic expansion to education spending both in the short and long-run, however, education spending only Granger causes long-run economic expansion in all the nations. The findings likewise demonstrated a positive effect of education spending on economic expansion. The study contended that education sector is one of the significant elements of economic expansion in each of the 14 Asian nations. A significant portion of government spending ought to be made on education by upgrading different essential, senior and technical educations in the respective countries to make available the skilled labour for long-term economic advancement. Ojewumi and Oladimeji (2016) explored government financing and Nigeria&rsquo;s education nexus. In the research work, public spending on education was arranged into two classes (recurrent and capital spending). The data covered the period 1981 to 2013 and were secondary in nature. The data were gotten for the most part from the publications of the World Bank, Central Bank of Nigeria and National Bureau of Statistics. BLUE-OLS estimator was utilized to study the data. The main results indicated that the effect of both capital and recurrent spending on education expansion were negative during the examination time frame. The study suggested that the elevated level of corruption common in the educational sector ought to be checked to guarantee that finances ear-marked for education particularly capital spending in the sector are prudently appropriated. Government at various levels ought to likewise increment both capital and recurrent spending to support the educational sector up to the United Nations standard. Obi, Ekesiobi, Dimnwobi, and Mgbemena, (2016) explored government education spending and Nigeria&rsquo;s education outcome nexus from 1970 &ndash; 2013. The investigation utilized BLUE-OLS estimator, and demonstrated that government spending on education has a cordial and notable impact on education. Public health spending and urban population expansion were likewise found to positively affect education outcome but are insignificant in influencing education outcome. Omodero, and Azubike, (2016), explored government spending on education and Nigeria&rsquo;s economic advancement nexus from 2000&ndash;2015. Multiple regression analysis and student t-test were applied for investigation. The outcome of the investigation showed that education spending is significant and affects the economy. Additionally, education enrolment demonstrated a significant relationship with GDP but minor effect on the economy. Muhammad and Benedict (2015) explored education spending and Nigeria&rsquo;s economic expansion nexus during the time covering 1981-2010. Co-joining and Granger causality tests were utilized so as to unveil the causal nexus between education spending and economic expansion. They found that there is co-integration between real growth rate of GDP, aggregated government spending on education, recurrent expenditure on education and elementary school enrolment. Adeyemi and Ogunsola (2016) explored advancement in human capital and Nigeria&rsquo;s economic expansion nexus from 1980-2013 on secondary school enrolment, life expectancy rate, government spending on education, gross capital formation and economic expansion rate. ARDL cointegration approach was utilized in the investigation and it uncovered a positive since a long-run nexus among secondary school enrolment, life expectancy rate, government spending on education, gross capital formation and economic expansion rate. Olalekan (2014) explored human capital and Nigeria&rsquo;s economic expansion nexus utilizing yearly data on education and health, from 1980 to 2011. The investigation made use of Generalized Method of Moment (GMM) techniques in the research and the estimated outcomes gave proof of positive connection between human capital and economic expansion. Oladeji (2015) explored human capital (through education and effective services in healthcare) and Nigeria&rsquo;s economic expansion nexus from 1980 to 2012. The investigation utilized BLUE-OLS estimator and uncovered that there is a significant functional and institutional connection between the investment in human capital and economic expansion. The work indicated that a long-term nexus existed between education and economic expansion rate. Hadir and Lahrech, (2015) explored human capital advancement and Morocco&rsquo;s economic expansion nexus utilizing yearly data from 1973 to 2011. The BLUE-OLS estimator was incorporated utilizing aggregated government spending on education and health, the enrolment data of tertiary, secondary and elementary educational institutions as a measure for human capital. The research uncovered a positive nexus between aggregated government spending on education, aggregated government spending on health, elementary education enrolment, secondary education enrolment and tertiary education enrolment. Obi and Obi (2014) explored education spending and Nigeria&rsquo;s economic expansion nexus as a method for accomplishing ideal socio-economic change required from 1981 to 2012. The Johansen co-integration method and BLUE-OLS estimator econometric methods were utilized to closely study the connection between GDP and recurrent education spending. The results showed that regardless of the fact that a positive relationship was obtainable between education spending and economic expansion, a long-term nexus was not obtainable over the period under examination. Jaiyeoba (2015) explored investment in education/health and Nigeria&rsquo;s economic expansion nexus from 1982 to 2011. He utilized trend analysis, the Johansen cointegration and BLUE-OLS estimator. The outcomes demonstrated that there was long-term connection between government spending on education, health and economic expansion. The factors: health and education spending, secondary and tertiary enrolment rate and gross fixed capital formation carried the speculated positive signs and were notable determinants (apart from government spending on education and elementary education enrolment rate). Sulaiman, Bala, Tijani, Waziri and Maji (2015) explored human capital /technology and Nigeria&rsquo;s economic expansion nexus. They utilized yearly time series covering 35 years (1975-2010) and applied autoregressive distributed lag method of cointegration to look at the connection between human capital, technology, and economic expansion. Two measures of human capital (secondary and university enrolment) were utilized in two different models. Their outcome uncovered that all the factors in the two separate models were cointegrated. Besides, the findings from the two assessed models indicated that human capital in measured by secondary and tertiary education enrolments have significant positive effect on economic expansion. Borojo and Jiang (2015) explored education/health (human capital) and Ethiopia&rsquo;s economic expansion nexus from 1980 to 2013. Human capital stock is measured by elementary, secondary and tertiary education enrolment. Human capital investment is proxied by spending on health and education. The Augmented Dickey Fuller test and Johansen&#39;s Co-integration method were utilized to test unit root and to ascertain co-integration among factors, respectively. Their investigation indicated that public spending on health as well as education and elementary as well as secondary education enrolments has positive and significant impacts on economic expansion both in the short-term and the long-term. Ekesiobi, Dimnwobi, Ifebi and Ibekilo (2016) explored public education investment and Nigeria&rsquo;s manufacturing yield nexus. The investigation utilized Augmented Dickey Fuller (ADF) unit root test and BLUE-OLS estimator to examine the connection between public educational spending, elementary school enrolment rate, per capita income, exchange rate, FDI and manufacturing yield (output) rate. The investigation discovered that public education spending has a positive but inconsequential impact on manufacturing yield (output) rate. Odo, Nwachukwu, and Agbi (2016) explored government spending and Nigeria&rsquo;s economic expansion nexus. Their finding demonstrated that social capital had inconsequential positive effect on economic expansion during the period under consideration. Jiangyi, (2016) explored government educational spending and China&rsquo;s economic expansion nexus bearing in mind the spatial third-party spill-over effects. The findings uncover that public educational spending in China has a significant positive effect on economic expansion, but spending in various educational level shows varying outcomes. Public educational spending beneath high-education is positively related with domestic economic expansion, while the impact of educational spending in high-education is inconsequential. Lawanson (2015) explored the importance of health and educational elements of human capital to economic expansion, utilizing panel data from sixteen West African nations over the period 1980 to 2013. He utilized Diff-GMM dynamic panel procedure. The empirical results show that coefficients of both health and education have positive and significant impacts on GDP per capita. The paper ascertains the importance of human capital to economic expansion in West Africa. He suggested that more assets and policies to persuade and improve access to both education and health by the populace ought to be sought after by policy makers. Ehimare, Ogaga-Oghene, Obarisiagbon and Okorie (2014), explored the connection between Nigerian government Expenditure and Human Capital Development. The level of human capital development, which is a measure of the degree of wellbeing (health) and educational achievement of a country influence the level of economic activities in that country. The unit root test was employed to ascertain if the stationary or non-stationary with the Phillip Peron test. So as to measure the efficiency of government spending on human capital development, the data analysis was performed with Data Envelopment Analysis including Input Oriented Variable Return to Scale. The discoveries of the study uncovered that there has been substantial decrease in the efficiency of government spending since 1990 up till 2011 which has been diminishing. Ajadi and Adebakin (2014), investigated the nature of association between human capital development and economic expansion. The descriptive survey method of research was incorporated and multi&ndash;stage sampling method was utilized to choose a size of 200 respondents utilized for the research. An adopted questionnaire with 0.86 reliability index was utilized for information gathering. Data gathered were examined utilizing the Pearson&#39;s Product Moment Correlation Coefficient. The results demonstrated that education has a predictive r-value of 0.76 on individual personal earnings and the type of occupation (job) is linked with individual personal earnings (r=0.64). It, subsequently, concluded that economic expansion rate is influenced by individual personal earnings and suggested that government ought to create adequate educational policy to avail the human capital need of the populace for economic prosperity. Harpaljit, Baharom and Muzafar (2014) examined the connection between education spending and economic expansion rate in China and India by utilizing yearly data from 1970 to 2005. This investigation used multi econometric methods including co-integration test, BLUE-OLS estimator, and VECM. The result uncovered that there is a long-term nexus between earnings (income) level, Gross Domestic Product per capital and education spending in both China and India. Also, a unidirectional causal relationship was obtained for the two nations, running from earnings (income) level to education spending for China, while for India, education spending Granger causes the level of earnings. Urhie (2014) analyzed the impacts of the components of public education spending on both educational achievement and Nigeria&rsquo;s economic expansion rate from 1970 to 2010. The investigation utilized Two Stage Least Squares estimation procedure to analyze the hypotheses. The result uncovered that both capital and recurrent spending on education affect education achievement and economic expansion rate differently. Recurrent spending negatively affected education while capital spending was found to have a positive effect. Conversely, recurrent education had a positive and notable effect on economic expansion while capital spending had a negative effect. Chude and Chude (2013) explored the impacts of public education spending on Nigeria&rsquo;s economic expansion over a time frame from 1977 to 2012, with particular focus on disaggregated and sectorial spending analysis. Error correction model (ECM) was utilized. The result uncovered that over the long-term, aggregated education spending is significant and has a positive relationship on economic expansion. Abdul (2013), analyzed Education and Economic expansion in Malaysia given the fact human capital or education has is now one of the focal issues in the research of economic advancement. The researcher contended that the current studies showed that human capital, particularly education, is a significant ingredient of economic expansion. Thus the researcher investigated the issue of Malaysia education data. Notwithstanding a few issues and data quality issues, Malaysian education datasets are heavily correlated for both secondary and tertiary education. The researcher further tests the impact of various datasets on education and economic expansion relationship. The results were fundamentally the same thereby indicating that Malaysian education datasets are not unreliable. The results were econometrically consistent irrespective of measure of education utilized. All datasets lead to the same conclusion; education is inversely associated with economic expansion. Alvina and Muhammad (2013), inspected the long-term connection between government education spending and economic expansion. The investigation utilized heterogeneous panel data analysis. Panel unit root test are applied for checking stationarity. The single equation approach of panel co-integration (Kao, 1999); Pedroni&#39;s Residual-Based Panel of co-integration Test (1997, 1999) was applied to ascertain the presence of long-term connection between public education spending and gross domestic production. Finally Panel fully modified OLS result uncovered that the effect of government public education on economic expansion is more prominent in developing nations as contrasted with the developed nations, which confirmed the &quot;catching-up effect&quot; in developing nations. Mehmet and Sevgi (2013), inspected the nexus between education spending and economic expansion in Turkey. The examination utilized econometric method as the principal investigation instrument. The result uncovered a positive connection between education spending and economic expansion in the Turkish economy for the period 1970-2012. Implying that, education spending in Turkey positively affected economic expansion. Edame (2014) researched the determinants of Nigeria&rsquo;s public infrastructure spending, utilizing ECM. He found that pace (rate) of urbanization, government income, population density, external reserves, and kind of government collectively or independently impact on public spending on infrastructure. Aregbeyen and Akpan (2013) examined the long-term determinants of Nigeria&rsquo;s government spending, utilizing a disaggregated approach. In their examination, they found that foreign aid is significantly and positively influencing recurrent spending to the detriment of capital spending; that income (revenue) is likewise positively influencing government spending; that trade transparency (openness) is adversely impacting government spending; that debt service obligation diminishes all parts of government spending over the long-term; that the higher the size of the urban population, the higher would be government recurrent spending on economic services; solid proof that Federal government spending is one-sided with regards to recurrent spending, which increments substantially during election times. In likewise manner, Adebayo et al. (2014) researched the effect of public spending on industrial expansion of Nigeria through co-integration and causality and discovered that public spending on administration, economic services, and transfers remained negatively related with industrial expansion while government spending on social services remained positively related in the long-term. They concluded in this manner that there is no crowding-out impact. From these studies reviewed, there is proof that all the investigations joined economic, social, and political determinants of government spending. Srinivasan (2013), analyzed the causal nexus between public spending and economic expansion in India utilizing co-integration approach and error correction model from 1973 to 2012. The co-integration test result uncovered the presence of a long-term equilibrium connection between public spending and economic expansion. The error correction model estimate indicated unilateral causality which runs from economic expansion to public spending in the short-term and long-term. Mohd and Fidlizan (2012), narrowed down on the long-term relationship and causality between government spending in education and economic expansion in Malaysian economy from 1970-2010. The investigation utilized Vector Auto Regression (VAR). The result indicated that economic expansion co-integrated with fixed capital formation (CAP), labour force participation (LAB) and government spending on education (EDU). The Granger cause for education variable and vice versa. In addition, the investigation demonstrated that human capital like education variable goes a long way in affecting economic expansion. Consensus from the above investigations demonstrates that government spending impacts positively on economic expansion. Notable theories that support this case include; Keynes, Wagner, Peacock and Wiseman. Keynes, in his hypothesis draws a connection between public spending and economic expansion and infers that causality runs from public spending to income, meaning that public sector spending is an exogenous factor and public instrument for expanding national income. Again, it holds that expansion in government spending prompts higher economic expansion. Wagner, Peacock and Wiseman and numerous economists have developed various theories on public spending and economic expansion. Wagner positioned public sector spending as a behavioral variable that positively indicates if an economy is prospering. Notwithstanding, the neo classical growth model created by Solow opined that the fiscal policy doesn&#39;t have any impact on the expansion of national income. These multifaceted results obtained from prior investigations show that in reality public spending and other inputs in the education system may have some innate heterogeneity, suggesting that what holds in a given area or country may not hold in another. In the light of the above, this investigation sees that it is necessary to revise the allotment of public spending on education, with regards to the type of impact this spending has on education outcomes. <strong>2.4 Theoretical Framework</strong> The endogenous growth theory has been adopted as the appropriate theoretical framework for this study. This owes much to the fact that, the theory emphasizes the critical role of human capital development, through public investments on education, as a major driver of aggregate productivity in the economy. This is also supported by the work of Ogunleye, Owolabi, Sanyaolu, and Lawal, (2017) who ascertained how economic expansion is influenced by advancement in human capital from 1981 to 2015. In this study it was discovered that economic expansion is greatly influenced by advancement in human capital. Also, economic expansion appeared to facilitated by secondary education enrolment, tertiary education enrolment, and aggregate spending on health and education by the government. <strong>2.5 Research Gap</strong> Though, so much research work has been carried out on the relationship between human capital development, Public Sector Expenditure on Education and Economic expansion in Nigeria, a lot still needed to be done to address some abnormities in these studies. Of note, is that methods adopted in most of these studies are faced with methodological limitations and policy carry-overs, not minding that no two economies are the same. This study therefore, seeks to fill these gaps created by previous researches. Importantly, time plays a vital role in research, making it imperative for continuous and up to date studies, so as to keep abreast with changes as quickly as possible. In the study carried out by Ojewumi and Oladimeji (2016), time series data covering from 1981-2013 was used, while Muhammad and Benedict (2015), used time series data from 1981-2010. These studies above used time series data of 1981 to 2013 and 1981 to 2010 respectively, while this study used updated data covering 1981-2018, thereby making the study current and up to date. &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; <strong>Chapter Three</strong> <strong>Research Methods</strong> <strong>3.1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Research Design</strong> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Ex post facto research design and econometric procedures of analysis will be employed for empirical investigation. <strong>3.2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Model Specification</strong> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Here, we specify a model which captures the relationship between real gross domestic product in per capita terms and the selected education enrolment variables. &nbsp; &nbsp; (3.1) In the above model, <em>Ln</em> denotes natural log, <em>PER_RGDP</em> denotes real gross domestic product in per capita terms,<em> PER_PEE</em> denotes public expenditure on education in per capita terms, <em>PENR</em> denotes percentage of primary education enrolment from population total, <em>SENR</em> denotes percentage of secondary education enrolment from population total, and <em>TENR</em> denotes percentage of tertiary education enrolment from population total. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; For further empirical analysis we can explicitly express the above model in the form of an autoregressive distributed lag (ARDL) model: &nbsp; &nbsp; (3.2) Here, based on economic theory and intuition all of the coefficients are expected to be positive. <strong>3.3&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Estimation Procedure</strong> <strong>3.3.1&nbsp;&nbsp;&nbsp; Unit Root Test with Breaks</strong> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Unlike the popularly used unit root tests (e.g. ADF and PP) which test the null of non-stationarity without accounting for possible breaks-points in data, the break-point unit root test of Peron (1989) tests the null of non-stationarity against other alternatives while accounting for a single break-point in the given data. The alternative hypotheses for this test are succinctly described in the following equations: &nbsp; &nbsp; (3.3) &nbsp; &nbsp; (3.4) &nbsp; &nbsp; (3.5) The first equation captures a break in the intercept of the data with the intercept-break dichotomous variable <em>I_t</em> which takes on values of 1 only when <em>t</em> surpasses the break-point <em>Br</em>; the second captures a break in the slope of the data with a regime-shift dichotomous variable <em>T_t*</em> which takes on values of 1 only when <em>t </em>surpasses the break-point <em>Br</em>; and the third equation captures both effects concurrently with the &ldquo;crash&rdquo; dichotomous variable <em>D</em> which takes on values of 1 only when <em>t</em> equals <em>Br</em>+1. <strong>3.3.2&nbsp;&nbsp;&nbsp; ARDL Bounds Cointegration Approach</strong> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; The popularly-used residual-based cointegration methods may not be very useful when the time-series under consideration attain stationarity at different levels. On the other hand, in addition to being econometrically efficient for small sample cases (<em>n</em> &lt; 30), the bounds cointegration method developed by Pesaran and Shin (1999) is particularly useful for combining time-series that attain stationarity at levels and first-difference. The bounds cointegration method makes use of upper bounds and lower bounds derived from 4 pairs of critical values corresponding to 4 different levels of statistical significance: the 1% level, the 2.5% level, the 5% level, and the 10% level. The null of &ldquo;no cointegration&rdquo; is to be rejected only if the computed bounds f-statistic surpasses any of the upper bounds obtained from a chosen pair of critical values, while the alternative hypothesis of cointegration is to be rejected only if the bounds f-statistic falls below any of the lower bounds obtained from a chosen pair of critical values. Therefore, in contrast to other cointegration tests, the bounds test can be inconclusive if the bounds f-statistic neither surpasses the chosen upper bound nor falls below the chosen lower bound. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; To obtain the bounds f-test statistic, an f-test is performed jointly on all of the un-differenced explanatory variables of the &ldquo;unrestricted&rdquo; error correction model (ECM) derived from any corresponding autoregressive distributed lag (ARDL) model such as the previously specified empirical ARDL model in (3.2). This takes the general form: &nbsp; &nbsp; (3.6) where &Delta;<em>i_t</em> denotes the chosen endogenous variable in first difference; &Delta;<em>j_t</em> and &Delta;<em>k_t</em> denote the chosen exogenous variables in first differences;&nbsp; and <em>e_t</em> denotes the stochastic component. Choosing the best lag-length to be included is made possible by information criteria such as the Akaike and the Schwarz Information Criterion. In the case where the bounds cointegration test disapproves the null, a &ldquo;restricted&rdquo; version of the error correction model can be estimated along-side a long-run model to capture the relevant short-run and long-run dynamics as seen in the following expressions: &nbsp; &nbsp; (3.7) &nbsp; &nbsp; (3.8) Here, the error correction term <em>_t</em>_-1 is non-positive and bounded between 0 and 1 (or 0 and 100) in order to capture the short-run rate of adjustment to long run equilibrium, while the coefficients <em>₁</em>,&hellip;,<em>_j</em>&nbsp; in (3.7) capture the state of long-run equilibrium and are obtained from <em>₁</em>=<em>b₂</em>/<em>b₁</em>,&hellip;, <em>_j</em>=<em>b_j</em>/<em>b₁</em> respectively. &nbsp; &nbsp; &nbsp; <strong>3.4&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Model Evaluation Tests and Techniques</strong> <strong>3.4.1&nbsp;&nbsp;&nbsp; R² and Adjusted R²</strong> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; The R² and the adjusted R² both provide measures of goodness-of-fit. However, the adjusted R² is preferably used because it is robust against redundant regressors which inflate the conventional R². They involve the following statistics: &nbsp; &nbsp; (3.9) &nbsp; &nbsp; (3.10) where <em>SS_r</em> denotes the sum of squares of the regression residuals, <em>SS_t</em> denotes the total sum of squares of the dependent variable, <em>n</em> denotes the number of observations, and <em>k</em> denotes the number of regressors (Verbeek, 2004). <strong>3.4.2&nbsp;&nbsp;&nbsp; T-Test and F-Test</strong> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; The t-test and the f-test can be utilized to evaluate hypotheses pertaining to statistical significance of the parameters in a regression. Particularly, the t-test may be applied to a single parameter while the f-test may be applied to multiple parameters. They involve the following statistics: &nbsp; &nbsp; (3.11) &nbsp; &nbsp; (3.12) where <em>a_k</em> denotes a single parameter-estimate, <em>se </em>denotes its standard error, <em>R²</em> denotes the coefficient of determination of the regression, <em>N</em> denotes the number of observations, and <em>J</em> denotes the number of regressors. For the t-test, the statistical insignificance null hypothesis is to be rejected only if <em>t_i</em> exceeds its 5% critical-value, while for the f-test the joint statistical insignificance null hypothesis is to be rejected only if <em>f</em> exceeds its 5% critical-value at <em>N-J</em> and <em>J-1</em> degrees of freedom (Verbeek, 2004). <strong>3.4.3&nbsp;&nbsp;&nbsp; Residual Normality Test</strong> The Jarque-Bera test statistic (Jarque and Bera, 1987) is useful in determining whether the residuals of a regression are normally distributed. The Jarque-Bera statistic is computed as: &nbsp; &nbsp; (3.13) where <em>S</em> is the skewness, <em>K</em> is the kurtosis, and <em>N</em> is the number of observations. Under the null hypothesis of a normal distribution, the Jarque-Bera statistic is distributed as <em>X²</em> with 2 degrees of freedom. Therefore, the null hypothesis is to be accepted if the absolute value of the Jarque-Bera statistic exceeds the observed value under the null hypothesis. Contrarily, the null hypothesis is to be rejected if the absolute value does not exceed the observed value. <strong>Heteroskedasticity Test</strong> The Breusch-Pagan-Godfrey test (Breusch and Pagan, 1979; Godfrey, 1978) evaluates the null hypothesis of &ldquo;no heteroskedasticity&rdquo; against the alternative hypothesis of heteroskedasticity of the form , where is a vector of independent variables. The test is performed by completing an auxiliary regression of the squared residuals from the original equation on . The explained sum of squares from this auxiliary regression is then divided by to give an LM statistic, which follows a chi square <em>X² </em>distribution with degrees of freedom equal to the number of variables in <em>Z </em>under the null hypothesis of no heteroskedasticity. Therefore, the null hypothesis is to be accepted if the LM statistic exceeds the observed value under the null hypothesis. Contrarily, the null hypothesis is to be rejected if the LM statistic does not exceed the observed value. <strong>3.4.5&nbsp;&nbsp;&nbsp; Serial Correlation Test</strong> The Godfrey (1978) Lagrange multiplier (LM) test is useful when testing for serial correlation in the residuals of a regression. The LM test statistic is computed as follows: First, assuming there is a regression equation: &nbsp; &nbsp; (3.14) where <em>&beta;</em> are the estimated coefficients and <em>&epsilon;</em> are the errors. The test statistic for the lag order <em>&rho;</em> is based on the regression for the residuals <em>&epsilon; = y - XḂ</em> which is given by: &nbsp; &nbsp; (3.15) The coefficients <em>𝛾</em> and <em>𝛼</em><em>_&delta;</em>are expected to be statistically insignificant if the null hypothesis of &ldquo;no serial correlation&rdquo; is to be accepted. On the other hand, the null hypothesis cannot be accepted if the coefficients <em>𝛾</em> and <em>𝛼</em><em>_&delta;</em>are found to be statistically significant. <strong>3.4.6&nbsp;&nbsp;&nbsp; Model Specification Test</strong> The Ramsey (1969) Regression Error Specification Test (RESET) is a general test for the following types of functional specification errors: Omitted variables; some relevant explanatory variables are not included. Incorrect functional form; some of the dependent and independent variables should be transformed to logs, powers, etc. Correlation between the independent variables and the residuals. Ramsey (1969) showed that these specification errors produce a non-zero mean vector for the residuals. Therefore, the null and alternative hypotheses of the RESET test are: &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (3.16) The RESET test is based on an augmented regression which is given as: &nbsp; &nbsp; (3.17) The test of the null hypothesis of a well-specified model is tested against the alternative hypothesis of a poorly specified model by evaluating the restriction <em>𝛾</em><em> = 0</em>. The null hypothesis is to be accepted if <em>𝛾</em><em> = 0</em>, whereas the null hypothesis is to be rejected if <em>𝛾</em><em> &ne; 0</em>. The crucial factor to be considered in constructing the augmented regression model is determining which variable should constitute the <em>Z</em> variable. If <em>Z</em> is an omitted variable, then the test of <em>𝛾</em><em> = 0</em> is simply the omitted variables test. But if <em>y</em> is wrongly specified as an additive relation instead of a multiplicative relation such as <em>y =</em><em>𝛽</em><em>₀</em> X^𝛽¹X^𝛽² + 𝜖 then the test of <em>𝛾</em><em> = 0 </em>is a functional form specification test. In the latter case the restriction <em>𝛾</em><em> = 0 </em>is tested by including powers of the predicted values of the dependent variables in <em>Z</em> such that . <strong>3.4.7&nbsp;&nbsp;&nbsp; CUSUMSQ Stability Test</strong> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; For the test of stability, cumulative sum of recursive residuals (CUSUM) and cumulative sum of recursive residuals squares (CUSUMSQ) tests as proposed by Brown, Durbin, and Evans (1975) was employed. The technique is appropriate for time series data and is recommended for use if one is uncertain about when a structural change might have taken place. The null hypothesis is that the coefficient vector &szlig; is the same every period. The CUSUM test is based on the cumulated sum of the residuals: &nbsp; &nbsp; &nbsp; (3.18) where &nbsp; &nbsp; (3.19) and &nbsp; &nbsp; (3.20) <strong>3.5&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Sources of Data</strong> The Central bank of Nigeria served as the main source of data collection. This implies also that the study adopted secondary data. <strong>Chapter Four</strong> <strong>Empirical Results</strong> <strong>4.1&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Descriptive Statistics</strong> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Before going into cointegration analysis, we will attempt to briefly examine the properties of the data with descriptive statistics. Table 4.1 and Figures 4.1 to 4.5 will be acknowledged for this purpose. Table 4.1: Descriptive Statistics &nbsp; PER_RGDP PER_PEE PENR SENR TENR Mean 264316.01 635.72 23096192.94 5796345.78 787115.08 Median 232704.55 361.03 19747039.31 4410684.33 755776.70 Maximum 385349.04 2340.12 46188979.59 11840028.21 1648670.36 Minimum 199039.15 7.38 9554076.94 1846106.82 49626.49 Std. Dev. 66113.04 681.06 9425336.46 3142601.76 592505.50 Skewness 0.65 0.77 0.59 0.79 0.17 Kurtosis 1.83 2.43 2.27 2.15 1.36 Jarque-Bera 4.88 4.24 3.01 5.14 4.47 Probability 0.09 0.12 0.22 0.08 0.11 Observations 38 38 38 38 38 &nbsp; Figure 4.1: Trend of Real Gross Domestic Product (RGDP) Per Capita &nbsp; &nbsp; &nbsp; Figure 4.2: Trend of Public Expenditure on Education (PEE) Per Capita &nbsp; &nbsp; Figure 4.3: Trend of Primary School Enrolment (PENR) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Figure 4.4: Trend of Secondary School Enrolment (SENR) &nbsp; &nbsp; Figure 4.5: Trend of Tertiary School Enrolment (TENR) &nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; From the second column of Table 4.1, RGDP per capita mean is NGN 264, 316.01 ($734.21). This critically lags behind RGDP per capita mean in all developed (OECD) countries and underscores the need for human and non-human capital development. Further, RGDP per capita maximum is NGN 385, 349.04 while its minimum is NGN 199, 039.15. Given that the trend of RGDP per capita is positively sloped as seen in Figure 4.1, the disparity between RGDP per capita maximum and its minimum indicates growth in RGDP per capita during the period under investigation. Lastly, the Jarque-Bera statistic (4.88) and probability value (0.09) of RGDP per capita simply suggest that it follows a normal distribution, with NGN 66, 113.04 as its standard deviation. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; From the third column of Table 4.1, PEE per capita mean is NGN 635.72 ($1.77). Just like RGDP per capita, this critically lags behind PEE per capita mean in all developed (OECD) countries and underscores the need for more government intervention in the education sector. Further, PEE per capita maximum is NGN 2, 340.12 while its minimum is NGN 7.38. Given that the trend of PEE per capita is positively sloped exponentially as seen in Figure 4.2, the disparity between PEE per capita maximum and its minimum indicates rapid growth in PEE per capita during the period under investigation. Lastly, the Jarque-Bera statistic (4.24) and probability value (0.12) of PEE per capita simply suggest that it follows a normal distribution, with NGN 681.06 as its standard deviation. From the fourth column of Table 4.1, PENR mean is 23096192.94. This represents about 18.33% of total population mean (126036036.63) and indicates high primary school enrolment during the period under investigation. Further, PENR maximum is 46188979.59 while its minimum is 9554076.94. Given that the trend of PENR is positively sloped linearly as seen in Figure 4.3, the disparity between PENR maximum and its minimum indicates consistent growth in PENR during the period under investigation. Lastly, the Jarque-Bera statistic (3.01) and probability value (0.22) of PENR simply suggest that it follows a normal distribution, with 9425336.46 as its standard deviation. From the fifth column of Table 4.1, SENR mean is 5796345.78. This represents about 4.60% of total population mean (126036036.63) and indicates relatively low secondary school enrolment during the period under investigation. Further, SENR maximum is 11840028.21 while its minimum is 1846106.82. Given that the trend of SENR is positively sloped exponentially as seen in Figure 4.4, the disparity between SENR maximum and its minimum indicates rapid growth in SENR during the period under investigation. Lastly, the Jarque-Bera statistic (5.14) and probability value (0.08) of SENR simply suggest that it follows a normal distribution, with 3142601.76 as its standard deviation. From the sixth column of Table 4.1, TENR mean is 787115.08. This represents about 0.63% of total population mean (126036036.63) and indicates very low tertiary school enrolment during the period under investigation. Further, TENR maximum is 1648670.36 while its minimum is 49626.49. Given that the trend of TENR is positively sloped concavely as seen in Figure 4.5, the disparity between TENR maximum and its minimum indicates slow growth in TENR during the period under investigation. Lastly, the Jarque-Bera statistic (4.47) and probability value (0.11) of TENR simply suggest that it follows a normal distribution, with 592505.50 as its standard deviation. From the descriptive statistics above, it is obvious that substantial disparities exist between the maximum and minimum values of the variables, especially for PEE per capita and TENR. This may distort the regression results of the cointegration analysis and may also lead to unnecessarily large regression coefficients. In order to avoid these problems, we have transformed the variables in two major ways. Firstly, we have reduced disparity among the variables by expressing PENR, SENR, and TENR as percentages of population total. Secondly, we have downsized all the variables to a smaller scale by expressing them in natural log form. Therefore instead of RGDP, PEE per capita, PENR, SENR, and TENR, we now have Ln_PER_RGDP, Ln_PER_PEE, Ln_PENR, Ln_SENR, and Ln_TENR respectively as our investigative variables. &nbsp; &nbsp; &nbsp; <strong>4.2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Break-Point Unit Root Test Results</strong> Table 4.2: Break-Point Unit Root Test Result Summary Variables Lags Included Specification Break Date ADF Test Statistic 5% Critical Value Summary <em>Ln_PER_RGDP</em><em>_t</em> 0 Intercept &amp; Trend 2001 -3.3506 -5.1757 Non-Stationary <em>∆Ln_PER_RGDP</em><em>_t</em> 2 Intercept &amp; Trend 2001 -5.4176 -5.1757 Stationary <em>Ln_PER_PEE</em><em>_t</em> 0 Intercept &amp; Trend 2004 -3.3665 -5.1757 Non-Stationary <em>∆Ln_PER_PEE</em><em>_t</em> 5 Intercept &amp; Trend 1995 -5.6226 -5.1757 Non-Stationary <em>Ln_PENR</em><em>_t</em> 7 Intercept &amp; Trend 2004 -7.6901 -5.1757 Stationary <em>Ln_SENR</em><em>_t</em> 3 Intercept &amp; Trend 1998 -5.0584 -5.1757 Non-Stationary <em>∆Ln_SENR</em><em>_t</em> 3 Intercept &amp; Trend 2016 -6.4199 -5.1757 Stationary <em>Ln_TENR</em><em>_t</em> 1 Intercept &amp; Trend 1998 -6.9768 -5.1757 Stationary Note(s): Lag selection based on Schwarz Information Criterion (SIC) &nbsp; As seen in the above table, there are different orders of integration for the time-series variables. Specifically, <em>Ln_PENR</em> and <em>Ln_TENR</em> are stationary at levels, while others are stationary only at the first difference. The bounds cointegration method is more appropriate in this case because it permits the combination of stationary and difference-stationary time series. <strong>4.3&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ARDL Bounds Cointegration Test Results</strong> Table 4.3: Lag/Model Selection Criteria Table Number of Models Evaluated: 16 Dependent Variable: <em>Ln_PER_RGDP</em> S|N Model AIC Specification 1 16 -4.0889 ARDL(1, 0, 0, 0, 0) 2 15 -4.0552 ARDL(1, 0, 0, 0, 1) 3 12 -4.0477 ARDL(1, 0, 1, 0, 0) 4 14 -4.0448 ARDL(1, 0, 0, 1, 0) 5 8 -4.0445 ARDL(1, 1, 0, 0, 0) 6 11 -4.0212 ARDL(1, 0, 1, 0, 1) 7 13 -4.0121 ARDL(1, 0, 0, 1, 1) 8 10 -4.0118 ARDL(1, 0, 1, 1, 0) 9 7 -4.0066 ARDL(1, 1, 0, 0, 1) 10 6 -3.9994 ARDL(1, 1, 0, 1, 0) 11 4 -3.9970 ARDL(1, 1, 1, 0, 0) 12 9 -3.9894 ARDL(1, 0, 1, 1, 1) 13 3 -3.9672 ARDL(1, 1, 1, 0, 1) 14 5 -3.9626 ARDL(1, 1, 0, 1, 1) 15 2 -3.9589 ARDL(1, 1, 1, 1, 0) 16 1 -3.9357 ARDL(1, 1, 1, 1, 1) Note(s): * indicates chosen optimal lag specification based on the Akaike Information Criterion The Akaike criterion shows that ARDL(1, 0, 0, 0, 0) is the best lag specification for the ARDL model, thereby indicating that it is best to include only a single lag of the endogenous variable (<em>Ln_PER_RGDP</em>), and 0 lags of the other exogenous variables (<em>Ln_PER_PEE, Ln_PENR, Ln_SENR, </em>and <em>Ln_TENR</em>). On this basis, an ARDL model was estimated and the bounds cointegration method was applied to test for cointegration as seen in the following tables. Table 4.4: Auto Regressive Distributed Lag (ARDL) Model Estimates Dependent Variable: <em>Ln_PER_RGDP</em><em>_t</em> Regressors Coefficient Standard Error t-statistic Prob. <em>Ln_PER_RGDP _t-1</em> 0.723844 0.063884 11.33053 0.0000 <em>Ln_PER_PEE</em><em>_t</em> 0.006558 0.014438 0.454194 0.6529 <em>Ln_PENR</em><em>_t</em> 0.166945 0.048731 3.425881 0.0017 <em>Ln_SENR_t</em> 0.105751 0.044395 2.382033 0.0235 <em>Ln_TENR</em><em>_t</em> 0.033421 0.036354 0.919326 0.3650 <em>C</em> 2.80666 0.598588 4.688802 0.0001 &nbsp; Table 4.5: Bounds Cointegration Test &nbsp; Computed Wald (F-Statistic): 8.5420 10% level 5% level 2.5% level 1% level <em>k </em>= 4 I(0) I(1) I(0) I(1) I(0) I(1) I(0) I(1) <em>F</em>* 2.45 3.52 2.86 4.01 3.25 4.49 3.74 5.06 Source: Pesaran et al. <em>k</em> signifies the number of regressors <em>F</em>* corresponds to the model with unrestricted intercept and trend In the above table, the bounds test statistic (8.5420) surpasses the upper-bound (4.01) at the 5% level of significance and therefore leads to the rejection of the null hypothesis of &ldquo;no cointegration&rdquo;. Based on this result, a &ldquo;restricted&rdquo; error correction model was estimated as well as a long-run &lsquo;equilibrium&rsquo; model as seen in the subsequent tables and equations. Table 4.6a: Error Correction Model Dependent Variable: &Delta;<em> Ln_PER_RGDP</em><em>_t</em> Regressors Coefficient Standard Error t-statistic Prob. <em>∆Ln_PER_PEE</em><em>_t</em> 0.0065 0.0144 0.4541 0.6529 <em>∆Ln_PENR</em><em>_t</em> 0.1669 0.0487 3.4258 0.0017 <em>∆Ln_SENR</em><em>_t</em> 0.1057 0.0443 2.3820 0.0235 <em>∆Ln_TENR</em><em>_t</em> 0.0334 0.0363 0.9193 0.3650 <em>ECT _t-1</em> -0.2761 0.0638 -4.3227 0.0001 &nbsp; Table 4.6b: Long-Run Model Dependent Variable: <em>Ln_PER_RGDP</em><em>_t</em> Regressors Coefficient Standard Error t-statistic Prob. <em>Ln_PER_PEE</em><em>_t</em> 0.0237 0.0489 0.4850 0.6310 <em>Ln_PENR</em><em>_t</em> 0.6045 0.1253 4.8213 0.0000 <em>Ln_SENR</em><em>_t</em> 0.3829 0.1106 3.4602 0.0016 <em>Ln_TENR</em><em>_t</em> 0.1210 0.1500 0.8064 0.4261 <em>C</em> 10.1633 0.5757 17.6524 0.0000 &nbsp; In the error correction model, the error correction term (<em>ECT_t-1</em>) is expectedly negative and statistically significant at the 5% level (based on its <em>p</em>-value (0.0001)). Its magnitude (-0.2761) indicates a low but significant rate of adjustment to long-run equilibrium and specifically implies that approximately 27.61% of all discrepancies in long-run equilibrium will be corrected in each period. On the other hand, in the long-run model, the first long-run coefficient (<em>Ln_PER_PEE</em><em>_t</em>) is expectedly positive but its <em>p</em>-value (0.6310) indicates that it is statistically insignificant at the 5% level of significance, thereby implying that increment in <em>Ln_PER_PEE</em> will not cause <em>Ln_PER_RGDP</em> to increase. . Similarly, the fourth long-run coefficient (<em>Ln_TENR</em>) is expectedly positive but its <em>p</em>-value (0.4261) indicates that it is statistically insignificant at the 5% level of significance, thereby implying that increment in <em>Ln_TENR</em> will not cause <em>Ln_PER_RGDP</em> to increase. On the other hand, the second long-run coefficient (<em>Ln_PENR</em>) is expectedly positive and its <em>p</em>-value (0.0000) indicates that it is statistically significant at the 5% level of significance, thereby implying that increment in <em>Ln_PENR</em> will cause <em>Ln_PER_RGDP</em> to increase by 0.6045. Similarly, the third long-run coefficient (<em>Ln_SENR</em>) is expectedly positive and its <em>p</em>-value (0.0016) indicates that it is statistically significant at the 5% level of significance, thereby implying that increment in <em>Ln_SENR</em> will cause <em>Ln_PER_RGDP</em> to increase by 0.3829. The intercept also appears to be positive and statistically significant thereby indicating that the long-run model has a positive autonomous component measuring up to 10.1633 units. <strong>4.4&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Model Evaluation Results</strong> <strong>4.4.1&nbsp;&nbsp;&nbsp; Test of Goodness-of-Fit</strong> Table 4.7: Test of Goodness-of-Fit Summary Model R² Adj. R² ARDL Model 0.9875 0.9854 ECM 0.6948 0.6567 &nbsp; The adjusted R² of the ARDL model has a magnitude of 0.9854 and therefore implies that the ARDL model explains as much as 98.54% of the variation in its endogenous variable. Further, the adjusted R² of the ECM has a magnitude of 0.6567 and therefore implies that the error correction model (ECM) explains as much as 65.67% of the variation in its endogenous variable. <strong>4.4.2&nbsp;&nbsp;&nbsp; T-Test and F-Test</strong> Table 4.8: F-Test Summary Model F-Statistic 5% Critical Value Prob. Remarks ARDL Model 490.1238 F(5,31) = 2.52 0.0000 Jointly Significant @ 5% ECM 15.2700 F(4,32) = 2.67 0.0000 Jointly Significant @ 5% &nbsp; The F-statistic (490.1238) for the ARDL model exceeds its 5% critical value (2.66), thereby implying that the parameters of the ARDL model are jointly significant at the 5% level of significance. Further, the F-statistic (15.2700) of the ECM also exceeds its 5% critical value (2.84), thereby implying that the parameters of the error correction model (ECM) are jointly significant at the 5% level of significance. Table 4.9: T-Test Summary T-Test for the Long-Run Estimates Regressors t-statistic 5% Critical Value Remarks <em>Ln_PER_PEE</em><em>_t</em> 0.4850 1.9600 Insignificant <em>Ln_PENR</em><em>_t</em> 4.8213 1.9600 Significant <em>Ln_SENR</em><em>_t</em> 3.4602 1.9600 Significant <em>Ln_TENR</em><em>_t</em> 0.8064 1.9600 Insignificant <em>C</em> 17.6524 1.9600 Significant &nbsp; T-Test for the Error Correction Model (ECM) Estimates &nbsp; Regressors t-statistic 5% Critical Value Remarks &nbsp; <em>∆Ln_PER_PEE</em><em>_t</em> 0.4541 1.9600 Insignificant &nbsp; <em>∆Ln_PENR</em><em>_t</em> 3.4258 1.9600 Significant &nbsp; <em>∆Ln_SENR</em><em>_t</em> 2.3820 1.9600 Significant &nbsp; <em>∆Ln_TENR</em><em>_t</em> 0.9193 1.9600 Insignificant &nbsp; <em>ECT _t-1</em> -4.3227 1.9600 Significant &nbsp; &nbsp; In the long-run model, the t-statistics for the first and fourth parameters are less than the 5% critical value (1.96), thereby indicating that the first and fourth parameters are statistically insignificant at the 5% level of significance, while the t-statistic for the second, third, and fifth parameters are greater than the 5% critical value (1.96), thereby indicating that they are statistically significant at the 5% level of significance. Similarly, in the ECM, the t-statistics for the first and fourth parameters are less than the 5% critical value (1.96), thereby indicating that the first and fourth parameters are statistically insignificant at the 5% level of significance, while the t-statistic for the second, third, and fifth parameters are greater than the 5% critical value (1.96), thereby indicating that they are statistically significant at the 5% level of significance. <strong>Normality Test</strong> Table 4.10: Jarque-Bera Normality Test Summary Model Skewness Kurtosis JB Statistic Prob. ARDL Model -0.5558 2.8731 1.9297 0.3810 ECM -0.7369 2.9430 3.3544 0.1868 &nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; In the ARDL model, the <em>p</em>-value (0.3810) of the J-B test exceeds the 0.05 benchmark, and therefore indicates that the residuals of the ARDL model are normally distributed. Further, in the ECM, the <em>p</em>-value (0.1868) of the J-B test also exceeds the 0.05 benchmark, and therefore indicates that the residuals of the error correction model (ECM) are normally distributed. <strong>4.4.4&nbsp;&nbsp;&nbsp; Heteroskedasticity Test</strong> Table 4.11: Breusch-Pagan-Godfrey Heteroskedasticity Test Summary Model BPG Statistic (Obs*R-sq) Prob. ARDL Model 4.3085 0.5059 ECM 7.2979 0.1209 &nbsp; In the ARDL model, the <em>p</em>-value (0.5059) of the BPG test exceeds the 0.05 benchmark, and therefore indicates that the residuals of the ARDL model are homoskedastic. Similarly, in ECM, the <em>p</em>-value (0.1209) of the BPG test also exceeds the 0.05 benchmark, and therefore indicates that the residuals of the error correction model (ECM) are homoskedastic. <strong>4.4.5&nbsp;&nbsp;&nbsp; Autocorrelation Test</strong> Table 4.12: Breusch-Godfrey Serial Correlation Test Summary Model BG Statistic (Obs*R-sq) Prob. ARDL Model 0.1021 0.7493 ECM 0.8776 0.3488 &nbsp; In the ARDL model, the <em>p</em>-value (0.7493) of the BG test exceeds the 0.05 benchmark, and therefore indicates that the residuals of the ARDL model are not serially correlated. Similarly, in ECM, the <em>p</em>-value (0.3488) of the BG test also exceeds the 0.05 benchmark, and therefore indicates that the residuals of the error correction model (ECM) are not serially correlated. <strong>4.4.6&nbsp;&nbsp;&nbsp; Functional Specification Test</strong> Table 4.13: RESET Model Specification Test Summary Model Test Statistics Value Degrees of Freedom Prob. ARDL Model t-statistic 0.805722 30 0.4267 F-statistic 0.649189 (1, 30) 0.4267 ECM t-statistic 0.533837 31 0.5973 F-statistic 0.284982 (1, 31) 0.5973 &nbsp; In the ARDL model, the F-statistic <em>p</em>-value (0.4267) of the RESET test exceeds the 0.05 benchmark, and therefore indicates that the ARDL model was adequately specified. Further, in the ECM, the F-statistic <em>p</em>-value (0.5973) of the RESET test exceeds the 0.05 benchmark, and therefore indicates that the error correction model (ECM) was adequately specified. <strong>4.4.7&nbsp;&nbsp;&nbsp; CUSUMSQ Stability Test</strong> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; The Cumulative Sum of Residuals (CUSUM) Squares test was used to examine the stability of the ARDL model. The result is captured in the following figure. Figure 4.6: CUSUMSQ Plot &nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; In interpreting the CUSUMSQ test, we may conclude that there is instability only if the CUSUMSQ plot falls outside the boundaries of the upper and lower dotted lines which signify the &ldquo;5% level of significance&rdquo;. In this regard, the plot of the CUSUMSQ test in the above figure shows that the ARDL model becomes momentarily unstable in year 2002. However, apart from 2002, the ARDL model appears to be stable in every other year as indicated by the confinement of the CUSUMSQ plot between the upper and lower dotted lines. Overall, considering the fact that this momentary period of instability does not coincide with any major event in Nigeria&rsquo;s education sector, we can conclude that instability is due to chance, and that the estimates of the model are reliable because apart from year 2002 the ARDL model appears to be stable.