The theoretical foundation is based on the Keynesian theory that advocates for equality of investment and savings at equilibrium level of national income complimented with the financial liberation hypothesis put forth by Mckinnon [25] and Shaw [26] , which postulated that financial liberation has potential of inducing high savings that can help to channel surplus fund to the need of the deficit unit, which in turn has potential of stimulating investment [24] [25] [26] [43] . The justification of this theoretical consideration is based on the fact that their studies are intuitively appealing and provide ground breaking approach to access the correlation between savings and investment in Nigeria.

Following Levy [43] and Coakley et al. [44] who earlier showed the possibility of an economy intertemporal budget constraint to be balanced, suggesting that zero frequency coherence and gain of savings and investment will equal one.

We assume that the time series of domestic investment and national saving are non-stationary at level. That is I t ~ I ( I ) , and S t ~ I ( I ) .

Thus,

I t = I t − 1 + μ t (1)

S t = S t − 1 + ϑ t (2)

where

u t ~ I ( 0 )     and     ϑ t ~ I (0)

We assumed that investment and savings are cointegrated which means that the process have a common stochastic trend.

Let

[ I t S t ] = [ 1 1 ] T + [ i t s t ] (3)

Where T t is common stochastic trend with property ( 1 − L ) T t = z t , z t ~ i i d ( 0 , σ 2 ) , i t ~ I ( 0 ) , s t ~ I ( 0 ) .

Applying a difference operator to yield a bivariate stationary process, we have,

[ ( 1 − L ) I t ( 1 − L ) S t ] = [ 1 1 ] z t + [ ( 1 − L ) i t ( 1 − L ) s t ] , (4)

With spectral matrix

f ( ω ) = [ f Δ I f Δ I , Δ s f Δ s , Δ I f Δ s ] (5)

where the element on the diagonal are spectral density functions of ( 1 − L ) I t and ( 1 − L ) S t , while the off diagonal elements are the cross spectral density function of ( 1 − L ) I t and ( 1 − L ) S t respectively [43] [44] .

To compute the spectral and cross spectral density function. Levy compute the autovariance and cross covariance function and then apply Fourier transformation to the resulting series.

Following Equation (4) above, ( 1 − L ) I t = z t + ( 1 − L ) I t , with the autocovariance function

γ Δ I ( τ ) = E [ ( z t + τ + Δ i t + τ ) ( z t + Δ i t ) ] = E ( z t + τ z t ) + E ( Δ i t + τ Δ i t ) + E ( z t + τ + Δ i t ) + E ( Δ i t + τ z t ) = γ z ( τ ) + γ Δ i ( τ ) + γ Δ i , z ( τ ) (6)

Applying Fourier transform to both sides of Equation (6), we have:

1 2π ∫ − ∞ ∞ γ Δ I ( τ ) e − i τ ω d τ = 1 2π ∫ − ∞ ∞ γ z ( τ ) e − i τ ω d τ + 1 2π ∫ − ∞ ∞ γ Δ i ( τ ) e − i τ ω d τ         + 1 2π ∫ − ∞ ∞ γ z , Δ i ( τ ) e − i τ ω d τ + 1 2π ∫ − ∞ ∞ γ Δ i , z ( τ ) e − i τ ω d τ (7)

Using the standard definitions of spectral and cross spectral density functions presented by Levy [43] , we have

f Δ I ( ω ) = f z ( ω ) + f Δ i ( ω ) + f z , Δ i ( ω ) + f Δ i , z ( ω ) . (8)

Realizing that the f ( ω ) is a complex function, apply Cartesian form, written as:

f z , Δ i ( ω ) = C z , Δ ι ( ω ) − i q z , Δ i ( ω ) . (9)

f Δ i , z ( ω ) = C Δ ι , z ( ω ) − i q Δ i , z ( ω ) . (10)

where c denotes the cospectral density function and q denotes the quadrature spectral function. Following the derivation results presented by Priestley [45] ,

f z , Δ i ( ω ) = f Δ i , z ( ω ) ¯

where bar denote complex conjugate. Thus, using Equation (9), we have:

f z , Δ i ( ω ) + f Δ i , z ( ω ) = f z , Δ i ( ω ) + f z , Δ i ( ω ) ¯ = 2 c z , Δ i ( ω ) . (11)

Therefore, Equation (8) can be rewritten as:

f Δ I ( ω ) = f z ( ω ) + f Δ i ( ω ) + 2 c z , Δ i ( ω ) . (12)

Similarly, deviation of f Δ s ( ω ) and f Δ I , Δ S ( ω ) is express as:

f Δ s ( ω ) = f z ( ω ) + f Δ s ( ω ) + 2 c z , Δ s ( ω ) (13)

f Δ I , Δ S ( ω ) = f z ( ω ) + f Δ i , Δ s ( ω ) + f z , Δ s ( ω ) + f Δ i , z ( ω ) (14)

Since z t is an error term/white noise process, it sis theoretical band is flat equals f z ( ω ) = σ 2 / 2π for all frequencies − π ≤ ω ≤ π . In addition, Δ i and, Δ s are, I ( − 1 ) and therefore their frequency cospectral density, Cospectral density function equal zero.

Thus, combining Equations (12) and (14), the spectral matrix in Equation (5) evaluated at zero frequency becomes:

f ( ω ) | ω = 0 = [ σ z 2 2 π σ z 2 2 π σ z 2 2 π σ z 2 2 π ] . (15)

From the polar representation of f ( ω ) we have

R y , x 2 ( ω ) = | f x , y ( ω ) | 2 f y ( ω ) f x ( ω ) = c y , x 2 ( ω ) + q y , x 2 ( ω ) f y ( ω ) f x ( ω ) (16)

and

Γ y , x ( ω ) = | f y , x ( ω ) | f y ( ω ) = | c y , x 2 + ( ω ) + q y , x 2 ( ω ) | 1 / 2 f y ( ω ) (17)

where R y , x 2 ( ω ) and Γ y , x ( ω ) denote the squared coherence and the gain of investment and saving, respectively (see Jenkin & Watts, 1968). Thus, using matrix in Equation (15) along with definition of squared coherence and gain provided in Equations (16) and (17), we show the zero frequency as

R y , x 2 ( ω ) | ω = 0 = | σ z 2 2π | 2 σ z 2 2π σ z 2 2π = 1 (18)

This study draws inspiration from the Keynesian theory of savings and investment as used by Feldtein and Horioka [38] . In particular, Felstein and Horioka combined the absolute income hypothesis and the life cycle hypothesis in developing their theoretical framework. The model is specified as:

| I Y | t = α + β | S Y | t + μ t (19)

where I denote domestic investment, S denote national savings, Y denote income and μ t denote error term. The coefficient α referred to as saving retention coefficient measured as the proportion of the incremental saving that is invest in the domestic economy.

Two major hypotheses are in support of this framework. First, the absolute income hypothesis postulated by Keynes [24] established the link between savings and income. Keynes suggested that savings is a function of income but the relationship is not linear as represented above.

Thus our model becomes:

I N V t = α + β 1 S A V t + β 2 C F t + β 3 L M t + β 4 F E D t + μ t (20)

where, I N V t is the ratio of non-government gross investment to GDP (the true rate aggregate business investment) in year t; S A V t ratio of national savings to GDP (domestic savings) in year t; C F t is proxy as the net capital flow as a percentage of GDP in year t (external finance); LM is share of broad money in GDP (level of monetization in the economy) in year t; Financial development and efficiency proxy as the credit to the private sector as a ratio of banks overhead cost to total asset in year t; α is constant; β 1 - 4 are slopes, μ t error term and t is time (Table 1).

1) Unit Root Test

The Dickey Fuller (DF)-GLS unit root test was adopted in this study to test the stationarity of each of the variables [49] . The null hypothesis was that the variable was non stationary. If the values of the DF-GLS statistic was less than or equal to the critical value, then the null hypothesis was rejected and it can be inferred that the variable was stationary at conventional level. The expression for the unit root is given as follows.

Δ Y t = β + ρ Y t − 1 + ∑ j = 1 n b j Δ Y t − s + v t (21)

It is important to include the lags of the dependent variable in Equation (1) to eliminate autocorrelation. The hypothesis for stationarity and non-stationarity are expressed in terms of p. When ρ = 0 , it implies that series is not stationary, hence it has unit root.

2) ARDL Bounds Cointegration Test

The study employs the Autoregressive Distributed Lag (ARDL) bounds test by Pesaran, Shin and Smith [50] to examine the effects of monetary policy on output growth in the long and the short run periods in Nigeria. With this approach,

Source: Authors computation.

output growth is expressed as a function of the lagged value of itself and the current and the lagged values of the explanatory variables.

Δ I N V t = a + ∑ p = 1 n b p Δ I N V t − p + ∑ p = 1 n c p S A V t − p + ∑ p = 1 n d p C F t − p     + ∑ p = 1 n e p L M t − p + ∑ p = 1 n f p F E D t − p + ρ 1 S A V t + ρ 2 C F t     + ρ 3 L M t + ρ 4 F E D t + e t (20)

where Δ is the first difference operator. The parameters ρ i , where i = 1, 2, 3, 4, 5, 6, 7 are the respective long run multipliers while the parameters b, c, d, e, f, g, h are the short run dynamic coefficients of the underlying ARDL model in the equation. ε t denotes the white noise error term. The Bounds cointegration test will involve estimating Equation (19) and restricting the parameters of the lag level variables to zero. Based on this equation, we tested the following null and alternative hypotheses:

H 0 = ρ 1 = ρ 2 = ρ 3 = ρ 4 = ρ 5 = 0 (i.e. no cointegration or level relationship) as against H 1 = ρ 1 = ρ 2 = ρ 3 = ρ 4 = ρ 5 ≠ 0 .

The existence of co-integrating relationship among the variables is determined by testing the significance of the lag levels of the variables using the F-test. The calculated F-statistic is compared with the two critical values for the upper and lower bounds tabulated by Narayan [51] .

3) Causality Test

Granger [52] proposed a time series procedure in order to determine causality among time series variables. In Granger sense, there are three possible situations in which a Granger-causality test can be applied. First, in a simple Granger Causality there are two variables and lag considered; second is a multivariate Granger Causality test were more than two variables are considered, while the third considered testing a VAR framework. In this present study the multivariate Granger Causality is used.

Owing to the fact that the direction of co-integration is not a priori established, then each variable is normalized as dependent variable while the existence of level relationship is tested. We study also conducted diagnostic tests such as serial correlation, normality, functional form and heteroscedasticity tests.

Prior to our cointegration tests, it is conventionally plausible to first carry out unit root test to probe the order of cointegration of the series data. The rationale behind the unit root test lies in the fact that the tests help to determine the nature of the series to avoid spurious regression results.

The unit roots estimates were based on Dickey Fuller-GLS test with the result presented in Table 2. From the table, it was observed that all the variables were I(1) and I(0) series.

*indicate 1%, **indicate 5%, level of significance. Source: Authors’ computation.

In order to empirically examine the long-run nexus and short-run dynamic relationships among our research variables, we explore the ARDL bounds test co-integration method developed by Pesaran and Shin [50] . Our choice of method was necessitated by the fact that the method is more explicit and reliable in probing the extent of the relationship among variables in comparison with other previous and traditional co-integration methods. Specifically, the ARDL is not preconditioned to the uniformity of co-integration order for all variables. In essence, the need for all the variables to be integrated in the same order and it can equally be applied when variables are either integrated at level or first difference. More importantly, Harris and Sollis [53] , noted that applying the ARDL technique enhance unbiased estimates of the long-run model.

Going by the underlining assumptions of the ARDL Model, one set assumes that all variables in the model are I(0) and the other set assumes they are all I(1). If the calculated F-statistic exceeds the upper critical bounds value, then the H₀ is rejected. If the F-statistic falls within the bounds, then the test is inconclusive. Lastly, if the F-statistic falls below the lower critical bounds value, it implies that there is no co-integration.

Hence, from the ARDL Bound Test co-integration results, the value of the F-static (12.51) exceeds the critical values at the upper bound (44.68 at 1%, 4.18 at 2.5%, 3.79 at 5% and 3.35 at 10%). Therefore, the empirical findings lead to the conclusion that a long run relationship exists among business investment ( I N V t ), national saving ( S A V t ), Capital flow ( C F t ), Level of monetization ( L M t ) and Financial development efficiency ( F E D t ).

Having established the existence of co-integration from Table 3, the conditional ARDL for the long run relationship can be estimated given the model as thus;

ln I N V t = a + ∑ p = 1 n   b p ln ( I N V t − p ) + ∑ p = 1 n   c p ln ( S A V t − p ) + ∑ p = 1 n   d p ln ( C F t − p )     + ∑ p = 1 n   e p ln ( L M t − p ) + ∑ p = 1 n   f p ln ( F E D t − p ) + e t (21)

Source: Authors’ computation.

where, all variables are as previously defined. The order of the ARDL ( p , q 1 , q 2 , q 3 , q 4 , q 5 ) model in five variables are selected by using AIC Equation (21) is estimated using the ARDL (1, 0, 0, 0, 0) specification (Table 4).

From the long run estimates results in Table 4, it was observably shown that except for negative relations in Capital flow ( C F t ), other variables are significant enough to explain business investment in Nigeria and all conform to theoretical expectation. Thus, null hypothesis of no significant relation between and independent variables are rejected. Specifically, 1% point increase in national saving ( S A V t ), Level of monetization ( L M t ) and Financial development efficiency ( F E D t ) increase business investment by 0.04, 0.2, 0.49 respectively. However, Capital flow ( C F t ) suggested a negative impact on business investment in Nigeria which conform to a prior expectation. Thus, an increase in capital flow ( C F t ) by 1%, business investment ( I N V t ) is expected to reduce by 0.2% correspondingly.

Taking inferences from the studies conducted by Odhiambo [54] and Narayan, Smyth [55] and Mounir (n.d.) [56] , we further estimate the short-run parameters through the error correction model in relation to the long-run parameters estimates. The stated hypothesis of no co-integration which is associated with the vector error correction model is stated thus:

ln ( Δ I N V t ) = a + ∑ p = 1 n   b p ln ( Δ I N V t − p ) + ∑ p = 1 n   c p ln ( Δ S A V t − p ) + ∑ p = 1 n   d p ln ( Δ C F t − p )     + ∑ p = 1 n   e p ln ( Δ L M t − p ) + ∑ p = 1 n   f p ln ( Δ F E D t − p ) + α E C T t − 1 + e t (22)

where, all variables are as previously defined. The order of the ARDL ( p , q 1 , q 2 , q 3 , q 4 , q 5 ) model in five variables are selected by using AIC Equation (22) is estimated using the ARDL (3, 2, 1, 2, 3) specification.

The short run dynamic relationship between saving and investment fundamentals in Nigeria is indicated in the second part of the estimated ARDL in Table 4. The estimated short run results suggested that first and second (lagged) of national saving ( S A V t ) exert positively on business investment ( I N V t ) in Nigeria. The results show that error correction term (ECT) was statistically significant

(*) (**) (***) indicate 1%, 5%, 10% level of significant. Source: Authors’ computation.

with negative sign as expected. Explicitly, the coefficient of the lagged error correction term (ECT) is (0.34) and negatively significant at 1%. The magnitude of the coefficient implies that 34% of the disequilibrium caused by previous shocks converges back to the long run equilibrium in the current period.

Causality is a critical issue when testing co-integration and in general macroeconomic model building. Below is the Pairwise Granger causality test that determines the cause effects of the Savings-Investment fundamentals. The results are analysed based on their causal direction. In econometric analysis, unidirectional Granger Causality is usually used to predict the possibility of a variable to influence another without possibility of reversed case. Bidirectional or feedback causality of the growth rate of variables has the possibility of predicting each other, while no direction or independence between two or more variables show no Granger causality [27] .

The pairwise Granger causality test presented aims to determine whether causality exist between savings and investment fundamentals. Precisely, Table 5 affirmed that bidirectional relationship exists between national savings ( S A V t ) and business investment ( I N V t ), which implies that domestic resource mobilization through savings can trigger business investment in Nigeria and as well business investment ( I N V t ) can enhance national savings ( S A V t ) respectively. Similarly, bidirectional link exists between capital flow ( C F t ) and business investment ( I N V t ). Interestingly, unidirectional relationship is observed between Level of monetization ( L M t ) and Financial development efficiency ( F E D t ) and business investment ( I N V t ), which suggests that the variables in the model can trigger and propel business investment during the period reviewed.

The estimated ARDL was tested for heteroscedasticity, serial correlation, function form misspecification, parameter stability and normality. The results from the test are shown in Table 6.

The model for the underlying ARDL fulfills the stated criteria examined by all the diagnostic tests observable from the serial correlation (Durbin Watson test and Breusch-Godfrey test) which suggests that the model is free from serial correlation. This indicates that the model is reliable in explaining the dynamics of

Source: Authors computation.

Note: *, ** and ***signify significant level at 1%, 5% and 10% respectively. Source: Authors’ computation.

inflation in Nigeria for the study period. Similarly, the Breusch-Pagan-Godfrey Heteroskedasticity test reveals that the disturbance term in the equation is equally homoscedastic. Going by the result of the Jarque-Bera (JB) test, the null hypothesis of normally distributed residuals cannot be rejected. While the Ramsey RESET test result shows that there is no specification error, the Wald test reinforces our standpoint about the validity and correctness of our obtained results. Finally, the Chow predictive failure test suggests that the model may possibly be useful for forecasting with 2009 as the breakpoint year.