Essay: Causal relationship of exports on economic growth in Nigeria and Ghana (Page 2 of 2)

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X_t=∝_1+ β_11 x_(t-1)+ β_(12 ) Y_(t-1)+ε_1t (1)

Y_t=∝_2+ β_21 x_(t-1)+ β_(22 ) Y_(t-1)+ε_2t (2)

You can see that in this framework both variables which we could suppose of as GDP and exports are assign to percussion each other simultaneously. If we suppose stationary of the data train, then in this frank figure appraisement can carelessly take spot worn equilibrium-by-equilibrium ordinary least-squares (OLS).

4.2.1 Stationarity
It already stated, inference based on coefficient estimates from a VAR model are only meaningful if the order of integration of each of the series has been properly accounted for. In the unchecked presence of variables containing unit roots (denoted I(1) variables) or higher order integration the standard asymptotic distributions of the estimates are no longer valid for straightforward testing. For this reason, it is crucial that each of the series included in the VAR model are examined for stationarity using an appropriate pre-test. The standard tests used in the literature are the ADF test of Dickey and Fuller (1979) and the test of Phillips and Perron (1988). The Phillips-Perron test is mainly used where heteroskedasticity or autocorrelation are believed to be present, but considerable evidence suggests that this test performs poorly compared to the ADF test in finite samples (Davidson & MacKinnon, 2004, p.623), so I use the ADF test as our primary method for all stationarity checks and resort to the Phillips-Perron only when the ADF strikes difficulties.

The ADF test regress a variable on its own lagged value, testing the coefficient on the lagged term to determine whether it could be equal to one. It does this in a roundabout way, transforming the model such that the first difference of the variable is regressed on the level of its lagged value, and the coefficient on the lagged value is equal to zero in the case of a unit root. Intercept and time-trend terms, as well as an indefinite number of differenced lags, are included as necessary so that the error term becomes a white noise process. A general form of the ADF test with a quadratic time-trend is as follows:
∆xᵼ = α₀ + α₁t + α₂t + β₁ x₋₁ + (β₂ ∆xᵼ₋₁ + β₃ ∆ᵼ₋₁ + …….) + Ԑᵼ (3)

Where: Additional trend terms may be added as necessary
Additional differenced lags may be added as necessary

The coefficient B_i is equal to (θ-1), where θ is the coefficient on the lagged variable in a simple auto-regression. Therefore, a test of β₁ = 0 is equivalent to a test of θ = 1. Dickey and Fuller (1979) demonstrate that the coefficients of the ADF tests do not follow the standard t-distribution and must be compared to critical values from an appropriate distribution. Estimates of this distribution are available, and Eviews 8 is able to calculate p-values based on simulated critical values. As with VARs, the choice of how many lags to include is important for the test results to be valid. Adding, rather than rely on an arbitrary lag choice this study use an information criterion to determine the appropriate order of the equation. In this case the SC is used, as this is the default criterion used by Eviews 8.
As most of our included series are indeed non-stationary, this study has to adjust our VAR model in order to take account of this. The literature review highlighted two main approaches to doing this: using the framework of the VECM, or conducting tests within an augmented-VAR. The VECM makes explicit provision for any long-run relationships between the variables and keeps the addition of lagged terms to a minimum, which makes it an efficient approach for testing causal relationships. However, in larger systems (involving more variables) identification issues can arise if the model is not estimated using a two-step method. The augmented-VAR does not strike such difficulties, although it is less efficient than the VECM. So while most papers rely on one or other of these approaches, this study use both frameworks where possible. This is another important dimension in helping us to evaluate the buxomness of the results.
4.2.1 Lag length selection
In order for a model to be most accurately the true relationship between the variables. A usual question arises, what is the necessary order of the VAR (i.e. how many lags should be included) This is a very important aspect of such tests, and authors such as Thornton and Batten (1985) have argued that because the order of the true VAR is unknown, it is usually possible to obtain entirely contrasting Granger-causality results by altering the order of the estimated VAR. They assert that in order to legitimize the resulting Granger-causality findings the choice of lag length needs to be justifiably identified, rather than exogenously determined or chosen in an ad hoc manner. Many studies do make the decision in an arbitrary fashion, but there are a number of different information criteria that can be employed in order to base the choice of lag-length on empirical foundations. The most common among these include three of those introduced earlier: the Hannan-Quinn Information Criterion (HQ), the Schwartz Information Criterion (SC) and the Akaike Information Criterion (AIC), each of which use the log-likelihood functions of the model at each possible lag-length but apply different weights in the bias/efficiency trade-off. Unfortunately this weighting is subjective, and there is no consensus as to which criterion performs best at choosing the correct lag length. Ultimately though, as Ivanov & Kilian (2005) assert, researchers will have to choose a lag length and take a stand on the validity or soundness of their results. In view of this, this study agree to use the appropriate lag length of each model based on reports of all three information criterion, giving slightly more emphasis to the AIC as our sample sizes are relatively small. This is as scientific as it is possible to be given the aforementioned difficulties. Liew (2004) performs a Monte Carlo simulation and finds that the AIC generally bring the best in small samples of 30 or 60, while in larger samples the HQ do better.
4.2.3 Vector Error Correction Models (VECMs)

A usual approach for correcting for the of unit roots in the data series, particularly in earlier of studies has been to put in the variables in first-differences and run a differenced-VAR (VARD) model. This exploits the property that I(1) variables are stationary (or I(0)) in their first differences, but in most cases it is not suitable and does not allow for any long-run relationship that may exist between the variables. Engle and Granger (1987) showed that when two or more variables are integrated of order 1 there may be an existence in a linear combination of them that is stationary. This linear combination is familiar as a cointegrating equation, and may be interpreted as a long-run relationship between the variables that has implications on their short-run movements. Referring to what Engle and Granger call Granger Representation Theorem, they view that the cointegrating equation can be incorporated into a meaningful model the VECM. This study will now outline the process that wil move with in constructing VECMs for this empirical study.
After determining that at least two variables are integrated of order 1, i test for cointegrating relationships where there may be more than one in a multivariate setting using the method described by Johansen (1991). This involves estimating the coefficient matrix of a rearranged but unrestricted VAR model and then fathoming the rank of the matrix. According to the Granger Representation Theorem, the rank of the coefficient equation or matrix is equal to the number of cointegrating relations present in the system. This study now make the standard assumption of a linear deterministic trend in the data and an intercept but no trend in the cointegrating equation and choose the lag length according to the criteria discussed previously. the research now use two tests to determine the significance of any cointegrating relationship(s) – the trace test and the maximum eigenvalue test both of which are derived from the eigenvalues of the coefficient matrix. These two tests can cause conflicting results, and where this occurs they have to make a judgement as to the likely number of relations based on what is theoretically feasible. Cointegrating relations are not uniquely identified without some arbitrary form of normalization, so in the bivariate case i restrict the coefficient on GDP to unity. The default approach in EViews is to estimate such models using a two-step approach; first estimating the cointegrating relation(s) by least squares and then estimating the VECM equations one by one with the cointegrating relations predetermined. Moreover, this does not allow us to test for Granger-causality in our desired manner, so in order to be able to test all the coefficients in the system together i conduct an repetitive maximum likelihood procedure without pre-fitting the cointegrating relations (it can easily be demonstrated that this provides identical coefficient estimates for the relevant parameters to the two-step approach). In the bivariate case with evidence for one cointegrating relation and a lag length of two in levels, i therefore estimate the following VECM:

∆xᵼ = α₁ + δ₁ (xᵼ₋₁ + θУᵼ₋₁) + β₁₁∆xᵼ₋₁ + β₁₂∆Уᵼ₋₁ + Ԑ₁ᵼ (4)

∆Уᵼ = α₂+ δ₂ (xᵼ₋₁ + θУᵼ₋₁) + β₂₁∆xᵼ₋₁ + β₂₂∆Уᵼ₋₁ + Ԑ₂ᵼ (5)

Where the intercept terms do actually composite terms comprise both the standard intercepts and the intercepts from the respective cointegration terms. In the trivariate case the largest case to which this study apply this model there are a number of possible forms depending on the number of cointegrating relations and the normalizations chosen, but it is important to note that the results are unaffected by the choice of normalization. The repetitive procedure used to estimate these VECM systems is inherently fragile to initial parameter values and can take a few attempts before they accurately match the correct values found in the two-step procedure. Then this study will discuss how Granger-causality is tested for in these models after first introducing the augmented-VAR approach in more detail.
4.2.4 Testing for Granger-causality
This study have already talked about the idea trailing Granger-causality which decide something more related or connected to predictability than causality, although possibly the best I can do statistically is to and I have discussed the extensive use of Granger-causality in the literature on exports and growth. These studies now turn to the specific ways in which I test for Granger-causality in the models just described. This involves testing the significance of coefficients of appropriate lagged terms of one variable on another variable determining, for example, the ability of previous values of exports to statistically explain the value of GDP in the following period. This is not undertaken in a uniform way across the literature mostly where VEC models are concerned because decisions have to be made around which lagged terms should be tested. As Toda and Phillips (1994) make clear; in the VECM case, significance of the coefficient on the cointegrating relation ( and in equation 4) implies long-run Granger-causality in the sense that the ‘endogenous’ variable is adjusting in line with the longer-term disequilibrium between it and the other variables in the system. But likewise, significance of the coefficients on the lags of the other variables (just in equation 4) implies short-run Granger-causality in the sense that the ‘endogenous’ variable is responding to previous values of the other variables. In order to reconcile these two approaches – and in line with discussions in Giles and Williams (2000b) and Ghali (2000), this study test for both long and short-run Granger-causality concurrently by testing the significance of all coefficients on the lags of the supposed causal variable. This is identical to a test for the complete forbidden of one variable in the equation for another. For example, in the VECM model depicted by equations 4 and 5 above, this study would test for Granger causality by the following exclusions under the null hypothesis:
H₀: β₁₄ = 0, β₁₅ = 0 to test the causality of y on x, and:

H₀: β₂₁ = 0, β₂₂ = 0 to test the causality of x on y

These restrictions are tested using the Wald test statistic. In the case of tests which contain more variables than just GDP and exports, this study also fathom the other causal relationships in the system only in order to determine whether there may be indirect effects on GDP or exports through the other variables. As discussing these findings can be quite extended mostly in the larger models and I only present these results where they are significant to our final ending.

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