Impact of ADP on GDP in Bangladesh: A Cointegration Approach

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All data are in US dollars

According to above table we can say that the frequency distributions of all variables are not normal. We know Skewness is a measure of a distribution, and skewness values of all variables are pretty much nearer to zero. The kurtosis values of all variables are closed to 3 that indicate that the distributions of all variables are normal. Kurtosis measures the peakedness or flatness of a distribution. Kurtosis value of ADP indicates that it is leptokurtic distribution and the other two variables (GDP and GCF) are platy kurtic distribution.

4. Analysis of Regression Results

Finally we consider the following model because of economic significance.

This study proceeds with the OLS method.

Estimated results with Ordinary Least Square method has been reported in Table 2.

Table 2. Regression Results

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Table 3. Summary of the above model

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From, E-views results we can say that the coefficients of all explanatory variables are positive which make economic sense. The constant coefficient is also making an economic sense although it is not necessary to consider statistically meaningful. The general meaning of constant coefficient indicates that if ADP and GCF are zero then we will enjoy 14439454875.46 units GDP. It is equivalent to autonomous GDP. Coefficient of ADP and GCF indicate that the 1 unit increases of ADP and GCF ensure the GDP also increase 1.292874 and 2.698719 unit respectively. It makes economic sense but we need to consider it statistically.

4.1. T-Test, F-Test, R² vs. Adjusted R²and R²& F
4.1.1. T Test

From t-statistics and probability value we can test ADP and GCF significantly affect GDP or not. Now, we can set null and alternate hypothesis as follows-

Here, we see that the probability value of coefficient of ADP and GCF are 0.03 and 0.00 respectively, which are less than 0.05 where the level of significance is 5%. We can reject null. So, ADP and GCF are significantly influencing GDP.

4.1.2. F-Test

Now, we proceed for joint significance test or overall significance test. Using F-test can check this. If the p-value of F statistic is less than 5% we reject null and accept alternate hypothesis.

Here,

In our result, p-value of F statistics is 0.00. So, we can reject null, it means that ADP and GCF jointly can influence GDP. By considering an important relationship between R² and F (Gujarati, p-258) we can draw same conclusion.

4.1.3. R²vs. Adjusted R²

R² Measures the proportion of the variation in the GDP that is explained by variations in the ADP and GCF. In our regression model, 99.7349% of the total variation was explained.

Adjusted R² is a measure the proportion of the variance in the GDP that is explained by variations in the ADP and GCF. Here, our regression model shows that 99.7201% of the variance is explained. It is need to report because it ‘corrects’ for adding more variables to a regression. Adjusted R² indicates that if we add more explanatory variables, it will lead greater R². So, it is reasonable that adjusted R²<R².

4.1.4. R²And F

In any regression process, another important matter is the coefficient of determination R² that is a measure of ‘goodness of fit’. In another word, it is a summery measure that tells how well the sample of our regression line fits the data. Here, the value of R² of 0.997349 means that about 99% of the variation in the GDP is explained by ADP and GCF. F test is also indicates the same direction. It suggests that the sample regression line fits the data very well which greater than conventional value of 0.60.

4.2. Actual, Fitted and Residual Graph

We can represent graphically the effective value of the dependent variable, its estimated value and errors in regression

Figure 1. Residual, Actual and Fitted Graph

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4.3. Stability Test

The most used tests of stability are CUSUM tests. The tests of equation stability and of estimated values are represented below-

Figure 2. Cusum Graph at 5% significance level

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CUSUM test is based on cumulative sum of the equation errors in regression. E-views directly represent graphically the cumulative sum of errors together with critical lines of 5%. Parameters of the equation are not considered stable if the whole sum of recursive errors gets outside the two critical lines. In our model, the equation parameters are stable because the whole sum of recursive errors does not cross the two critical lines.

4.4. Variance Analysis Test

According to ANOVA (Analysis Of Variance) table, the value of F statistic is 96.90957 and probability value is 0.00. It is indicate that R² is statistically significant.

Table 4. Analysis of Variance

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4.5. Muticollineaity Test

Multicollinearity is a statistical phenomenon in which two or more predictor variables in a multiple regression model are highly correlated. In this situation the coefficient estimates may change erratically in response to small changes in the model or the data.

Correlation matrix is one of the best techniques to detect multicollinearity. Now let’s have a look at the following correlation matrix.

Table 5. Correlation Matrix for multicollieaity identification

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The correlation matrix shows a very high collinearity among all variables. This suggests that it may be difficult to separate effects of one variable from another variable. So, we can conclude from our correlation matrix the variables are highly correlated and all values are greater than conventional level.

According to Gujarati, do nothing is the main solution. But formally to solve this problem we can use different ways such as avoid dummy variable trap, get more data, restrictions on parameters, dropping variables etc. But for simplicity we do not go through that way.

4.6. Normality Test (Jarque –Bera Test)

With the normality assumption, the OLS estimators are minimum variance estimators in the entire class of unbiased estimators.

A histogram of residuals is a simple graphic device that is used to learn something about the shape of the probability density function (pdf) of a random variable. Frequency and residuals are shown in vertical axis and horizontal axis respectively. If we mentally superimpose the bell shaped normal distribution curve on the histogram, then we can assume about normality. The following graph (figure 3), we can say our considering distribution is normal. According to Gujarati, it is always a good practice to plot the histogram of the residuals as a rough and ready method of testing for the normality assumption

We can compute Jarque-Bera test statistic using the following rule: JB=n[S ²/6+ (K-3)²/24].

Where, n= sample size, s= skewness coefficient and k= kurtosis coefficient. If s=0 and k=3 then the value of the J-B statistic is expected to be 0. In our model, the JB value is 1.51. The 5% critical value from a chi-square distribution with 3 degrees of freedom is 7.815 and 1% critical value is 11.345. because of 1.51< 7.815 and 1.51<11.345 so there is insufficient evidence from residuals to conclude that the normal distribution assumption is unreasonable at the 5% and 1% level of significance

Figure 3. E-views output residuals histogram and summary statistics

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4.7. Tests for Heteroskedasticity

The presence of heteroscedasticity is potentially a serious problem and the researcher need to whether it is present in a given situation. The main consequences of heteroskadasticity are losing BLUE property, increasing type-2 error and may lead wrong prediction. So, it is important to find out whether there is any heteroscedasticity problem. To test heteroscedasticity, we use graphical method, Breusch Pagan Godfrey test and the white test.

4.7.1. Graphical Method

If the errors are homoscedastic, there should be no patterns of any sort in the residuals. If the errors are heteroscedastic, they may tend to exhibit grater variation in some systematic way. In our model, we suspect that the variance increases as ADP and GCF increase. Two plots of the least squares residuals against ADP and GCF appears in below-

Figure 4. Least square GDP residuals plotted against ADP

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Figure 5. Least square GDP residuals plotted against GCF

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4.7.2. Breusch–Pagan-Godfrey Test

Here the null and alternative hypotheses are;

Ho: There is no heteroskedasticity

Ha: There is heteroskedasticity problem

The formula of the Breusch-Pagan-Godfrey test shows as follows:

Table 6. Breusch–Pagan-Godfrey test (Heteroskedasticity)

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Where χ² shows chi-square distribution with (s-1) degrees of freedom. Our observed χ² = 13.32. If the computed value of χ² exceeds the critical value of Chi- Square at the chosen level of significance, we can reject the hypothesis of homoscedasticity; otherwise does not reject it. In our model, chi-square value is 13.31590 with 2 df the 5% and 1% critical chi square value are 5.99147 and 9.21034 which are less than computed chi-square. Therefore, we reject the hypothesis of homoscedasticity.

4.7.3. White Test

Table 7. White Heteroskedastcity test

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In our model, df=5 since there are 5 regressors in the auxiliary regression which is shown in table. If the calculated chi-square value exceeds the critical value at the chosen level of significance then we can conclude there is heteroscedasticity. Our calculated value is 21.48(obs* R²) With 5 df. The 5% and 1% critical value for degrees of freedom 5 are 11.0705 and 15.086. So, we can conclude on the basis of the white test, that there is heteroscedasticity.

4.7.4. Remedy of Heteroscedasticity

We use white Heteroscedasticity -Consistent Standard Errors and Covariance. Now, we compare our estimation output from the uncorrected OLS regression with the heteroscedasticity consistent covariance output. Note that in our model the coefficients are the same but uncorrected standard error is smaller. It means that the heteroscedasticity consistent covariance method has reduced the size of the t-statistics for the coefficients. It helps us to avoid incorrect values for test statistics in the presence of heteroscedasticity.

4.8. Test for Autocorrelation

The presence of autocorrelation the OLS estimators are still linear unbiased as well as consistent and asymptotically normally distributed, but they are no longer efficient. Therefore, we have to consider autocorrelation is present or not in our model.

4.8.1. Durbin Watson Test

Table 8. Durbin Watson Test

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In the absence of software that computes a p-value, a test known as the bounds test, can be used partially overcome the problem. Durbin & Watson considered two other statics d_l& d_u whose probability distribution do not depend on the explanatory variables.

That is, irrespective of the explanatory variables in the model under consideration will be bounded by an upper bound d_u and 0 a lower bound d_l.If d <d_L,then H₀ is rejected and d >d_u indicates null is not rejected. Our regression model is qualified by all these assumptions. So, we can use Durbin-Watson test. The test procedure is as follows:

(No Autocorrelation)

And, H₁: p>0 (Positive Autocorrelation)

In our model, calculated value of d=0.561985, against for n=39, k=2 and α=5%, the lower bound =1.382 and an upper bound =1.597 at 5% level of significance. So we can reject H0 because of d<, At 1% level of significance also gives same decision.

We conclude that, there is an evidence of positive first order serial correlation. For reducing this problem, we can apply “Cochrane-Orcutt” iterative procedure to estimate ρ, where, ρ is known as the coefficient of auto-covariance. After that we can have a conclusion with the help of EGLS technique.

4.8.2. Breusch- Godfrey Serial Correlation LM Test

Due to some limitations of Durbin-Watson d test, statistician T.S. Breusch and L.G. Godfrey jointly introduce a general test for testing autocorrelation, which is known as BG, or LM test. We consider for testing for serially correlated errors are derived from a general set of hypothesis testing principles that produce Lagrange Multiplier (LM) tests. In this test whether errors are one period apart are correlated. Here, we test the

Significantly different from zero. If e_t and e_t-1 correlated then,

Here, p is an unknown parameter and v_t is another random error term. In our model, regression equation

By replacing

Assuming that is independent of , one-way to test whether and are correlated is to test the null hypothesis and alternative hypothesis H₁: p≠0

Rejecting null at conventional significance levels (here, 5% and 1%) indicate that the errors are serially correlated.

LM test can be written as the simple expression t*R2 where t is the number of observations and R2 is the goodness of fit statistic from an auxiliary regression. If null hypothesis is true then LM has an approximate chi square distribution. We include lag 2 in our E-views operation. In our model, LM= obs*R2 =25.60890 and chi square critical values are 7.815 and 11.345 at the 5% and 1% level of significance. Here, LM > chi square critical. So, we can reject the null hypothesis on the basis of our result. It means that autocorrelation exists. Alternatively, we can reject null by examining the p-value for LM= 25.60890 which is 0.00. Since, p-cal< 0.05 and 0.01. So, we can reject the null and we can conclude that residuals are serially correlated.

4.9. Specification Test
4.9.1. Reset Test

This test is proceed as follows-

We have specified and estimated the equation-

Let, (b₁, b₂, b₃) be the LS estimates and let,

Is obtain and then consider the following artificial model-

Now, if δ₁ is statistically insignificant then it can be conclude that there is no misspecification with omitted variables and wrong functional form. Rejection of null hypothesis implies that the original model is inadequate and can be improved when

Table 9. Test of specification error (Ramsey Reset Test)

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From our model the calculated value of F is 0.1630 and at the 5% and 1% level of significance with J=1 and df=n-k=39-4=35 the critical values are 4.12 and 7.42 respectively. Here,. So, we cannot reject the null of no misspecification.

4.9.2. Wald Test

According to Wald test we can draw the same conclusion. Since the p-values are 0.00. Therefore, we have no sufficient evidence to accept the hypothesis that the coefficients of these variables are zero.

5.Time Series Analysis

To ensure a reliable regression analysis we have to consider the used time series data whether stationary or not. To avoid a spurious regression result, we need to know that the variables are cointegrated at all. This probability is assessed in this analysis.

5.1.Stationary Issues

A random time series is said to be stationary if its mean and variance constant over time and the value of covariance between two time periods depends only the distance between the two time periods and not on the actual time at which the variance is computed (Gujarti, 1995). In order to check for time series properties stationary of the variables, the widely applied unit root test such as Augmented Dccky –Fuller (1981) and Philips-Perrron (1988) tests have been used.

5.2. Nature of Our Variables

In our model we use time series data. To make a proper regression we need to cheek the stationary and nonstationary properties of our variables. There are several way to find out it. The main ways are-

1. Graphical presentation

2. Correlogram

3. Unit root test

5.2.1. Graphical Presentation

From graph we can say a series is stationary or not. If the graph shows the mean reversion property then we can say this series is nonstationary. Graphical presentations of ADP, GCF and GDP are given below

Figure 6. Trend of ADP series

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Figure 7. Trend of GCF series

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Figure 8. Trend of GDP series

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Observing all the variables we can conclude that all variables are non stationary

5.2.2. Correlogram

For a specific series, correlogram shows the ACF (Augmented Correlation Function) values at many lags. ACF shows the correlations of different lags. Such as, GDP_t may be correlated with GDP_t_-₁ but not with directly GDP_t_-2. With the help of ACF, we can identify the nature of the series. For example, long decaying ACF values indicate that the series is non stationary

5.2.3. Unit Root Test
5.2.3.1. Augmented Dickey Fuller Test(ADF)

In the Dickey-Fuller test it is assumed that the error term is uncorrelated. But in case the error terms are correlated, Dickey and Fuller(1979) have developed a test, known as the Augmented Dickey-Fuller test. In ADF we still test null hypothesis what is “there is an unit root” and the ADF test follows the same asymptotic distribution as the DF statistic, so the same critical values can be used. So, we can reject the null hypothesis when |t-value|>|τcri|, otherwise H₀ can be accepted. In general, by testing null hypothesis against alternative, the unit root test can be rejected if the t test statistic is negatively less than the critical value tabulated. The significance levels for all critical values are 1%, 5% and 10%. We cannot reject null of a unit root since; the p value is too high. To figure out the order of integration and to test the stationarity, unit root test will be carried out using the ADF test for all respective variables.

Table 10. Augmented Dickey-Fuller Test for FDI, GCF and GDP at First Difference

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5.3. Trade-off between Loss of Efficiency and Loss of Information

Due to mixed result of different unit root tests the decision about model and the question of stationarity of data comes next. The answer to this question involves an assessment of the trade-off between the loss of efficiency and loss of information. A model specified with levels, when time serieses are nonstationary, will generate estimate that may be spurious. On the other hand a model specified with difference, when serieses are nonstationary will generate estimates that are efficient but will ignore potential long run relationships. Sims (1980a) and Doan (2000), recommend against differencing even if the variable contains a unit root because it throws away information concerning the co-movement of variables. Fuller (1976) shows that differencing produces no gain in asymptotic efficiency even if it is appropriate. Although we conduct unit root tests and got mixed result but following Sims and Doan, the present study uses levels rather than difference of the variables involved.

5.4. Co-Integration Test
5.4.1. Unit Root Test on Residuals/ Engle Granger Test/ Augmented Engle Granger Test

If the residual series of the regression has a unit root then this regression result will spurious where used variables are not integrated of order zero, I(0). We will operate different test for unit root on the residuals that can be used to know cointegrating relationship. Here, we consider ADF test only.

We can set the null and alternate hypotheses are-

H₀: The series are not cointegrated=residuals are non-stationary.

H₁: The series are cointegrated=residuals are stationary

Augmented Dickey-Fuller test for residual series of regression at level

Table 11. Test for co-integration of all the variables series

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In our model, we use nonstationary variables and the estimated residual shows the non stationary property at 1% and 5% critical value. But at 10% critical value it is stationary. So, we can draw a conclusion that GDP, ADP and GCF are not co-integrated at 1% and 5% critical value and regression result is spurious. There is no long run equilibrium relationship among GDP, ADP and GCF. But if we consider the 10% level critical value all variables are co-integrated and there is a long run equilibrium relationship among them that indicates that our regression is not spurious.

5.4.2. Visual Examination

Cointegration means the long run relationship between non-stationary time series. There If two or more series are individually non-stationary (have to be integrated of the same order), but there exists their linear combination which is stationary - we called these series are cointegrated. Visual inspection of the series are expected to be cointegrated is always a good idea.

Figure 9. Trend of ADP, GCF and GDP series

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In this graph all variables show the same upward trend. We can say all variables are cointegrated.

5.4.3. Johansen Cointegration Test

Table 12. Johansen Cointegration Test Result

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