The Classical Linear Regression Model - Coursework Example

? 2. OLS and hypothesis tests Part OLS estimation Table OLS results   Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Constant 0.764386 0.408977 1.869019 0.068604 -0.06096 1.589736 Per-capita GDP (1990) -0.20231 0.07579 -2.66937 0.010758 -0.35526 -0.04936 Enrollment 0.25454 0.097956 2.598522 0.01286 0.056857 0.452223 Share of Government in GDP 0.009735 0.006343 1.534827 0.132326 -0.00307 0.022534 Openness 0.000263 0.001627 0.161524 0.872456 -0.00302 0.003545 Inflation 0.114668 0.107046 1.071209 0.290193 -0.10136 0.330695 Ratio of private credit 0.212199 0.141023 1.50471 0.139882 -0.0724 0.496795 Multiple R 0.446225 R Square 0.199117 Adjusted R Square 0.084705 Standard Error 0.279137 F( 6, 42) 1.74 Prob > F 0.1354 Part 2: Hypothesis tests: H0:?2=?3=?4=?5=?6=?7=0 H1:?j?0 (significance level of 0.05) The p-value is: 0.1354. Thus, it is not significant. Therefore, the test fails to reject the null hypothesis. H0:?2=0 H1:?2?0 (significance level of 0.05) The alternative hypothesis is that the coefficient on the first independent variable, per-capita GDP in 1990 is not equal to zero. Since the alternative is of non-equality, the rejection zones will be in both ends of the distribution. Therefore the test will be two tailed. The p-value is 0.01070 using a significance level of 0.05 The alternative hypothesis is that the true coefficient is positive. So the rejection zone is on the right tail of the distribution and thus the test is right tailed. The computed t-value is 2.598522> 1.682 = the critical one sided 5% t value. Therefore, we reject the null hypothesis at 5% level of significance. Therefore, the test provides evidence of the fact that secondary enrollment has a significant positive impact on the growth of GDP. H0:?7=0 H1:?7>0 using a significance level of 0.1 Similar to the test conducted above, the alternative hypothesis is that the coefficient is positive. Therefore, the test will be right tailed once more. The computed t-value =1.50471< 2.418 = the critical 1% t-value. So, we fail to reject the null hypothesis at the 1% level. This implies there is no evidence of private credit ratio having any observable impact on the dependent variable, growth of GDP. Evidently a contradiction arises between the conclusions obtained in (i) and (ii). In (i), the test fails to reject the null hypothesis that all the coefficients are jointly zero. But in (ii), the null hypothesis that the coefficient on the 1990 percapita GDP is zero is rejected. So, (ii) accepts the hypothesis (null in (ii), alternative in (i)) that is rejected by (i). To reconcile between these contradictions, it should be noted that tests of joint significance may fail to reject the null of joint insignificance if majority of the explanatory variables included are found to be insignificant. Usually, such cases arise if the basic assumptions of OLS are violated. Certainly if the presence of outliers skews the distribution of errors, such contradictions should be anticipated. 3. Primary recommendation for the finance minister on the basis of results From the results of the OLS model in part 1, we find that the coefficient on secondary enrolment as well as private credit ratio is positive. The coefficient for secondary enrolment is 0.254 and that for private credit ratio is 0.212. Thus, these are quite close to one another although that for private credit ratio is slightly lower. But critically, it should be noted that only the coefficient of secondary enrolment is statistically significant. So, only secondary enrolment among our variables of interest seems to have any effect on growth of GDP. There is no evidence that private credit has any impact on growth GDP. Thus my recommendation will be to invest the entire sum of $2 Billion on the policy measures that will increase the country’s rate of enrolment in secondary education. 4. Diagnostics This section will evaluate the validity of the standard assumptions of linearity, homoscedasticity and normality of the classical linear regression model in the present context. (i) Linearity We use the standard RESET test using powers of the independent variables to evaluate the validity of the assumption of linearity. The null hypothesis is that the model is linear (or that no higher orders of the included variables are omitted). If the model is non-linear, in that case, including higher orders (quadratic or cubic) of the independent variables would improve the fit of the model. The computed test statistic under the alternative hypothesis is F(18, 24) = 1.461 and the associated probability: Prob > F = 0.1898. Therefore, we fail to reject the null hypothesis. So, the linearity assumption is not violated. (ii) Homoscedasticity To evaluate the validity of the assumption of homoscedasticity, we simply use the Breush-Pagan test2. The null hypothesis states that the errors are distributed with a constant variance. The p-value for the test is 0.526>0.05. So, we fail to reject the null hypothesis. Therefore, we find no evidence of heteroscedasticity. Thus, the assumption of homoscedasticity holds true. (iii) Normality To evaluate normality, first a histogram of the fitted residuals along with a normal density estimate plot is drawn (figure 1). Figure 1 Although the histogram resembles the normal density plot closely, there is a noticeable deviation in the left tail. This can be the result of outliers as well as due to a small sample size. In table 2, we use a Shapiro-Wilk test for a numerical diagnosis of the normality of the distribution. Table 2 The null hypothesis is that the errors follow a normal distribution. The p-value here is 0.00867 F 0.0003 From the table above, we see that the coefficient on the enrolment variable has increased to 0.26 from approximately 0.25 and the p-value has fallen to 0.002 from 0.012 implying that it has become more significant. The coefficient on second variable of interest, credit, has fallen to 0.155 from approximately 0.21 and the p-value has increased to 0.19 from about 0.14. Thus, neutralizing the impact of the outlier shows even more clearly that it is secondary enrolment which has a positive significant impact on GDP growth. No evidence of any impact of credit on GDP growth is found. Another important point that should be noted from the table above is that the contradiction between the test of joint significance and that of the individual significance tests is no longer there. The p-value for the test of joint significance is0.0003 Read More