Gretl Econometrics - Math Problem Example

Gretl Econometrics Name Course Tutor Unit Code Date Model 1 1. PHGt = 1105.88 + 0.216Lnct – 11.084Gpt + 0.577Pnct – 5.875Puct + 6.907Pptt + 1.229Pdt – 28.028Pstt + 72.504Tt. Summary Results Table 1.1: Summary Results Variable const Inc Gp Pnc Puc Ppt Pd Pn Ps T Expected sign Positive Positive Negative Negative Negative Positive Positive Positive Negative Positive Coefficient 1105.88 0.215749 -11.0839 0.577349 -5.87463 6.90726 1.22888 12.6905 -28.0278 72.5037 Versus expected sign Positive Expected Positive Expected Negative Expected Positive Not Expected Negative Expected Positive Expected Positive Expected Positive Expected Negative Expected Positive Expected Std. Error 569.379 0.051762 3.97812 12.8441 4.87032 4.83613 11.8818 12.598 7.99625 14.1828 t-ratio 1.9423 4.1681 -2.7862 0.045 -1.2062 1.4283 0.1034 1.0073 -3.5051 5.1121 const Inc Gp Pnc Puc Ppt Pd Pn Ps T Coefficient 1105.88 0.215749 -11.0839 0.577349 -5.87463 6.90726 1.22888 12.6905 -28.0278 72.5037 Std. Error 569.379 0.051762 3.97812 12.8441 4.87032 4.83613 11.8818 12.598 7.99625 14.1828 t-ratio 1.9423 4.1681 -2.7862 0.045 -1.2062 1.4283 0.1034 1.0073 -3.5051 5.1121 N = 52, R2 = 0.991, and F (9, 42) = 530.818 2. Individual Significance of the Slope Coefficients To test the individual significance of the slope coefficients, the computed t-ratio is compared to the critical t-ratio. Below is the hypothesis testing for individual significance of the slope coefficients; : : Test statistic used is: The table below summarises the individual significance of the slope coefficients. Table 1.2: Individual Significance of the Slope Coefficients Coefficient Inc Gp Pnc Puc Ppt Pd Pn Ps T Computed t-ratio (gretl) 4.1681 -2.7862 0.045 -1.2062 1.4283 0.1034 1.0073 -3.5051 5.1121 Critical t-ratio () 2.018 2.018 2.018 2.018 2.018 2.018 2.018 2.018 2.018 Comparison 4.1681 > 2.018 -2.7862 < -2.018 0.045 < 2.018 -1.2062 > -2.018 1.4283 < 2.018 0.1034 < 2.018 1.0073 < 2.018 -3.5051 < -2.018 5.1121 > 2.018 Null hypothesis Reject Do not reject Do not reject Reject Do not reject Do not reject Do not reject Do not reject Reject Conclusion Significant Not significant Not significant Significant Not significant Not significant Not significant Not significant Significant 3. To determine whether the price of new cars (Pnc) and the price of used cars (Puc) have the same effect, : Pnc and Puc have the same effect : Pnc and Puc do not have the same effect Test statistic used is: The computed t-ratio from gretl for Pnc is 0.045 and for Puc is -1.2062. 0.045 is less than the critical t-ratio whereas -1.2062 is greater than the critical t-ratio. Reject the null hypothesis and conclude that the price of new cars (Pnc) and the price of used cars (Puc) do not have the same effect. 4. Overall Significance of the Model Testing the overall significance of the model 1; : , : ,, , , ,,,, or all are nonzero Test statistic: Critical value = = = 2.112 Computed value of F from gretl = 530.818 530.818 > 2.112, we reject the null hypothesis at 5% significance level and conclude that all and/or each of the variables have an influence on the per household gas consumption. 5. Elasticity The elasticity will be given by; The own-price elasticity = = -0.223 = 0.223. This indicates that a 1% increase in the price of gasoline will lead, on average, to a 0.223% increase in the per household gas consumption. Seeing as the elasticity is less than 1, this indicates that gasoline is a necessity, not a luxury. The income elasticity = = 0.949. This indicates that a 1% increase in income will lead, on average, to a 0.949% increase in the per household gas consumption. Seeing as the elasticity is less than 1, this indicates that gasoline is a necessity, not a luxury. Model 2 lnPhgt = -0.379 + 0.993lnLnct + 0.061lnGpt – 0.155lnPnct – 0.489lnPuct +0.019lnPptt + 1.732lnPdt – 0.73lnPnt – 0.868lnPst + 0.038Tt. 6. Comparison Table 2.1: Sign and Magnitude of Coefficients Model 1 const Inc Gp Pnc Puc Ppt Pd Pn Ps T Coefficient 1105.88 0.215749 -11.0839 0.577349 -5.87463 6.90726 1.22888 12.6905 -28.0278 72.5037 Model 2 const l_Inc l_Gp l_Pnc l_Puc l_Ppt l_Pd l_Pn l_Ps T Coefficient -0.37944 0.992991 0.060518 -0.15472 -0.48909 0.01927 1.73206 -0.72954 -0.86798 0.037972 Sign 1 +ve, and 2 -ve Both +ve 1 -ve, and 2 +ve 1 +ve, and 2 -ve Both -ve Both +ve Both +ve 1 +ve, and 2 -ve Both -ve Both +ve Magnitude 1 is larger 1 is smaller 1 is smaller 1 is larger 1 is smaller 1 is larger 1 is smaller 1 is larger 1 is smaller 1 is larger Table 2.2: Individual Significance of the Slope Coefficients Coefficient l_Inc l_Gp l_Pnc l_Puc l_Ppt l_Pd l_Pn l_Ps T Computed t-ratio (gretl) -0.1505 3.966 1.1205 -0.5795 -5.7405 0.1412 6.6647 -2.7523 -2.4595 Critical t-ratio () 2.018 2.018 2.018 2.018 2.018 2.018 2.018 2.018 2.018 Comparison -0.1505 > -2.018 3.966 > 2.018 1.1205 < 2.018 -0.5795 > -2.018 -5.7405 < -2.018 0.1412 < 2.018 6.6647 > 2.018 -2.7523 < -2.018 -2.4595 < - 2.018 Null hypothesis Reject Reject Do not reject Reject Do not reject Do not reject Reject Do not reject Do not reject Conclusion Significant Significant Not significant Significant Not significant Not significant Significant Not significant Not significant Significant Not significant Model 1 Lnc, Puc, T Gp, Pnc, Ppt, Pd, Pn, Ps Model 2 Ln(Lnc), ln(Puc), ln(Gp), ln(Pn) Ln( Pnc), ln(Ppt),ln( Pd), ln(Ps),T Testing the overall significance of the model 2; : , : ,, , , ,,,, or all are nonzero Test statistic: Critical value = = = 2.112 Computed value of F from gretl = 351.326 351.326 > 2.112, we reject the null hypothesis at 5% significance level and conclude that all and/or each of the variables have an influence on the per household gas consumption. Table 2.3: Goodness of Fit and Overall Significance of Model Goodness of Fit Overall Significance of Model Model 1 R2 = 0.991, is of good fit All and/or each of the variables have an influence on the per household gas consumption Model 2 R2 = 0.987, is of good fit All and/or each of the variables have an influence on the per household gas consumption 7. Elasticity The own-price elasticity = = 0.033. This indicates that a 1% increase in the price of gasoline will lead, on average, to a 0.033% increase in the per household gas consumption. Seeing as the elasticity is less than 1, this indicates that gasoline is a necessity, not a luxury. The income elasticity = = 1.162. This indicates that a 1% increase in income will lead, on average, to a 1.162% increase in the per household gas consumption. Seeing as the elasticity is less than 1, this indicates that gasoline is a luxury, not a necessity. 8. Individual Significance of the Three Aggregates : The three aggregates are not significant : The three aggregates are significant Table 2.4: Individual Significance of the Three Aggregates Coefficient l_Pd l_Pn l_Ps Computed t-ratio (gretl) 0.1412 6.6647 -2.7523 Critical t-ratio () 2.018 2.018 2.018 Comparison 0.1412 < 2.018 6.6647 > 2.018 -2.7523 < -2.018 Null hypothesis Do not reject Reject Do not reject Conclusion Not significant Significant Not significant 9. Residual Plots Figure 2.1: Model 1 Residual Plot Figure 2.2: Model 2 Residual Plot The residual plots have a somewhat similar pattern. They do not show any trend, and are disordered indicating that the regressions are of good fit. 10. Model Comparison Choosing between model 1 and model 2, model 2 is preferable. A log-linear model 2 is more sensitive and it is much easier to interpret. Also, generally, in statistics transformation of variables in a model produces more meaningful and accurate results and this is true for model 2. References Freedman, D.H et al, 2007, Statistics, 4th edn, New York, W.W Norton & Company. Read More