Mathematical Calculations for the Model Prediction - Math Problem Example

Model 1 1. PHGt = 1105.88 + 0.22Lnct – 11.08Gpt + 0.58Pnct – 5.88Puct + 6.91Pptt + 1.23Pdt – 28.03Pstt + 72.50Tt. Table 1: Summary Estimate Results Variable const Inc Gp Pnc Puc Ppt Pd Pn Ps T Expected sign +ve +ve -ve -ve -ve +ve +ve +ve -ve +ve Coefficient 1105.88 0.22 -11.09 0.58 -5.87 6.91 1.23 12.69 -28.03 72.50 Versus expected sign Expected Expected Expected Not Expected Expected Expected Expected Expected Expected Expected Std. Error 569.38 0.05 3.98 12.84 4.87 4.84 11.88 12.60 7.99 14.18 t-ratio 1.94 4.17 -2.79 0.05 -1.21 1.43 0.10 1.01 -3.51 5.11 N = 52, R2 = 0.99, and F (9, 42) = 530.82 2. The t-ratio is used to test the individual significance of the slope coefficients : : Test statistic used is: Table 2: Individual Significance of the Slope Coefficients Coefficient Inc Gp Pnc Puc Ppt Pd Pn Ps T Gretl t-ratio 4.17 -2.79 0.05 -1.21 1.43 0.10 1.01 -3.51 5.11 Critical t-ratio () 2.02 2.02 2.02 2.02 2.02 2.02 2.02 2.02 2.02 Computed t-ratio versus critical t-ratio Greater than critical Less than critical Less than critical Greater than critical Less than critical Less than critical Less than critical Less than critical Greater than critical Null hypothesis Reject Accept Accept Reject Accept Accept Accept Accept Reject Conclusion Significant Not significant Not significant Significant Not significant Not significant Not significant Not significant Significant 3. : the price of new cars (Pnc) and the price of used cars (Puc) have the same effect : the price of new cars (Pnc) and the price of used cars (Puc) do not have the same effect Test statistic used is: From Gretl, the computed t-ratio for Pnc 0.05 is and for Puc is -1.21. The critical t-ratio is 2.18. 0.05 < 2.18 and -1.21 > 2.18. Therefore, we 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. Testing the overall significance of the model 1; : , : ,, , , ,,,, or all are nonzero Test statistic: Critical value = = = 2.11 Computed value of F from gretl = 530.82 530.82 > 2.11, therefore, 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. The elasticity will be given by; The own-price elasticity = = -0.22 = 0.22. This indicates that a 1% increase in the price of gasoline will lead, on average, to a 0.22% 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 = = 0.95. This indicates that a 1% increase in income will lead, on average, to a 0.95% 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 = -0.38 + 0.99lnLnct + 0.06lnGpt – 0.16lnPnct – 0.49lnPuct +0.02lnPptt + 1.73lnPdt – 0.73lnPnt – 0.87lnPst + 0.04Tt. 6. Table 3: Sign and Magnitude of Coefficients Model 1 const Inc Gp Pnc Puc Ppt Pd Pn Ps T Coefficient 1105.88 0.22 -11.08 0.58 -5.87 6.91 1.23 12.69 -28.03 72.50 Model 2 const l_Inc l_Gp l_Pnc l_Puc l_Ppt l_Pd l_Pn l_Ps T Coefficient -0.38 0.99 0.06 -0.15 -0.49 0.02 1.73 -0.73 -0.87 0.04 Sign Model 1 +ve, and model 2 -ve Both +ve Model 1 +ve, and model 2 –ve Model 1 +ve, and model 2 –ve Both -ve Both +ve Both +ve Model 1 +ve, and model 2 -ve Both -ve Both +ve Magnitude Model 1 is bigger Model 1 is bigger Model 1 is bigger Model 1 is bigger Model 1 is bigger Model 1 is bigger Model 1 is bigger Model 1 is bigger Model 1 is bigger Model 1 is bigger Table 4: Individual Significance of the Slope Coefficients Coefficient l_Inc l_Gp l_Pnc l_Puc l_Ppt l_Pd l_Pn l_Ps T Gretl t-ratio -0.15 3.97 1.12 -0.58 -5.74 0.14 6.66 -2.75 -2.46 Critical t-ratio () 2.02 2.02 2.02 2.02 2.02 2.02 2.02 2.02 2.02 Computed t-ratio versus critical t-ratio Greater than critical Greater than critical Less than critical Greater than critical Less than critical Less than critical Greater than critical Less than critical Less than critical 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 variables Not significant variables 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 = = = 2.11 Computed value of F from gretl = 351.33 351.33 > 2.11, therefore, 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 5: Goodness of Fit and Overall Significance of Model Goodness of Fit Overall Significance of Model Model 1 R2 = 0.99, is of good fit All and/or each of the variables have an influence on the per household gas consumption Model 2 R2 = 0.99, is of good fit All and/or each of the variables have an influence on the per household gas consumption 7. The own-price elasticity = = 0.03. This indicates that a 1% increase in the price of gasoline will lead, on average, to a 0.03% 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 = = 1.16. This indicates that a 1% increase in income will lead, on average, to a 1.16% 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. : 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 Gretl t-ratio 0.14 6.66 -2.75 Critical t-ratio () 2.02 2.02 2.02 Computed t-ratio versus critical t-ratio Less than critical Greater than critical Less than critical Null hypothesis Do not reject Reject Do not reject Conclusion Not significant Significant Not significant 9. Figure 1: Model 1 Residual Plot Figure 2: Model 2 Residual Plot The residual plots have a similar pattern. They do not show any trend, and are disordered. This indicates that the regression models are of good fit. 10. Model 2 is preferable. A log-linear model is more sensitive and it is much easier to interpret. Cowan (1988) indicates that statistical transformation of variables in a model produces more meaningful and accurate results, which applies in model 2. Reference Cowan, G 1998. Statistical Data Analysis New York, Oxford University Press Read More