Econometrics Data Analysis - Assignment Example

Part A - Report (a) From the descriptive statistics (see table we find that the suggested retail price for cars by manufacturers (MSRP) has considerable variability (high standard deviation relative to mean). From the correlation matrix (table 2) we find that the MSRP has a correlation coefficient of -0.6 with miles per gallon in the city (MPGC) implying a moderately strong negative association. MSRP has a correlation coefficient of 0.84 with horsepower (BHP) implying that higher horsepower in the sample is strongly associated with higher prices. Finally, BHP and MPGC have a negative association of approximately 70%. Therefore, if horsepower is higher by one unit, the millage of the car is expected to drop by 0.7. The scattergrams graphically shows that there is a negative association between MSRP and MPGC and a positive association between MSRP and BHP (see part B for the graphs). (b) The estimates of the constant in this section represent the mean suggested price when millage per gallon in the city is zero, while the coefficient on MPGC reflects the sensitivity of the suggested price to millage in the city. For all specifications, MPGC has a negative coefficient while the intercept is positive. However, MPGC is not significant for compact cars. For the rest of the specifications it is significant, though the magnitude of the coefficient differs considerably. The coefficient is -0.067 for the specification with all car types included. It is roughly -0.09 for four wheel drive vehicles and -0.10 for sport cars (see tables 3 – 6). Therefore, we find that drivers of compact cars are least sensitive to millage while drivers of sport cars are most sensitive to millage. However, the impact is negative, so lower the millage, higher the price. This is seemingly counterintuitive. (c) We include the dummy “sport” and the interaction term “bhp*sport” along with a constant and bhp in the regression (see table 7 for results). The effect will be the same if the “sport” has an insignificant coefficient so that the intercepts are the same for both all cars and sport cars and if the coefficient on the interaction term is statistically not different from the coefficient on bhp. The first part is satisfied as can be seen from the table. Visibly, while the coefficient on bhp is positive, it is negative for bhp_sport, the interaction term. To formally test the next part we use an F-test with the null hypothesis: bhp - bhp_sport = 0 The computed statistic is: F( 1, 247) = 87.78 and Prob > F = 0.0000 so, the test rejects the null even at 1%. Thus, there is evidence to support that bhp has a different effect on MSRP for sport cars compared to its effect on other cars (see table 8 for an alternative paired t-test, the results are the same). (d) For subcompact vehicles there is no statistical evidence of any relationship. Although the coefficient is negative, it is insignificant. For compact and sport vehicles there is a negative relationship, since larger tanks defeat the fundamental purpose of these vehicles. For all other types, the relationship is positive (see table 9). (e) The quadratic specification shows that the coefficient on weight squared although very small is negative and significantly different from zero (table 10). Additionally, the log-linear model shows that the log of msrp is significantly positively dependent upon the log of weight. Therefore, we conclude that the relationship between suggested price and weight of the vehicle is non-linear. Thus, the exact weight of the vehicle determines the magnitude of the coefficient for that weight. (f) The estimated relationship is (using only significant variables): MSRP = - 29175.3 + 148*BHP + 1870.41*CYL- 771.28*ECAP + 141.79*TANK + 118.25*COMPACT +857.93*LUXURY (see table 12) Then if weight = 2200, Luxury=1, compact=0, BHP=270, CYL=6, Ecap=2.8 and Tank =90, the MSRP can be estimated to be = $ 33466.606. (g) The estimated relationship is: Ln(MSRP) = 8.50 + 0.009*MPGC + 0.0003*Weight + 0.0038*BHP – 0.06*ECAP- 0.114*COMPACT +0.295*LUXURY +0.13*SPORT (see table 13) Since four wheel drive is not significant, replacing the rear wheel drive will not matter for the suggested retail price. From the given specifications, the estimated value of ln(MSRP) is: 10.463. Therefore, the estimated MSRP = exp(10.463) = 34996.38 (h) Sum of squared fitted residuals = 12155273806.93 Denominator = (1/251)* 12155273806.93 = 48427386 So, the dependent variable for the secondary regression is: = The explained sum of squares from the secondary regression is 193.369 Therefore, the Chi square statistic is 193.369/2 = 96.68 which is highly significant (p-value of 0.00). Therefore, the null hypothesis of homoscedasticity is rejected in favour of the alternative of multiplicative heteroscedasticity even at the 1% level. (i) Yes the dependence does seem to cause trouble. The trouble results from the fact that this dependence violates the OLS assumption of homoscedasticity. Consequently, although OLS estimates are still unbiased and consistent, they are no longer variance minimizing. The GLS estimator in the presence of heteroscedasticity leads to more efficient estimates in that the error variances are minimized. Since in model A, we use OLS inspite of heteroscedasticity, the error variances are large. In Model B, the GLS procedure is used and this leads to smaller error variances. (j) To understand how GLS works, it is imperative to understand the nature of the problem. Essentially, the error variance is not constant and instead we have evidence to support that the error variance is a multiplicative function of bhp. The reason that the problem is absolved through the GLS methodology is that in the process of dividing the model throughout by bhp, the error term is also divided by bhp. Consequently, the variance of this modified error term is no longer a multiplicative function of bhp. Appendix to Part A, tables and figures Table 1: Descriptive Statistics Variable Obs Mean Std. Dev. Min Max msrp 251 29190.64 14468.02 9590 93315 mpgc 251 20.00797 4.548399 11 52 bhp 251 198.6892 63.5854 70 460 Table 2: Correlation Matrix msrp mpgc bhp msrp 1 mpgc -0.6011 1 Bhp 0.8405 -0.6939 1 Figure 1: Scattergram of msrp and mpgc – negative association Figure 2: Scattergram of msrp and bhp – positive association Table 3: Sensitivity of MSRP to MPGC for all car types Regression Diagnostics Source SS df MS Number of obs 251 F( 1, 249) 213.37 Model 23.22374 1 23.22374 Prob > F 0 Residual 27.10187 249 0.108843 R-squared 0.4615 Adj R-squared 0.4593 Total 50.32562 250 0.201302 Root MSE 0.32991 Regression results lmsrp Coef. Std. Err. t P>|t| [95% Conf. Interval] mpgc -0.06701 0.004588 -14.61 0 -0.07604 -0.05797 _cons 11.51833 0.094118 122.38 0 11.33296 11.70369 Table 4: Sensitivity of MSRP to MPGC for compact cars Regression Diagnostics Source SS df MS Number of obs 49 F( 1, 47) 0.87 Model 0.040286 1 0.040286 Prob > F 0.3566 Residual 2.184838 47 0.046486 R-squared 0.0181 Adj R-squared -0.0028 Total 2.225124 48 0.046357 Root MSE 0.21561 Regression results lmsrp Coef. Std. Err. t P>|t| [95% Conf. Interval] mpgc -0.00578 0.006209 -0.93 0.357 -0.01827 0.006711 _cons 9.755185 0.16409 59.45 0 9.425079 10.08529 Table 5: Sensitivity of MSRP to MPGC for sport cars Regression Diagnostics Source SS df MS Number of obs 13 F( 1, 11) 16.1 Model 1.516992 1 1.516992 Prob > F 0.002 Residual 1.036615 11 0.094238 R-squared 0.5941 Adj R-squared 0.5572 Total 2.553606 12 0.212801 Root MSE 0.30698 Regression results lmsrp Coef. Std. Err. t P>|t| [95% Conf. Interval] mpgc -0.1072 0.026719 -4.01 0.002 -0.16601 -0.04839 _cons 12.50069 0.514757 24.28 0 11.36772 13.63367 Table 6: Sensitivity of MSRP to MPGC for four wheel drive vehicles Regression Diagnostics Source SS df MS Number of obs 17 F( 1, 15) 15 Model 0.738873 1 0.738873 Prob > F 0.0015 Residual 0.739101 15 0.049273 R-squared 0.4999 Adj R-squared 0.4666 Total 1.477974 16 0.092373 Root MSE 0.22198 Regression results lmsrp Coef. Std. Err. t P>|t| [95% Conf. Interval] mpgc -0.08678 0.02241 -3.87 0.002 -0.13454 -0.03901 _cons 11.71772 0.331319 35.37 0 11.01153 12.42391 Table 7: OLS with sports dummy and interaction term Regression Diagnostics Source SS df MS Number of obs 251.00   F( 3, 247) 217.52 Model 37962000000 3 12654000000 Prob > F 0.00 Residual 14369000000 247 58173133.4 R-squared 0.73   Adj R-squared 0.72 Total 52331000000 250 209323468 Root MSE 7627.10             Regression results msrp Coef. Std. Err. t P>t [95% Conf. Interval] sport 6978.013 6827.46 1.02 0.308 -6469.45 20425.48 bhp 207.0445 8.57804 24.14 0 190.149 223.9399 bhp_sport -54.4295 23.62503 -2.3 0.022 -100.962 -7.89725 _cons -11507.2 1736.199 -6.63 0 -14926.9 -8087.59 Table 8: Paired t-test for mean differences Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] bhp 251 198.6892 4.013475 63.5854 190.7847 206.5938 bhp_sport 251 14.71713 4.215987 66.7938 6.413751 23.02051     diff 251 183.9721 4.471439 70.84091 175.1656 192.7786     Results of paired t-test mean(diff) = mean(bhp - bhp_sport) t = 41.1438 Ho: mean(diff) = 0 degrees of freedom = 250     Ha: mean(diff) < 0 Ha: mean(diff) != 0 Ha: mean(diff) > 0 Pr(T < t) = 1.0000 Pr(T > t) = 0.0000 Pr(T > t) = 0.0000 Table 9 Regression Diagnostics Source SS df MS Number of obs 251 F( 5, 245) 35.17 Model 28413.23 5.00 5682.65 Prob > F 0 Residual 39581.86 245.00 161.56 R-squared 0.4179 Adj R-squared 0.406 Total 67995.10 250.00 271.98 Root MSE 12.711 Regression results tank Coef. Std. Err. t P>t [95% Conf. Interval] subcom -23.1275 12.77526 -1.81 0.071 -48.2908 2.035895 compact -21.9777 2.223887 -9.88 0 -26.358 -17.5973 fullsize 14.8013 3.42724 4.32 0 8.050687 21.55191 luxury 4.362145 1.957494 2.23 0.027 0.506482 8.217809 sport -8.79206 3.751819 -2.34 0.02 -16.182 -1.40213 _cons 83.13745 1.283962 64.75 0 80.60844 85.66646 Table 10 : a quadratic specification Regression Diagnostics Source SS df MS Number of obs 251 F( 2, 248) 88.8 Model 421792.29 2.00 210896.15 Prob > F 0 Residual 588983.47 248.00 2374.93 R-squared 0.4173 Adj R-squared 0.4126 Total 1010775.76 250.00 4043.10 Root MSE 48.733 Regression results bhp Coef. Std. Err. t P>|t| [95% Conf. Interval] weight 0.419678 0.0720428 5.83 0 0.277784 0.561572 weight2 -0.0001 0.0000214 -4.14 0 -0.00013 0.00 _cons -234.473 59.47413 -3.94 0 -351.612 -117.334 Table 11: a logarithmic specification Regression Diagnostics Source SS df MS Number of obs 251 F(1, 249) 251.78 Model 12.44 1.00 12.44 Prob > F 0 Residual 12.31 249.00 0.05 R-squared 0.5028 Adj R-squared 0.5008 Total 24.75 250.00 0.10 Root MSE 0.22231 Regression results lbhp Coef. Std. Err. t P>t [95% Conf. Interval] lweight 1.13 0.0713562 15.87 0 0.991705 1.27 _cons -3.07092 0.5241278 -5.86 0 -4.10321 -2.03863 Table 12 Regression Diagnostics Source SS df MS Number of obs 251 F( 14, 236) 77.24 Model 42956000000 14.00 3068300000 Prob > F 0 Residual 9375200000 236.00 39725549.80 R-squared 0.8208 Adj R-squared 0.8102 Total 52331000000 250.00 209323468 Root MSE 6302.8 Regression results msrp Coef. Std. Err. t P>t [95% Conf. Interval] mpgc 292.353 175.4534 1.67 0.097 -53.3019 638.008 weight 4.477061 3.769196 1.19 0.236 -2.94851 11.90263 bhp 148.4532 15.1287 9.81 0 118.6486 178.2577 cyl 1870.41 775.5174 2.41 0.017 342.5886 3398.231 ecap -2771.28 1206.424 -2.3 0.022 -5148.02 -394.547 tank 141.7905 58.5816 2.42 0.016 26.38085 257.2002 subcom 2160.595 6440.084 0.34 0.738 -10526.8 14847.99 compact 3118.253 1432.848 2.18 0.031 295.4471 5941.059 fullsize -1459.22 1808.821 -0.81 0.421 -5022.72 2104.279 luxury 8857.931 1275.388 6.95 0 6345.331 11370.53 sport 2903.874 2350.789 1.24 0.218 -1727.34 7535.085 awd 897.0138 1535.115 0.58 0.56 -2127.27 3921.294 fwd -1594.05 1163.624 -1.37 0.172 -3886.47 698.3646 fourwd 542.0779 1989.757 0.27 0.786 -3377.88 4462.033 _cons -29175.3 7033.652 -4.15 0 -43032.1 -15318.6 Table 13 Regression Diagnostics Source SS Df MS Number of obs 251 F( 14, 236) 128.88 Model 44.50 14.00 3.18 Prob > F 0 Residual 5.82 236.00 0.02 R-squared 0.8843 Adj R-squared 0.8775 Total 50.33 250.00 0.20 Root MSE 0.15705 Regression results lmsrp Coef. Std. Err. T P>t [95% Conf. Interval] mpgc 0.009086 0.0043719 2.08 0.039 0.000473 0.017699 weight 0.000379 0.0000939 4.04 0 0.000194 0.000564 bhp 0.003779 0.000377 10.02 0 0.003036 0.004521 cyl 0.032519 0.0193242 1.68 0.094 -0.00555 0.070589 ecap -0.06251 0.0300615 -2.08 0.039 -0.12174 -0.00329 tank 0.001283 0.0014597 0.88 0.381 -0.00159 0.004158 subcom 0.046141 0.1604731 0.29 0.774 -0.27 0.362283 compact -0.11434 0.0357035 -3.2 0.002 -0.18468 -0.04401 fullsize 0.019296 0.0450719 0.43 0.669 -0.0695 0.108091 luxury 0.294746 0.0317799 9.27 0 0.232138 0.357355 sport 0.133116 0.0585766 2.27 0.024 0.017716 0.248516 awd 0.011716 0.0382518 0.31 0.76 -0.06364 0.087075 fwd -0.04638 0.028995 -1.6 0.111 -0.1035 0.010741 fourwd 0.002566 0.0495805 0.05 0.959 -0.09511 0.100243 _cons 8.50468 0.1752635 48.53 0 8.1594 8.849961 Table 14 Table 15 Regression Diagnostics Source SS df MS Number of obs 251 F( 6, 244) 19.47 Model 109317.55 6.00 18219.59 Prob > F 0 Residual 228378.63 244.00 935.98 R-squared 0.3237 Adj R-squared 0.3071 Total 337696.19 250.00 1350.78 Root MSE 30.594 Regression results msrp_bhp Coef. Std. Err. t P>t [95% Conf. Interval] mpgc_bhp 432.952 78.8316 5.49 0 277.6748 588.2293 weight_bhp 14.09132 3.01989 4.67 0 8.142934 20.0397 cyl_bhp 2182.996 665.2211 3.28 0.001 872.6873 3493.305 ecap_bhp -5641.86 1036.164 -5.44 0 -7682.82 -3600.89 tank_bhp 50.53955 55.24422 0.91 0.361 -58.2769 159.356 bhp_inv -36031.6 3832.397 -9.4 0 -43580.4 -28482.8 _cons 176.9396 13.05329 13.56 0 151.2281 202.6511 Part B: Stata log – commands and output Read More