Question

1. Samirah has been appointed by a research institute to investigate what determined the price of automobiles in Ghana in 2019. She collected data on different variables including the price, mileage, repair record, weight and whether the car is a foreign or domestic one. Using Stata econometric software, she fitted a multiple linear regression and reported the following results to her superiors.

. reg price mpg rep78 foreign weight

      Source |       SS           df       MS      Number of obs   =        69

-------------+----------------------------------   F(4, 64)        =     15.82

       Model |   286761158         4 71690289.6   Prob > F        =    0.0000

    Residual |   290035800        64 4531809.38   R-squared       =    0.4972

-------------+----------------------------------   Adj R-squared   =    0.4657

       Total |   576796959        68 8482308.22   Root MSE        =    2128.8

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       price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]

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         mpg |   27.32371   77.53757     0.35   0.726    -127.5754    182.2228

       rep78 |   121.1322   334.3828     0.36   0.718    -546.8742    789.1387

     foreign |   3520.324    857.318     4.11   0.000     1807.634    5233.013

      weight |   3.565247   .6582976    5.42   0.000     2.250146    4.880347

       _cons |   -6729.56   3450.835    -1.95   0.056     -13623.4    164.2752

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1. Does mileage have a significant effect on the price of automobile? How do you know?
2. Interpret the coefficient on foreign. Is it statistically different from zero?
3. Do all the regressors have significant impact on the price of automobiles?
4. Interpret the value of the coefficient of determination.
5. Identify the relevant variables in determining the price of automobile.

Answer 1

(a) The mileage does not have a significant effect on the price, sinc ewe have the p-value on the coefficient of the mpg is more than usual 5% (0.05) and even 10% (0.10). The mileage variable is mpg, and test for the significance of effect of mpg on price would have the null hypothesis as and the alternate hypothesis as . The t-statistic would be as , which follows the t-distribution with df=n-k=69-5=64 (considering k being the number of parameters to be estimated, including the intercept). We have the t-statistic as or . The critical-t would be as . Since we have , we fail to reject the null and conclude that coefficient of mpg is not significant (ie the coefficient of mpg is not statistically different from zero).

(b) The test for the significance of effect of foreign on price would have the null hypothesis as and the alternate hypothesis as . The t-statistic would be as , which follows the t-distribution with df=n-k=69-5=64 (considering k being the number of parameters to be estimated, including the intercept). We have the t-statistic as or . The critical-t would be as . Since we have , we can reject the null and conclude that coefficient of mpg is significant (ie the coefficient of mpg is indeed statistically different from zero).

(c) The test of significance of all the independent variables would have the null that , and the alternate hypothesis that at least one of the coefficient is significant. The F-statistic would be as , which would follow the F-distribution with df=k-1,n-k=4,65 (considering k be the number of parameters to estimate, including the intercept). We have the F-statistic is given as . The critical F would be as . Since , we reject the null hypothesis, concluding that at least one of the coefficient is statistically significant.

(d) The coefficient of determination or R-squared is basically the proportion of variation in the dependent variable explained by the variation in the the independent variable. This means that, in this case, about 0.4972 or 49.75% of the variation in price is explained by the variations in the mpg, rep78, foreign and weight.

(e) The relevant variables would be foreign and weight. The test of individual significance is concluded on the basis of the p-value of the coefficient. If the p-value is less than the chosen significance level (usually of 5%), then we reject the null and conclude that the coefficient is significant. However, if the p-value is more than the chosen significance level, then we fail to reject the null and conclude that the coefficient is not-significant.

We have the p-value less than 0.05 for the coefficients of foreign and weight variables, while mpg and rep78 are not significant at 5% (and even at 10%). Note that the intercept coefficient may be meaningless or may have meaning, depending on the meaning of the value of price for the rest of the variables being zero. Hence, the relevant variables in determining the price of automobile would be foreign and weight variables.