. Compute The Multiple Regression Of Per Capita Consumption Of Gasoline G Pop On All The Other Explanatory Variables Including The Time Trend And Report All Results. Do The Signs Of The Estimates Agree With Your Expectations
The true model underlying these data is y = xi + x2 + X3 + e.
a. Compute the simple correlations among the regressors.
b. Compute the ordinary least squares coefficients in the regression of y on a constant xi,x2, and X3.
c. Compute the ordinary least squares coefficients in the regression of y on a constant x\ and x2, on a constant x\ and x$, and on a constant x2 and x$.
d. Compute the variance inflation factor associated with each variable.
e. The regressors are obviously collinear. Which is the problem variable?
8. Consider the multiple regression of y on K variables X and an additional variable z. Prove that under the assumptions A1 through A6 of the classical regression model, the true variance of the least squares estimator of the slopes on X is larger when z is included in the regression than when it is not. Does the same hold for the sample estimate of this covariance matrix? Why or why not? Assume that X and z are nonstochastic and that the coefficient on z is nonzero.
9. For the classical normal regression model y = X/i + e with no constant term and K regressors, assuming that the true value of ft is zero, what is the exact expected value of F[K, n - K] = (R2/K)/[( 1 - R2)/(n - A")]?
10. Prove that £[b'b] = /}'/? + Ylk=i0-/kk) where b is the ordinary least squares estimator and k* is a characteristic root of X'X.
11. Data on U.S. gasoline consumption for the years 1960 to 1995 are given in Table F2.2.
a. Compute the multiple regression of per capita consumption of gasoline, G/pop, on all the other explanatory variables, including the time trend, and report all results. Do the signs of the estimates agree with your expectations?
b. Test the hypothesis that at least in regard to demand for gasoline, consumers do not differentiate between changes in the prices of new and used cars.
c. Estimate the own price elasticity of demand, the income elasticity, and the cross-price elasticity with respect to changes in the price of public transportation.
d. Reestimate the regression in logarithms so that the coefficients are direct estimates of the elasticities. (Do not use the log of the time trend.) How do your estimates compare with the results in the previous question? Which specification do you prefer?
e. Notice that the price indices for the automobile market are normalized to 1967, whereas the aggregate price indices are anchored at 1982. Does this discrepancy affect the results? How? If you were to renormalize the indices so that they were all 1.000 in 1982, then how would your results change?
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