Question

1. Evaluate this regression with respect to its economic meaning, overall fit, and the signs and significance of the individual coefficients. What econometric problems does this regression have? Explain. Suggest possible remedies.

A modern consumption function is estimated using quarterly data for 1959:2 to 2000:4, a total of 166 observations.

Consumption Real Personal Consumption Expenditures (Billions of 1996 dollars)

Income                        Real Disposable Income (Billions of 1996 dollars)

Sentiment       Index of Consumer Sentiments (1966q1=100)

RealR              Real 3 Month Treasury Bill rate (percent)

Wealth            Household net worth (Billions of dollars)

Consumption	Coefficient	Std. Error	t-Statistic
Income	0.781254	0.009032
Sentiment	1.597824	0.329168
RealR	-10.95494	1.493983
Wealth _cons	0.020927 52.97639	0.001179 42.76011
R-squared	0.999204
Adjusted R-squared	0.999184
F-statistic	50518.28
Durbin-Watson stat	0.607651

Answer 1

The estimated model is given. The individual statistical significance and economic meaning is given as below. The t-test for individual significance would have the test statistic as , which will follow t-distribution with df=n-k=161. For 5% significance level, the critical t would be as . Note that the null is and alternate is , and if , then we would fail to reject the null, and otherwise if , we may reject the null.

The model suggests that the consumption

• increased on average by $0.781254 (approx $0.78) for a unit increase in income (since both C and I are measured in billions). The estimated mpc is hence 0.781254. The sign of the coefficient seems justified since increase in income is increasing the income. The t-statistic would be as . Since , we may reject the null and conclude that the coefficient is statistically significant.
• increased on average by $1.597824 billion for a unit increase in consumer sentiment (index). The sign of the coefficient seems justified as the consumer sentiment, ie the consumer confidence on market and economy increases, so does the consumption. The t-statistic would be as . Since , we may reject the null and conclude that the coefficient is statistically significant.
• decreased on average by $10.95494 billion for a unit increase in Real 3 Month Treasury Bill rate (1% increase in the rate). The sign of the coefficient seems justified, since the variable is a proxy of the interest rate, and increase in interest rate would decrease the demand for liquid money and hence the consumption (ceteris paribus). The t-statistic would be as . Since , we may reject the null and conclude that the coefficient is statistically significant.
• increased on average by $0.020927 (approx $0.02) for a unit increase in household wealth. The t-statistic would be as . Since , we may reject the null and conclude that the coefficient is statistically significant.
• is $52.97639 billion if all the variable are zero. This coefficient is not economically meaningful since index of sentiment of index cannot be zero. Also, zero T-bill rate, welath and income are not practically feasible (yet theoretically they might be, but only theoretically). The t-statistic would be as . Since , we fail to reject the null and conclude that the coefficient is not statistically significant.

The overall fit is suggested by R-squared and (a bit better by) adjusted R-squared. The R-square are high enough, suggesting that a major proportion (more than 99%) of the variation in consumption is explained by variations in the independent variables. The test of overall significance would have the null and alternate that at least one of the coefficient is statistically significant (statistically different from zero). The F-statistic is given as , and would follow the F-distribution with df=k,n-k-1=4,161. The critical F at 5% significance would be as . Since , we may reject the null, and conclude that the model is overall significant (not all or at least one of the variables are useful in explaining the dependent variable).

The econometric problem posed by the model is that, the model suffers from (serial) autocorrelation (among residuals). This is suggested by the low durbin-watson test statistic d, since d<2. The autocorrelation means that regressors are related as (w is normal - IID with zero mean), and as a consequence, the R-square and individual t-statistics would be useless, and usually overestimated (more than the true parameters). The parameters are also inconsistent, ie the biasness is not removed even if we increase the sample size.

A remedy is to estimate the rho, which can be approxed as , for d be the DW statistic. Then, we have or or or or or or . In this case, we have the residual wt, which is normal - IID with mean zero. Hence, transforming the dependent variable by and the independent variables by after estimating the rho, we may run the regression as , and would have the required true coefficients and their individual significance, along with the true overall significance.