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Interactive Exercise 1 |
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Heteroskedasticity. Heteroskedasticity is extremely unlikely in this example for a number of reasons. First, it is a time-series model, meaning that huge cross-sectional differences in size do not exist. Second, while there certainly was growth in passbook deposits during the 1970s, that growth was not exceptional. Indeed, competition from money market certificates and other assets actually decreased passbook deposits in the late 1970s. Finally, there is no indication of any dramatic change in the quality of measurement of the data.
As a result, many econometricians would not run a Park test on this example, feeling that even a significant result would indicate impure, rather than pure, heteroskedasticity. However, it would be good to get into the habit of examining the residuals of the final specification plotted against a suspected proportionality factor and then running a Park test if the plot indicates heteroskedasticity. Look over your independent variables; which of them would be candidates to be a potential proportionality factor Z? One good choice is to use disposable income because income is the best measure of size available to us. The interest rate variables are much less likely to be proportionality factors, mainly because of their fluctuations. Fortunately, plots of the residuals of the final specifications most frequently chosen by previous users of this example give no indications of pure heteroskedasticity with respect to disposable income, and so we should not seriously consider adjusting for heteroskedasticity. As it turns out, a Park test using the log of QYDUS to explain the log of the squared residuals from regression run 4 shows no evidence of heteroskedasticity at all (tZ = 0.80).
Serial Correlation. As you have surely noted, almost all of the regression runs produce Durbin-Watson statistics that indicate positive serial correlation (or inconclusive DW statistics quite close to the lower limit). Does this mean that there is serial correlation in the error term? How do we tell whether we have pure or impure serial correlation? Should we adjust the equation with Generalized Least Squares?
One way to determine whether the serial correlation indicated by the Durbin-Watson statistic is pure or impure is to examine the degree to which the model matches the underlying theory. The last part of the decade was marked by turmoil in the S & L industry (deregulation, high interest rates, some closures of S & Ls, and increasing competition for passbook accounts from other kinds of savings vehicles), but only the MMC dummy attempts to account for this turmoil. To most observers, such a set of circumstances is an indication that the residual pattern picked up by the Durbin-Watson statistic might be due to impure serial correlation. That is, the model has not properly captured all the factors that were influencing passbook deposits in the last few years of the decade. Such a result is a strong indication that the introduction of a more sophisticated explanatory variable (somehow measuring increasing competition for passbook accounts from other kinds of savings assets) should be attempted before GLS is run.
Since such a variable is not available in the present data set, we went ahead and ran GLS on the most frequently chosen "best" equation, regression run 4. The results of that estimation are in Table 3.
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Dependent Variable: QDPASS Method: Least Squares Sample(adjusted): 1970:2 1979:4 Included observations: 39 after adjusting endpoints Convergence achieved after 143 iterations
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Note that the estimated coefficient of the spread variable is very similar in both the GLS and the OLS versions of run 4. In addition, note that even when the supposed serial correlation is "corrected," the Durbin-Watson d is only 0.97, even though it is biased toward 2 after GLS is run. This is evidence of an omitted variable or higher-order serial correlation. Finally, note that all the slope coefficients are still significantly different from zero in the hypothesized directions after the GLS adjustment except for the coefficient of MMCDUM, which is just barely insignificant in the expected direction. As a result, we would prefer sticking with the OLS estimate at present and attempting to find the possible omitted variable. Even if that variable is not found, it appears that serial correlation is doing little harm to the OLS results. Document such situations, including both the OLS and GLS results.
Reporting Your Final Equation. When reporting your "final" equation, remember also to report (say, in an appendix or a footnote) the equations estimated previously and subsequently (the order of estimation doesn't matter) to the one you chose. This practice allows readers to make up their own minds as to whether your choice was one with which they would agree. Such complete reporting will also let readers be able to judge for themselves the extent to which the reported t-scores are likely to follow the t-distribution and be comparable to the critical t-values found in the t-table.
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