Wooldridge IE AISE SSM ch12 - CHAPTER 12 SOLUTIONS TO...

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This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 65 CHAPTER 12 SOLUTIONS TO PROBLEMS 12.1 We can reason this from equation (12.4) because the usual OLS standard error is an estimate of / x SST σ . When the dependent and independent variables are in level (or log) form, the AR(1) parameter, ρ , tends to be positive in time series regression models. Further, the independent variables tend to be positive correlated, so ( x t x )( x t + j x ) – which is what generally appears in (12.4) when the { x t } do not have zero sample average – tends to be positive for most t and j . With multiple explanatory variables the formulas are more complicated but have similar features. If < 0, or if the { x t } is negatively autocorrelated, the second term in the last line of (12.4) could be negative, in which case the true standard deviation of 1 ˆ β is actually less than / x SST . 12.3 (i) Because U.S. presidential elections occur only every four years, it seems reasonable to think the unobserved shocks – that is, elements in u t – in one election have pretty much dissipated four years later. This would imply that { u t } is roughly serially uncorrelated. (ii) The t statistic for H 0 : = 0 is .068/.240 .28, which is very small. Further, the estimate ˆ = .068 is small in a practical sense, too. There is no reason to worry about serial correlation in this example. (iii) Because the test based on ˆ t is only justified asymptotically, we would generally be concerned about using the usual critical values with n = 20 in the original regression. But any kind of adjustment, either to obtain valid standard errors for OLS as in Section 12.5 or a feasible GLS procedure as in Section 12.3, relies on large sample sizes, too. (Remember, FGLS is not even unbiased, whereas OLS is under TS.1 through TS.3.) Most importantly, the estimate of is practically small, too. With ˆ so close to zero, FGLS or adjusting the standard errors would yield similar results to OLS with the usual standard errors. 12.5 (i) There is substantial serial correlation in the errors of the equation, and the OLS standard errors almost certainly underestimate the true standard deviation in ˆ EZ . This makes the usual confidence interval for EZ and t statistics invalid. (ii) We can use the method in Section 12.5 to obtain an approximately valid standard error. [See equation (12.43).] While we might use g = 2 in equation (12.42), with monthly data we might want to try a somewhat longer lag, maybe even up to g = 12.
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Wooldridge IE AISE SSM ch12 - CHAPTER 12 SOLUTIONS TO...

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