745 part 15 generalized regression applications

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Part 15: Generalized Regression Applications Behavior of OLS Implications for conventional estimation technique and hypothesis testing: 1 . b is still unbiased. Proof of unbiasedness did not rely on homoscedasticity 2. Consistent? We need the more general proof. Not difficult. 3. If plim b = , then plim s2 = 2 (with the normalization). ™  8/45
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Part 15: Generalized Regression Applications Inference Based on OLS What of s2( XX )-1 ? Depends on XX - XX . If they are nearly the same, the OLS covariance matrix is OK. When will they be nearly the same? Relates to an interesting property of weighted averages. Suppose i is randomly drawn from a distribution with E[i] = 1. Then, (1/n)ixi2  E[x2] and (1/n)iixi2  E[x2]. This is the crux of the discussion in your text. ™  9/45
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Part 15: Generalized Regression Applications Inference Based on OLS VIR : For the heteroscedasticity to be substantive wrt estimation and inference by LS, the weights must be correlated with x and/or x2. (Text, page 272.) If the heteroscedasticity is important. Then, b is inefficient. The White estimator. ROBUST estimation of the variance of b . Implication for testing hypotheses. We will use Wald tests. Why? ( ROBUST TEST STATISTICS ) ™  10/45
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Part 15: Generalized Regression Applications Finding Heteroscedasticity The central issue is whether E[2] = 2i is related to the xs or their squares in the model. Suggests an obvious strategy. Use residuals to estimate disturbances and look for relationships between ei2 and xi and/or xi2. For example, regressions of squared residuals on xs and their squares. ™  11/45
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Part 15: Generalized Regression Applications Procedures White’s general test : nR2 in the regression of ei2 on all unique xs, squares, and cross products. Chi- squared[P] Breusch and Pagan’s Lagrange multiplier test . Regress {[ei2 /( ee /n)] – 1} on Z (may be X ). Chi-squared. Is nR2 with degrees of freedom rank of Z . (Very elegant.) Others described in text for other purposes. E.g., groupwise heteroscedasticity. Wald, LM, and LR tests all examine the dispersion of group specific least squares residual variances. ™  12/45
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Part 15: Generalized Regression Applications Estimation: WLS form of GLS General result - mechanics of weighted least squares. Generalized least squares - efficient estimation. Assuming weights are known. Two step generalized least squares: p Step 1: Use least squares, then the residuals to estimate the weights. p Step 2: Weighted least squares using the estimated weights. p (Iteration: After step 2, recompute residuals and return to step 1. Exit when coefficient vector stops changing.) ™  13/45
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Part 15: Generalized Regression Applications Autocorrelation The analysis of “autocorrelation” in the narrow sense of correlation of the disturbances across time largely parallels the discussions we’ve already done for the GR model in general and for heteroscedasticity in particular. One difference is that the relatively crisp results for the model of heteroscedasticity are replaced with relatively fuzzy, somewhat imprecise results here.
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