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How do I know the specification in (3.2) is a chi-square distribution with df k-1? and why is it the case that the asymptotic variance of $\hat{\beta}_{FD}- \hat{\beta}_{OLS}$ is simply the difference of asymptotic variance of $\hat\beta_{FD}$ and that of $\hat\beta_{OLS}$?

*FD refers to first-difference method

Could someone help me with it? Still struggling figuring it out. Many thanks.

The following is the material from William H. Greene's Econometric analysis 7th edition, p.379

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  • $\begingroup$ I think most of the answers to your question are here: en.wikipedia.org/wiki/Durbin%E2%80%93Wu%E2%80%93Hausman_test $\endgroup$ – Andrew M Jun 2 '19 at 14:05
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    $\begingroup$ Two things to clarify: (i) What is $k$? (ii) What does $\hat{avar}$ mean? About (i), in (3.2), $k-1$ is supposed to be the rank of the avar matrix. For (ii), it looks like $\hat{avar}$ is the estimated asymptotic variance of $\sqrt{nT}$ times the argument. Regarding your last question, "$avar(a-b) = avar(a)-avar(b)$" is the heart of Hausman's finding, where $b$ is an efficient estimator. $\endgroup$ – chan1142 Jun 2 '19 at 14:30
  • $\begingroup$ First of all, I want to thank you both for answering me, especially when I'm new to this platform. And I would like to clarify that in the context of my question, k refers to the number of variables in the regression model, with the constant included, while avar hat refers to the estimated asymptotic variance of square root of nT multiplied by the argument mentioned in the question, which is to be used when performing Hausman test. $\endgroup$ – Jeff Jun 3 '19 at 3:37
  • $\begingroup$ As from the the wiki page describing the Hausman test, I've checked, but confused. For this reason, I read William H. Greene's Econometric analysis 7th edition, p.379, in which Hausman test is performed to determine whether to adopt random effect or fixed effect method, to see if it helps. What I find out is that the author of the book also mentioned the heart of Hausman's finding, the covariance of an efficient estimator with its difference from an inefficient estimator is zero. In this sense, "b" should be an inefficient estimator one, right? $\endgroup$ – Jeff Jun 3 '19 at 4:13
  • $\begingroup$ Also, by comparing the results shown in Greene's book and the wiki page, I just failed to connect the concepts conveyed in both sides. Namely, in the covariance matrix of the difference vector, which estimator should be the first one? Efficient or inefficient one? In the variance part of the wald statistic, the order should be the same as the the covariance matrix of the difference vector, but it turns out not the case in the wiki page? $\endgroup$ – Jeff Jun 3 '19 at 4:42

The first screenshot (reference?) assumes that OLS is efficient when $\alpha_i=0$ for all $i$. That is, it is assumed that the idiosyncratic errors are $iid$ so OLS achieves asymptotic efficiency under the null hypothesis. Let $(\hat\alpha_{ols}, \hat\beta_{ols}')'$ be the OLS estimator, where the dimension of $\hat\beta_{ols}$ is $k-1$. Then Hausman's results imply that $$avar(\hat\beta_{fd} - \hat\beta_{ols}) = avar(\hat\beta_{fd}) - avar(\hat\beta_{ols}).$$ [Algebra is given in Greene's textbook. You shouldn't switch the order of the RHS terms as it should be positive semidefinite.] Note that $$avar(\hat\beta_{fd} - \hat\beta_{ols}) = avar(\hat\beta_{ols} - \hat\beta_{fd})$$ so it does not matter which one comes first. The key implication of Hausman's result is that the variance of the difference equals the difference of the variances (if one is efficient). The result that $nT (\hat\beta_{fd} - \hat\beta_{ols})' \hat{avar}(\hat\beta_{fd} - \hat\beta_{ols})^{-1} (\hat\beta_{fd} - \hat\beta_{ols}) \to \chi^2_{k-1}$ is rather straightforward because the dimension of $\hat\beta$'s is $(k-1)\times 1$. [If $\xi_n \to N(0, C)$, where $\xi_n$ is $m\times 1$, then $\xi_n' \hat{C}^{-1} \xi_n \to_d \chi^2_{m}$, where $\hat{C}$ is a consistent estimator of $C$.] The $nT$ factor is there because $avar(z)$ is the asymptotic variance of $\sqrt{nT}$ times $z$.

Hausman's test is a general tool, which can be used for any efficient estimator. In your original question, OLS is efficient (because it is assumed so by the author of the document though it is not clearly stated so) so Hausman's test can be used for the comparison of an inefficient estimator (FD) and an efficient estimator (OLS). In the RE/FE testing, RE is efficient by assumption so Hausman's test can be used. There are lots of other examples of Hausman tests.

BTW, the assumption that the idiosyncratic errors are $iid$, which is generally required for the efficiency of OLS or RE, is very strong, and is not satisfied by typical panel applications. For the panel model $y_{it} = \alpha_i + x_{it}\beta + e_{it}$, where $x_{it}$ is strictly exogenous, $e_{it}$ represents factors other than time-invariant factors ($\alpha_i$) and $x_{it}$. I see no strong reasons why $e_{it}$ should be serially independent. (This is similar to a time-series model with strictly exogenous regressors. The Newey-West (heteroskedasticity and autocorrelation consistent) estimator is developed to handle this case.) The Hausman test using OLS and FD is based on the strong assumption that there is no heteroskedasticity or autocorrelation in $e_{it}$. If $e_{it}$ is serially correlated or heteroskedastic, OLS is not efficient under the null hypothesis that $\alpha_i$ is constant, and the mentioned Hausman test fails.

Simlarly, the Hausman test of RE vs FE relies on the efficiency of RE, which again requires that $e_{it}$ is $iid$ (loosely speaking). But again there are no strong reasons why $e_{it}$ should be so. (It's different for dynamic models which require serial independence for identification.)

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  • $\begingroup$ Thank you for great insight on my question, you really help me a lot. But just one last question. I understand the fact that Hausman's heart finding is that the covariance of an efficient estimator with its difference from an inefficient estimator is zero, and the algebra is given in the textbook, but how to interpret it intuitively in plain language. Is it just because it look simpler without having to take into consideration of the interaction terms? $\endgroup$ – Jeff Jun 5 '19 at 14:53
  • $\begingroup$ In Hausman's original paper published in 1978, he mentioned that if an efficient estimator is correlated with its difference from an inefficient estimator, then a linear combination of an efficient estimator and its difference from an inefficient estimator would lead to an unbiased estimator, say beta star, which would have smaller variance than the originally assumed efficient estimator. What does this imply? How do I convince a friend who does not know Hausman well that this finding is so important that it is Hausman's heart finding? $\endgroup$ – Jeff Jun 5 '19 at 15:02
  • $\begingroup$ Many thanks again! $\endgroup$ – Jeff Jun 5 '19 at 15:02
  • $\begingroup$ What my concern here is that I may be able to show the algebra, the derivation, and tell people from the textbook I know this part is Hausman's heart finding, but can't interpret anything more than this. $\endgroup$ – Jeff Jun 5 '19 at 15:20
  • $\begingroup$ You seem to be interested in intuitively understanding Hausman’s result. I think your second comment on Hausman (1978) is nice. If $a$ and $b-a$ are correlated, then $a+h\times (b-a)$ should have a smaller variance than $a$ for some $h$. In order for $a$ to be efficient, $a$ and $b-a$ should be uncorrelated. $\endgroup$ – chan1142 Jun 5 '19 at 15:52

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