Estimation and Inference in Econometrics by Davidson and MacKinnon (1993 edition, the older one) on page 552, ch 16.3 'Covariance Matrix Estimation' states:

"Consequently, the matrix \begin{eqnarray} (n^{-1} \hat{X}'\hat{X})^{-1}(n^{-1}\hat{X}' \hat{\Omega} \hat{X})(n^{-1}\hat{X}'\hat{X})^{-1} \end{eqnarray} consistently estimates the probability limit of the OLS heteroskedastic covariance matrix. Of course, in practice one ignores the factors of $n^{-1}$ and uses the matrix \begin{eqnarray} ( \hat{X}'\hat{X})^{-1}(\hat{X}' \hat{\Omega} \hat{X})(\hat{X}'\hat{X})^{-1} \end{eqnarray} to estimate the covariance of $\hat{\beta}$.

Now, why can we ignore the factors of $n^{-1}$? (They don't even cancel out.)

As a second point, what do the authors mean by $\hat{X} \equiv X(\hat{\beta})$?

Can someone help me understand this?

  • $\begingroup$ It would help if you mentioned what acclaimed text you had in hand, preferably with a page number. $\endgroup$ – Dimitriy V. Masterov Mar 3 '15 at 19:53
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    $\begingroup$ You might also want to give it a more informative title. Ed Leamer had something completely different in mind when he wrote Let's take the con out of econometrics :-) $\endgroup$ – Andy Mar 3 '15 at 20:01
  • $\begingroup$ Oh ok what would be a better title? Well, I remember reading about the "whimsical" by which Leamer I think referred to something including this variable and sometimes not. I find the dropping of the n just as random and frustrating as all the handwaviness that abounds. But am happy for a better title! ;-) $\endgroup$ – Hirek Mar 3 '15 at 20:04
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    $\begingroup$ It will be necessary to have a clear account of what "$\hat X$" and "$\hat \Omega$" are in order to answer this question. (My suspicion is that the "factors of $n^{-1}$" to be ignored are involved in asymptotic expressions for these estimates, and not the ones in the matrix given in the first formula.) $\endgroup$ – whuber Mar 3 '15 at 20:27
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    $\begingroup$ @Hirek good job! By the way, I really appreciate your questions here. I wish we had more people ask questions about econometrics and yours are good ones as well. $\endgroup$ – Andy Mar 5 '15 at 10:29

You have the following NLS model:

$$y=x(\beta_0) + u \\ E(u)=0 \\ E(uu^T)=\Omega$$

The proofs are for the asymptotic variance matrix of $$n^{1/2}(\hat \beta -\beta_0),$$ but if we were to actually calculate this with software for the purpose of testing some hypotheses, we don't need the $\sqrt n$, so we leave them out. In other words, we have shown that $$\sqrt n (\hat \beta -\beta_0)$$ has variance matrix $B_{0}^{-1}M_0 B_{0}'^{-1},$ where $B$ is the "bread" and $M$ is the "meat" of the sandwich. It follows that $\hat \beta$ has an asymptotic variance matrix $n^{-1}B_{0}^{-1}M_0 B_{0}'^{-1},$ so all the $n$s would cancel. Another difference is that software would typically use some sort of finite sample correction, though that is more of an aside.

I have the newer edition, where $\hat X$ is written as $X_0$, which is the matrix of derivatives of $x(\beta)$ with respect to $\beta$, evaluated at the true value $\beta_0$. It's defined on p. 551.

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  • $\begingroup$ Thanks so much @Dimitriy V. Masterov but why again can we leave out the factors of n? I.e. is there a compelling reason to do so other than saying we don't need them? $\endgroup$ – Hirek Mar 3 '15 at 23:49
  • $\begingroup$ Practitioners care about $V(\hat \beta)$, not $V(f(\hat \beta))$. $\endgroup$ – Dimitriy V. Masterov Mar 4 '15 at 0:14
  • $\begingroup$ Right, I see that but still why can we leave out n? Isn't it an important normalization to achieve some type of convergence? $\endgroup$ – Hirek Mar 4 '15 at 0:17
  • $\begingroup$ I tried to clarify in the body of the answer. Maybe it would be helpful to consider a linear constant-only model, so your only parameter is the intercept, or $\hat \beta = \bar y$. You want to test the hypothesis that $\bar y = 10,$ so you need $V(\bar y)$. $\endgroup$ – Dimitriy V. Masterov Mar 4 '15 at 0:32
  • $\begingroup$ Yes, I understand that I need a variance estimate to test the null hypothesis you suggest to compute a t statistic that can be judged against a standard normal but my question here is about the mathematical intuition: Why is it that I write down an estimator 1/n {matrix operations} and conclude its plim is the desired parameter and then, when using it in C or MATLAB I exclude the factor 1/n? Like am I not programming up a different thing then? $\endgroup$ – Hirek Mar 4 '15 at 0:38

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