I'm running a couple of regressions and, as I wanted to be on the safe side, decided to use HAC (heteroskedasticity & autocorrelation consistent) standard errors throughout. There might be a few cases where serial correlation is not present. Is this anyways a valid approach? Are there any drawbacks?
Solved – Using HAC standard errors although there might be no autocorrelation
least squaresrobustrobust-standard-errorstandard errortime series
Related Solutions
Seeing as how I had a similar question earlier and came across this long-unanswered question through a simple web search, I'll take a stab and post what I think is one possible solution to your situation that others may also be encountering.
According to SAS Support, you can take the time-series you have and fit an intercept-only regression model to the series. The estimated intercept for this regression model will be the sample mean of the series. You can then pass this intercept-only regression model through the SAS commands used to retrieve Newey-West standard errors of a regression model.
Here is the link to the SAS Support page: http://support.sas.com/kb/40/098.html
Look for "Example 2. Newey-West standard error correction for the sample mean of a series"
In your case, simply try the same approach with Matlab.If someone has a better approach, please enlighten us.
Question 3)
In notation to be understood as matrix-vector, assume that the correct specification is
$$y = X\beta + \gamma y_{-1}+ e$$
(where $X$ contains the constant and the $X_1$ variable and $e$ is white noise, and $E(e\mid X) =0$), but you specify and estimate instead
$$y = X\beta + u$$ i.e. without including the LAD, and so in reality $u =\gamma y_{-1}+ e$.
Then OLS estimation will give
$$\hat \beta = (X'X)^{-1}X'y = (X'X)^{-1}X'(X\beta + \gamma y_{-1}+ e) $$ $$= \beta + (X'X)^{-1}X'y_{-1}\gamma +(X'X)^{-1}X'e$$
The expected value of the estimator is
$$E(\hat \beta) = \beta + E\Big[(X'X)^{-1}X'y_{-1}\gamma\Big] +E\Big[(X'X)^{-1}X'e\Big]$$ and using the law of iterated expectations
$$E(\hat \beta) = \beta + E\Big(E\Big[(X'X)^{-1}X'y_{-1}\gamma\Big]\mid X\Big) +E\Big(E\Big[(X'X)^{-1}X'e\Big]\mid X\Big)$$
$$= \beta + E\Big((X'X)^{-1}X'E\Big[y_{-1}\gamma\mid X\Big]\Big) +E\Big((X'X)^{-1}X'E\Big[e\mid X\Big]\Big)$$
$$=\beta + E\Big((X'X)^{-1}X'E\Big[y_{-1}\gamma\mid X\Big]\Big) + 0 $$ the last term being zero per our assumptions. But $E\Big[y_{-1}\gamma\mid X\Big] \neq 0$, because $X$ contains all the regressors (from all time periods), and so there is correlation with the LAD vector. Therefore $E(\hat \beta) \neq \beta$. In other words, ignoring the lag dependent variable will not make the estimator unbiased, as long as $\gamma \neq 0$, i.e. as long as the LAD does belong to the regression.
Question 1)
Assume now that you specify correctly, and denote $Z$ the matrix containing also the LAD.
Here (using the same steps as before)
$$\hat \beta = \beta + (Z'Z)^{-1}Z'e$$
and $$E(\hat \beta) = \beta + E\Big((Z'Z)^{-1}Z'E\Big[e\mid Z\Big]\Big)$$
But is $e$ (the vector) independent of $Z$? No, because $Z$ contains the LAD from all time periods bar the most recent, while $e$ contains the errors from all time periods bar the first. So even if $e$ is not serially correlated, it is correlated with the vector $y_{-1}$. So indeed, the last term is not zero and $$E(\hat \beta) \neq \beta$$ the OLS estimator is biased.
But the OLS estimator will be consistent if indeed the inclusion of the LAD eliminates serial correlation, because (using the properties of the plim operator)
$$\operatorname{plim}\hat \beta = \beta + \operatorname{plim}\left(\frac 1{n-1} Z'Z\right)^{-1}\cdot \operatorname{plim}\left(\frac 1{n-1}Z'e\right)$$
Part of the standard assumptions (and rather "easily" satisfied), is that the first plim of the product converges to something finite. The second plim written explicitly is (and using the stationarity assumption to invoke the LLN)
$$\operatorname{plim}\left(\frac 1{n-1}Z'\mathbf e\right) = \left[\begin{matrix} \operatorname{plim}\frac 1{n-1}\sum_{i=2}^ne_i \\ \operatorname{plim}\frac 1{n-1}\sum_{i=2}^nx_{i}e_i \\ \operatorname{plim}\frac 1{n-1}\sum_{i=2}^ny_{i-1}e_i \\ \end{matrix}\right] \rightarrow\left[\begin{matrix} E(e_i) \\ E(x_{i}e_i) \\ E(y_{i-1}e_i) \\ \end{matrix}\right]\; \forall i$$
$E(e\mid X) = 0 \Rightarrow E(e_i) = 0$, and also that $E(x_{i}e_i)=0$, for all $i$.
Finally, IF serial correlation has been removed, then $E(y_{i-1}e_i) =0$ also. So this plim goes to zero and therefore
$$\operatorname{plim}\hat \beta = \beta$$ i.e. the OLS estimator is indeed consistent in this case. So the "summary" is correct.
Question 2)
The full sentence from Wooldridge is
"It is also valid to use the SC-robust standard errors in models with lagged dependent variables assuming, of course, that there is good reason for allowing serial correlation in such models".
meaning, when we have good reasons to believe that the inclusion of lagged dependent variables does not fully remove autocorrelation. And it seems we got ourselves a Catch-22: if serial correlation (SC) has been removed, why use SC-robust std errors? And if serial correlation has not been removed, our OLS estimator will be inconsistent, so in such a case is it meaningful/useful/appropriate to use asymptotic inference? Well, it appears that if we do suspect that SC still exists, it is better to try to do something about it, regardless. But your comment has merit, and I would suggest to contact Wooldridge directly on the matter, in order to get an authoritative answer.
Best Answer
Loosely, when estimating standard errors:
If you have enough data, you should be entirely safe since the estimator is consistent!
As Woolridge points out though in his book Introductory Econometrics (p.247 6th edition) a big drawback can come from small sample issues, that you may be effectively dropping one assumption (i.e. no serial correlation of errors) but adding another assumption that you have enough data for the Central Limit Theorem to kick in! HAC etc... rely on asymptotic arguments.
If you have too little data to rely on asymptotic results:
See this answer here to a related question: https://stats.stackexchange.com/a/5626/97925