# Comparison between Newey-West (1987) and Hansen-Hodrick (1980)

Question: What are the main differences and similarities between using Newey-West (1987) and Hansen-Hodrick (1980) standard errors? In which situations should one of these be preferred over the other?

Notes:

• I do know how each of these adjustment procedures works; however, I have not yet found any document that would compare them, either online or in my textbook. References are welcome!
• Newey-West tends to be used as "catch-all" HAC standard errors, whereas Hansen-Hodrick comes up frequently in the context of overlapping data points (e.g. see this question or this question). Hence one important aspect of my question is, is there anything about Hansen-Hodrick that makes it more suited to deal with overlapping data than Newey-West? (After all, overlapping data ultimately leads to serially correlated error terms, which Newey-West also deals with.)
• For the record, I am aware of this similar question, but it was relatively poorly posed, got downvoted and ultimately the question that I am asking here did not get answered (only the programming-related part got answered).
• Aren't NW-type HAC estimators superseded by the fixed smoothing HAC estimators of Kiefer & Vogelsang (2002) & the subsequent literature? Nov 11, 2017 at 7:02
• In particular, you might want to read the opinion posts of Frank Diebold here & here. Nov 12, 2017 at 13:35
• @tchakravarty That's an interesting thought, thanks for sharing! I'll have to back up a bit and first look into Kiefer, Vogelsang and Bunzel (2000). If you want to expand on your point in an answer, also explaining what this implies for Hansen-Hodrick-type estimators that deal with overlapping data, you'd have a very good chance of being awarded the bounty. (It would not be honest from me to guarantee it, obviously, since somebody else might write a competing answer, but so far my bounty has not proven very popular.) Nov 15, 2017 at 14:39
• @tchakravarty, the theoretical literature seems to settle on that, but in practice, these estimators are not yet in widespread use, I would say. Sep 10, 2018 at 17:04

Consider a class of long-run variance estimators

$$\hat{J_T}\equiv\hat{\gamma}_0+2\sum_{j=1}^{T-1}k\left(\frac{j}{\ell_T}\right)\hat{\gamma}_j$$ $k$ is a kernel or weighting function, the $\hat\gamma_j$ are sample autocovariances. $k$, among other things must be symmetric and have $k(0)=1$. $\ell_T$ is a bandwidth parameter.

Newey & West (Econometrica 1987) propose the Bartlett kernel $$k\left(\frac{j}{\ell_T}\right) = \begin{cases} \bigl(1 - \frac{j}{\ell_T}\bigr) \qquad &\mbox{for} \qquad 0 \leqslant j \leqslant \ell_T-1 \\ 0 &\mbox{for} \qquad j > \ell_T-1 \end{cases}$$

Hansen & Hodrick's (Journal of Political Economy 1980) estimator amounts to taking a truncated kernal, i.e., $k=1$ for $j\leq M$ for some $M$, and $k=0$ otherwise. This estimator is, as discussed by Newey & West, consistent, but not guaranteed to be positive semi-definite (when estimating matrices), while Newey & West's kernel estimator is.

Try $M=1$ for an MA(1)-process with a strongly negative coefficient $\theta$. The population quantity is known to be $J = \sigma^2(1 + \theta)^2>0$, but the Hansen-Hodrick estimator may not be:

set.seed(2)
y <- arima.sim(model = list(ma = -0.95), n = 10)
acf.MA1 <- acf(y, type = "covariance", plot = FALSE)\$acf
acf.MA1[1] + 2 * acf.MA1[2]
## [1] -0.4056092


which is not a convincing estimate for a long-run variance.

This would be avoided with the Newey-West estimator:

acf.MA1[1] + acf.MA1[2]
## [1] 0.8634806


Using the sandwich package this can also be computed as:

library("sandwich")
m <- lm(y ~ 1)
kernHAC(m, kernel = "Bartlett", bw = 2,
prewhite = FALSE, adjust = FALSE, sandwich = FALSE)
##             (Intercept)
## (Intercept)   0.8634806


And the Hansen-Hodrick estimate can be obtained as:

kernHAC(m, kernel = "Truncated", bw = 1,
prewhite = FALSE, adjust = FALSE, sandwich = FALSE)
##             (Intercept)
## (Intercept)  -0.4056092


See also NeweyWest() and lrvar() from sandwich for convenience interfaces to obtain Newey-West estimators of linear models and long-run variances of time series, respectively.

Andrews (Econometrica 1991) provides an analysis under more general conditions.

As to your subquestion regarding overlapping data, I would not be aware of a subject-matter reason. I suspect tradition is at the roots of this common practice.

• I appreciate your answer but will probably only be able to review and hopefully accept over the weekend. Thanks again. Sep 11, 2018 at 2:13
• Thanks again for your answer. Just to clarify, your answer in effect says that Newey-West should be preferred over Hansen-Hodrick in all cases since the latter might "behave poorly", which "interferes with asymptotic confidence interval formation and hypothesis testing" (both quotes from Newey-West, 1987)? Sep 16, 2018 at 2:12
• PS. Could you please also clarify the source for "Andrews"? Sep 16, 2018 at 2:12
• I linked the papers to Jstor. As for the previous comments, indeed, when a variance estimate is not even guaranteed to be positive, we should also not expect it to be a good ingredient into confidence intervals and test statistics. Sep 17, 2018 at 6:33