# A Statistical Question, from Bruno Ombreux

September 13, 2007 |

I have two time series A and B with 120 monthly observations each. I want to test whether A's yearly changes predict B's yearly changes. But there are only 10 non-overlapping years. What is the least horrible method that would use overlapping 12-months changes? I am thinking of a bootstrap but looking around, I found mention of the Generalized Method of Moments (aka Generalized Estimating Equations) which looks complicated. Do readers have other suggestions?

## Alex Castaldo replies:

The traditional approach used in the literature (by Shiller among others) is to do a rolling (i.e. overlapping) predictive regression and then correct for the overlap by using Newey-West standard errors (rather than the usual standard errors that regression software normally uses).

Victor and Laurel do not like the Newey-West approach, and the literature has been coming around to their point of view. The problem is that Newey-West is correct asymptotically (that is, as the number of data points goes to infinity) but in these problems we do not have a large amount of data (that is why we are resorting to using overlap). Simulation studies show that in small samples the Newey_West method can be biased.

What is the solution? I don't know; it is an open research problem. There is something called the Hodrick (1992) method which is said to be free from small sample bias. (It is different from the Hansen-Hodrick method). Also you might try to read recent papers on the subject, such as Ang and Bekaert "Stock Return Predictability" (2006) and the references therein.

This is what Stata throws up: package lomackinlay from RePEc

TITLE
'LOMACKINLAY': module to perform Lo-MacKinlay variance ratio test

DESCRIPTION/AUTHOR(S)

lomackinlay computes a overlapping variance-ratio test on a
timeseries. The timeseries should be in level form; e.g., to
test that stock returns vary randomly around a constant    mean,
you consider the null hypothesis that the log price series is a
random walk with    drift. The log price series would then be
given in the varlist. If the assumption of homoskedastic
errors in the process generating the differenced series is not
reasonable,  the robust option may be used to calculate a
variance ratio test statistic robust to    arbitrary
heteroskedasticity. This is version 1.0.5, corrected for errors
in logic    identified by Allin Cottrell.

KW: variance ratio test
KW: random walk
KW: heteroskedasticity
KW: time series

Requires: Stata version 9.2

Distribution-Date: 20060804

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