Sep

13

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.

Adi Schnytzer writes:

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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2 Comments so far

  1. Erling Skorstad on September 12, 2007 3:10 am

    Look up “An unbiased variance estimator for overlapping returns” by Bod, Blitz Franses and Kluitman
    Also “Statistics of variables over overlapping intervals” by Müller

  2. Kostas Savvidis on September 13, 2007 7:20 pm

    The simple answer is that you should use the monthly data, with 120 observations you will be able to obtain a fairly good estimate of the correlation coefficient, and whatever else you may care about.

    The deeper answer is that, you are probably interested in the yearly movement because you think that the relation between series X and Y might be more “clean” and free of noise when looking at the yearly moves. This may indeed be the case if the two series are cointegrated. In this case it is known that not only the rolling regression gives unbiased estimate of the cointegrating coefficient (aka hedging ratio) , but luckily it is also superconvergent ie the speed with which the error of the estimate goes to zero is 1/N instead of 1/sqrt(N).

    Strictly speaking the theory of cointegration applies when there is a relation of the form
    Y_i = a*X_i + b + e_i
    where X and Y are the raw series, not changes; conintegration applies when the error series e_i is stationary.

    This theory is usually applied to economic timeseries. In financial series it is highly unlikely that you will be able to establish true cointegration, however
    I have encountered a special case, where rolling regression revealed a very good conformance to the idea of cointegration, ie
    Y_i - Y_{i-d} = a* ( X_i - X_{i-d}) + e_i
    where d is the delay such that we are doing the regression on d-period rolling *changes*. The error term e_i had the behavior that it did not increase appreciably for d between a few days and a few months, and this is precisely the kind of behavior suggestive of cointegration.

    As I alluded to earlier, for cases when there is indeed cointegration between the series the coefficient obtained in this way is free of bias, and has very good convergence properties.

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