Overlapping Returns and the Hansen-Hodrick Correction

Suppose you want to know whether dividend yield predicts the next twelve months of stock returns. You have monthly data, so the tempting move is to compute a 12-month forward return starting every single month and regress all of them on that month's yield. You now have almost as many rows as you have months of history. You have not gained nearly that much information, because your February-to-January return and your March-to-February return share eleven of their twelve months in common. Consecutive rows are telling you almost the same story twice, and ordinary regression standard errors have no idea.

Where the dependence comes from

This is a self-inflicted version of ordinary autocorrelation. If you sampled non-overlapping 12-month returns — January to January, then the next January to January — the rows would be genuinely separate observations, but you would only have one row per year, throwing away most of your sample. Overlapping lets you keep every month as a data point, at the cost of manufacturing a very specific, very predictable kind of dependence: a 12-month overlapping return series behaves like a moving-average process of order 11, because each observation is built from eleven of the same underlying monthly returns as its neighbour.

Ordinary least squares standard errors assume each row's error is unrelated to the next row's error. Overlap violates that assumption by construction, not by accident, so the fix has to know the overlap length.

The correction

Hansen and Hodrick's fix widens the regression's standard error to account for the known correlation structure. For a forecast horizon of $k$ periods, the corrected variance of the slope estimate sums the ordinary variance of the regression residual with $k-1$ extra covariance terms, weighted equally rather than tapered:

$$\widehat{\text{Var}}_{HH} = \frac{1}{T}\left(\hat\gamma_0 + 2\sum_{j=1}^{k-1}\hat\gamma_j\right),$$

where $\hat\gamma_j$ is the sample autocovariance of the regression residual at lag $j$. In words: take the usual residual variance, then add back twice the covariance between each pair of residuals that are close enough together to share overlapping months, all the way out to $k-1$ lags — beyond that, two windows no longer share any months and the extra terms stop. This is the same idea as the Newey-West HAC Standard Errors correction, except Hansen-Hodrick was built specifically for the overlap problem and does not taper the distant lags down, since every lag inside the window is equally "real" overlap rather than a fading nuisance correlation.

Worked example

A predictive regression of 12-month forward equity returns on dividend yield, run on $T = 240$ months of overlapping data, gives a naive OLS slope standard error of $0.045$ (ignoring overlap). The regression residual's own lag-1 through lag-11 autocovariances, relative to its lag-0 variance $\hat\gamma_0$, sum to $2\sum_{j=1}^{11}\hat\gamma_j = 5.8 \times \hat\gamma_0$ — a realistic figure for heavily overlapping 12-month windows.

Step 1. The corrected variance is $\hat\gamma_0(1 + 5.8) = 6.8\,\hat\gamma_0$, versus $\hat\gamma_0$ for the naive calculation.

Step 2. Standard errors scale with the square root of variance, so $\text{SE}_{HH} = 0.045 \times \sqrt{6.8} = 0.045 \times 2.61 = 0.117$.

Step 3. A slope estimate of $0.09$ that looked like $t = 0.09/0.045 = 2.0$ (barely significant) becomes $t = 0.09/0.117 = 0.77$ once corrected — not distinguishable from noise. The naive regression was implicitly claiming 240 independent years' worth of evidence about a 12-month effect from 20 years of data.

In practice

• The correction never fixes the point estimate, only the error bar. The slope itself is unchanged; overlap only inflates how confident you should be in it.
• The rule of thumb before running any correction: if your forecast horizon is $k$ periods and you sample every period, expect your effective sample size to fall to roughly $T/k$. See Effective Sample Size With Autocorrelated Observations for the general version of this shrinkage.
• Do not just widen the sample instead. Extending the history to get more overlapping rows does not solve the problem; the true information content still grows only with non-overlapping years, not months.
• This is a first cousin of the panel problem that Clustered Standard Errors and Fama-MacBeth Regression solve in the cross-section: in every case, naive OLS counts rows that are not independent as if they were.