Overlapping Returns and Inference in Forecasting Regressions
Summary
The document examines a proposed monthly regression of year-over-year stock index returns on lagged returns and macroeconomic variables, including credit spreads, inflation, and industrial production. It describes a forecasting setup that uses ARIMA forecasts for some explanatory variables and subjective estimates for credit spreads. The central statistical concern is that rolling year-over-year returns overlap across months, potentially making regression errors serially correlated and undermining standard OLS inference.
The question raises heteroscedasticity-consistent and autocorrelation-consistent covariance estimators, including Newey–West and Hansen–Hodrick approaches. The accepted response does not resolve the model’s validity or recommend a specific correction; it points to research on regression with overlapping observations. Thus, the material is useful as a framing of inference and forecasting issues, but offers no empirical results, worked correction, or assessment of the forecast design.
Key ideas
- Monthly year-over-year returns overlap because adjacent observations share much of the same return window.
- Overlapping outcomes can induce serial correlation in regression errors.
- The document asks whether conventional OLS inference needs a robust covariance correction.
- Forecasting the specified returns also requires forecasts or estimates of future macroeconomic predictors.
- The answer refers readers to research on overlapping-observation inference but provides no direct resolution.
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Full text
# Time Series Regression with Overlapping Data
# Time Series Regression with Overlapping Data
I am seeing a regression model which is regressing Year-on-Year stock index returns on lagged (12 months) Year-on-Year returns of the same stock index, credit spread (difference between monthly mean of risk-free bonds and corporate bond yields), YoY inflation rate and YoY index of industrial production.
It looks thus (though you'd substitute the data specific to India in this case):
```
SP500YOY(T) = a + b1*SP500YOY(T-12) + b2*CREDITSPREAD(T)
+ b4*INDUSTRIALPRODUCTION(T+2) + b3*INFLATION(T+2)
+ b4*INFLATIONASYMM(T+2)
```
`SP500YOY` is the year-on-year returns for S&P500 index To compute this, monthly average of S&P500 values is computed and then converted to year-on-year returns for each month (i.e. Jan'10-Jan'11, Feb'10-Feb'11, Mar'10-Mar'11, . . ). On the explanatory variables side, a 12-month lagged value of the `SP500YOY` is used along with the `CREDITSPREAD` at time `T` and `INFLATION` and `INDUSTRIALPRODUCTION` two period AHEAD. The `INFLATIONASYMM` is a dummy for whether the Inflation is above a threshold value of 5.0%. The index in the parenthesis shows the time index for each variable.
This is estimated by standard OLS linear regression. To use this model for forecasting the 1,2 and 3-months ahead YOY returns of S&P500, one has to generate 3,4 and 5-month ahead forecasts for Inflation and the Index of Industrial Production. These forecasts are done after fitting an ARIMA model to each of the two individually. The CreditSpread forecasts for 1,2 and 3 month ahead are just thrown in as mental estimates.
I'd like to know whether this OLS linear regression is correct/incorrect, efficient/inefficient or generally valid statistical practice.
The first problem I see is that of using overlapping data. i.e. the daily values of the stock index are averaged each month, and then used to compute yearly returns which are rolled over monthly. This should make the error term autocorrelated. I would think that one would have to use some 'correction' on the lines of one of the following:
- White's heteroscedasticity consistent covariance estimator
- Newey & West heteroscedasticity and autocorrelation consistent (HAC) estimator
- heteroscedasticity-consistent version of Hansen & Hodrick
Does it really make sense to apply standard OLS linear regression (without any corrections) to such overlapping data, and more so, use 3-period ahead ARIMA forecasts for explanatory variables to use in the original OLS linear regression for forecasting `SP500YOY`? I have not seen such a form before, and hence cannot really judge it, without the exception of correcting for the use of overlapping observations.
## Answer by Tal Fishman (score 7, accepted)
https://quant.stackexchange.com/a/1874
This question was ultimately answered on Cross Validated
> Here are a couple of articles that deal with this subject: Britten-Jones and Neuberger, Improved inference and estimation in regression with overlapping observations Harri & Brorsen, The Overlapping Data ProblemShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.