Modeling Lagged Residuals in Equity Return Forecasts

Summary

The document asks how to include yesterday’s forecast error in a regression intended to predict future equity returns. Its replies connect lagged residuals to moving-average terms: an MA model uses past errors, while combining lagged observations and lagged errors gives an ARMA structure. When explanatory variables are also included, the discussion uses ARX or ARIMAX terminology, where the additional inputs are treated as exogenous predictors.

These labels describe related but distinct model components, so the precise name depends on the full specification. A regression with external predictors and lagged errors is commonly described as an ARMAX-type model; an ARX model alone does not necessarily include residual lags. The document offers terminology rather than guidance on estimating the model, checking residual behavior, or avoiding leakage in a forecasting setup. It also does not establish that adding a lagged forecast error improves out-of-sample equity return predictions; that would require empirical validation.

Key ideas

• Lagged residuals represent moving-average terms in a time-series model.
• Lagged observations and lagged errors together form an ARMA structure.
• With explanatory inputs, models may be described using ARX or ARIMAX terminology.
• The exact model label depends on which lagged observations, errors, and predictors are included.
• The discussion does not show whether lagged errors improve forecast performance.

Tags

# Lagged Residual as Independent Variable


# Lagged Residual as Independent Variable












I am building a factor model to estimate future equity returns. I'd like to include an autoregressive residual term in this model. I'd like to have yesterday's error (the difference between yesterday's predicted return and actual return) be included in the regression as an independent variable. What type of autoregressive model is this called? I've searched through various time series econometrics texts and have not found this particular model described.

## Answer by PlantFox (score 1)

https://quant.stackexchange.com/a/40692

This type of model is called an ARIMAX or ARX. The "X" stands for exogenous inputs or explanatory inputs.

Here is a good reference:

https://robjhyndman.com/hyndsight/arimax/

## Answer by compilation-error (score 1)

https://quant.stackexchange.com/a/41206

This is a MA(1) model. if you keep the lagged time-series observation as well as the lagged residual, it would be an ARMA(1, 1) model. Basically, p lagged observations and q lagged residuals will form ARMA(p, q) model.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.