Fit an AR(1) to Discrete-Time Regression Residuals

Summary

The document considers a statistical-arbitrage workflow that applies principal component analysis to stock log returns, regresses each stock’s returns on leading components, and models the resulting residuals for mean reversion. The researcher tries to fit an Ornstein-Uhlenbeck process by regressing residual sums on their lagged values, but sometimes obtains negative intercept or slope estimates.

The answer focuses on the sampling interval: for observations in discrete time, it recommends fitting an autoregressive model of order one, which serves as the discrete-time analogue of the continuous-time Ornstein-Uhlenbeck process. The suggested calibration is a regression using the residual series. The short exchange does not give the regression equation, explain how to interpret negative estimates, or demonstrate that the residuals are stationary or mean reverting. It therefore offers a model choice and basic fitting direction, rather than a complete validation procedure or evidence that the proposed trading strategy is profitable.

Key ideas

• For discretely sampled residuals, an AR(1) model is the discrete-time counterpart of an Ornstein-Uhlenbeck process.
• The proposed calibration regresses residuals on their lagged values.
• The workflow first removes leading principal-component exposures from stock returns.
• The answer does not establish residual stationarity or explain how negative parameter estimates should be interpreted.

Tags

# Calibrating an Ornstein Uhlenbeck process on residuals of regression


# Calibrating an Ornstein Uhlenbeck process on residuals of regression












I am trying a basic statistical arbitrage strategy as follows:

- Perform PCA on a log return series of a basket of stocks

- Regress returns against top principal components identified

- Calculate the residuals of regression for each stock

- Fit a OU process on the residuals

To fit an OU process, calculated the sum of residuals for each stock and regressed them on the lagged sum of residuals. However sometimes the intercept and slope are negative.

How do I calibrate this to an OU process when intercept or slope is negative?

## Answer by Richi Wa (score 4, accepted)

https://quant.stackexchange.com/a/14939

You work in discrete time so you should not fit an OU-process but simply an AR(1) process which is its analogon in discrete time. Look here to see why this is true.

Calibrating the AR(1) boils down to do a regression on your residuals.

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