Interpreting Expected Returns in a Regression with Interaction Terms

Summary

The document explains how to interpret a multiple linear regression for future returns when predictors include a historical return, a news indicator, and their interaction. The conditional expected future return is the full fitted expression: intercept plus each coefficient multiplied by its corresponding predictor, including the interaction. It is therefore not generally the sum of the coefficients alone; the prediction depends on the values of the inputs.

For the unconditional expectation given only the news indicator, the answer gives a simplified expression under the additional assumption that the unconditional expected historical return is zero. It cautions that return regressions may violate ordinary least squares assumptions, including homoscedasticity and uncorrelated errors, which can undermine inference and standard errors. The response is brief and does not discuss alternative estimators or how to diagnose these assumptions in a particular dataset.

Key ideas

• A conditional prediction uses the intercept and each coefficient multiplied by its predictor value.
• An interaction term makes the effect of news depend on the historical return.
• The fitted expected return is not generally the sum of regression coefficients.
• A simplified expectation given only the news indicator assumes zero unconditional expected historical return.
• Return data may violate OLS error assumptions, affecting estimates or standard errors.

Tags

# Expected return from a multiple linear regression?


# Expected return from a multiple linear regression?












How can I compute the predicted return from a linear regression that includes a number of different terms. For instance, suppose my equation is:

$r_{future} = \alpha + \beta_1 r_{history} + \beta_2 x_{news} + \beta_3 r_{history} * x_{news} $

Where $r$ is the geometric return, and $x$ is a news dummy variable (0 or 1 depending on whether news existed).

Can I still conclude that the expected return $r_{future} = \sum \beta_i$?

## Answer by Tal Fishman (score 5)

https://quant.stackexchange.com/a/1769

If the equation satisfies all the assumptions of OLS, particularly homoscedasticity and no autocorrelation in the errors, then the expected return for the equation you laid out is

$E[r_{future}|r_{history},x_{news}]=\alpha+\beta_1r_{history}+\beta_2x_{news}+\beta_3r_{history}*x_{news}$

If the unconditional expected return is zero (as is likely to be approximately true for short horizon returns), then

$E[r_{future}|x_{news}]=\alpha+\beta_2x_{news}$

These types of return regressions usually do not satisfy the conditions for OLS, so your coefficients estimated using OLS or (more likely) your standard errors may be biased.

Shown 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.