# 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 $r_{future} = \sum \beta_i$?

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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.

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