# Average effect of coefficients across multiple linear models?

I have several OLS models with robust s.e.'s that predict an outcome variable Y. For instance:

Model 1:

$Y=B_0 +B_1X_1$

Model 2:

$Y=B_0 + B_1X_1 + B_2X_2$

Model 3:

$Y=B_0 +B_1X_1 + B_2X_2 +B_3X_3$

I am interested in giving an average effect for $B_1$ across Models 1-3 with an accompanied 95% CI.

Can I just take the average of $B_1$'s across Models 1-3 and the average of standard errors to construct my confidence interval? What is this called?

• Are you fitting all three models to one data set? Why would youbwant to average those parameters? – Harvey Motulsky Dec 17 '10 at 7:14

If these 3 models are estimated from independent samples, then you can assume that $\beta_1$ are independent for these 3 models. Then you can average them. The standard error of the average then will be the square root from the average of the squares of the standard errors.
However you should check if you do not have omited-variables problem. If you do, then one of the $\beta_1$ in your models is biased, so it should be discarded.
Of course if you have all the data for these 3 models, I suggest estimate Model 3 with all the data and then use the coefficient $\beta_1$ with its standard error, assuming that your model is adequate.
• +1. But what would the interpretation of the average of the $\beta_1$ be? You might just as well average the price of bananas in Costa Rica with the answers to question #1 in the latest Gallup poll. And the "CI" would require careful interpretation indeed; it certainly doesn't reveal anything about the relationship between $X_1$ and $Y$! – whuber Dec 19 '10 at 6:27