# 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? Dec 17, 2010 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.

• Thanks Mike. That's what I thought. It's just in common practice in econometrics literature to present several models to show that your result is robust across specifications. Your point about omitted variables is well taken.
– Thomas
Dec 17, 2010 at 7:18
• @Thomas, if it is the same data for all three models, then I do not suggest averaging. The common practice you mention is used to show that there is no omited variable problem. But since the same data is used to estimate all three models, then you cannot assume independence. If there is no omitted variable bias, then the average will be unbiased, but the standard error of such average cannot be calculated easily, since the coefficients will be clearly correlated. Dec 17, 2010 at 7:36
• +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, 2010 at 6:27

This is an interesting paper one may want to read before doing model averaging: On Model Averaging Regression Coefficients. Even though it is counter intuitive to average the coefficients as the interpretation is no longer Effect of X1 on Y in presence of X2, X3, ...., there are some ongoing investigations on how and when to do averaging of coefficients.