3
$\begingroup$

I am conducting an analysis, where I examine the size of the intercepts of three regression models (time-series). The models look something like this:

$y_1=\alpha+\beta_1x_1+\varepsilon$

$y_2=\alpha+\beta_1x_1+\beta_2x_2+\beta_3x_3+\varepsilon$

$y_3=\alpha+\beta_1x_1+\beta_2x_2+\beta_3x_3+\beta_4x_4+\beta_5x_5+\varepsilon$

When I run the regressions, I then examine values of $\alpha$ and $t(\alpha)$ for all three models. I find that as I add factors (going form model 1 to 3), the $R^2$ increases (as expected), so it seems that the added factors add some explanation. However, I find that the $t(\alpha)$ values increase as I add factors.

Maybe this is something that I am missing, but shouldn't $t(\alpha)$ decrease as I add factors, since more of the explanatory output is now explained by the factors, and is not put on the intercept?

Edit: $t(\alpha)$ is the estimated $t-$statistic of the $\alpha$ intercept from the regression output.

$\endgroup$
  • $\begingroup$ This could be a problem of overfitting. $R^2$ will go up regardless of whether or not the new variable improves the model. That is why we look at other measures such as adjusted $R^2$, AIC, BIC and Mallows' $C_p$. $\endgroup$ – Michael R. Chernick Apr 22 '18 at 19:05
  • 4
    $\begingroup$ What does the notation $t(\alpha)$ mean? $\endgroup$ – Jake Westfall Apr 22 '18 at 19:30
6
$\begingroup$

It's not true that adding predictors should generally cause the estimate of the intercet $\alpha$ to decrease.

The intercept is the predicted $y$ value when all the $x$ predictors are equal to 0. So adding new predictors can cause the intercept to increase or decrease, by pretty much any amount, based on the mean of the $x$ predictor you're adding and the size/direction of its corresponding regression coefficient.

This becomes especially clear when we note that we can write the estimate for the intercept as $$ \hat{\alpha} = \bar{y} - \hat{\beta_1}\bar{X_1} - \hat{\beta_2}\bar{X_2} - \dots, $$ where $\bar{y}$ denotes the sample mean of $y$, $\hat{\beta_j}$ is the sample estimate of $\beta_j$, and $\bar{X_j}$ is the sample mean of $X_j$.

$\endgroup$
  • $\begingroup$ Thank you very much, Mr. @JakeWestfall! I guess that it was a hole in my regression terminology. This is a very nice explanation. $\endgroup$ – Andreas Apr 22 '18 at 19:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.