# Intercept increases in regression when adding explanatory variables

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.

• 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' $C_p$. – Michael R. Chernick Apr 22 '18 at 19:05
• What does the notation $t(\alpha)$ mean? – Jake Westfall Apr 22 '18 at 19:30

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 $X_j$.