# Solved – 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.

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#### Best Answer

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\$.

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