# How do I compute the standard error of X1 in a regression also featuring X2?

There is a closed form expression for this IF X1 and X2 are *both* centred. That is, have their means subtracted from them.

Centering a variable can reduce collinearity but still give an equivalent fit to the response variable as using the raw data. Centering predictor variables gives the same regression coefficient for the *last* predictor entered into a regression as using the raw data but a more reliable standard error.

let

A = $$\sum_text{obs i}(X_text{1i}-\mbox{mean})^text{2}$$

B = $$\sum_text{obs i}(X_text{2i}-\mbox{mean})^text{2}$$

C = $$\sum_text{obs i}(X_text{1i}-\mbox{mean})(X_text{2i}-\mbox{mean})$$

D = Mean square residual (amount not explained by X1 and X2)

then

s.e of b(X1) = $$\frac{BD}{AB-C^text{2}}$$

s.e of b(X2) = $$\frac{AD}{AB-C^text{2}}$$

Reference

Aiken and West (1991) Multiple Regression: Testing and Interpreting Interactions.