Stepwise linear regression: unexpected result of form X1*X2 and X1:X2. How to interpret it?

Have a model with two independent variables ( X1 and X2) of the form Y ~ 1 + X1 + X2.

Would expect stepwise regression to either leave Y ~ 1 + X1, Y ~ 1 + X2 or Y ~ 1 + X1 + X2.

What I get is Y ~ 1 + X1*X2. How do I interpret that. That does not seem like a multiple linear model; more like some kind of quadratic. The coefficients I get are listed as:

(Intercept)  X1          X2          X1:X2
0.38500977  0.002251122  0.291180937  -0.005529228

How do I interpret the X1:X2; is this division?

Answer 1

[Roger Wohlwend]

Yes, it is a multiple linear Regression. The model consists of three variables: X1, X2 and X1*X2. The third variable is just the multiplication of X1 and X2. When Matlab presents the results, it writes X1:X2 but means in this case X1*X2 (This is Wilkinson Notation. You'll find an Explanation on the Matlab help page for the class stepwiselm).

[A.]

But isn't stepwise regression about simplifying the model so that only the necesary variables remain (say only X1, because X2 turns out to have t-stat close to zero). The addition of X1*X2 seems to only complicate the model.

[Roger Wohlwend]

Yes, stepwise Regression tries to build a model which is as simple as possible. However, that does not mean that it always excludes at least one of the x-variables. Sometimes the best model includes all variables.

As the algorithm includes all three variables in the model, that means that the model does not improve if one of the three variables is excluded.

In your case all three variables (X1, X2 and the product X1*X2) seem to be statistically insignifcant (The t-value is very small for all of them, which is a bad sign). Try building the model with new x-Variables.