## Thursday, September 18, 2014

### "Inverting" Singular Matrices

You can only invert a matrix if that matrix is non-singular. Right? Actually, that's wrong.

You see, there are various sorts of inverse matrices, and most of them apply to the situation where the original matrix is singular

Before elaborating on this, notice that this fact may be interesting in the context of estimating the coefficients of a linear regression model, y = Xβ + ε. Least squares estimation leads to the so-called "normal equations":

X'Xb = X'y  .                                                                (1)

If the regressor matrix, X, has k columns, then (1) is a set of k linear equations in the k unknown elements of β. You'll recall that if X has full column rank, k, then (X'X) also has full rank, k, and so (X'X)-1 is well-defined. We then pre-multiply each side of (1) by (X'X)-1, yielding the familiar least squares estimator for β, namely b = (X'X)-1X'y.

So, as long as we don't have "perfect multicollinearity" among the regressors (the columns of X), we can solve (1), and the least squares estimator is defined. More specifically, a unique estimator for each individual element of β is defined.

What if there is perfect multicollinearity, so that the rank of X, and of (X'X), is less than k? In that case, we can't compute (X'X)-1, we can't solve the normal equations in the usual way, and we can't get a unique estimator for the (full) β vector.

Let's look carefully at the last sentence above. There are two parts of it that bear closer scrutiny: