 ## C. Derivation of OLS Estimates for the Multiple Regression Model

### Definition of variables used in OLS estimation OLS estimates of minimize the scalar function SS. In class we derive the OLS estimator for the MLRM: This expression is only meaningful if the kxk matrix X'X is invertible. We rely on the following result which we will not prove:

#### X'X is invertible if and only if

The columns of X (corresponding to variables in the data set) are linearly independent of each other.

Linear independence means that no column can be written as a linear combination of the others. You might recall that this is the same as saying that X has full rank. You may be tempted to distribute the inverse operator through the matrix multiplication: If this is possible, then reduces to . But note that one can distribute the inverse operator only if each matrix in the multiplication has an inverse. X is a k x N, and can only have an inverse if k=N, because only square matrices have inverses. Therefore, if then it is not possible to simplify (***) any more.

### OLS estimates in 3 Special Cases of the MLRM

1. k=1 (only a constant term appears on the right hand side)

Now X is simply a column of 1's and must have full rank. X'X then equals the scalar N (because it equals 1*1+1*1+...+1*1 = N. The matrix X'y simply equals the sum of Y values in the sample. So the OLS estimate reduces to when only a constant term is included in the regression.

2. k=2 (simple LRM)

This is the case studied as the Simple LRM. You should verify that: Then you should use the formula for the inverse of a 2x2 matrix to get . With some manipulation it is possible to show that (***) gives back exactly the scalar formulas for and for the case k=2.

3. Let k=N

In this case we have as many parameters to estimate as we have observations. We have N equations in N unknowns. As long as X (which is now square N x N and k x k since N=k) has an inverse, then OLS will find the estimates that satisfy the equation , which is simply . But we already have seen that (***) collapses to this expression if X is invertible. The prediction error is identically 0 for each observation because we can perfectly explain each observation. If X is not invertible, then we cannot find N different values of . We know longer have a system of linearly independent equations because X is no longer full rank.

Using the derived formulas for the OLS estimates we get the OLS prediction and error formulas: Note that each of these expressions except the last takes the form "some matrix that involves only X" X the vector y. In other words, each is a linear function of the Y observation in the sample. The final expression is different since it includes y' and y. The last step in deriving the matrix expression for is proved in class.

Exercise: use these formulas to derive computational properties of OLS estimates for the MLRM that are analogous to those derived for the simple LRM.

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