Previous: Nonlinear Programming, Up: Optimization

### 24.4 Linear Least Squares

— Function File: [beta, v, r] = gls (y, x, o)

Generalized least squares estimation for the multivariate model y = x b + e with mean (e) = 0 and cov (vec (e)) = (s^2) o, where y is a t by p matrix, x is a t by k matrix, b is a k by p matrix, e is a t by p matrix, and o is a t p by t p matrix.

Each row of y and x is an observation and each column a variable. The return values beta, v, and r are defined as follows.

beta
The GLS estimator for b.
v
The GLS estimator for s^2.
r
The matrix of GLS residuals, r = y - x beta.

— Function File: [beta, sigma, r] = ols (y, x)

Ordinary least squares estimation for the multivariate model y = x b + e with mean (e) = 0 and cov (vec (e)) = kron (s, I). where y is a t by p matrix, x is a t by k matrix, b is a k by p matrix, and e is a t by p matrix.

Each row of y and x is an observation and each column a variable.

The return values beta, sigma, and r are defined as follows.

beta
The OLS estimator for b, beta` = pinv (`x```) * ```y, where `pinv (`x`)` denotes the pseudoinverse of x.
sigma
The OLS estimator for the matrix s,
```               sigma = (y-x*beta)'
* (y-x*beta)
/ (t-rank(x))
```

r
The matrix of OLS residuals, r` = `y` - `x``` * ```beta.