Linear model

In statistics the linear model is given by


 * $$Y = X \beta + \varepsilon$$

where Y is an n&times;1 column vector of random variables, X is an n&times;p matrix of "known" (i.e. observable and non-random) quantities, whose rows correspond to statistical units, &beta; is a p&times;1 vector of (unobservable) parameters, and &epsilon; is an n&times;1 vector of "errors", which are uncorrelated random variables each with expected value 0 and variance &sigma;2.

Much of the theory of linear models is associated with inferring the values of the parameters &beta; and &sigma;2. Typically this is done using the method of maximum likelihood, which in the case of normal errors is equivalent (by the Gauss-Markov theorem) to the method of least squares.

Multivariate normal errors
Often one takes the components of the vector of errors to be independent and normally distributed, giving Y a multivariate normal distribution with mean X&beta; and co-variance matrix &sigma;2 I, where I is the identity matrix. Having observed the values of X and Y, the statistician must estimate &beta; and &sigma;2.

Rank of X
We usually assume that S is of full rank p, which allows us to invert the p &times; p matrix $$X^{\top} X$$. The essence of this assumption is that the parameters are not linearly dependent upon one another, which would make little sense in a linear model. This also ensures the model is identifiable.

&beta;
The log-likelihood function (for $$\epsilon_i$$ independent and normally distributed) is


 * $$l(\beta, \sigma^2; Y) = -\frac{n}{2} \log (2 \pi \sigma^2) - \frac{1}{2\sigma^2} \sum_{i=1}^n \left(Y_i - x_i^{\top} \beta \right)^2$$

where $$x_i^{\top}$$ is the ith row of X. Differentiating with respect to &beta;j, we get


 * $$\frac{\partial l}{\partial \beta_j} = \frac{1}{\sigma^2} \sum_{i=1}^n x_{ij} \left( Y_i - x_i^{\top} \beta \right) $$

so setting this set of p equations to zero and solving for &beta; gives


 * $$X^{\top} X \hat{\beta} = X^{\top} Y.$$

Now, using the assumption that X has rank p, we can invert the matrix on the left hand side to give the maximum likelihood estimate for &beta;:


 * $$ \hat{\beta} = (X^{\top} X)^{-1} X^{\top} Y$$.

We can check that this is a maximum by looking at the Hessian matrix of the log-likelihood function.

&sigma;2
By setting the right hand side of


 * $$ \frac{\partial l}{\partial \sigma^2} = -\frac{n}{2\sigma^2} + \frac{1}{2 \sigma^4} \sum_{i=1}^n \left(Y_i - x_i^{\top} \beta \right)^2$$

to zero and solving for &sigma;2 we find that


 * $$ \hat{\sigma}^2 = \frac{1}{n} \sum_{i=1}^n \left(Y_i - x_i^{\top} \hat{\beta} \right)^2 = \frac{1}{n} \| Y - X \hat{\beta} \|^2. $$

Accuracy of maximum likelihood estimation
Since we have that Y follows a multivariate normal distribution with mean X&beta; and co-variance matrix &sigma;2 I, we can deduce the distribution of the MLE of &beta;:


 * $$ \hat{\beta} = (X^{\top} X)^{-1} X^{\top} Y \sim N_p (\beta, (X^{\top}X)^{-1} \sigma^2 ).$$

So this estimate is unbiased for &beta;, and we can show that this variance achieves the Cramér-Rao bound.

A more complicated argument shows that


 * $$ \hat{\sigma}^2 \sim \frac{\sigma^2}{n} \chi^2_{n-p}; $$

since a chi-squared distribution with n &minus; p degrees of freedom has mean n &minus; p, this is only asymptotically unbiased.

Generalized least squares
If, rather than taking the variance of &epsilon; to be &sigma;2I, where I is the n&times;n identity matrix, one assumes the variance is &sigma;2M, where M is a known matrix other than the identity matrix, then one estimates &beta; by the method of "generalized least squares", in which, instead of minimizing the sum of squares of the residuals, one minimizes a different quadratic form in the residuals &mdash; the quadratic form being the one given by the matrix M&minus;1:


 * $${\min_{\beta}}\left(y-X\beta\right)'M^{-1}\left(y-X\beta\right)$$

This has the effect of "de-correlating" normal errors, and leads to the estimator


 * $$\widehat{\beta}=\left(X'M^{-1}X\right)^{-1}X'M^{-1}y$$

which is the best linear unbiased estimator for $$\beta$$. If all of the off-diagonal entries in the matrix M are 0, then one normally estimates &beta; by the method of weighted least squares, with weights proportional to the reciprocals of the diagonal entries. The GLS estimator is also known as the Aitken estimator, after Alexander Aitken, the Professor in the University of Otago Statistics Department who pioneered it.

Generalized linear models
Generalized linear models, for which rather than


 * E(Y) = X&beta;,

one has


 * g(E(Y)) = X&beta;,

where g is the "link function". The variance is also not restricted to being normal.

An example is the Poisson regression model, which states that


 * Yi has a Poisson distribution with expected value e&gamma;+&delta;xi.

The link function is the natural logarithm function. Having observed xi and Yi for i = 1, ..., n, one can estimate &gamma; and &delta; by the method of maximum likelihood.

General linear model
The general linear model (or multivariate regression model) is a linear model with multiple measurements per object. Each object may be represented in a vector.