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```Estimating AR(p) models by ESTIMATE

The AR(p) model is
(1) Y = X*beta + u
where Y is an n-vector of dependent variable, X is an n*m-matrix of
regressors, beta is an m-vector of regression coefficients and
u = (u(1),u(2),...,u(n))' are residuals of the form

u(j)=r(1)*u(j-1)+...+r(p)*u(j-p)+e(j)

where e(j), j=1,2,...,n are independent N(0,sigma^2) variables.

The easiest method for estimation of beta and r(1),...,r(p)
is to estimate the nonlinear regression model (please note that u=Y-X*beta)
p
(2) y(j) = x(j)*beta +  sum r(i)*[y(j-i)-x(j-i)*beta]  + e(j),
i=1
j=p+1,2,...,n,
where x(j) is a row vector of regressors x1,x2,...,xm in the j'th
observation.

Least squares estimates for the model (2) are obtained by the ESTIMATE
operation as follows (the scheme is here presented for p=3 in a case
where we have Y as the regressand and X1,X2,X3,X4 as regressors).
.......................................................................
{A0}=b0+b1*X1+b2*X2+b3*X3+b4*X4
{A1}=r1*(Y[-1]-b0-b1*X1[-1]-b2*X2[-1]-b3*X3[-1]-b4*X4[-1])
{A2}=r2*(Y[-2]-b0-b1*X1[-2]-b2*X2[-2]-b3*X3[-2]-b4*X4[-2])
{A3}=r3*(Y[-3]-b0-b1*X1[-3]-b2*X2[-3]-b3*X3[-3]-b4*X4[-3])

MODEL AR3
Y={A0}+{A1}+{A2}+{A3}

ESTIMATE <data>,AR3,<line_for_results>
.......................................................................
ESTIMATE automatically recognizes the observed variables Y,X1,X2,X3,X4
from the data and interprets unidentified notations
b0,b1,b2,b3,b4 and r1,r2,r3 as parameters to be estimated.

The components of the model are presented by shorthand notations
{A0},{A1},... permitting a brief notation for the model itself.
Lagged values are notated as Y[-2] (Y lagged by 2) as in the VAR operation.
It is easy to extend the model for larger p and even drop some {Ai}
components.
Good starting values for b's are obtained from OLS estimates (LINREG,
REGDIAG, or ESTIMATE with model Y={A0}).

Since ARMA(p,q) models can be represented as A(infinity), in many cases
the regression coefficients may be estimated for such more general
models from AR(p) when p is large enough.