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generate AR(1) data

From: Heber Farnsworth
Subject: generate AR(1) data
Date: Wed, 16 Jul 2003 21:39:19 -0500

This is probably a simple question but I'm stumped and if someone knows the answer off the top of their head it would greatly speed up my code.

I need to generate a large number of random (Gaussian) vectors that have an AR(1) correlation structure. Obviously I could do this in a loop but this is slow. I would rather rotate a vector of IID normal draws so that the result will have the right covariance matrix. So what I need is the matrix square root of a symmetric toeplitz matrix

1   .5  .25  .125 ...
.5   1  .5    .25 ...
.25 .5  1   .5 ...

etc (here the correlation is .5). It seems inefficient to try to compute this square root numerically. Isn't it known in closed form?

Heber Farnsworth

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