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st: Variance of first 2 observations in simulated AR(2) process


From   Bert Lloyd <bert.lloyd.89@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   st: Variance of first 2 observations in simulated AR(2) process
Date   Wed, 4 Jan 2012 08:57:36 -0500

Hi Statalist,

I would like to simulate an AR(2) error process for a Monte Carlo
study. I would like to get the variance of the first two observations
right.

For an AR(1) process, this is easy to find, for example Baltagi,
Econometric Analysis of Panel Data, 4th ed (2008), page 92 points out
that for an AR(1) process defined as
y_t = rho*y_{t-1} + epsilon_t, with abs(rho)<1 and epsilon_t ~ iid(0,sigma^2),
the variance of the first observation is sigma^2 / (1 - rho^2).

I would like to know if there is a similar analytical expression for
an AR(2) model.

I have looked at Jeff Pitblado's very nice sim_arma, but this is
cannot be combined with by, which is inconvenient since I am trying to
generate a panel. I suppose a similar "burn-in" approach could be
constructed by hand, but I worry this would slow down the simulation,
and also I find the analytical approach somewhat cleaner.

Many thanks in advance for any suggestions.

 - BL
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