# How is the prediction variance defined?

I wonder how the output $var.pred returned by ar.ols() in R is defined? It is not the variance of a prediction value, is it? I guess no, because the variances of$n$-step ahead forecasts may be different for different$n$, correct? Does it equal the covariance matrix of the forecast error? Does it equal the residual covariance matrix? From ?ar.ols() var.pred: The prediction variance: an estimate of the portion of the variance of the time series that is not explained by the autoregressive model.  Following is an example (x is a bivariate time series): > output = ar.ols(x, aic = F, order.max = 2, demean = F, intercept = T) > output Call: ar.ols(x = x, aic = F, order.max = 2, demean = F, intercept = T)$ar
, , 1

TS1    TS2
TS1  0.5592 0.3561
TS2 -0.1555 0.1147

, , 2

TS1    TS2
TS1 -0.3455 0.5119
TS2  0.2105 0.4648

$x.intercept TS1 TS2 2.195 3.891$var.pred
TS1    TS2
TS1 1.583  1.582
TS2 1.582 18.446t


Thank you so much!

$X_t = AX_{t-1} + E_{t}$
then it's the variance matrix of $E$.