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Consider the model $$x_i=\rho_1x_{i-1}+\rho_2x_{i-2}+\dots+\rho_px_{i-p}+\omega_i,\:\mathbf{\omega}\sim N(\vec0,\mathbf{I}\sigma^2)$$

In the case of order 1 autocorrelation (i.e. where $\rho_2$ and up are 0), I am able to make a covariance matrix of $\mathbf{x}$ quite easily (see here). But I need to be able to compute the variances and covariances for any order of autocorrelation, and I have not been able to find any literature that covers the math behind it. Does anyone know where I can find out how to compute this matrix?

Dylan Way
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  • The "math behind it" is matrix algebra. Using it will reveal the inherent patterns and make easy work of the calculations (compared to writing out all the sums). But I notice your notation for $\omega$ strongly implies $x_i$ is vector valued: is that what you intend? – whuber Feb 03 '24 at 22:50
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    I meant for $\mathbf{\omega}$ to denote the vector of iid error values, $\mathbf{x}$ to denote the vector of values coming out of the AR process, and $\omega_i$ and $x_i$ to denote individual entries within those vectors – Dylan Way Feb 04 '24 at 16:52

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