4

If an AR(2) model is stationary, how to prove that $$\rho_1^2<\frac{\rho_2+1}{2}$$

I know that $$\rho_1=\frac{\phi_1}{1-\phi_2}$$ and $$\rho_2=\frac{\phi_1^2+\phi_2(1-\phi_2)}{1-\phi_2}$$ according to Yule-Walker equation, but when I try to prove it by the two equations above it just doesn't make sense.

Christoph Hanck
  • 33,180
Lily
  • 43
  • 4

1 Answers1

8

This is a consequence of the (general) fact that (auto)correlation matrices like $$ \begin{pmatrix} 1&\rho_1&\rho_2\\ \rho_1&1&\rho_1\\ \rho_2&\rho_1&1\\ \end{pmatrix} $$ are positive semi-definite, i.e. their determinant satisfies $$ 1-2\rho_1^2-2\rho_1^2\rho_2-\rho_2^2\geq0 $$ Write the determinant as $$ (1-\rho_2)(1+\rho_2-2\rho_1^2) $$ Since $\rho_2\leq1$ and thus $1-\rho_2\geq0$, we must have $$1+\rho_2-2\rho_1^2\geq0$$ or $$\rho_1^2\leq\frac{1+\rho_2}{2}$$ for the determinant to be nonnegative.

Hence, this appears to be general property of (auto)correlations and has nothing to do with an AR(2) process specifically. (Also, my argument suggests that the inequality is only weak, but see whuber's comment below for more on this point.)

Christoph Hanck
  • 33,180
  • 1
    +1 -- very clear and to the point. Concerning whether the inequality is weak, it will reduce to an equality only when the matrix is of lower rank, which implies there is some linear combination that is zero. Stationarity means the same windowed linear combination in the process itself is constantly zero -- but that, in turn, implies the random terms in the AR(2) definition could not be a white noise process. – whuber Mar 07 '23 at 14:18
  • 2
    Thanks, I had a feeling that equality was likely to be some kind of degeneracy, and it is useful for me to see precisely how! – Christoph Hanck Mar 07 '23 at 14:32
  • Thanks!It helps a lot. – Lily Mar 10 '23 at 06:39
  • 2
    What @ChristophHanck intends to say is that if the post helped you provide sufficient insight to clear your confusion, you can accept the answer: What does it mean when an answer is "accepted"?. – User1865345 Mar 10 '23 at 08:03