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I'm currently reading through Analysis of Financial Time Series by Ruey Tsay. The AR model is introduced in chapter 2 and its properties in 2.4.1. The difference equations are explained and then its stated (for an AR(2)) that if $\phi^2_1 - 4\phi_2 <0$, a complex conjugate pair exists. The average length of the stochastic cycle is defined as:

$k = \frac{2\pi}{cos^{-1}[\phi_1/2\sqrt{-\phi_2}]}$

Are the $\phi_1$ and $\phi_2$ in the average length equation the same as the coefficients found for AR(2) model using least squares? If not, how can we find $\phi_1$ and $\phi_2$ for any p?

It seems im overlooking something obvious. Thanks in advance.

IDontKnowCode
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  • Hi: You should estimate the AR(2) using maximum likelihood. It's not quite the same as using least squares because least squares is slightly biased. If you are using software such as R or Python, they both have ML routines for estimating AR(p) models. – mark leeds Jan 18 '21 at 17:39
  • Thanks for the suggestion, any idea about average length of business cycles using higher order AR models? – IDontKnowCode Jan 19 '21 at 01:27
  • Hi: business cycles are quite tricky but maybe google for "estimating the length of business cycles". I would be kidding myself if I tried to say anything intelligent here. Good luck. – mark leeds Jan 19 '21 at 17:52
  • @markleeds, if I remember correctly, the maximum likelihood estimator is biased as well. – Richard Hardy Jan 21 '21 at 10:54
  • Hi Richard: You may be correct. I'm not sure about that. What I really should have said was use an R or Python AR function that builds the likelihood and optimizes over it. I don't think that there's a better way because, assuming you're correct, then there is definitely is no closed form unbiased estimator so optimization seems like the best approach. – mark leeds Jan 22 '21 at 17:22

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