Hi, guys this is the first time I use this forum, I hope I could get some answer from experts in this quantitative finance field.My problem is that I'm looking for the formula for computing zero-crossing rate of ARMA. I know there is a Rice formula, but it is not available freely on website( at least I didn't find it). Is there someone knows it and could provide some information on it?Another question on ARMA: when I got a AR(2) from PACF of one time series, the theoretical formula should be Yt=Yt-1-0.9Yt-2+u, however the series I simulated from this AR(2) is quite different from the series observed. Can you point out same drawbacks in it?And how about the volatility feature of ARMA(1,3)?Thanks, I'm waiting

No one knows it? I can't believe it. Any suggestion could make sense. Did anyone here knows some formula related to zero-crossing rate?

okay, ive done a quite a lot of time series work so im going to try and help you.first off you are going to have to clarify a few things for me, a) what is the zero crossing rate?your simulated AR(2) series is different from the series observed, if you are talking about your ACF, this shouldnt be the case? what do the two look like?also when you ask about the "volatility feature" of an ARMA(1,3) what do you mean exactly? this is a homoskedastic model, do you want to derive the ACF?James

The characteristic equation from your estimation result is .9B^2 +B +1 = 0 where B is backshift operator. Since the discriminant of the equation is less than zero, B has imaginary roots. This means that the underlying time series has seasonal component. In this case one should consider seasonal lags.

first of all, thanks for all your replies.1.zero-crossing rate is the number of times a time series in a certain time periods cross its mean which is zero in white noise case.2.When I tested the PACF, I found the series could be described as an AR(2) with many in ACF. However, after I simulated AR(2) series with random numbers I found the result exhibit a statioinary series instead of a fluctuate series in the real observation. And I test many observed time series with the same length, some of them behave like a AR(2), some are completely different,For instance : a random walk. Even though ACF and PACF test on them are all AR(2)!3. ARMA(1,3) in the long run will have a constant volatility, but in short term I found the MA(3) part would make short term volatility unstable,for example given 100 time points the volatility change significantly. Well with 1000 time interval or more, the volatility is constant4. To weare, thanks a lot,this is a good point I didn't take into account before, since the series I was dealing with shouldn't have seasonal factor based on intuition.

first of all, thanks for all your replies.1.zero-crossing rate is the number of times a time series in a certain time periods cross its mean which is zero in white noise case.2.When I tested the PACF, I found the series could be described as an AR(2) with many in ACF. However, after I simulated AR(2) series with random numbers I found the result exhibit a statioinary series instead of a fluctuate series in the real observation. And I test many observed time series with the same length, some of them behave like a AR(2), some are completely different,For instance : a random walk. Even though ACF and PACF test on them are all AR(2)!3. ARMA(1,3) in the long run will have a constant volatility, but in short term I found the MA(3) part would make short term volatility unstable,for example given 100 time points the volatility change significantly. Well with 1000 time interval or more, the volatility is constant4. To weare, thanks a lot,this is a good point I didn't take into account before, since the series I was dealing with shouldn't have seasonal factor based on intuition.