# Test of significance between autocorrelation coefficients for two time series

I have a time series measured at monthly intervals, and I want to determine if the first half of the series is less persistent than the second half of the series. My initial thinking was to the estimate coefficients for the autocorrelation function using each half of the initial time series.

For each series I have estimated coefficient for lags 1-25. As I initially imagined, the coefficient on each lag is higher in the second series and this difference between the estimates increases with the lag. What I want to know is whether the difference is statistically significant.

I imagine there is a straightforward way to do this, but I am new to statistics and none of the methods I understand well seem to fit. I am also using R if that matters.

## 1 Answer

Form a useful ARIMA model . Estimate separate coefficients for each each time interval (two groups). Estimate the coefficients globally ( using all the data ). Perform the CHOW TEST for purposes of testing the equality of coefficients over the two groups .

• I wonder, does this approach properly address the aspect of interest, i.e. persistence? Is it possible to deduce statistical significance of difference in persistence from the Chow test suggested above? – Richard Hardy Dec 25 '15 at 19:37
• Thanks for your response. I have tried to do some reading on ARIMA models, but I don't fully understand the motivation for them. If it is possibly to briefly summarize the beneficial properties or direct to me a thorough introduction to the subject I would greatly appreciate it. – Borges Dec 25 '15 at 20:12
• Persistence ( a.k.a.) is reflected in the coeficients. Two or more sets of coefficients can be be tested for commonality of persistence via The CHOW test . – IrishStat Dec 25 '15 at 23:49
• ARIMA models are based on the ACF/PACF and reflect the way the past relates to future values. An arima model is simply a weighted sum of the past. For example a twevce period average of the past is a simple ARIMA model where the lenght of memory is 12 and the weights are equal to 1/12th .The form of the ARIMA model will tell you how to optimize these two specifications ( i.e. the length and the actual weights. – IrishStat Dec 25 '15 at 23:54
• ARIMA models based on the ACF/PACF reflect the way the past relates to future values. arima model is simply a weighted sum of the past.For example a 12 period average of the past is an ARIMA where the lengtht of memory is 12 and the weights are = 1/12 .The form of the ARIMA model will tell you how to optimize these two specifications ( i.e. the length and the actual weights.You can pursue ARIMA model educational info at autobox.com/cms/index.php/blog or here autobox.com/cms/index.php/afs-university/intro-to-forecasting or here people.duke.edu/~rnau/411arim.htm – IrishStat Dec 26 '15 at 0:02