# Statistical Comparison of Time Series

I have two-time series of daily temperatures for two locations for 30 years. Is there any way to compare them statistically? To see if the differences are statistically significant?

I considered commenting but apparently I haven't got enough reputation. I can't answer your question per se, but here are some recommendations:

In general, autocorrelation plots reveal a considerable amount of information to characterize a time series. If you obtain two quite different autocorrelation plots, it is sufficient to claim that two series are different.

You can fit a (S)ARIMA(X) model and compare statistically significant estimates for both series. If any of these are different, the series would not have a similarity.

You can perhaps do a regression using lags of one series as explanatory variables of the other series (Xt-1 to predict Yt and Yt-1 to predict Xt). I think if either one of these appear to be statistically significant, the series would not be similar.

If you had more than two time series, you could take an experimental design approach, considering (S)ARIMA(X) parameters as individual observations.

Suppose $$X_{t}$$ and $$Y_{t}$$ are your time series. You can obtain their difference as:

$$Z_{t} = X_{t} - Y_{t}$$

Then, you can test whether Z_{t} is white noise which implies that the two time series are not significantly different.

You can find how to do such a test here: https://ccrma.stanford.edu/~jos/sasp/Testing_White_Noise.html#:~:text=To%20test%20whether%20a%20set,successive%20noise%20samples%20are%20uncorrelated.

You can also use change point models to see if there is a significant change point in $$Z_{t}$$ indicating the starting point of divergence between the two.

• Thanks for that. I do not understand why Z_{t} being white noise implies that the two time series are not significantly different.
– lola
Commented Jan 30, 2023 at 15:26
• Note that if you compare their differences $Z_t=X_t-Y_t$, your $Z_t$ will likely have autocorrelation. You can then test whether the mean of $Z_t$ is significantly different than zero by dividing the mean of $Z_t$ by the long-run variance of $Z_t$. Commented Jan 30, 2023 at 15:43
• I should have added that you should put a threshold on the variance of the Z. Having Xt = Yt + Zt, obviously a white noise with high variance destroys Y's signal. So as long as Zt is mean zero with negligible variance I would say X and Y are not significantly different. Commented Jan 30, 2023 at 16:05
• @DavidVeitch Not if Zt is white noise and that is the whole point of testing for it. Commented Jan 30, 2023 at 16:05