Using 2 sample T test in time series data

Question

I have two data series (not stationary) and I would like to see if the mean of series 1 is significantly different when a certain condition (on the other series) is met. The theory is that when series 2 reaches a value greater than 100, the value of series 1 declines.

I've done the following: I've broken the data down into chunks, where each chunk represents a time period during which all the data in series 2 were either below or above 100. I then compare the means of series 1 in each chunk with the mean in the next chunk using the two sample T test to see if the mean of series 1 is lower when series 2 is greater than 100, and higher when series 2 is less than 100.
For example:

Obs Series1 Series2  
1    0.05     50  
2    0.03     80  
3    -0.4     30  
4    0.1      110  
5    0.03     105  
6    0.12     90  
7    -0.3     92  
8    0.11     100 
9    0.2      120

The first chunk would be the first 3 observations (all less than 100, and the second chunk the next 2 (both greater than 100). I would compare the means of series 1 in these two chunks to one another to see if they are different using the two sample t test. I would then compare the mean of series 1 from observations 6&7 with the mean from observations 8&9.

Then, if I find that in the majority of the chunks the means are significantly different, I can conclude that series 1 is significantly lower when series 2 exceeds 100.

Does this methodology even make any sense? I'm sure there is a much better way, if anyone has suggestions for me, I'm just fairly new to this kind of analysis.

Thanks

Mike

Answer 1

First, a comment on your approach. The standard setting in which a t-test is used would require independence across observations. Meanwhile, your observations may have some time patterns (autocorrelations etc), if I understand it correctly. Then you would need some adjustment to account for that instead of using the plain-vanilla t-test.

Second, here is how I would try solving this problem. Call the first series $y$ and the second series $x$. I would build a model for $y$ incorporating a feature that you are interested in. Then I would test whether the estimated feature is statistically significant.

From the limited information you have provided, I am not able to suggest what kind of model that would be. But for the sake of an example, let me assume two things:
(1) $x$ is exogenous to $y$ ($y$ does not determine $x$);
(2) $y$ follows an ARIMA(p,d,q) model with an exogenous regressor.

Since you say that $y$ (and $x$) is not stationary, let $y$ ~ $I(1)$ as an example. Then $y$ ~ ARIMA(p,1,q) with an exogenous regressor. Here is how you incorporate the feature of interest: make the exogenous regressor the indicator of $x$ being above/below 100.

Now choose suitable AR and MA orders for your model (by using AIC, BIC, looking at model residuals etc. – follow the standard guidelines). Once done, you only need to look at the coefficient on the exogenous regressor and the associated p-value. This should answer you question of interest.

If $x$ is not exogenous to $y$, you would need to model $y$ and $x$ jointly. However, the logic is still the same. Once you have built a decent model that incorporates the feature of interest, just check the relevant coefficient and its significance.