I have daily return data of 11 sectors of MSCI World Index and the MSCI ACWI index. I want to know the stationarity of correlations between the sectors and MSCI ACWI index.

This is what I have done:

1) aggregate the daily returns into month and calculate correlation between each of the sectors and the MSCI ACWI index (so for each monthly correlation, around 20 daily returns were used). Then I used the Augmented Dickey-Fuller test to see if the monthly correlation time series is stationary or not.

2) aggregate the daily returns into week and calculate correlation between each of the sectors and the MSCI ACWI index (so for each weekly correlation, around 5 daily returns were used). Then I used the Augmented Dickey-Fuller test to see if the weekly correlation time series is stationary or not.

The result is reproduced here: enter image description here

As you can see, the stationary result is different for monthly and weekly correlations.

My question is:

1) which model is a better one?

2) what other kinds of tests or useful stuff can I do with this stationary test?

Thank you in advance!

  • $\begingroup$ Augmented Dickey–Fuller (ADF), Ljung-Box, Kwiatkowski-Phillips-Schmidt-Shin (KPSS) all test time series for stationarity. $\endgroup$
    – ERT
    Jul 6, 2018 at 22:17
  • $\begingroup$ @E.Trauger Yea they are all great tests for stationarity. But I wasn't asking for the tests. I was asking for something else. $\endgroup$ Jul 9, 2018 at 11:29

1 Answer 1


To answer your first question, it depends on the purpose of the analysis.

If you are performing the analysis for a trading strategy, you could create two models each using a different statistics. Evaluate the models and then pick the model that has better performance statistics.

If your purpose is purely exploratory, then it might be useful to dig deeper into each of these series. Decompose each correlation series into the sub-components, e.g noise, trend and seasonality. Trend and seasonality might be more present in the monthly series, so that could account for the difference in results. You could also the decompose the returns using wavelets and correlate the detail components. Then you perform the stationary analysis on each, This will give a more detailed picture of the correlations between the index and the components.

Both answers are woefully brief, let me know if you want elaboration on either.


Understanding the decomposition plot can be a challenge. Mainly what you are interested in is trend and seasonality. These will tell you much about the relationship between the sectors and index. On the non-stationary vectors, you will observe some trend or seasonality. On stationary there should be none. The plot helps you explain the behavior of the correlation series. A strong seasonal or trend component suggests that something is happening with the sector and index. An example of why trend might exist is correlations increase when markets enter corrections. So the correlations might exhibit a strong trend component around market corrections.

Furthermore, I was thinking and you can also try to cointegrate the sectors and the index. This is an alternative to correlation. A nice thing about cointegration is that making the series stationary is easy. Just determine the lag period (if any needed) and the method of cointegration (difference, beta coef, etc.) You can do a bunch with the cointegrated series, like measure half-life (speed of mean reversion).

  • $\begingroup$ Hi PlantFox, thank you very much for your reply. My goal is purely explanatory (given a timeframe, what happened between the sector and MSCI ACWI index?) I decomposed the time-series correlation into trend, seasonality, and residual, but I can't read the plots. What kinds of useful information can I get from the plots? Thanks! $\endgroup$ Jul 11, 2018 at 20:16
  • $\begingroup$ Just send an update. $\endgroup$
    – PlantFox
    Jul 12, 2018 at 19:58

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