# How can I estimate the confidence interval of correlations possibly dependend with time?

I have a multivariate problem (with solar data from different meteorological stations) that I am working on my engineering master thesis. I would like to estimate the correlations of different variables pairs. However I would also like to know if the values from the sample time (monthly data for some 20 years) is representative of the real values. The question involves how can I calculate confidence interval on such cases.

The two methods I considered was based on fisher transformation and on bootstrapping. However, for both cases I would like to know:

1. How can I verify if correlations are time-independent?
2. If they are not, how could I apply those methods?

My Idea so far was to calculate correlations for groups of n months (I was thinking in using 6) and see if there was time-dependency by using a test such as Ljung-Box. Than the sample size used on fishers transformation would be the original one divided by n. Using bootstrap, The permutations would be of these groups with n months.

Is my idea fine? How could I make this analysis better?

• when you ask about time-dependent correlation, are you asking about $\text{corr}(X_s,Y_s)$ changing as $s$ changes, or are you asking about lagged correlations $\text{corr}(X_s,Y_{s-d})$ (i.e. the cross correlation function)? Dec 22 '18 at 23:07
• @Glen_b In my opinion he is interested in computing ( and comparing ) simple correlations for different time ranges.e.g. month 1-24 , month 25-48 .....month non-23 -month nob BUT I have been wrong before.. Dec 23 '18 at 21:47
• Different simple correlations across time (as you suggest) is also my impression but I wanted to make it more explicit. Dec 23 '18 at 23:20
• Yes, it is that, simple correlation that might vary with time. Dec 24 '18 at 2:06
• Similar Q (without answer): stats.stackexchange.com/questions/13515/… Dec 24 '18 at 12:13

This is an interesting question, I have seen very little written about time-varying (cross-)correlations in time series. But your question would be helped by some more context, what are your different variables measuring? That would allow for some more useful answers!

So for now (its Christmas!) a very short answer. I would start with some visualizations, in R there is a useful function coplot for conditioning plots, some examples in following posts: Investigate correlation conditional on a threshold, Can I analyze or model a conditional correlation? or What is the physical significance of cumulative correlation coefficient?.

I don't know about formal tests ... maybe some others can churn in? The following google search gives some interesting hits. You could also search for "Garch dynamic correlation" for some ideas.

There are two assumptions underlying the significance test associated with a Pearson correlation coefficient between two variables. Assumption 1: The variables are bivariately normally distributed. ... The significance test for a Pearson correlation coefficient is not robust to violations of the independence assumption. See course material here http://oak.ucc.nau.edu/rh232/courses/eps525/handouts/pearson%20correlation%20coefficient%20-%20handout.doc

Rather than simply focus on the correlation coefficient http://www.math.mcgill.ca/dstephens/OldCourses/204-2007/Handouts/Yule1926.pdf (possibly flawed by both anomalies the I's and the need for lag structure or differencing in either Y or X ) focus on the strength of regression structure between the two series after you have accounted for arima structure and any latent deterministic structure (the I's) such as pulses, level/step shifts , seasonal pulses and or local time trends.

excerpt from Yule's paper

To Question 1: Estimate the General Linear Model both globally and locally in order to test whether or not the REGRESSION COEFFICIENTS ( on X ) are constant over time slices (intervals) using the CHOW formal F test for constancy of parameters while parametric tests for correlations are quite problematic.