How to determine if two time series are significantly related to each other

Question

Based on our knowledge of other characteristics of these two variables, we have reason to believe that changes in admits to a ward has an impact on a certain bad outcome on that ward (these are counts collected monthly):

> dput(admits)
structure(c(3L, 4L, 3L, 22L, 54L, 74L, 35L, 58L, 59L, 45L, 38L, 
52L, 37L, 29L, 39L, 27L, 14L, 4L, 6L, 15L, 31L, 10L, 12L, 14L, 
11L, 18L, 36L, 33L, 42L, 35L, 20L, 28L, 22L, 54L, 26L, 41L, 26L, 
41L, 40L, 34L, 31L, 23L, 34L, 22L, 21L, 11L, 29L, 13L, 27L, 40L, 
41L), .Tsp = c(2010, 2014.16666666667, 12), class = "ts")

> dput(badOutcome)
structure(c(12L, 14L, 13L, 12L, 42L, 55L, 47L, 29L, 25L, 28L, 
17L, 22L, 54L, 30L, 31L, 25L, 26L, 9L, 12L, 7L, 14L, 17L, 13L, 
13L, 14L, 12L, 15L, 20L, 17L, 30L, 35L, 41L, 18L, 19L, 26L, 15L, 
12L, 5L, 15L, 12L, 21L, 13L, 18L, 22L, 19L, 21L, 12L, 8L, 7L, 
15L, 12L), .Tsp = c(2010, 2014.16666666667, 12), class = "ts")

Since the data are serially correlated, my understanding is that the assumption of independence for regular regression techniques is violated. What are the steps then, and techniques, to determine if "admits" is significantly related to "badOutcome" ?

Answer 1

There are several ways that you can use specialized techiniques to account for serial correlation, such as autoregressive models with lags, generalized least squares, and HAC (heteroskedasticity and autocorrelation consistent) standard errors.

Once you have used these techniques to determine parameter coefficients/standard errors, you can then use standard hypothesis tests to determine if your variables are statistically significantly correlated.

Answer 2

Cross recurrence plot (CRP) will be the equivalent of cross phase analysis in wavelets. CRP is a graph which shows all those times at which a state in one dynamical system occurs simultaneously in a second dynamical system. With other words, the CRP reveals all the times when the phase space trajectory of the first system visits roughly the same area in the phase space where the phase space trajectory of the second system is.