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I've used R's "chisq.test()" function to calculate the chi square statistic between two time series, the first is a stock market series of a bank with a length 4262, and the second is a simulation of the first using some model with the same length. I used chisq.test as a measure of independence of the two series. Using R, I get the following output.

    Pearson's Chi-squared test

data:  as.numeric(banks[, 2]) and as.numeric(banks[, 3])
X-squared = 16117000, df = 16094000, p-value = 3.434e-05

Regardless of whether this method is correct, I don't understand where the high degrees of freedom comes from. I assumed the chi square test measures the statistic between two series using a 2 X length contingency table, with one column as observed and the other as expected. Can anyone tell me what is actually being calculated in this test?

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    $\begingroup$ You messed up the code somehow. But coding questions are off topic here. And, as an aside, I can't imagine a reason to do what you are trying to do. $\endgroup$ – Peter Flom - Reinstate Monica Aug 3 '17 at 0:04
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Chi-square stats is used to compare the difference between observed value and the expected value. One Example can be the following case:

| category | Observed | Expected |
+------------+--------------+-------------+
| True | 4 | 3 |
| False | 6 | 7 |
+-------+---+--+

chi2 = (4-3)/3 + (6-7)/7. The according degree of freedom = n_cat - 1 = 2 - 1 = 1.

In your example, the number of categories is not 2 (Category), but the number of time series points. Take a daily time series data in a year as an example, where observed values are consecutively increasing numbers 1 to 365, and the expected values are consecutively decreasing numbers 365 to 1.

| category | Observed | Expected |
+------------+--------------+-------------+
| 2017-01-01 | 1 | 365
| 2017-01-02 | 2 | 364
| ... | ... |
| 2017-12-31 | 365 | 1
+---------------+------+--+

chi2 = (1-365)^2/365 + (2-364)^2/364 + ... + (365-1)^2/365. The according n_cat in this case is 365 (there are 365 days in 2017). Thus the dof = 365 - 1 = 364 instead of 2.

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    $\begingroup$ There are many other applications of chi square besides being an approximate test for differences between groups (from a contingency table). $\endgroup$ – Michael R. Chernick Aug 3 '17 at 0:47
  • $\begingroup$ Yes. I've never mentioned that comparison between observed and expected value is the ONLY application for chi square. The main purpose of my post is to discuss how to solve Arman's questions in details. $\endgroup$ – Jm M Sep 17 '17 at 3:29

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