How to find effective number of degrees of freedom for time series I have 10 years time series of daily precipitation and temperature data (3652 records) as given below:




Date
Precipitation(mm)
Temperature (°C)




01.01.2010
20
12


02.01.2010
5
6


03.01.2010
0
9


...
...
...


01.01.2011
32
15


02.01.2011
0
02


...
...
...




so on.
These timeseries data will have autocorrelation due to seasonal cyclic variation of temperature and precipitation. The degrees of freedom will be different from (n-2).  I have calculated the correlation coefficient between the series. However, for significance test I want to calculate the effective degrees of freedom. How to calculate the effective number of degrees of freedom for this data?
I have tried going through a few research articles (Bretherton et. al., 1999; Wang,1999) but not able to figure out how to calculate effective degrees of freedom as I am new in this field and required simplified explanation. I will be thankful if someone could suggest some software/tools to for such analysis.
In northern hemisphere the temperature is low in January and starts increasing from February. Temp remain high in summer and starts decreasing from Autumn. It shows a cyclic fluctuations (decorrelation time of less than a year) if multiple years are considered. This will reduce the effective sample size and I want to find the effective degrees of freedom. I have calculated the Pearson correlation between two series(temp and precipitation) and found high correlation. The scattered plot is shown below. I am proposing to use Student's t test for significance testing. I just want to test this hypothesis and don't want to use it for forecast. I have averaged the data from multiple nearby locations so not worried about spatial degrees of freedom.

I have reduced the data in time series and now considering only 4 months June to September (122 days) of every year. The correlation(r) at different time lags is shown below. From the graph the r decreases and falls to lowest negative value in around 60 days. But it increases and reaches highest maximum value in around 120 days. So can we find the autocorrelation time from this graph? will it be 60 or 120 days?


After googling and reviewing a few papers, I want to post the solution for people who may be stuck and have difficulty in finding the effective degrees of freedom. The time series can be made stationary by different methods like differencing etc. If all other conditions as mentioned by @ClosedLimelikeCurves & @EdM, are satisfied there are following methods adopted especially for climate data to calculate effective degrees of freedom.

*

*Bretherton et. al., 1999 explained different methods of finding effective sample size(ESS) or temporal degrees of freedom of a time series. According to which ESS can be calculated:


where T* can be regarded as the effective sample size of the T observations after accounting for their temporal correlation.
If rx is the lag-one autocorrelation of Xi and similarly for Y, and if rx, ry ≪ 1, then

This is appropriate for significance tests involving second-order moments of the original time series (variance, covariance, and correlation).


*For computing the variance of the sample mean (i.e., the first-order moment) of T observations from a time series can be expressed as (Thiébaux and Zwiers):




*To test the hypothesis that two populations have equal means Welch's t-test is utilized. The effective degrees of freedom is approximated using the Welch–Satterthwaite equation.

where v is effective degrees of freedom and s is sample variance.
