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?


This question sounds like an XY problem to me.
There's no "Effective degrees of freedom" for seasonal/cyclical data. The fundamental problem here is that seasonal data is nonstationary: the values of your parameters are changing over time, while any estimates of your model are going to have to assume the parameters are constant. Until you correct for that, your estimates are not going to be consistent, regardless of what you use as your degrees of freedom. I'd recommend going through a good resource on time series modeling, either a textbook or the documentation for the R forecast package. Pay close attention to the parts on seasonal time series and ARFIMA models, which should address your concerns. As for p-values, the package will automatically calculate them for you.

I have averaged the data from multiple nearby locations so not worried about spatial degrees of freedom

I wouldn't recommend doing that, unless you're very certain that measurements are almost independent; it invalidates your p-values if the data aren't independent.
