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?

 A: 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.
A: 
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?

Bivariate normality assumption. Putting aside for a moment both the time-series and the "effective degrees of freedom" issues, a standard t-test of the Pearson correlation coefficient wouldn't be appropriate here. That test assumes bivariate normality of the two variables. Many precipitation values are 0 and they cannot be negative, so you can't make that normality assumption about precipitation.
As an example, I obtained 10 years' data starting from 2012-08-21 at Houston International Airport (IAH) via NOAA. Under an independence assumption, the Pearson correlation coefficient of 0.04 was "statistically significant" at p = 0.017, but the Kendall correlation test (under the same independence assumption but not assuming bivariate normality) showed p = 0.56. For this particular application, the normality assumption might be much more of a problem than the "effective degrees of freedom."
Stationarity. As @ClosedLimelikeCurves correctly notes, the notion of "effective degrees of freedom" or "effective sample size" (ESS) strictly holds only for stationary data (constant distribution over time). When you work with time series data you need to understand the issues and use appropriate tools. The Hyndman and Athanasopoulos text and its associated fpp3 and fable packages in R provide a good way to start.
Effective sample size (ESS). Even when an ESS value is appropriate you can't always use it as a simple correction for significance testing. Thiébaux and Zwiers examined this problem decades ago. They noted:

regardless of the extent of time coherence, a sample of N observations from any nonsingular
Gaussian stochastic process has N degrees of freedom. A suitable linear transformation of the observation vector, for which the transformation is determined by the covariance structure of the process, has N independent components. The phrase "effective sample size" is less troublesome; it manages to convey the notion that the N pieces of information are "smeared" across the N observations by the time durations of their influences.

They evaluated several ways to estimate the ESS for a stationary process.* When you find the ESS, however, you can't always plug that into the degrees of freedom for a t-test and get a valid result.
Thiébaux and Zwiers illustrated this with simulated data, and showed why in their Appendix. It is possible to use the ESS to estimate the variance of the estimate of sample mean. A t-test, however, assumes that the numerator (normal distribution) and denominator (chi-square distribution) are statistically independent. To meet that assumption with serially correlated data you must take the variance/covariance matrix of the data into account.
What is the correlation estimating? This page describes why it can be difficult to interpret correlations between time series. This page and this page discuss the related issue of so-called "spurious correlation." Read those pages to see how, depending on the situation, a correlation between time series can be anywhere between completely misleading and highly informative.
In your situation, there are presumably underlying seasonal patterns and long-term trends, around which there is random variability that might in turn be correlated in time. You might, for example, be most interested in long-term trends after subtracting out seasonal patterns and accounting for the time-correlated random variability. Tools for time-series analysis can allow you to tease apart those three contributions to the observations in a way that is much more informative than just calculating the raw correlation between the series.
H. J. Thiébaux and F. W. Zwiers,
"The Interpretation and Estimation of Effective Sample Size," Journal of Climate and Applied Meteorology 23(5): 800-811, 1984.

*As the page recommended by @usεr11852 illustrates, even today there isn't complete agreement about how to do this. The papers to which you linked are for evaluating spatial degrees of freedom, not temporal.
A: 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.
