correlation coefficient between two time series

[hardmann]

Dear Tom:

May I ask a fundamental question. How to compare the similarity between two time series? Directly calculating the correlation coefficient, due to the common trend, the coefficient is very high, which is obviously a spurious correlation. After differential or logarithmic difference, calculate the correlation coefficient to be 0.64. Can this coefficient indicate moderate similarity? Even if the cointegration equation and ECM are estimated, it seems impossible to explain the degree of correlation between the two? More generally speaking, we have estimated a time series and compared it with a benchmark. How can we determine whether our results are acceptable or excellent?

Best regard
Hardmann

[TomDoan]

That is and has been a problem for decades. At one point people looked into frequency domain methods (such as cross-spectral analysis) to try to break down the correlation by frequency, where the longer (multi-year) cycles were the main focus. The problem there is that first-differencing series largely eliminates the contribution of most of those frequencies and leaving them undifferenced produces series which (in practice) have infinite variance and thus no actual correlations. Plus, the spurious regression analysis shows that what correlation there seems to be at low frequencies can be completely random when you have unit roots in both series.