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Do you remove the seasonal component, the trend component, or both, when adjusting a time series to avoid finding spurious correlations?

I have two time series x and y with trends and yearly seasonality and I would like to test for correlation. I read this article about spurious correlation in time series https://svds.com/avoiding-common-mistakes-with-time-series/ which recommends taking the trendline out by first differencing or decomposition. However I have also seen sources suggesting instead to remove the seasonality component.

What is the correct way to adjust the time series to avoid detecting spurious correlations? If I removed both the trend and seasonality would I not be left with random noise? Should I only remove the trend or seasonality if they are the same in both series?

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Sometimes neither. You might need to take differences or trends or seasonal differencing or multiple trends. For causal problems, you would want to use the cross-correlation to determine if you have a relationship. The transfer function process has you build a model for x and then use the residuals to correlate the X with the Y. Don't forget outliers and changes in trend....and seasonality and parameters and variance.

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