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I am building a VAR model with two variables.

When I used it with non-stationary data with strong trend and seasonality, the correlation between variables was 0.48.

I did the first order differencing and one variable looks stationary now (the second looks stationary after the 2nd order differencing), but the correlation dropped to 0.16.

Interestingly enough, when I did the differencing of the second order, the correlation decreased slightly - from 0.16 to 0.13.

Cannot find anything about it. What is the intuition behind? Thanks

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One possibility is that the original correlation was due to the trend term.

Suppose for example,

$$ x_t = t + \epsilon_t $$ $$ y_t = t + \eta_t $$

where $\epsilon_t$ and $\eta_t$ are independent noise. Then these two series are non-stationary due to the trend term and they are correlated, with the degree depending on the variance of the noise.

When you take the first difference, then

$$ \Delta x_t = 1 + \Delta \epsilon_t $$ $$ \Delta y_t = 1 + \Delta \eta_t $$

The two series are now stationary (under some appropriate assumptions about the noise). Clearly, the two series are not correlated anymore.

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  • $\begingroup$ Thanks for your time and explanation! $\endgroup$ – Anakin Skywalker Dec 1 '20 at 1:46

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