# Changes in correlation between two variables in VAR model after differencing

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

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.

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