# 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

## 1 Answer

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