# Reverse causality

I have 4 time series, $y, x_1, x_2$ and $x_3$, and I'm running the following reduced form VAR:

$\Delta y_{t} = A_1 \,x_{1,\,t-1} + A_2 \,x_{2,\,t-1} + A_3 \,x_{3,\,t-1} + \varepsilon_t$

Is there any chance the regression estimates may suffer from reverse causality? My guess would be no, since it's unlikely that the future can affect the past, right?

• Staying outside the realm of exotic physics, it's safe to assume that the future doesn't cause the past. But, common causes and spurious correlations are still possible, of course. So, non-zero coefficients wouldn't imply that $x_{t}$ causes $y_{t+1}$. – user20160 Aug 31 '18 at 21:15

For example, if I know that I am going to go on vacation and eat out a lot at time $t$ (so $y_t$ is restaurant expenditure, so $(y_t - y_{t-1})>0$), my expenditures on wine, meat, and cheese at $t-1$ may reflect that ($x_{1,t-1},x_{2,t-1},x_{3,t-1}$ would be lower). That does not mean that lowered grocery expenditures cause the restaurant expenditure.