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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?

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    $\begingroup$ 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 $y_{t+1}$. $\endgroup$ – user20160 Aug 31 '18 at 21:15
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If people have accurate expectations about the future, then that can certainly happen.

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.

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  • $\begingroup$ What if x_1, x_2 and x_3 are the volumes at time t-1 of a particular stock that is traded in three different exchanges, and y is the price of that stock in one of the three exchanges? Can the A coefficients be regarded as price impacts? Is this model any similar to the Kyle (1985) model? $\endgroup$ – Mary Sep 1 '18 at 9:33
  • $\begingroup$ I am not a finance person, so I can't really say, and I have not even glanced at this paper. I expect all the coefficients to be one, so that would not be a a very interesting impact to interpret. $\endgroup$ – Dimitriy V. Masterov Sep 1 '18 at 16:33

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