# Possible simultaneity issue in time series analysis

Let $Y_t$ be the amount of sales at time $t$ and $X_t$ be the amount of expenses in marketing activities. I want to estimate the effect of $X_t$ on $Y_t$.

However, we suppose that $X_t$ is determined by the sales in previous period, so $X_t$ depends on $Y_{t-1}$. Does this problem involve a case of simultaneity? Consider that $Y_t$ and $Y_{t-1}$ are independent. Can I use normal OLS or other methods are needed?

Specific references would be appreciated.

• I am not an expert in this particular task, but I found this to be an interesting reading: statisticshowto.com/instrumental-variable As for me I suggest measuring the obsrved correlation between X t and Y t-1 to be sure the effect is significant to even consider it in the design. – Alexey Burnakov Oct 3 '17 at 10:53
• It's hard to consider Y not autocorrelated. Sales data is very persistent and almost always autocorrelated. You may wont to look at differences both in Y and X, and lag dY on dX. Do you have marketing expense budget variable? You could look at the diff between expense and its budget as another predictor – Aksakal Oct 3 '17 at 11:37
• Thank you both for your comments. @Aksakal, I know it seems unusual, but in my specific case Y is not autocorrelated (there would be a lot to write to justify this, but trust me). That's way I'm interested to know if in this specific case there is a simultaneity problem or not... – Andrea Oct 3 '17 at 12:32
• Can you write down your equations? – Aksakal Oct 3 '17 at 12:35
• $Sales_t$= $\alpha$ + $\beta_1$ $price_t$ + $\beta_2$ $mkt expenses_t$ + $\beta'$ $X'$ + $\epsilon_t$ where $X'$ is a vector of control variables don't change over time. On the other hand: $mkt expenses_t$=$\alpha$ + $Sales_{t-1}$ +$\beta'$ $Z'$ + $\epsilon_t$. I want to underline that this work is at a very preliminary stage..actually I'm only investigating what I can do with my dataset. – Andrea Oct 3 '17 at 12:58