Suppose I have 4 years worth of monthly panel data on:
- exports of widgets $y$ from home country to 12 different nations (in US dollars)
- nominal exchange rates $x$ for those 12 countries (in US dollars per unit of local currency)
- domestic consumption of widgets $z$ (in US dollars)
- nominal exchange rate $r$ of home country (in US dollars per unit of home currency)
I am primarily interested in the effect of exchange rate volatility on exports. I estimated the following seasonally-differenced equation using OLS with standard errors clustered on country and time:
$$ \ln y_{i,t} - \ln y_{i,t-52} = \alpha + \beta \cdot(\ln x_{i,t}-\ln x_{i,t-52})+\gamma \cdot(\ln z_{i,t}-\ln z_{i,t-52}) +\eta \cdot(\ln r_{i,t}-\ln r_{i,t-52}) + (\varepsilon_{i,t}-\varepsilon_{i,t-52})$$
The subscript $i$ indexes countries and $t$ indexes months. Exports are all strictly positive, so taking logs is not a problem.
Two Questions:
My estimates are somewhat sensitive to whether I control for own exchange rate $r$, and $\hat \beta \approx -\hat \eta$. Does $r$ belong in the equation?
Are the any robustness checks that I should consider? Are there any problems that jump out? I am not familiar with this literature, so any references would be most welcome.