# Time series and instrumental variables

I have a time series model that suffers from endogeneity. In other contexts it would be reasonable to use instrumental variables. However, I have not seen this done before with time series. Can I ask if it is valid to use an IV in this context and if there are any examples in the economics literature?

• I suggest writing out instrumental variables in the title, as iv means independent variable to many people on this forum. – Christoph Hanck Jun 4 '16 at 14:46

Consider a series $Y_t$ generated as an $ARMA(1,1)$ process $$Y_t=\phi Y_{t-1}+\epsilon_t+\theta\epsilon_{t-1}$$ Suppose our interest centers on estimating $\phi$. We have an endogeneity issue here, as the error term $\epsilon_t+\theta\epsilon_{t-1}$ is correlated with the regressor $Y_{t-1}$, so OLS of $Y_{t}$ on $Y_{t-1}$ would not consistently estimate $\phi$: $$\hat{\phi}_{OLS}=\frac{\sum_tY_{t-1}Y_{t}}{\sum_tY_{t-1}^2}=\frac{\frac{1}{T}\sum_tY_{t-1}Y_{t}}{\frac{1}{T}\sum_tY_{t-1}^2}\to_p\frac{\gamma_1}{\gamma_0},$$ where the convergence in probability follows from standard arguments about plims of $\frac{1}{T}\sum_tY_{t-j}Y_{t-l}$ and the continuous mapping theorem. Now, it is known that $\gamma_0=\sigma^2\frac{1+\theta^2+2\phi\theta}{1-\phi^2}$ and $\gamma_1=\sigma^2\frac{(\phi+\theta)(1+\phi\theta)}{1-\phi^2}$. Hence, \begin{eqnarray*} \hat{\phi}&\to_p&\frac{\sigma^2\frac{(\phi+\theta)(1+\phi\theta)}{1-\phi^2}}{\sigma^2\frac{1+\theta^2+2\phi\theta}{1-\phi^2}}\\ &=&\frac{(\phi+\theta)(1+\phi\theta)}{1+\theta^2+2\phi\theta}\neq\phi, \end{eqnarray*} unless the process is an $AR(1)$, i.e. unless $\theta=0$.
Instrumental variables estimation of $\phi$ using $Y_{t-2}$ as an instrument for $Y_{t-1}$, in turn, is consistent for $\phi$: the IV estimator is $$\hat{\phi}_{IV}=\frac{\sum_tY_{t-2}Y_{t}}{\sum_tY_{t-2}Y_{t-1}}=\frac{\frac{1}{T}\sum_tY_{t-2}Y_{t}}{\frac{1}{T}\sum_tY_{t-2}Y_{t-1}}\to_p\frac{\gamma_2}{\gamma_1}$$ We furthermore know that the autocovariance function of an $ARMA(1,1)$ is such that $\gamma_2=\phi\gamma_1$. Hence, $$\hat{\phi}_{IV}\to_p\phi$$ This works because the error term in this IV model, $\epsilon_t+\theta\epsilon_{t-1}$, is uncorrelated with the instrument, which itself is correlated with the regressor $Y_{t-1}$ due to the autoregressive structure of the process.
While this simple example (and I think simple examples are useful) shows how to use instruments in time series analysis, it is somewhat artificial in that if one knew that the process is $ARMA(1,1)$ one could estimate such a process directly. And it is somewhat fragile in that if the process were $ARMA(1,2)$, $Y_{t-2}$ would no longer be a valid instrument, as it would now be correlated with the new error $\epsilon_t+\theta_1\epsilon_{t-1}+\theta_2\epsilon_{t-2}$.