# Solved – 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?

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Consider a series \$Y_t\$ generated as an \$ARMA(1,1)\$ process \$\$ Y_t=phi Y_{t-1}+epsilon_t+thetaepsilon_{t-1} \$\$ Suppose our interest centers on estimating \$phi\$. We have an endogeneity issue here, as the error term \$epsilon_t+thetaepsilon_{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_pfrac{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^2frac{1+theta^2+2phitheta}{1-phi^2}\$ and \$gamma_1=sigma^2frac{(phi+theta)(1+phitheta)}{1-phi^2}\$. Hence, begin{eqnarray*} hat{phi}&to_p&frac{sigma^2frac{(phi+theta)(1+phitheta)}{1-phi^2}}{sigma^2frac{1+theta^2+2phitheta}{1-phi^2}}\ &=&frac{(phi+theta)(1+phitheta)}{1+theta^2+2phitheta}neqphi, 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_pfrac{gamma_2}{gamma_1} \$\$ We furthermore know that the autocovariance function of an \$ARMA(1,1)\$ is such that \$gamma_2=phigamma_1\$. Hence, \$\$hat{phi}_{IV}to_pphi\$\$ This works because the error term in this IV model, \$epsilon_t+thetaepsilon_{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_1epsilon_{t-1}+theta_2epsilon_{t-2}\$.

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# Solved – 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?