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 $epsilon_t+theta_1epsilon_{t-1}+theta_2epsilon_{t-2}$.

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 $epsilon_t+theta_1epsilon_{t-1}+theta_2epsilon_{t-2}$.