Consider the geometric distributed model in equation (18.8),...

Consider the geometric distributed model in equation (18.8), written in estimating equation form as in equation (18.11): $$y_{t}=\alpha_{0}+\gamma z_{t}+\rho y_{t-1}+v_{t}$$ where $v_{t}=u_{t}-\rho u_{t-1}$ i. Suppose that you are only willing to assume the sequential exogeneity assumption in (18.6). Why is $z_{t}$ generally correlated with $v_{t} ?$ ii. Explain why estimating (18.11) by IV, using instruments $\left(z_{t}, z_{t-1}\right),$ is generally inconsistent under $(18.6) .$ Using the IV estimator, can you test whether $z_{t}$ and $v_{t}$ are correlated? iii. Evaluate the following proposal when only (18.6) holds: Estimate (18.11) by IV using instruments $\left(z_{t-1}, z_{t-2}\right)$ iv. Explain what you gain by estimating (18.11) by 2 SLS using instruments $\left(z_{t}, z_{t-1}, z_{t-2}\right)$ v. In equation $(18.16),$ the estimating equation for a rational distributed lag model, how would you estimate the parameters under (18.6) only? Might there be some practical problems with your approach?

Consider the geometric distributed model in equation (18.8), written in estimating equation form as in equation (18.11):
$$y_{t}=\alpha_{0}+\gamma z_{t}+\rho y_{t-1}+v_{t}$$
where $v_{t}=u_{t}-\rho u_{t-1}$
i. Suppose that you are only willing to assume the sequential exogeneity assumption in (18.6). Why is $z_{t}$ generally correlated with $v_{t} ?$
ii. Explain why estimating (18.11) by IV, using instruments $\left(z_{t}, z_{t-1}\right),$ is generally inconsistent under $(18.6) .$ Using the IV estimator, can you test whether $z_{t}$ and $v_{t}$ are correlated?
iii. Evaluate the following proposal when only (18.6) holds: Estimate (18.11) by IV using instruments $\left(z_{t-1}, z_{t-2}\right)$
iv. Explain what you gain by estimating (18.11) by 2 SLS using instruments $\left(z_{t}, z_{t-1}, z_{t-2}\right)$
v. In equation $(18.16),$ the estimating equation for a rational distributed lag model, how would you estimate the parameters under (18.6) only? Might there be some practical problems with your approach?

Key Concepts

Distributed Lag Models

Distributed lag models involve current and lagged values of an independent variable to model dynamic responses. Rational distributed lag models simplify the lag structure by parameterizing the lag coefficients, yet estimation under weak exogeneity conditions (such as sequential exogeneity) can still be challenging. Practical problems in estimation may arise from multicollinearity, weak instruments, and the loss of efficiency when the necessary strong exogeneity conditions are not satisfied.

Two-Stage Least Squares (2SLS) Estimation

Two-Stage Least Squares is an extension of IV estimation that uses all valid instruments in two stages to obtain consistent parameter estimates. In dynamic settings, employing 2SLS with multiple instruments (including both current and lagged instruments) can potentially improve efficiency. However, if some instruments are invalid due to correlation with the error term, the consistency of the 2SLS estimator will be compromised, and careful instrument selection and testing is required.

Lagged Instruments

Lagged instruments are past values of explanatory variables used as instruments in dynamic models. Their validity hinges on the assumption that lagged variables are not correlated with the current error term. When only sequential exogeneity holds, instruments such as z??? or z??? may still be problematic if the model’s error structure induces correlation between these lagged instruments and the error term.

Correlation Between Regressors and Error Terms

In dynamic models where the error term is defined with serial correlation (for example, as a difference between current and lagged shocks), regressors such as z? may inadvertently be correlated with the composite error term. This happens because past shocks, which may influence both z? and subsequent disturbances, enter the error term via lagged values, thereby violating the strict exogeneity required for unbiased estimation.

Sequential Exogeneity

Sequential exogeneity is the assumption that the error term at time t is uncorrelated with current and past values of the explanatory variables, but not necessarily with future values. In dynamic models, even though this condition provides limited exogeneity, it is insufficient to guarantee that lagged instruments or regressors entering the error term through lagged disturbances are uncorrelated with the error term.

Instrumental Variables (IV) Estimation

Instrumental Variables estimation is a technique used when some regressors are endogenous—that is, correlated with the error term. The idea is to employ instruments that are correlated with the endogenous regressors but uncorrelated with the error term. However, when the instruments themselves are correlated with the error due to the structure of the model (like lagged errors transferring correlation), IV estimators become inconsistent.

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