Question

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?

Short answer

i. \( z_{t} \) correlates with \( v_{t} \) because it affects \( u_t \). ii. IV with \( (z_{t}, z_{t-1}) \) is inconsistent if \( z_{t} \) is correlated with \( v_{t} \), tested by IV estimates. iii. Lagged \( (z_{t-1}, z_{t-2}) \) instruments can be more consistent. iv. 2SLS with more instruments reduces bias. v. IV estimation faces instrument validity and strength issues.

Step-by-step solution

Understanding the Correlation in Part i

When considering the sequential exogeneity assumption, which implies that past information up to time \( t-1 \) does not affect the mean of \( v_t \), we notice that \( v_t = u_t - \rho u_{t-1} \). Since \( z_t \) is part of the contemporaneous regressors, it's potentially correlated with the error term \( v_t \) if \( z_t \) can influence \( u_t \) directly or indirectly. Therefore, \( z_t \) may be correlated with \( v_t \) because the influence of \( z_t \) affects the current equation through \( u_t \).

Examining IV Estimation in Part ii

When estimating using IV with instruments \( (z_{t}, z_{t-1}) \), the prerequisite is that instruments must be uncorrelated with the error term \( v_t \). However, under only the assumption in (18.6), \( z_{t} \) is potentially correlated with \( v_{t} \), making the estimation inconsistent. The IV estimator can be used to test the correlation between \( z_{t} \) and \( v_{t} \) by checking whether the estimated coefficients are significantly different from zero, suggesting inconsistency due to endogeneity.

Evaluation of Alternative Instruments in Part iii

Using lagged instruments \( (z_{t-1}, z_{t-2}) \) under the assumption (18.6) can be more consistent because these instruments are less likely to be contemporaneously correlated with the error term \( v_t \). They rely on past information, which is assumed uncorrelated with \( u_t \) at time \( t \), addressing the issue seen in Step 2.

Benefits of 2 SLS in Part iv

By employing more instruments \( (z_{t}, z_{t-1}, z_{t-2}) \), the 2 Stage Least Squares (2SLS) accounts for more variation in the regressors while potentially reducing the bias from omitted variables. This approach enhances the robustness of the estimation by addressing any weak instrument problems when more information is incorporated.

Estimating Parameters in Part v with the Rational Distributed Lag Model

In this scenario, estimating the rational distributed lag model parameters under only assumption (18.6) might involve using IV techniques with lagged variables as instruments. However, practical issues include ensuring that these instruments are valid (uncorrelated with the error term) and strong, having sufficient explanatory power, which can be challenging if there are weak instruments leading to imprecise estimates.

Key concepts

Sequential Exogeneity

Sequential exogeneity is a crucial concept that helps us understand how past information relates to current outcomes in a model. Under this assumption, the mean of the error term, such as \(v_t\), is not influenced by information available at time \(t-1\). It means that the past cannot affect the present unexpected shocks.

However, in the context of a geometric distributed model, sequential exogeneity suggests that instruments like \(z_t\) can still be contemporaneously correlated with the error term \(v_t\) if \(z_t\) directly or indirectly affects \(u_t\). Thus, ensuring instruments that adhere to this assumption is key in having reliable estimations. Understanding when this assumption holds can aid researchers and students to design models that minimize biases in estimates.

Geometric Distributed Model

The geometric distributed model is a way to model relationships where past values influence current data in a gradually decreasing pattern. In such models, influence from past input values diminishes geometrically, following the equation form \(y_{t}=\alpha_{0}+\gamma z_{t}+\rho y_{t-1}+v_{t}\).

This model accounts for how past values affect the current outcome \(y_t\), estimating how quickly past effects dissipate. The parameter \(\rho\) determines the slow decline of past influences.

• If \(\rho\) is close to 1, past values heavily influence the present.
• If \(\rho\) is near 0, past dependencies quickly vanish.

For students studying econometrics, this model is pivotal in capturing time-related dependencies.

Rational Distributed Lag Model

The rational distributed lag model provides a more nuanced way to understand how inputs influence outputs over time. Unlike the geometric model, which assumes a geometric decay of past influences, this model allows for a more flexible approach. It adjusts the lag structure based on a rational function, often providing fits that better reflect economic or real-time data.

In practice, estimating the parameters of a rational distributed lag model requires using techniques that handle its complexities, like employing instrumental variables. However, one practical challenge is selecting appropriate instruments that maintain their validity over time without introducing errors themselves. Weak instruments can pose a problem, leading to unreliable estimates.

Correlation

Correlation is key in understanding relationships between variables in statistical models. It measures how two variables move together.

• Positive correlation means they increase together.
• Negative correlation means one increases when the other decreases.

In instrumental variable contexts, it's crucial that chosen instruments don't correlate with the error term; otherwise, bias and inconsistent estimates may arise.

In the geometric distributed model, \(z_t\) may correlate with \(v_t\), complicating the estimation. Instrumental Variable estimation relies heavily on avoiding such correlation, ensuring that instruments accurately depict the causal relationship in the model.

Two-Stage Least Squares

Two-Stage Least Squares (2SLS) is an econometric technique designed to address endogeneity in models, which occurs when regressors correlate with the error term. This method involves taking two steps.

• First, regressors are predicted using instruments that are assumed uncorrelated with the error term.
• Second, predicted regressors are used to estimate the actual model.

This mitigates biases from endogenous variables and allows for more accurate parameter estimation.

Using 2SLS with more instruments—like \((z_{t}, z_{t-1}, z_{t-2})\) in the mentioned exercise—can enhance estimation precision by incorporating additional information, addressing both weak instrument issues and potential omitted variable biases.