An interesting economic model that leads to an econometric...

An interesting economic model that leads to an econometric model with a lagged dependent variable relates $y_{t}$ to the expected value of $x_{t},$ say, $x_{t}^{*},$ where the expectation is based on all observed information at time $t-1:$ $$y_{t}=\alpha_{0}+\alpha_{1} x_{t}^{*}+u_{t}$$ A natural assumption on $\left\{u_{t}\right\}$ is that $\mathrm{E}\left(u_{t} | I_{t-1}\right)=0,$ where $I_{t-1}$ denotes all information on $y$ and $x$ observed at time $t-1 ;$ this means that $\mathrm{E}\left(y_{t} | I_{t-1}\right)=\alpha_{0}+\alpha_{1} x_{t}^{*} .$ To complete this model, we need an assumption about how the expectation $x_{i}^{*}$ is formed. We saw a simple example of adaptive expectations in Section $11-2$, where $x_{i}=x_{t-1}$. A more complicated adaptive expectations scheme is $$x_{i}^{*}-x_{t-1}^{*}=\lambda\left(x_{t-1}-x_{t-1}^{*}\right)$$ where $0<\lambda<1 .$ This equation implies that the change in expectations reacts to whether last period's realized value was above or below its expectation. The assumption $0<\lambda<1$ implies that the change in expectations is a fraction of last period's error. i. Show that the two equations imply that $$y_{t}=\lambda \alpha_{0}+(1-\lambda) y_{t-1}+\lambda \alpha_{1} x_{t-1}+u_{t}-(1-\lambda) u_{t-1}$$ [Hint: Lag equation (18.68) one period, multiply it by ( $1-\lambda$ ), and subtract this from ( 18.68 ). Then, use (18.69).] ii. Under $\mathrm{E}\left(u_{t} | I_{t-1}\right)=0,\left\{u_{t}\right\}$ is serially uncorrelated. What does this imply about the new errors, $$ v_{t}=u_{t}-(1-\lambda) u_{t-1} ? $$ iii. If we write the equation from part (i) as $$y_{t}=\beta_{0}+\beta_{1} y_{t-1}+\beta_{2} x_{t-1}+v_{t}$$ how would you consistently estimate the $\beta_{j}$ ? iv. Given consistent estimators of the $\beta_{j}$, how would you consistently estimate $\lambda$ and $\alpha_{1}$ ?

An interesting economic model that leads to an econometric model with a lagged dependent variable relates $y_{t}$ to the expected value of $x_{t},$ say, $x_{t}^{*},$ where the expectation is based on all observed information at time $t-1:$
$$y_{t}=\alpha_{0}+\alpha_{1} x_{t}^{*}+u_{t}$$
A natural assumption on $\left\{u_{t}\right\}$ is that $\mathrm{E}\left(u_{t} | I_{t-1}\right)=0,$ where $I_{t-1}$ denotes all information on $y$ and $x$ observed at time $t-1 ;$ this means that $\mathrm{E}\left(y_{t} | I_{t-1}\right)=\alpha_{0}+\alpha_{1} x_{t}^{*} .$ To complete this model, we need an assumption about how the expectation $x_{i}^{*}$ is formed. We saw a simple example of adaptive expectations in Section $11-2$, where $x_{i}=x_{t-1}$. A more complicated adaptive expectations scheme is
$$x_{i}^{*}-x_{t-1}^{*}=\lambda\left(x_{t-1}-x_{t-1}^{*}\right)$$
where $0<\lambda<1 .$ This equation implies that the change in expectations reacts to whether last period's realized value was above or below its expectation. The assumption $0<\lambda<1$ implies that the change in expectations is a fraction of last period's error.
i. Show that the two equations imply that
$$y_{t}=\lambda \alpha_{0}+(1-\lambda) y_{t-1}+\lambda \alpha_{1} x_{t-1}+u_{t}-(1-\lambda) u_{t-1}$$
[Hint: Lag equation (18.68) one period, multiply it by ( $1-\lambda$ ), and subtract this from ( 18.68 ). Then, use (18.69).]
ii. Under $\mathrm{E}\left(u_{t} | I_{t-1}\right)=0,\left\{u_{t}\right\}$ is serially uncorrelated. What does this imply about the new errors,
$$
v_{t}=u_{t}-(1-\lambda) u_{t-1} ?
$$
iii. If we write the equation from part (i) as
$$y_{t}=\beta_{0}+\beta_{1} y_{t-1}+\beta_{2} x_{t-1}+v_{t}$$
how would you consistently estimate the $\beta_{j}$ ?
iv. Given consistent estimators of the $\beta_{j}$, how would you consistently estimate $\lambda$ and $\alpha_{1}$ ?

Key Concepts

Consistency of Estimators

Consistency is a fundamental property of estimators in econometrics, ensuring that as the sample size increases, the estimates converge in probability to the true parameter values. In the context of dynamic models with adaptive expectations, achieving consistency typically involves appropriate model transformation and the use of valid instruments to control for issues such as autocorrelation and endogeneity.

Instrumental Variables Estimation

Instrumental variables (IV) estimation is an econometric technique used to obtain consistent parameter estimates when a regressor is correlated with the error term, or when there is potential endogeneity. In dynamic models with lagged variables, IV methods help to address the problems that arise due to simultaneity or omitted variable bias by using instruments that are correlated with the endogenous regressors but uncorrelated with the error term.

Adaptive Expectations

Adaptive expectations refer to a process where agents form their future expectations based on past observations and errors in previous forecasts. This mechanism, often modeled as a weighted average where the current update is a fraction of the previous forecast error, is crucial in economic models attempting to capture how expectations evolve over time, influencing decisions and subsequent outcomes in the model.

Lagged Dependent Variable Models

These are dynamic models where the current value of the dependent variable is influenced by its own past values. This setup is key in time series analysis and econometrics because the past behavior of a variable can help explain its current behavior, but also introduces complications such as potential endogeneity and persistence that must be addressed during estimation.

Error Structure and Serial Correlation

Understanding the error structure is essential in econometric modeling. When errors are assumed to be serially uncorrelated, as in this context, any transformation of these errors (such as combining current and lagged terms) must be carefully examined for induced serial correlation. This aspect of the modeling ensures that the properties assumed for the error terms are maintained after any algebraic manipulation or model transformation.

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