Question

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 \(\alpha_{1}\) ?

Short answer

Transform the adaptive model, verify new errors are uncorrelated, and estimate using OLS. Estimate \(\lambda\) via \(1-\beta_1\) and \(\alpha_1\) from \(\beta_2/\lambda\).

Step-by-step solution

Lag and Subtract (Original Equations)

Start by taking equation (18.68) which is the adaptive expectations equation: \[x_{t}^{*} - x_{t-1}^{*} = \lambda (x_{t-1} - x_{t-1}^{*})\]. Lag this equation by one period to get: \[x_{t-1}^{*} - x_{t-2}^{*} = \lambda (x_{t-2} - x_{t-2}^{*})\]. Multiply the lagged equation by \(1 - \lambda\): \[(1-\lambda)(x_{t-1}^{*} - x_{t-2}^{*}) = (1-\lambda)\lambda(x_{t-2} - x_{t-2}^{*})\]. Subtract the adjusted lagged equation from the original: \[(x_{t}^{*} - x_{t-1}^{*}) - (1-\lambda)(x_{t-1}^{*} - x_{t-2}^{*}) = \lambda(x_{t-1} - x_{t-1}^{*}) - (1-\lambda)\lambda(x_{t-2} - x_{t-2}^{*})\]. This simplifies to create a relationship between \(x_t^*\) and past values.

Derive Equation with Lag Dependent Variable

Insert the adaptively expected value \(x_t^*\) into the original equation: \[y_{t} = \alpha_{0} + \alpha_{1}x_{t}^{*} + u_{t}\]. Use the previous calculation to replace \(x_{t}^* = x_{t-1}^* + \lambda(x_{t-1} - x_{t-1}^*)\), find: \[x_{t}^* = x_{t-1} + \lambda(x_{t-1} - x_{t-1}^*)\]. This converts into: \[y_{t} = \alpha_{0} + \alpha_{1}(x_{t-1} + \lambda(x_{t-1} - x_{t-1}^*)) + u_{t}\], simplify further. Finally, factor the roles of \(\lambda\) across terms, producing the form: \[y_{t} = \lambda\alpha_0 + (1-\lambda)y_{t-1} + \lambda\alpha_1 x_{t-1} + u_{t} - (1-\lambda)u_{t-1}\]. This demonstrates the model specified in the exercise.

Analyze Error Structure

Given \(E(u_{t}|I_{t-1}) = 0\) implies \(u_{t}\) are serially uncorrelated. Substitute to define the new error \[v_{t} = u_{t} - (1-\lambda)u_{t-1}\]. To analyze \(v_{t}\), consider its expectation given past information \(I_{t-1}\): \[E(v_{t} | I_{t-1}) = E(u_{t} | I_{t-1}) - (1-\lambda)E(u_{t-1} | I_{t-1}) = 0\] because \(u_t\) are serially uncorrelated. Thus, \(v_t\) is white noise, maintaining zero expectation, indicating no serial correlation.

Estimating Coefficients of New Model

The equation from part (i) is rewritten as: \[y_{t} = \beta_0 + \beta_1 y_{t-1} + \beta_2 x_{t-1} + v_{t}\], where \(\beta_0 = \lambda\alpha_0\), \(\beta_1 = 1 - \lambda\), and \(\beta_2 = \lambda\alpha_1\). To estimate \(\beta_j\), any consistent method that handles linear regression can be used, such as Ordinary Least Squares (OLS), provided that \(v_t\) is uncorrelated over time.

Estimating Original Model Parameters

With the consistent estimations of \(\beta_j\), analyze them to extract \(\lambda\) and \(\alpha_1\): \[\lambda = 1 - \beta_1\] and \[\alpha_1 = \frac{\beta_2}{\lambda}\]. The sequential calculation allows consistent estimators for original model parameters.

Key concepts

Adaptive Expectations

Adaptive expectations refer to how individuals, businesses, or models adjust their expectations based on past experiences or information. In econometrics, this often involves predicting future values of a variable using past data. The basic idea is that people adjust their expectations slowly based on what has happened before rather than making sudden changes.
In the provided exercise, adaptive expectations are captured by the equation: \[x_{i}^{*} - x_{t-1}^{*} = \lambda(x_{t-1} - x_{t-1}^{*})\]where \(\lambda\) is a parameter between 0 and 1. This parameter governs the speed of adjustment of expectations. A higher value of \(\lambda\) means quicker adjustments and vice versa. When empirical data shows a less responsive pattern to changes, \(\lambda\) would be small. This equation implies that each period's expectation adjusts by some fraction of last period's prediction error.

Lagged Dependent Variable

A lagged dependent variable is a variable from a previous time period that is used as a predictor in a regression model. It's important in capturing dynamics in economic relationships because it accounts for the effects of past outcomes on current outcomes.
In the model presented, the lagged dependent variable is \(y_{t-1}\). This variable helps to capture elements like momentum or persistence over time. For instance, a person's current financial status might depend heavily on their financial status in the previous period. Incorporating \(y_{t-1}\) helps the model to more accurately reflect such dynamics by allowing one to control for past time effects.

Serial Correlation

Serial correlation, also known as autocorrelation, occurs when error terms in a regression model are correlated across time. In other words, it suggests the presence of patterns not captured by the model. This can be a major issue in time-series data, leading to inefficient estimates. In the exercise, errors \(u_{t}\) are assumed to be serially uncorrelated when given past information. However, when errors are adjusted as \(v_{t} = u_{t} - (1 - \lambda)u_{t-1}\), it's crucial to maintain that \(v_{t}\) remains free from serial correlation to ensure the validity of regression estimates. This property confirms that newly adjusted error terms do not carry any dependency from past error terms, implying they've been correctly filtered.

Ordinary Least Squares

Ordinary Least Squares (OLS) is a method for estimating the parameters in a linear regression model. It finds the line (or hyperplane, in higher dimensions) that minimizes the sum of the squares of the vertical deviations from each data point to the line. It's popular because of its simplicity and interpretability.In the context of this model, OLS can be used to estimate \(\beta_{j}\)s in the equation: \[y_{t} = \beta_{0} + \beta_{1} y_{t-1} + \beta_{2} x_{t-1} + v_{t}\]Given that \(v_{t}\) is uncorrelated and homoscedastic, OLS provides consistent and unbiased estimates of the parameters \(\beta_{j}\). Thus, OLS is suitable as long as the assumptions about the nature of \(v_{t}\) hold.

Error Structure Analysis

Error structure analysis involves assessing the nature and patterns of errors in an econometric model. This analysis aims to ensure that errors are random and do not show systematic patterns that could obscure the interpretation of results.In this exercise, the original error \(u_{t}\) and transformed error \(v_{t}\) were analyzed. Given \(E(u_{t}|I_{t-1}) = 0\) ensures that \(u_{t}\) are originally serially uncorrelated. The transformation leading to \(v_{t} = u_{t} - (1-\lambda)u_{t-1}\) was crucial to maintaining error properties and constructing correct model specifications. In doing so, it eliminates autocorrelation within the errors, thus freeing the estimates from bias related to serial correlation.
Errors without patterns imply that the interpretations drawn from the regression model are more reliable and robust.