Question

Consider a simple time series model where the explanatory variable has classical measurement error: $$ \begin{array}{l} y_{t}=\beta_{0}+\beta_{1} x_{t}^{*}+u_{t} \\ x_{t}=x_{t}^{*}+e_{t} \end{array} $$ where \(u_{t}\) has zero mean and is uncorrelated with \(x_{t}^{*}\) and \(e_{t} .\) We observe \(y_{t}\) and \(x_{t}\) only. Assume that \(e_{t}\) has zero mean and is uncorrelated with \(x_{t}^{*}\) and that \(x_{i}^{*}\) also has a zero mean (this last assumption is only to simplify the algebra). i. Write \(x_{i}^{*}=x_{t}-e_{t}\) and plug this into \((15.58) .\) Show that the error term in the new equation, say, \(v_{t}\), is negatively correlated with \(x_{t}\) if \(\beta_{1}>0 .\) What does this imply about the OLS estimator of \(\beta_{1}\) from the regression of \(y_{t}\) on \(x_{t} ?\) ii. In addition to the previous assumptions, assume that \(u_{t}\) and \(e_{t}\) are uncorrelated with all past values of \(x_{i}\) and \(e_{t}\); in particular, with \(x^{*}_{t-1}\) and \(e_{t-1}\). Show that \(\mathrm{E}\left(x_{t-1} v_{t}\right)=0\) where \(v_{t}\) is the error term in the model from part (i). iii. Are \(x_{t}\) and \(x_{t-1}\) likely to be correlated? Explain. iv. What do parts (ii) and (iii) suggest as a useful strategy for consistently estimating \(\beta_{0}\) and \(\beta_{1} ?\)

Short answer

Use an instrumental variable like \( x_{t-1} \) to consistently estimate \( \beta_1 \). This compensates for the correlation between error term \( v_t \) and \( x_t \).

Step-by-step solution

Express \( x_t^* \) in terms of \( x_t \) and \( e_t \)

Start with the measurement error model, where we know that \( x_t = x_t^* + e_t \). Rewriting \( x_t^* \, \), we have \( x_t^* = x_t - e_t \). This will let us substitute into the equation for \( y_t \).

Substitute \( x_t^* \) into the equation for \( y_t \)

Substitute \( x_t^* = x_t - e_t \) into the equation \( y_{t} = \beta_{0} + \beta_{1} x_{t}^* + u_{t} \). This gives \( y_{t} = \beta_{0} + \beta_{1} (x_{t} - e_t) + u_{t} \). Simplifying, \( y_t = \beta_0 + \beta_1 x_t - \beta_1 e_t + u_t \).

Identify and analyze \( v_t \)

Identify the new error term \( v_t \) in the equation \( y_t = \beta_0 + \beta_1 x_t + v_t \), where \( v_t = u_t - \beta_1 e_t \). Analyze this error term: since \( e_t \) is part of \( v_t \), \( v_t \) and \( x_t \) (which includes \( e_t \)) are negatively correlated if \( \beta_1 > 0 \).

Implication for OLS Estimator of \( \beta_1 \)

Because the error term \( v_t \) is negatively correlated with \( x_t \), the OLS estimator is biased and inconsistent. This is because one of the classical OLS assumptions is violated: the error term must be uncorrelated with the independent variables.

Show \[ \mathrm{E}(x_{t-1}v_{t})=0 \]

Since \( u_t \) and \( e_t \) are uncorrelated with past values, \( x_{t-1} \) is uncorrelated with \( u_t \) and \( e_t \). Therefore, any correlation term, like \( \mathrm{E}(x_{t-1} u_t) \) or \( \mathrm{E}(x_{t-1} e_t) \), is zero. Thus, \( \mathrm{E}(x_{t-1} v_t) = 0 \).

Correlation between \( x_t \) and \( x_{t-1} \)

Since both include their own error terms, \( x_t \) and \( x_{t-1} \) might be correlated, primarily due to potential autocorrelation in the underlying true values \( x_t^* \).

Strategy for Consistent Estimation

Given the results of parts (ii) and (iii), one potential strategy is to use instrumental variables (IV), where \( x_{t-1} \) serves as an instrument for \( x_t \) to provide consistency in estimating \( \beta_0 \) and \( \beta_1 \). This would help eliminate the bias caused by the correlation between \( x_t \) and the error term \( v_t \).

Key concepts

Time Series Analysis

Time series analysis is an essential part of econometrics that involves studying datasets where observations are collected sequentially over time. The primary purpose is to identify patterns or trends and use them to forecast future values. In our exercise, the model considers the time dependence of variables like \( y_t \) and \( x_t \).

• Finding patterns or trends in data collected over consistent time intervals.
• Using these patterns to make predictions about future data points.
• Recognizing how lag values, such as \( x_{t-1} \), can serve as useful instruments for understanding dependencies over time.

Time dependence structures may exhibit trends, seasonal patterns, cyclical movements, or random variations. Practicing time series analysis involves techniques like autoregression, moving averages, and the use of lagged variables to improve model accuracy and predictability.

Measurement Error

Measurement error occurs when the observed data \( x_t \) does not perfectly reflect the true value \( x_t^* \). In our exercise, this appears as classical measurement error, where \( x_t = x_t^* + e_t \). This kind of error is typically uncorrelated with the true values and has a mean of zero.

• Understanding that measurement errors can lead to biased and inconsistent results unless handled properly.
• Recognizing the importance of identifying and controlling measurement errors in econometric models.

This type of error directly impacts the values of the dependent variable \( y_t \) and the error term \( v_t \) in our adjusted equation. Proper handling of measurement error is crucial, as it ensures more reliable estimations in econometric models.

Ordinary Least Squares

Ordinary Least Squares (OLS) is one of the most common estimation techniques in econometrics. It aims to find a line that best fits a set of observations by minimizing the sum of squared errors between observed and predicted values. The OLS estimator relies on several key assumptions, such as the error term being uncorrelated with the explanatory variables.

• OLS provides unbiased estimates when all assumptions are satisfied.
• Reliance on the assumption that error terms should not correlate with the independent variables.
• Measurement errors can violate OLS assumptions, leading to biased and inconsistent estimators.

In the exercise, because the error term \( v_t \) is negatively correlated with \( x_t \), OLS becomes biased and inconsistent for estimating \( \beta_1 \) when \( \beta_1 > 0 \). This violation prompts the need to consider alternative methods like Instrumental Variables to obtain consistent estimators.

Instrumental Variables

Instrumental Variables (IV) estimation is used when a model has endogenous explanatory variables, often due to measurement errors or omitted variable bias. An instrument is a variable that is correlated with the endogenous variable but uncorrelated with the error term. In our exercise, using \( x_{t-1} \) as an instrument for \( x_t \) helps provide consistent estimates of \( \beta_0 \) and \( \beta_1 \).

• IV helps resolve bias in estimation caused by endogeneity.
• Choosing an appropriate instrument is crucial, where it should relate to the endogenous variable but should not be a direct part of the error term.
• The proper use of IV helps restore the validity of regression analysis when classical OLS assumptions fail.

Instrumental variables are especially useful in time series settings where prior values or lagged variables can serve as potential instruments, thus letting researchers retrieve unbiased and consistent parameter estimates.