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In the context of Introductory Econometric Methods, when an explanatory variable is measured with error, the classical linear regression model assumptions are violated, leading to biased and inconsistent parameter estimates. This situation is known as the errors-in-variables problem, or measurement error in the independent variable. The specification of the model and analysis of its consequences is discussed below. --- ### Model Specification with Measurement Error Let us consider the simple linear regression model: $$ Y_i = \beta_0 + \beta_1 X_i^* + u_i $$ Where: * $Y_i$ is the dependent variable. * $X_i^*$ is the true (but unobserved) value of the independent variable. * $u_i$ is the classical error term, assumed to satisfy the usual OLS assumptions. However, suppose we do not observe $X_i^*$ directly. Instead, we observe $X_i$, which is a measured version of $X_i^*$, containing a measurement error. The relationship is: $$ X_i = X_i^* + v_i $$ Where: * $v_i$ is the measurement ____ ____ _____ ______ __________ ____ ____ ___ _________.
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