New York University ECON-SHU ECON-SHU 360 Measurement Error In this problem, we’ll go through the basics of what happens with measurement error in a model. Suppose that one knows a model to…

New York University ECON-SHU ECON-SHU 360 Measurement Error In this problem, we’ll go through the basics of what happens with measurement error in a model. Suppose that one knows a model to…

Measurement Error In this problem, we’ll go through the basics of

what happens with measurement error in a model. Suppose that one knows a model to be, Yi = β0 + β1Xi + Ui and that the OLS assumptions are satisfied. In fact, we will go further and assume that U and X are independent. Suppose that you only have a noisy measure of Xi : X∗ i = Xi + ei (1) Let ei be independent of Xi and ui . Also, denote variance by σ 2 for each random variable. 1 Measurement error is pervasive in the social sciences. For example, when asked to report their compensation, people will often round to a whole number, report last year’s income, or forget parts of their income (e.g., return on savings). Our question is what happens in the situation that you, an econometrician, run this model: Yi = β0 + β1X∗ i + Ui (2) a. Prove that, Cov(X∗ , Y ) = Cov(X, Y ) Hint: If e is independent of X and U it must be independent of Y . What’s this mean for the covariance? b. Now show, V ar(X∗ ) = σ 2 X + σ 2 e c. Conclude that for model (2): βˆOLS 1 = β1 × σ 2 X σ 2 X + σ 2 e Hint: Remember that the OLS estimator is the covariance of the independent and dependent variables, over the variance of the independent variable. You’ll use this fact twice.