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Class Notes in Statistics and Econometrics Part 27

CHAPTER 53 Errors in Variables. 53.1. The Simplest Errors-in-Variables Model We will explain here the main principles of errors in variables by the example of simple regression, in which y is regressed on one explanatory variable with a constant term. | CHAPTER 53 Errors in Variables 53.1. The Simplest Errors-in-Variables Model We will explain here the main principles of errors in variables by the example of simple regression in which y is regressed on one explanatory variable with a constant term. Assume the explanatory variable is a random variable called x and the disturbance term in the regression which is a zero mean random variable independent of x will be called v. In other words we have the following relationship between random variables 53.1.1 y a x fl v. 1099 1100 53. ERRORS IN VARIABLES If n observations of the variables y and x are available one can obtain estimates of a and p and predicted values of the disturbances by running a regression of the vector of observations y on x 53.1.2 y ia x ß v. But now let us assume that x can only be observed with a random error. I.e. we observe x x u. The error u is assumed to have zero mean and to be independent of x and v. Therefore we have the model with the latent variable x 53.1.3 53.1.4 y a x ß v x x u This model is sometimes called regression with both variables subject to error. It is symmetric between the dependent and the explanatory variable because one can also write it as 53.1.5 53.1.6 53.1.7 y a x ß x x u y y v and as long as p 0 y a x p is equivalent to x a p y p. 53.1. THE SIMPLEST ERRORS-IN-VARIABLES MODEL 1101 What happens if this is the true model and one regresses y on x Plug x x u into 53.1.2 53.1.8 y ia xp v up x xz z The problem is that the disturbance term is correlated with the explanatory variable 53.1.9 cov x cov x u v up pvar u . Therefore OLS will give inconsistent estimates of a and p t E yi y xi x 53.1.10 p0LS --------------------- X Xj x 2 cov y x r var u 53.1.11 plim pOLS -------- p 1------- r . 0LS var x var x Since var u var x pOLS will have the right sign in the plim but its absolute value will understimate the true p. Problem 468. 1 point SM86 A3.2 3 Assume the variance of the measurement error au is 10 of the variance of the .