tailieunhanh - Class Notes in Statistics and Econometrics Part 27

CHAPTER 53 Errors in Variables. . 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 . 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 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 y ia x ß v. But now let us assume that x can only be observed with a random error. . 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 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 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. . 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 y ia xp v up x xz z The problem is that the disturbance term is correlated with the explanatory variable 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 p0LS --------------------- X Xj x 2 cov y x r var u 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 3 Assume the variance of the measurement error au is 10 of the variance of the .