# Variance of a corrected regression coefficient

#### Arttemus

##### New Member
Ppl I really need help!
Consider the model
Yi = b0 + b 1Xi +ei
e_i~N(0,s^2_e) iid
X_i_N(m_x,s^2_x) iid
Z_i = X_i + u_i,
u_i~ N(0,s^2_u) independent of Xi

Suppose we have the following data:
a. A sample M of n units where Y_i, X_i, Z_i obey the above model but we observe only (Y_i;Z_i).
b. An independent sample V of m units with data on (X_i;Z_i) following the above model, with s^2_u
being the same for M and V, but m_x and s^2_x possibly different, so that s^2_u is the only parameter that is portable from V to M.
Let c_1 be the estimate of the slope of the regression of Y on Z based on the data in M. Consider the estimate
b1= (s^2_z/(s^2_z-s^2_u))*c1
where s^2_z is the usual estimator based on the data in M and s^2_u
is the corresponding estimator based on the data in V. Using the delta method, derive the asymptotic distribution of b1. Give an expression for the ratio between the asymptotic variance of ^ 1 and the variance of the estimate of b1 that would be used if we had a sample of n units with observations on X and Y . Compute the value of this ratio for various values of n, m, and s^2_u
, assuming s^2_x=1.