Question

1. Consider the following unobserved effects regression model: Suppose that the idiosyncratic error u, t 1,..,T are serially correlated with constant variance σ2. Show that the correlation between adjacent differences, Δυ" and Δυ,-l is -0.5. Therefore under the ideal FE assumptions, first differencing induces negative serial correlation of a known value. [Note:

Answer 1

Now first differencing rightarrow (1) 0y_it = beta_0 + beta_1 x_it, 1 + beta_2 x_it, 2 + ... + beta_k x_it, k + alpha_i t + u_it (2) Now y_it - 1 = beta_0 + beta_1 x_it - 1, 1 + beta_2 x_it - 1, 2 + ... +_k x)it - 1, k + alpha i + u_i, t - 1 Now differencing 1 & 2 y_it - y_it - 1 = beta_1 x_it, 1 - x_it - 1, 1 + .... + beta_k x_it, k - beta_k x_it - 1, k + u_it - u_i, t - 1 thus delta y_it = beta_1 delta x_it,1 + beta_2 delta x_it, 2 + ... + beta_k delta x_it, k + delta u_it similarly delta y_it + 1 = beta_1 delta x_it + 1, 1 + .... + beta_k delta x_it + 1, k + delta u_it + 1 [Thus as error term for delta y_it (u_it - u_t - 1, i)] & for delta y_it + 1 u_i_1 t + 1 - u_it, so two