Least squares estimator - calculating whether it's unbiased, consistent

[nxcvb]

Hello!

I'm struggling a bit with my econometrics exercise - we only had one example and I'm not really sure how to transfer this to the exercise. The general task is to calculate whether the least squares estimator is unbiased and consistent.

The following is given:

x_t is exogenous, and ε_t ~ IID (0, σ ²). Exogeneity is defined as E(ε|x) = 0, with x= x_t and ε=ε_t as n*1 vectors.

1) y_t = βx_t + ε_t; t = 1,...,n

Here my result was that it's both unbiased and consistent.

For 2-4, I just don't really know how I need to change my calculations, and I would be really happy about some starting points or explanations so that I get on the right track as to what I need to do...

2) y_t = βy_t−1 + ε_t; |β|<1, t=1,...,n

3) y_t = βx_t + u_t; ut=pu_t−1+ ε_t; |p| < 1; t=1,...,n

4) y_t = βy_t−1 + u_t; |β| < 1 ; u_t=pu_t−1 + ε_t; |p| < 1; t=1,... ,n

I think for 3 and 4 I'm mostly confused about the new variables. For 2, I don't exactly know what I need to do, now that |β|<1 and given that it's y t-1 now and not x anymore.

Thank you so much in advance!

 

[JeffM]

I have not studied this stuff for fifty years. I am not going to try to answer it. I suggest that you repost it in the probability and statistics section to get a quicker answer.