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!