a) Define stationarity and weak stationarity. How are they related? Does one ever imply the other? b) What is the role of stationarity in a spurious regression? Is a regression involving non-stationar

a) Define stationarity and weak stationarity. How are they related? Does one ever imply the other?

b) What is the role of stationarity in a spurious regression? Is a regression involving non-stationary variables always spurious? Discuss.

c) Consider the following three stochastic processes:

i. ????(????) = 0.2 ????(????−1) ????(????) , ????(????)~????????????(0, 4)

ii. ????(????) = 0.5 0.4????(????−1) ???????? 0.3????(????−1), ????????~????????????(0, 2)

iii. ????(????) = 2 ????(????) 0.2????(????−1) 0.1????(????−2), ????(????)~????????????(0, 0.5)

For each process, derive and calculate its mean, e.g. ????[???????? ] = ????. Classify all processes as ARIMA(?, ?, ?) and explain which ones are stationary or why they fail to be stationary.

d) For process (iii) in the previous part, derive the 2- and 4-step ahead prediction.

e) Explain necessary conditions for the OLS estimator of an AR(p) process to be consistent. To which processes in part (c) could it be applied to bring about consistent estimates?

f) Carefully define the root mean squared prediction error. What does it measure?