1. Consider the simple random walk (et is a stationary white noise process) yt = yt1 + et Using back substitution (start with y1 = y0 + e1), rewrite the previous equation so that yt is a function of y0 and of the error term.

2. Consider the random walk with drift (et is a stationary white noise process) yt = 0 + yt1 + et Using back substitution (start with y1 = 0 + y0 + e1), rewrite the previous equation so that yt is a function of y0, a time trend and of the error term.

3. Consider the stochastic trend model yt = 0 + 1t + t with t defined as a simple random walk t = t1 + et (et is a stationary white noise process). Show that this model can be written as yt = 1 + yt1 + et 3. The stochastic trend model (random walk with drift) is given by yt = 0 + 1t + t with t defined as a simple random walk t = t1 + et . The deterministic trend model is given by yt = 0 + 1t + et where et is a stationary white noise process. Explain the difference between these two types of trend model. Will they give similar forecasts? Why? Which one will generate forecasts with more volatility?

1.3 Stationarity When a variable yt is stationary we require that the mean and variance are constant (do not vary in time). You will need to take the expectation of yt , Etyt , to compute the mean and the variance of yt , V art(yt). Using the equation you obtained above in [1.] show that yt is nonstationary. Show your steps and any assumptions you have used. Using the equation you obtained above in [2.] show that yt is nonstationary. Show your steps and any assumptions you have used.