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.

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?