1. Stochastic (Variable) Trend
Problems of fixed trend modeling
Well known stochastic trend examples – RW and RW with Drift
Stochastic trend modeling of GDP
Comparison of the long run forecasts of two models – Nelson-Plosser article
Unit root tests

2. Problem of Linear Trend Model
See the attached graph
From Stock, J. H. and Watson, M. W. “Variable Trends in Economic Time Series.” Journal of Economic Perspectives, Vol. 2, No. 3, Summer P. 147

3. Random Walk Definition: Yt = Y(t-1) + et et is Random N(0, s)
DYt = Yt - Y(t-1) = et
The best known example: Stock Prices

4. Random Walk With Drift as a Trend Model
Yt = d + Yt-1 + et et WN(0, s)
DYt = Yt - Y(t-1) = d + et
Key Properties:
(1) Random shock at each t affects the path.
(2) Uncertainty in the future is not bounded.

5. Comparison of Two Popular Competing Trend Models
Fixed Trend Yt = b0 + b1 t + et t=1, 2, …
et is WN(0, s)
Stochastic Trend Yt = d + Yt-1 + et t=1, 2, …
et is WN (0, s)
Interpretation of d d = b1 and b0 = Y0

6. Behavior of Two Trends in a Long Run - Simulation Study
For t = h
Fixed Trend Yh = b0 + b1 h + eh eh is WN(0, s)
Variable Trend Yt = Y0 + d h + e1 + e2…. eh …
SD of (e1 + e2…. eh) =

7. Stochastic Trend Modeling
Yt = Y(t-1) + ut
ut is ARMA (p, q)
Yt is called an I(1) process, or ARIMA(p, 1, q) process.

8. Stochastic Trend With Seasonality
(1-L) (1-Ls)Yt = ut
ut is ARMA (p, q) (P, Q)s