Stochastic Variable Trend 1 Problems of fixed trend

Stochastic (Variable) Trend 1. Problems of fixed trend modeling 2. Well known stochastic trend examples – RW and RW with Drift 3. Stochastic trend modeling of GDP 4. Comparison of the long run forecasts of two models – Nelson-Plosser article 5. Unit root tests

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 1988. P. 147

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

Random Walk With Drift as a Trend Model • 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.

Comparison of Two Popular Competing Trend Models • Fixed Trend Yt = b 0 + b 1 t + et t=1, 2, … et is WN(0, s) • Stochastic Trend Yt = d + Yt-1 + et et is WN (0, s) • Interpretation of d d = b 1 and b 0 = Y 0 t=1, 2, …

Behavior of Two Trends in a Long Run - Simulation Study For t = h • Fixed Trend Y h = b 0 + b 1 h + eh eh is WN(0, s) • Variable Trend Yt = Y 0 + d h + e 1 + e 2…. eh … SD of (e 1 + e 2…. eh) =

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.

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