1. Forecasting with non-stationary data series
Econ 427 lecture 22 slides
Forecasting with non-stationary data series
Byron Gangnes

2. Unit Roots Last time we talked about the random walk
Because the autoregressive lag polynomial has one root equal to one, we say it has a unit root.
Note that there is no tendency for mean reversion, since any epsilon shock to y will be carried forward completely through the unit lagged dependent variable.
But the time difference of a random walk is stationary.
Byron Gangnes

3. Integrated series
Terminology: we say that yt is integrated of order 1, I(1) “eye-one”, because it has to be differenced once to get a stationary time series.
In general a series can be I(d), if it must be differenced d times to get a stationary series.
Some I(2) series occur (the price level may be one), but most common are I(1) or I(0) (series that are already cov. stationary without any differencing.)
Byron Gangnes

4. ARIMA processes
An integrated process may also have AR and MA behavior.
Recall the general ARMA(p,q) model (see Ch. 8)
If yt is integrated, we call this an ARIMA model, autoregressive integrated moving average model.
ARIMA(p,d,q)
Where d is the number of times that you have to difference y to get a stationary process
Byron Gangnes

5. ARIMA processes
If the ARMA model Φ(L)yt = θ(L)εt has a unit root, it can be transformed into a form that is stationary in differences:
where phi-prime is of degree p-1. Or equivalently,
This is an ARMA(p-1,q) model in the covariance stationary variable, Δyt
(It does not appear easy to prove this result.)
Byron Gangnes

6. Statistical properties of integrated series
Random walk:
Note that if it started at some past time 0, we can rewrite it as:
What is the unconditional mean?
Byron Gangnes

7. Statistical properties of integrated series
What is the unconditional variance?
Note that the variance grows without bound: Why?
look at the graph of the random walk; since it does not revert to a mean value, it can deviate infinitely far from its starting point over time.
Byron Gangnes

8. Forecasting with ARIMA models
Unit roots mean that series will not revert to mean and it will have a variance that grows continuously.
So optimal forecasts of series with unit roots will not revert to mean and they will have ever-expanding confidence intervals.
What will be the optimal h-step ahead forecast given info at time T?
Just today’s value
Byron Gangnes

9. Forecasting with ARIMA models
Why?
Remember that for an AR(1) process
The optimal forecast is
For RW, phi=1.
Or just look at the y process and think about it:
Byron Gangnes

10. Forecasting with ARIMA models
What is the forecast error variance of a random walk?
hσ2
Why?
Recall (Ch 8) that h-step ahead forecast error for AR(1) is
with variance
So in RW case we sum h terms that are all σ2.
Byron Gangnes

11. Forecasting with ARIMA models
For models with ARMA terms, we model the differenced data with ARMA model and then “integrate” back up to get the levels of the series:
Byron Gangnes