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Econ 427 lecture 22 slides

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

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Econ 427 lecture 22 slides

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

  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.

  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.)

  4. ARIMA processes • An integrated process may also have AR and MA behavior. • Recall the general ARMA(p,q) model (see Ch. 8) • If ytis 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

  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.)

  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?

  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.

  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

  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:

  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.

  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:

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