Time Series Analysis: Forecasting a Single Variable from Its Own History, Slides of Economics

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Forecasting a Single Variable from

its own History, continued

What Do We Need From Our

Data (recap)

 History is available, and has implications for the present / future

 Errors have finite, time-independent variance

• How could you test?

 No data is perfect. The question is how imperfect the data can be, and still be useable

What Should We Do With the

Data? II

 How do you choose between AR and MA representations?

• Theory (should shocks die away?)
• Data (do long-lags seem relevant)
• Deciding between low-order ARs and high- order MAs is difficult

What Should We Do With the

Data? III

 What if the shocks are permanent, but caused by temporary shocks in the growth rate? These are referred to as Unit Root problems.

 Do you model a permanent shock to GDP (for example), or the temporary shock to the growth rate of GDP that caused it?

 E.g.: the effect of being in the Army on your lifetime earnings...

Unit Root Models II

 If the autocorrelations of first differences (changes over time) are not significantly different from zero, then we cannot predict future changes of a variable from past changes

• such models are called random walk models
• theoretical models of organized market price behavior (such as stock market prices) frequently predict such random walk behavior

Mixed Models

 Possible to have time series processes that are mixtures of AutoRegressive and Moving Average factors.

• such models known as ARMA processes.

 Example: ARMA(1,1) Process:

X (^) t  a 0  a X 1 t  1  e (^) t  b e 1 t 1

Forecasting from AR Models

 One period ahead forecasts:

• multiply coefficients of model by most recently observed values of time series and add the terms up = forecast of next period value (tXt+1)

 Multiple period ahead forecasts:

• most recent data values are not available
• for (^) tXt+2 use predicted values, (^) tXt+1 for Xt-1, Xt for Xt-2, etc.
• for (^) tXt+3 use predicted values, (^) tXt+2 for Xt-1, tXt+1 for Xt-2, Xt for Xt-3, etc.

Forecasting from AR Models II

 Procedure of using forecasted values for actual past values in making multiple period forecasts is known as the chain rule of forecasting.

Forecasts from AR Models II

 Effect of shock at time = t persists to affect observations at t+3, t+4, etc. (a 13 , a 14 , etc.)

 Size of the effect on future observations becomes smaller as long as absolute value of autoregressive coefficient is < 1.

 Shocks in AR model carry forward to affect future observations indefinitely

How Long a Lag or How Many

Parameters?

 Maximize Adjusted R^2 (or minimize standard error of estimate[s.e.e.])?

 Akaike and Schwartz Information Criteria?

• Diebold, p. 26; pp. 85-91.
• Applies a heavier penalty than the s.e.e. to using up degrees of freedom.
• Smaller numbers are better.

Forecasts from MA(1) Model II

 Assume et = 1.

 Xt = et + c 1 et-1 = 1.0 +c 1 et-

 Xt+1 = et+1+ c 1

 Xt+2 = et+2 + c 1 et+1 (shock has died out)

 Xt+3 = et+3 + c 1 et+

Forecasts from Unit Root Models I

 Random Walk models

• Xt - Xt-1 = et or: Xt = Xt-1 + et
• Assume et = 1. » Xt= Xt-1 + 1. » Xt+1 = Xt + et+1 = Xt-1 + 1.0 + et+ » Xt+2 = Xt+1 + et+2 = Xt-1 + 1.0 +et+1 + et+
• Shock to X at time = t is transmitted in full to all future value of X.