Module 3 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? In a perfect world, the Wold Representation (given infinite data) conveys all of the meaning we need In the real world, we need to approximate the Wold model with some simpler (i.e. non-infinite) representation, such as (so far): Auto-Regressive patterns (ARs) Moving Average Patterns (MAs)

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 I Variables which contain Unit Roots are also called Integrated or Difference Stationary Variables When we have an AR or MA model which produces transitory effects on changes (or first differences) in variables, such effects accumulate into permanent effects on the level of the variable e.g. a MA(1) model in real growth (changes in the logarithm of real GDP) implies that part of any shock has a permanent effect on level of real GDP

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:

Mixed Models with Permanent Shocks » Box-Jenkins Models When first difference of a time series follows an ARMA process, then we get a mixed model with permanent shocks known as an ARIMA (Integrated AutoRegressive Moving Average model (a.k.a Box-Jenkins Model) Example: ARIMA(1,1,1) model

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 III How does anything that occurs today (time = t) carry forward to influence future observations? Assume shock et = 1.0 Assume AR(1) model with coefficient of c1 Xt = c1Xt-1 +et = c1X t-1 +1.0 Shock of 1.0 at time = t carries forward to generate an increase in Xt+1 of c1 Xt+1 = c1Xt +ee+1 = c1[c1Xt-1+ 1.0] + et+1 = c12Xt-1+c1+et+1 Shock of c1 to Xt+1 carries forward to generate and increase in Xt+2 of c12

Forecasts from AR Models II Effect of shock at time = t persists to affect observations at t+3, t+4, etc. (a13, a14, etc.) Size of the effect on future observations becomes smaller as long as absolute value of autoregressive coefficient is < 1.0 Shocks in AR model carry forward to affect future observations indefinitely

How Long a Lag or How Many Parameters? Maximize Adjusted R2 (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) Models I How does anything that occurs today (time = t) carry forward to influence future observations? In MA(1) model the outcome at t+1 depends on et (multiplied by the MA coefficient) outcome in t+2 depends only on the shocks in t+2 and t+1. in contrast to AR Models, effects of shocks in MA models persist into the future only up to the order of the MA model

Forecasts from MA(1) Model II Assume et = 1.0 Xt = et + c1et-1 = 1.0 +c1et-1 Xt+1 = et+1+ c1 Xt+2 = et+2 + c1et+1 (shock has died out) Xt+3 = et+3 + c1et+2

Forecasts from Unit Root Models I Random Walk models Xt - Xt-1 = et or: Xt = Xt-1 + et Assume et = 1.0 Xt= Xt-1 + 1.0 Xt+1 = Xt + et+1 = Xt-1 + 1.0 + et+1 Xt+2 = Xt+1 + et+2 = Xt-1 + 1.0 +et+1 + et+2 Shock to X at time = t is transmitted in full to all future value of X.

Forecasts from Unit Root Models II Integrated MA(1) Model Xt - Xt-1 = et - c1et-1 Assume et = 1.0 Xt = Xt-1 +1.0 - c1et-1 Xt+1 = Xt +et+1 - c1 = Xt-1 + [1.0 - c1] - c1et-1+et+1 Xt+2 = Xt+1 + et+2 - c1et+1 = Xt-1+[1.0-c1]+ [1.0-c1]et+1 +et+2 Fraction of shock at time = t is carried forward permanently in all future realizations of X