1. BASIC CONCEPTS Financial Economics and Uncertainty Stochastic Process, Stationarity and Autocorrelation Stochastic process (e.g., nondeterministic discrete time series) two features: dependency and lack of replication Realizations and statistics of probability distribution: mean,variance,autocovariance stationarity: a particular state of statistical equilibrium strict stationarity: distribution properties unaffected by a change of time origin weak stationarity: the first and second moments do not depend on time. Ergodicity: the conditions about the consistency between sample statistics and population statistics Autocorrelation function (correlogram) and partial autocorrelation ACF[ ] and structure of the random process PACF: indirect correlation eliminating the other past effects
Stationary linear stochastic process White noise model ; ; Autoregressive model [AR(p)] p: lag order, :innovation(white noise process) stationarity(characteristic roots of must lie outside of the unit circle) to calculate the second moments based on Yule=Walker equation for an AR(p), there is no partial autocorrelation between and for s>p. Moving average model[MA(q)] stationary and non-deterministic process: to calculate each statistics based on their definition for an MA(q), there is no autocorrelation between and for s>p.
MA invertibility and AR stationarity the PACF coefficients exhibit a geometrically decaying pattern. ARMA(p,q) model stationarity [same as AR(p)];invertibility[same as MA(q)] the ACF will begin to decay at lag q,while PACF to decay at lag p. Autoregressive integrated moving average Model[ARIMA(p,d,q)] trend elimination Wolds decomposition theorem
2. Box-Jenkins Methodology Identification Estimation Diagnostic Checking( Forecasting) Principle of Parsimony Identification(Model Building) Plotting time series data Pattern of the ACF and PACF Test on Sample ACF and PACF (t, Q test) Nonstationarity and seasonality adjustment (integrated process) trend(mean by difference,variance by log transformation); seasonality(regular by difference,irregular by additive or multiplicative SARIMA) Estimation General method covariance matrix ML function Estimator(ML,QML,CML) long period needed Special Method AR(p): OLS, Yule=Walker equation MA(q) and ARMA(p,q): Gauss=Newton method(grid-search)
Diagnostic Checking(Model Selection) Residuals plot Information criteria[AIC(1969),SBIC(1978),etc.] AIC= ; :estimator of var( u t ) SBIC= They will be as small as possible(comparable with the same period) SBIC has superior large sample properties(asymptotically consistent). Overfitting and splitting analysis Forecast adequacy
3. FORECASTING Basic Concept Optimal forecast and prediction error stationarity and convergency of forecast weight and error variance The role of forecast model AR( ): make a forecast;MA( ):forecast error analysis Significant level and confidence intervals Forecast Function Iterative method Solution methodology Some Comments Efficacy of forecast(short period) Conditional forecast (start period) Large sample needed
4. SUMMARY AND CONCLUSIONS By definition, an ARMA model is weak stationary in that it has a finite and time-invariant mean and covariances.For an ARMA model to be stationary,the characteristic roots of the difference equation must lie inside the unit circle. Moreover,the process must have started infinitely far in the past or the process must always be in equilibrium. A well estimated model (1) is parsimonious; (2) has coefficients that imply stationarity and invertibility;(3) fits the data well;(4) has residula that approximate a white-noise process; (5) has coefficients that do not change over the sample period; and (6) has good out-of-sample forecasts.
Appendix : TSP Programs to Accompany Chapter 2 BJIDENT (option) variables; Option: NDIFF, NSDIFF ( NSPAN), NLAG, NLAGP Plot: series, ACF+PACF (20), Q (s-p-q-1) Output value: ACF, PACF, Q @AC, @PAC BJEST (option) variables (start values); Option (unnecessary to specify if same): NBACK (start condition)---MA: small value(5), AR: large (10) Start: value specification (orderAR,MA,CONST) previous estimation results Residuals: Q, p, periodogram (45 degree line) AIC=2logL + 2(p+q) BJFRCST (option) variables S start variables values; Option: CONBOUND(95%), NHORIZ, ORGBEG, ORGEND)