Modeling Cycles By ARMA Specification Identification (Pre-fit) Testing (Post-fit) Forecasting

Yt = Cyclet + Irregulart Definitions Data =Trend + Season+Cycle + Irregular Cycle + Irregular = Data – Trend – Season (curves) (dummy variables) For this presentation, let: Yt = Cyclet + Irregulart

Stationary Process For Cycles Cycle + Irregular =(A) Stationary Process =(A) ARMA(p, q) =(A) : Approximation

Stationary Process Series Yt is stationary if: mt = m, constant for all t st = s, constant for all t r(Yt, Yt+h) = rh does not depend on t WN is a special example of a stationary process

Models For a Stationary Process Autoregressive Process, AR(p) Moving Average Process, MA(q) Autoregressive Moving Average Process, ARMA(p, q)

Parameters of ARMA Models Specification Parameters fk Autoregressive Process Parameter qk Moving Average Process Parameter Characterization Parameters rk Autocorrelation Coefficient fkk Partial Autocorrelation Coefficient

AR Process AR (1) : (Yt - m ) = f1 (Y(t-1) - m ) + e t (stationarity condition) AR (2) : (Yt - m) = f1 (Y(t-1) - m) + f2 (Y(t-2) - m ) + e t f2 + f1 < 1, f2 - f1 < 1 , -1 < f2 < 1 e t is a WN (s)

MA Process - 1 < q 1 < 1 MA (1) : Yt - m = et + q 1 e(t-1) (invertibility condition) MA (2) : Yt - m = et + q 1 e (t-1) + q2 e (t-2) q2 + q1 >-1, q2 - q1 >- 1 , -1 < q2 < 1 e t is a WN (s)

ARMA (p, q) Models ARMA(1, 1): (Yt - m ) = f1 (Y(t-1) - m ) + e t + q 1 e(t-1) ARMA(2, 1): (Yt - m ) = f1 (Y(t-1) - m ) + f2 (Y(t-2) - m ) + e t + q 1 e(t-1) ARMA(1, 2): (Yt - m ) = f1 (Y(t-1) - m ) + e t + q 1 e(t-1) + q 2 e(t-2)

Wold Theorem Any “stationary process” can be defined as a linear combination of a WN series, et means: with: sum( ) < inf.

Lag Operator, L Lag Operator, L Then, the Wold Theorem can be written as:

Approximation Approximation of B(L) by a Simple Rational Polynomial of L

Generating AR(1) Let:

Generating MA(1) Let:

Generating ARMA(1,1) Your Exercise

AR, MA or ARMA? Pre-Fitting Model Identification Using ACF and PACF

Partial Autocorrelation Function: PACF Notation: The partial autocorrelation of order k is denoted as f kk Interpretation: f kk = Correlation (Yt, Y(t-k) Y(t-1) ,..., Y(t-k+1) ) Yt, {Y(t-1), Y(t-2), ... , Y(t-k+1)}, Y(t-k)

Patterns of ACF and PACF AR processes MA processes ARMA processes

Model Diagnostics – Post Fit Residual Check: Correlogram of the Residual QLB Statistic (m - # of parameters) SE Test of Significance of Coefficients AIC, SIC

AIC and SIC (Maximized) (Minimized)

Truth is Simple Parsimony Use a minimum number of unknown parameters

Importance of Parsimony In-Sample RMSE (SE) of Model Prediction vs. B. Out-of-Sample RMSE The two should not differ much.

Eview Commands AR MA ARMA ls series_name c ar(1) ar(2).. ls series_name c ma(1) ma(2).. ARMA ls series_name c ar(1) ar(2)….ma(1) ma(2)….

Forecasting Rules Sample range: 1 to T. Forecast T+h for h=1,2,… Write the model, with all unknown parameters replaced by their estimates. Write the information set WT (only necessary part) The unknown errors are given 0. Use the chain rule.

Interval Forecast h=1 h=2 Use SE of Regression for setting the upper and the lower limits h=2 a) AR(1) b) MA(1) c) ARMA(1,1)