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

DefinitionsData =Trend + Season+Cycle + Irregular

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

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

Stationary ProcessSeries Yt is stationary if: mt = m, constant for all tst = s, constant for all tr(Yt, Yt+h) = rh does not depend on t

WN is a special example of a stationary process

Models For a Stationary ProcessAutoregressive Process, AR(p)

Moving Average Process, MA(q)

Autoregressive Moving Average Process, ARMA(p, q)

Parameters of ARMA ModelsSpecification ParametersfkAutoregressive Process Parameter

qkMoving Average Process Parameter

Characterization ParametersrkAutocorrelation Coefficient

fkkPartial Autocorrelation Coefficient

AR ProcessAR (1) : (Yt - m ) = f1 (Y(t-1) - m ) + e t

-1 < f1 < 1 (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(stationarity condition) e t is a WN (s)

MA ProcessMA (1) : Yt - m = et + q 1 e(t-1) - 1 < q 1 < 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 (invertibility condition) e t is a WN (s)

ARMA (p, q) ModelsARMA(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 TheoremAny stationary process can be defined as a linear combination of a WN series, et

means: with: sum( ) < inf.

Lag Operator, LLag Operator, L

Then, the Wold Theorem can be written as:

ApproximationApproximation 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 IdentificationUsing ACF and PACF

Partial Autocorrelation Function:PACFNotation: The partial autocorrelation of order k is denoted asf 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 PACFAR processes

MA processes

ARMA processes

Model Diagnostics Post FitResidual Check:Correlogram of the ResidualQLB Statistic (m - # of parameters)

SE

Test of Significance of Coefficients

AIC, SIC

AIC and SIC(Maximized)(Minimized)

Truth is SimpleParsimonyUse 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 CommandsARls series_name c ar(1) ar(2)..

MAls series_name c ma(1) ma(2).. ARMAls series_name c ar(1) ar(2).ma(1) ma(2).

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