Modeling Cycles By ARMA

• Specification

• Identification (Pre-fit)

• Testing (Post-fit)

• Forecasting

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:

t = , constant for all t

t = , constant for all t

(Yt, Yt+h) = h 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

k Autoregressive Process Parameter

kMoving Average Process Parameter

Characterization Parameters

k Autocorrelation Coefficient

kk Partial Autocorrelation Coefficient

AR Process

• AR (1) : (Yt - ) = 1 (Y(t-1) -) + t

-1 < 1 < 1

(stationarity condition)

• AR (2) : (Yt - ) = 1 (Y(t-1) -) + 2 (Y(t-2) - ) +t

2 + 1 < 1, 2 - 1 < 1 , -1 < 2 < 1

(stationarity condition)

t is a WN ()

MA Process• MA (1) : Yt - = t + 1 (t-1)

- 1 < 1 < 1

(invertibility condition)

• MA (2) : Yt - = t + 1 (t-1) + 2 (t-2)

2 + 1 >-1, 2 - 1 >- 1 , -1 < 2 < 1

(invertibility condition)

t is a WN ()

ARMA (p, q) Models

• ARMA(1, 1):

(Yt - ) = 1 (Y(t-1) - ) + t + 1 (t-1)

• ARMA(2, 1):

(Yt - ) = 1 (Y(t-1) - ) + 2 (Y(t-2) - ) + t + 1 (t-1)

• ARMA(1, 2):

(Yt - ) = 1 (Y(t-1) - ) + t + 1 (t-1) + 2 (t-2)

Wold Theorem

• Any “stationary process” can be defined as a linear combination of a WN series, t

means:

with: sum( ) < inf.

1 ( 1) 2 ( 2)t t t tY b b 2jb

Lag Operator, L

• Lag Operator, L

• Then, the Wold Theorem can be written as:

1 21 21 .....t tY b L b L B L

1t tL

22t tL

Approximation

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

t t t

LY B L

L

21 21 q

qL L L L

21 21 p

pL L L L

Generating AR(1)

• Let: 11

1 1

11t t

LY

LL L

1 1

1 ( 1)

1 ( 1)

1 1 ( 1)

1

1

t t t t t

t t t

t t t

t t t

L Y Y L Y

Y Y

Y Y

Y Y

Generating MA(1)

• Let:

1

1

11

1t t

L LY L

L

1 1t t tY

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

kk

• Interpretation:

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

_L log likelihood

2 2L K

AICT T

2 log log2

K TLSIC

T T

(Maximized)

(Minimized)

Truth is Simple

• Parsimony– Use a minimum number of unknown

parameters

Importance of Parsimony

A. In-Sample RMSE (SE) of Model Prediction

vs.

B. Out-of-Sample RMSE

The two should not differ much.

Eview Commands

• AR– ls series_name c ar(1) ar(2)..

• MA– 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 T (only necessary part)

• The unknown errors are given 0.

• Use the chain rule.

Interval Forecast

• h=1– Use SE of Regression for setting the upper and the

lower limits

• h=2– a) AR(1)

– b) MA(1)

– c) ARMA(1,1)

211 SE

211 SE

21 11 SE