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

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

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

4. 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

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

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

7. 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)

8. 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)

9. 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)

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

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

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

13. Generating AR(1)
Let:

14. Generating MA(1)
Let:

15. Generating ARMA(1,1)
Your Exercise

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

17. 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)

18. Patterns of ACF and PACF
AR processes
MA processes
ARMA processes

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

20. AIC and SIC
(Maximized)
(Minimized)

21. Truth is Simple
Parsimony
Use a minimum number of unknown parameters

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

23. 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)….

24. 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.

25. 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)