Modeling Cycles By ARMA

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Modeling Cycles By ARMA

• Specification
• Identification (Pre-fit)
• Testing (Post-fit)
• Forecasting

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Definitions

• Data Trend SeasonCycle Irregular
• Cycle Irregular Data Trend Season
  (curves) (dummy variables)
• For this presentation, let
• Yt Cyclet Irregulart

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

• Series Yt is stationary if mt m, constant
  for all t
• st s, constant for all t
• r(Yt, Yth) rh does not depend on t
• WN is a special example of a stationary process

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Models For a Stationary Process

• Autoregressive Process, AR(p)
• Moving Average Process, MA(q)
• Autoregressive Moving Average Process, ARMA(p, q)

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Parameters of ARMA Models

• Specification Parameters
• fk Autoregressive Process Parameter
• qk Moving Average Process Parameter
• Characterization Parameters
• rk Autocorrelation Coefficient
• fkk Partial Autocorrelation Coefficient

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AR Process

• AR (1) (Yt - m ) f1 (Y(t-1) - m ) e
  t
• -1 lt f1 lt 1
• (stationarity condition)
• AR (2) (Yt - m) f1 (Y(t-1) - m) f2
  (Y(t-2) - m ) e t
• f2 f1 lt 1, f2 - f1 lt 1 , -1 lt f2 lt 1
• (stationarity condition)
• e t is a WN (s)

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MA Process

• MA (1) Yt - m et q 1 e(t-1)
• - 1 lt q 1 lt 1
• (invertibility condition)
• MA (2) Yt - m et q 1 e (t-1) q2 e
  (t-2)
• q2 q1 gt-1, q2 - q1 gt- 1 ,
  -1 lt q2 lt 1
• (invertibility condition)
• e t is a WN (s)

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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)

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Wold Theorem

• Any stationary process can be defined as a
  linear combination of a WN series, et
• means
• with sum( ) lt inf.

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Lag Operator, L

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

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Approximation

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

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Generating AR(1)

• Let

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Generating MA(1)

• Let

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Generating ARMA(1,1)

• Your Exercise

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AR, MA or ARMA?Pre-Fitting Model Identification

• Using ACF and PACF

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Partial Autocorrelation FunctionPACF

• 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-k1) )
• Yt, Y(t-1), Y(t-2), ... , Y(t-k1), Y(t-k)

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Patterns of ACF and PACF

• AR processes
• MA processes
• ARMA processes

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Model Diagnostics Post Fit

• Residual Check
• Correlogram of the Residual
• QLB Statistic (m - of parameters)
• SE
• Test of Significance of Coefficients
• AIC, SIC

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AIC and SIC
(Maximized)
(Minimized)
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Truth is Simple

• Parsimony
• Use a minimum number of unknown parameters

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Importance of Parsimony

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

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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).

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Forecasting Rules

• Sample range 1 to T. Forecast Th for h1,2,
• 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.

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Interval Forecast

• h1
• Use SE of Regression for setting the upper and
  the lower limits
• h2
• a) AR(1)
• b) MA(1)
• c) ARMA(1,1)