Lecture 2: ARMA(p,q) models (part 3)

1 Lecture 2: ARMA(p,q) models (part 3)Florian PelgrinUniversity of Lausanne, Ecole des HECD epartment of mathematics (IMEA-Nice)Sept. 2011 - Jan. 2012 Florian Pelgrin (HEC)Univariate time seriesSept. 2011 - Jan. 20121 / 32 IntroductionMotivationCharacterize the main properties of ARMA(p,q) of ARMA(p,q) modelsFlorian Pelgrin (HEC)Univariate time seriesSept. 2011 - Jan. 20122 / 32 IntroductionRoad map1 ARMA(1,1) modelDefinition and conditionsMomentsEstimation2 ARMA(p,q) modelDefinition and conditionsMomentsEstimation3 Application4 AppendixFlorian Pelgrin (HEC)Univariate time seriesSept.

2 2011 - Jan. 20123 / 32 ARMA(1,1) modelDefinition and conditions1. ARMA(1,1) Definition and conditionsDefinitionA stochastic process (Xt)t Zis said to be a mixture autoregressive movingaverage model of order 1, ARMA(1,1), if it satisfies the followingequation :Xt= + Xt 1+ t+ t 1 t (L)Xt= + (L) twhere 6= 0, 6= 0, is a constant term, ( t)t Zis a weak white noiseprocess with expectation zero and variance 2 ( t WN(0, 2 )), (L) = 1 Land (L) = 1 + Pelgrin (HEC)Univariate time seriesSept. 2011 - Jan. 20124 / 32 ARMA(1,1) modelDefinition and conditions0100200300-4-2024 Simulation of an ARMA(1,1) with ( ; )0100200300-10-50510 Simulation of an ARMA(1,1) with ( ; )0100200300-10-50510 Simulation of an ARMA(1,1) with ( ; )0100200300-4-2024 Simulation of an ARMA(1,1) with ( ; ) Florian Pelgrin (HEC)Univariate time seriesSept.

3 2011 - Jan. 20125 / 32 ARMA(1,1) modelDefinition and conditionsThe properties of an ARMA(1,1) process are a mixture of those of anAR(1) and MA(1) processes :The (stability) stationarity condition is the one of an AR(1) process (orARMA(1,0) process) :| |< invertibility condition is the one of a MA(1) process (orARMA(0,1) process) :| |< representation of an ARMA(1,1) process is fundamental or causalif :| |<1 and| |< representation of an ARMA(1,1) process is said to be minimal andcausal if :| |<1,| |<1 and 6= .Florian Pelgrin (HEC)Univariate time seriesSept.

4 2011 - Jan. 20126 / 32 ARMA(1,1) modelDefinition and conditionsIf (Xt) is stable and thus weakly stationary, then (Xt) has an infinitemoving average representation (MA( )) :Xt= 1 + (1 L) 1(1 + L) t= 1 + k=0ak t kwhere :a0= 1ak= k+ k Pelgrin (HEC)Univariate time seriesSept. 2011 - Jan. 20127 / 32 ARMA(1,1) modelDefinition and conditionsIf (Xt) is invertible, then (Xt) has an infinite autoregressiverepresentation (AR( )) :(1 L) 1(1 L)Xt= 1 + = 1 + k=1bkXt k+ twhere = , and :bk= k k 1 .Florian Pelgrin (HEC)Univariate time seriesSept.

5 2011 - Jan. 20128 / 32 ARMA(1,1) MomentsDefinitionLet (Xt) denote a stationary stochastic process that has a fundamentalARMA(1,1) representation,Xt= + Xt 1+ t+ t 1. Then :E[Xt] = 1 m X(0) V(Xt) =1 + 2 + 21 2 2 X(1) Cov[Xt,Xt 1] =( + )(1 + )1 2 2 X(h) = X(h 1) for|h|> : See Appendix Pelgrin (HEC)Univariate time seriesSept. 2011 - Jan. 20129 / 32 ARMA(1,1) modelMomentsDefinitionThe autocorrelation function of an ARMA(1,1) process satisfies : X(h) = 1 ifh= 0( + )(1+ )1+2 + 2if|h|= 1 X(h 1) if|h|> Pelgrin (HEC)Univariate time seriesSept.

6 2011 - Jan. 201210 / 32 ARMA(1,1) modelMomentsThe autocorrelation function of an ARMA(1,1) process exhibitsexponential decay towards zero : it does not cut off but gradually diesout autocorrelation function of an ARMA(1,1) process displays theshape of that of an AR(1) process for|h|> Pelgrin (HEC)Univariate time seriesSept. 2011 - Jan. 201211 / 32 ARMA(1,1) modelMomentsPartial Autocorrelation :The partial autocorrelation function of an ARMA(1,1) process willgradually die out (the same property as a moving average model).Florian Pelgrin (HEC)Univariate time seriesSept.

7 2011 - Jan. 201212 / 32 ARMA(1,1) modelEstimationEstimationSame techniques as before, especially those of MA estimator : the extended Yule-Walker equations could beused in principe to estimate the AR coefficients but the MAcoefficients need to be estimated by other the presence of moving average components, the least squaresestimator becomes nonlinear and the corresponding estimator is theconditional nonlinear least squares estimator (see estimation ofMA(q) models ). It has to be solved with numerical explicit distributional assumption for the error term, theconditional or exact maximum likelihood estimator can be computed(using also numerical or optimization methods).

8 Other methods are also available : the Kalman filter, the generalizedmethod of moments, Pelgrin (HEC)Univariate time seriesSept. 2011 - Jan. 201213 / 32 ARMA(1,1) modelEstimationEstimation of ARMA(p,q) models (true DGP: = 0, = , and = ) Coefficient Std. Error t-Statistic p-value. Akaike info criterion Schwarz criterion ARMA(1,1) C AR(1) MA(1) ARMA(2,2) C AR(1) AR(2) MA(1) MA(2) ARMA(2,1) AR(1) AR(2) MA(1) AR(2) C AR(1) AR(2) AR(1) C AR(1) MA(2) C MA(1) MA(2) MA(4) C MA(1) MA(2) MA(3) MA(4) Note.

9 C, AR(j), and MA(j) are respectively the estimate of the constant term, the jth autogressive term, and the jth moving average term. Florian Pelgrin (HEC)Univariate time seriesSept. 2011 - Jan. 201214 / 32 ARMA(p,q) modelDefinition and conditions2. ARMA(p,q) Definition and conditionsDefinitionA stochastic process (Xt)t Zis said to be a mixture autoregressive movingaverage model of orderpandq, ARMA(p,q), if it satisfies the followingequation :Xt= + 1Xt 1+ + pXt p+ t+ 1 t 1+ + q t q t (L)Xt= + (L) twhere q6= 0, p6= 0, is a constant term, ( t)t Zis a weak white noiseprocess with expectation zero and variance 2 ( t WN(0, 2 )), (L) = 1 1L pLpand (L) = 1 + 1L+ + Pelgrin (HEC)Univariate time seriesSept.

10 2011 - Jan. 201215 / 32 ARMA(p,q) modelDefinition and conditionsMain idea of ARMA(p,q) modelsApproximate Wold form of stationary time series by parsimoniousparametric modelsAR and MA models can be cumbersome because one may need ahigh-order model with many parameters to adequately describe thedata dynamics (see the effective Fed fund rate application)By mixing AR and MA models into a more compact form, the numberof parameters is kept Pelgrin (HEC)Univariate time seriesSept. 2011 - Jan. 201216 / 32 ARMA(p,q) modelDefinition and conditionsThe properties of an ARMA(p,q) process are a mixture of those of anAR(p) and MA(q) processes :The (stability) stationarity conditions are those of an AR(p) process (orARMA(p,0) process) :zp (z 1) = 0 zp 1zp 1 p= 0 |zi|< 1, , invertibility conditions are those of an MA(q) process (orARMA(0,q) process).