6.3.3 Parameter Estimation

Transcription

1 130 CHAPTER 6. ARMA MODELS Parameter Estimatio I this sectio we will iscuss methos of parameter estimatio for ARMAp,q assumig that the orers p a q are kow. Metho of Momets I this metho we equate the populatio momets with the sample momets to obtai a set of equatios whose solutio gives the require estimators. For example, the first populatio momet is µ 1 EX a its sample couterpart is m 1 X. This immeiately gives ˆµ 1 X. The metho of momets gives goo estimators for AR moels but less efficiet estimators for MA or ARMA processes. Hece we will preset the metho for AR time series. As usual we eote a ARp moel by X t φ 1 X t φ p X t p + Z t. This is a zero-mea moel, but the estimatio of the mea is straightforwar a we will ot iscuss it further. Here we use the ifferece equatios, we replace the populatio autocovariace cetral momet of orer two with the sample autocovariace. The first p + 1 ifferece equatios are γ0 φ 1 γ φ p γp + σ 2 γτ φ 1 γτ φ p γτ p, τ 1, 2,..., p. Note, that q 0, so the sum o the right ha sie of 6.15 is zero. I matrix otatio we ca write σ 2 γ0 φ T γ p Γ p φ γ p Replacig γτ by the sample ACVF Γ p {γi j} i,j1,...,p φ φ 1,...,φ p T γ p γ1,...,γp T. γτ 1 τ X t+τ XX t X t1

2 6.3. FORECASTING ARMA PROCESSES 131 we obtai the solutio σ 2 γ0 γ p T φ Γ p γ p. Γ p γ p 6.34 These equatios are calle Yule-Walker estimators. They are ofte expresse i terms of autocorrelatio fuctio rather tha autocovariace fuctio. The we have σ 2 γ0 1 ρ T p φ p ρ p, p ρ p p { ρi j} i,j1,2,...,p is the matrix of the sample autocorrelatios a is the vector of sample autocorrelatios. ρ p ρ1,..., ρp T 6.35 Propositio 6.3. The istributio of the Yule-Walker estimators φ of the moel parameters of a causal ARp process X t φ 1 X t φ p X t p + Z t. is asymptotically as ormal, i the sese that φ φ N0, σ 2 Γ p, a σ 2 p σ 2. Remark Note that the matrix equatio 6.22 is of the same form as Hece, we ca use the Durbi-Lewiso algorithm to calculate the estimates. This will give us the values of the sample PACF as well as the estimates of φ. Propositio 6.4. The istributio of the sample PACF of a causal ARp process is asymptotically ormal, that is φττ N0, 1 for τ > p. Example 6.8. Cosier a AR2 zero-mea causal process X t φ 1 X t + φ 2 X t 2 + Z t.

3 132 CHAPTER 6. ARMA MODELS The the Yule-Walker estimators are a σ 2 γ0 φ 2 ρ 2, 1 ρ T 2 2 ρ0 ρ1 ρ1 ρ0 ρ 2 ρ1, ρ2 T φ φ 1, T. 2 ρ 2 We ca easily ivert a 2 2 matrix a calculate the estimators, or we ca use the Durbi-Leviso algorithm irectly to obtai φ 11 ρ1 φ ρ2 ρ2 1 1 ρ ρ11 2 ] φ 1. Also, we get σ 2 γ0 1 ρ1, ρ2 φ1 ] γ01 ρ1 φ 1 + ρ2 ] Furthermore, from Propositio 6.3 we ca erive the cofiece iterval for φ i. The propositio says that φ φ N0, σ 2 Γ p, that is the variace of φ i φ i is the i-th iagoal elemet of the matrix σ 2 Γ p, say v ii, v ii var φ i φ i ] var φ i φ i var φ i. Hece, a the cofiece iterval is var φ i 1 v ii φ i u α 1 v ii, φi + u α 1 v ii ].

4 6.3. FORECASTING ARMA PROCESSES 133 To calculate the cofiece iterval for a give ata set we replace v ii by its estimate ˆv ii. Also, from Propositio 6.4 we have that is φττ This gives the asymptotic result N0, 1 for τ > p, var φ ττ 1 for τ > p. var φ ττ 1. However, we kow that the PACF for τ > p is zero. It meas that with probability 1 α we have u α < φ ττ 0 < u α. 1 It ca be iterprete that the estimate of the PACF iicates a o-sigificat value of φ ττ if it is i the iterval u α /, u α / ]. The, the iterval covers zero. φ ττ u α /, φ ττ + u α / ]. We will o the calculatios for the simulate AR2 process give i Figure 6.2. For these ata we have the followig values of the sample variace γ0 a the sample autocorrelatios ρ1 a ρ2 γ ρ ρ The, matrix 2 is equal to a its iverse is

5 134 CHAPTER 6. ARMA MODELS Hece, we obtai the followig Yule-Walker estimates of the moel parameters φ The estimate of the white oise variace is σ ] The series was simulate for φ a φ a a Gaussia White Noise with zero mea a variace equal to 1. These estimates are ot far from the true values. Ha we ot kow the true values we woul have like to calculate the cofiece itervals for them. There are 200 observatios, i.e. 200, which is big eough to use the asymptotic result give i the Propositio 6.3. To calculate v ii ote that Γ γ0r, which gives Hece σ 2 Γ σ 2 1 γ0 Γ 1 γ0 R a we obtai the estimate of the variace of the parameter estimators var φ i 1 v ii The 95% approximate cofiece itervals for the moel parameters φ 1 a φ 2 are, respectively , ] , ] , ] , ]

6 6.3. FORECASTING ARMA PROCESSES 135 Maximum Likelihoo Estimatio The metho of Maximum Likelihoo Estimatio applies to ay ARMAp,q moel X t φ 1 X t... φ p X t p Z t + θ 1 Z t θ q Z t q. This metho requires a assumptio o the istributio of the raom variable X X 1,..., X T. The usual assumptio is that the process is Gaussia. Let us eote the p..f. of X by f X X 1,...,X ; β, σ 2, β φ 1,...,φ p, θ 1,...,θ q T. Give the values of X the p..f. becomes a fuctio of the parameters. It is the eote by Lβ, σ 2 x 1,..., x a for the Gaussia process it is Lβ, σ 2 1 x 1,..., x { 2π etγ exp 1 } 2 XT Γ X. A more coveiet form ca be obtaie after takig atural logarithm. The lβ, σ 2 x 1,..., x l Lβ, σ 2 x 1,..., x 2 l2π 1 2 l etγ 1 2 XT Γ X. The Maximum likelihoo Estimates are the values of β a σ 2 which maximize the fuctio lβ, σ 2 x 1,...,x. Ituitively, a maximum likelihoo estimate is the parameter value for which the observe sample is most likely. The estimates are usually fou umerically usig some iterative umerical optimizatio routies. We will ot iscuss them here. The MLE have the property of beig asymptotically ormally istribute. It is state i the followig propositio. Propositio 6.5. The istributio of the MLE β of a causal a ivertible ARMAp,q process is asymptotically ormal i the sese that β β N0, σ 2 Γ p+q, 6.36 the p + q p + q-imesioal matrix Γ p+q epes o the moel parameters.

7 136 CHAPTER 6. ARMA MODELS Some Specific Asymptotic Distributios AR1: X t + φx t Z t φ AN φ, 1 ] 1 φ2 AR2: X t + φ 1 X t + φ 2 X t 2 Z t φ1 φ1 AN φ 2, 1 1 φ 2 2 φ φ 2 φ φ 2 1 φ 2 2 ] MA1: X t Z t + θz t θ AN θ, 1 ] 1 θ2 MA2: X t Z t + θ 1 Z t + θ 2 Z t 2 θ1 θ1 AN θ 2 θ 2, 1 1 θ2 2 θ θ 2 θ θ 2 1 θ2 2 ] ARMA1,1: X t φx t Z t + θz t φ φ 1 + φθ 1 φ AN, φθ θ 2 1 φ 2 θ θ φ + θ 2 θ 2 1 φ 2 1 θ φθ Usig these results we ca costruct approximate cofiece itervals for the moel parameters as i the metho of momets. ]