Properties of Estimates for some simple ARMA models
Petrutza Caragea
Fall 2009, Iowa State University
P. Caragea (ISU)
Stat 551 (Handout 9)
Fall 2009
1/7
AR models
AR(1): Var(φ̂) ≈
AR(2):
1−φ2
n


Var(φ̂1 ) ≈ Var(φ̂2 ) ≈






 Corr(φ̂ , φ̂ ) ≈
1 2
P. Caragea (ISU)
1−φ22
n
φ1
1−φ2
Stat 551 (Handout 9)
Fall 2009
2/7
MA models
MA(1): Var(θ̂) ≈
MA(2):
1−θ2
n


Var(θ̂1 ) ≈ Var(θ̂2 ) ≈







P. Caragea (ISU)
Corr(θ̂1 , θ̂2 ) ≈
1−θ22
n
θ1
1−θ2
Stat 551 (Handout 9)
Fall 2009
3/7
ARMA(1,1) model

h
ih
i2
1−φ2
1+φθ


Var(
φ̂)
≈

n
φ+θ









h 2ih
i2
1+φθ
1−θ
Var(
θ̂)
≈
n
φ+θ









√


(1−φ2 )(1−θ2 )

Corr(φ̂, θ̂) ≈
1+φθ
P. Caragea (ISU)
Stat 551 (Handout 9)
Fall 2009
4/7
Large-Sample Standard Deviations
MA(1) model: Comparison
MM vs. MLE
AR(1) model with various
n and φ
φ
0.4
0.7
0.9
50
0.13
0.10
0.06
n
100
0.09
0.07
0.04
200
0.06
0.05
0.03
θ
0.25
0.50
0.75
0.90
SDMM /SDMLE
1.07
1.42
2.66
5.33
since large sample variance for
the MoM estimator of θ is
Var(θ̂) ≈
P. Caragea (ISU)
Stat 551 (Handout 9)
1+θ2 +4θ4 +θ6 +θ8
n(1−θ2 )2
Fall 2009
5/7
Some examples in R. AR(1) case
>phi=0.4
>ar1.s=arima.sim(model=list(order=c(1,0,0),ar=phi),n=200)
>M1=ar(ar1.s,order.max=1,AIC=F,method=’yw’) # method of moments
>M2=ar(ar1.s,order.max=1,AIC=F,method=’ols’) # conditional sum of squares
>M3=ar(ar1.s,order.max=1,AIC=F,method=’mle’) # maximum likelihood
# The AIC option is set to be False otherwise the function will choose
the AR order by minimizing AIC, so that zero order might be chosen.
#summarize the information from the above functions:
>c(phi,M1$ar,M2$ar,M3$ar)
#to compute asymptotic std. dev:
>(1-M3$ar*M3$ar) / 200
From the help file for function ar:
ar
Estimated autoregression coefficients for the fitted model.
var.pred The prediction variance: an estimate of the portion
of the variance of the time series that is not explained by the
autoregressive model.
P. Caragea (ISU)
Stat 551 (Handout 9)
Fall 2009
6/7
Some examples in R. ARMA(1,1) case
#ARMA(1,1)
>phi=0.6;theta=0.3
>arma11.s=arima.sim(model=list(order=c(1,0,1),ar=phi,ma=theta),n=200)
>M2=arima(arma11.s, order=c(1,0,1),method=’CSS’) # conditional sum of
squares
>M3=arima(arma11.s, order=c(1,0,1),method=’ML’)# maximum likelihood
#summarize the three methods
>cbind(c(phi,theta),M2$coef[1:2],M3$coef[1:2])
P. Caragea (ISU)
Stat 551 (Handout 9)
Fall 2009
7/7