STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7)

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1 STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7)

2 Outline 1 AR Bootstrap 2 ARIMA 3 Homework 4b Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 2/ 20

3 Outline 1 AR Bootstrap 2 ARIMA 3 Homework 4b Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 3/ 20

4 Bootstrap Uses of the bootstrap. Measure properties of estimates like variance Construct hypothesis tests When to use the bootstrap. When the statistic is complex and its distribution is difficult to derive When assumptions, like being normally distributed, aren t satisfied...typical with small sample sizes Variants for dependent data: Subsampling Block Bootstrap AR Bootstrap Nonparametric AR Bootstrap Sieve Bootstrap Others (like Wild, Local, Markov, Frequency Domain, etc.) Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 4/ 20

5 Bootstrap Uses of the bootstrap. Measure properties of estimates like variance Construct hypothesis tests When to use the bootstrap. When the statistic is complex and its distribution is difficult to derive When assumptions, like being normally distributed, aren t satisfied...typical with small sample sizes Variants for dependent data: Subsampling Block Bootstrap AR Bootstrap Nonparametric AR Bootstrap Sieve Bootstrap Others (like Wild, Local, Markov, Frequency Domain, etc.) Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 4/ 20

6 Bootstrapping AR(1) Suppose {x t } 100 t=1 follows the following AR(1) model where w t iid Laplace(0, 2), i.e. x t =.95x t 1 + w t f w (x) = 1 4 exp( x /2) Note the variance of w t is 2b 2 = 8. Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 5/ 20

7 Asymptotic Distribution of φ YW Last time we saw ( d n ˆφYW φ) N ( 0, σ 2 Γ 1 ) and in the AR(1) case ( n ˆφYW φ) N ( 0, 1 φ 2) p Therefore we would approximate φ YW with the distribution ( ˆφ N φ, 1 ( 1 φ 2 )) ( 1 ( = N.95, )) = N (.95, ) n 100 Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 6/ 20

8 Simulation AR(1) with Laplace Innovations > e=rexp(150,rate=.5) > u=runif(150,-1,1) > de=e*sign(u) > x=50+arima.sim(n=100,list(ar=.95), innov=de, n.start=50) > ts.plot(x) Since φ =.95 1, the above instance looks nonstationary, but indeed it is. Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 7/ 20

9 Estimation of φ in R > ar.yw(x,aic=f,order.max=1) Call: ar.yw.default(x = x, aic = F, order.max = 1) Coefficients: Order selected 1 sigma^2 estimated as > ar(x,aic=f,order.max=1, method="burg") Coefficients: sigma^2 estimated as > ar(x,aic=f,order.max=1, method="ols") Coefficients: Intercept: (0.3109) sigma^2 estimated as 9.57 > ar(x,aic=f,order.max=1, method="ml") Coefficients: sigma^2 estimated as Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 8/ 20

10 Simulated Distribution of φ YW > phis<-double(10^4) > for(r in 1:10^4){ + e=rexp(150,rate=.5) + u=runif(150,-1,1) + de=e*sign(u) + x=50+arima.sim(n=100,list(ar=.95), innov=de, n.start=50) + phis[r]<-ar.yw(x,aic=f,order.max=1)$ar + } > plot(density(phis),xlim=c(.65,1.1), ylim=c(0,13),lwd=3) > t<-seq(.65,1.1,.01) > x<-dnorm(t,.95,.0312) > lines(t,x,col="red",lwd=3) Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 9/ 20

11 Bootstrap Estimate We wish to bootstrap the innovations which are assumed to be iid. Since we have And since x1 0 = µ = 0, we have However, w t = x t φx t 1, w t = x t x t 1 t = x t φx t 1 t = 2,..., 100 w 1 = x t x t 1 t = x 1 var(w 1 ) = var(x 1 ) = σ2 1 φ 2 var(w t ) = P t 1 t = σ 2 where the second equality comes from exercise 3.14 in your homework. Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 10/ 20

12 Bootstrap Estimate Therefore standardizing the innovations to have equal variances produces Inverting these equations gives ε 1 = w 1 1 φ 2 = x 1 1 φ 2 ε t = w t = x t φx t 1 t = 2,..., 100 x 1 = ε 1 1 φ 2 x t = φx t 1 + ε t Estimate the standardized residuals as ε 1 = x 1 1 φ 2 ε t = x t φx t 1 t = 2,..., 100 Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 11/ 20

13 Bootstrap Algorithm Algorithm (AR(1) BOOTSTRAP ALGORITHM) 1 Obtain a bootstrap sample, { ε 1,..., ε 100 } by sampling with replacement from { ε 1,..., ε 100 }. 2 Reconstruct the AR(1) process from the bootstrapped residules, i.e. compute x 1 = ε 1 1 φ 2 x t = ˆφx t 1 + ε t 3 Compute φ YW from the reconstructed time series. 4 Repeat steps 1 through 3 a total of B times yielding estimates (1) (B) φ YW,..., φ YW Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 12/ 20

14 AR Bootstrap in R > ### Data Simulation > e = rexp(150, rate =.5) > u = runif(150,-1,1) > de = e*sign(u) > x = arima.sim(n = 100, list(ar =.95), innov = de, n.start = 50) > ### Initializations > fit = ar.yw(x, order=1) > phi = fit$ar > nboot = 2000 # number of bootstrap replicates > resids = fit$resid > resids = resids[2:100] # the first resid is NA > x.star = x # initialize x.star > phis2 = double(nboot) #creates phis vector Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 13/ 20

15 AR Bootstrap in R (cont.) > ### Bootstrap Iterations > for (i in 1:nboot) { + resid.star = sample(resids, replace=true) + for (t in 1:99){ + x.star[t+1] = phi*(x.star[t]) + resid.star[t] + } + phis2[i] = ar.yw(x.star, order=1)$ar + } > plot(density(phis),xlim=c(.65,1.1), ylim=c(0,13),lwd=5) > t<-seq(.65,1.1,.001) > x<-dnorm(t,.95,.0312) > lines(t,x,col="red",lwd=3) > lines(density(phis2), lwd=3, col="orange") > x<-dnorm(t,phi,sqrt((1-phi^2)/100)) > lines(t,x,col="blue",lwd=3) Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 14/ 20

16 AR Bootstrap in R (cont.) Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 15/ 20

17 Outline 1 AR Bootstrap 2 ARIMA 3 Homework 4b Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 16/ 20

18 ARIMA Definition Definition (ARIMA) A process is said to be ARIMA(p,d,q) if is ARMA(p, q). d x t = (1 B) d x t Therefore an ARIMA(p,d,q) model (with mean zero) can be written as φ(b)(1 B) d x t = θ(b)w t Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 17/ 20

19 Outline 1 AR Bootstrap 2 ARIMA 3 Homework 4b Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 18/ 20

20 Textbook Reading Read the following sections from the textbook 3.9 (Multiplicative Seasonal ARIMA Models) Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 19/ 20

21 Textbook Problems Do the following exercise from the textbook 3.26 Arthur Berg STA 6857 Estimation and ARIMA Models ( 3.6 cont. and 3.7) 20/ 20

STA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9)

STA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9) Outline 1 Building ARIMA Models 2 SARIMA 3 Homework 4c Arthur Berg STA 6857 ARIMA and SARIMA Models ( 3.8 and 3.9) 2/ 34 Outline 1 Building ARIMA Models