Model-Building Strategy

Finding appropriate models for time series is a nontrivial task. We will

develop a multistep model-building strategy espoused so well by Box

and Jenkins (1976). There are three main steps in the process, each of

which may be used several times:

1. model specification (or identification)

2. model fitting, and

3. model diagnostics

Chapter 5, Models for Nonstationary Time Series

Yt = μt + Xt E Xt = 0

E Yt = μt is nonconstant

Yt is nonstaionary

5.1 Stationarity Through Differencing.

▼Yt = Yt - Yt-1 = (1 – B) Yt

outlier - atypical observation, different from the rest

Explosive behavior.

Yt = φ Yt-1 + et |φ| > 1, e.g. φ = 3

Y0 = 0, {e t } IN(0,1)

Stationary from nonstationary

Random walk.

Yt = φ Yt-1 + et φ = 1

▼Yt = Yt - Yt-1 = et stationary

Mt = Mt-1 + εt

Yt = Mt + et {ε t} and {et} independent

▼Yt = ▼Mt + ▼et = εt + et - et-1 stationary

Wt = Wt-1 + εt

Mt = Mt-1 + Wt

Yt = Mt + et

▼Yt = ▼Mt + ▼et = Wt + ▼et

▼2Yt = εt ▼Wt + ▼

2et = Wt FIX

= εt + et -2 et-1 + et-2 stationary

5,2 ARIMA models

{Yt } integrated autoregressive moving average if

Wt = ∇dYt stationary ARMA process

φ(B)Wt = θ(B)et BWt = Wt-1

Wt stationary if zeroes of φ(x) = 0 outside unit circle |x| = 1

(unit roots satisfy x| = 1)

In other words ∇dYt = Wt where φ(B)Wt = θ(B)et

A random walk, (1 – B)Vt = 0, is not stationary. Zero is x =1

Assume ARIMA(p,d,q)’s start at = -m < 1, where first observed series.

Take Yt = 0 for t < −m.

Consider ARIMA(p,1,q), Yt - Yt-1 = Wt

Can get means and variances

IMA(1,1).

Weights do not die out

IMA(2,2).

The ARI(1,1) nodel

Process not stationary. Characteristic equation, x2 – (1+ φ)x + φ = 0

One root x = 1.

It can be useful to obtain ψ-weights. Here

Constant term in ARIMA

arima package:stats R Documentation

ARIMA Modelling of Time SeriesDescription:

Fit an ARIMA model to a univariate time series.

Usage:

arima(x, order = c(0, 0, 0),

seasonal = list(order = c(0, 0, 0), period = NA),

xreg = NULL, include.mean = TRUE,

transform.pars = TRUE,

fixed = NULL, init = NULL,

method = c("CSS-ML", "ML", "CSS"),

n.cond, optim.method = "BFGS",

optim.control = list(), kappa = 1e6)

Arguments:

x: a univariate time series

order: A specification of the non-seasonal part of the ARIMA model:

the three components (p, d, q) are the AR order, the degree

of differencing, and the MA order.

seasonal: A specification of the seasonal part of the ARIMA model, plus

the period (which defaults to âfrequency(x)â). This should

be a list with components âorderâ and âperiodâ, but a

specification of just a numeric vector of length 3 will be

turned into a suitable list with the specification as the

âorderâ.

xreg: Optionally, a vector or matrix of external regressors, which

must have the same number of rows as âxâ.

New or Enhanced Functions in the TSA Library Function Description

acf Computes and plots the sample autocorrelation function starting with lag 1

.

arima This command has been amended to compute the AIC according to our

definition.

arima.boot Bootstraps time series according to a fitted ARMA(p,d,q) model.

arimax Extends the arima function, allowing the incorporation of transfer

functions and innovative and additive outliers.

ARMAspec Computes and plots the theoretical spectrum of an ARMA model.

armasubsets Finds “best subset” ARMA models.

BoxCox.ar Finds a power transformation so that the transformed time

series is approximately an AR process with normal error terms.

detectAO Detects additive outliers in time series.

detectIO Detects innovative outliers in time series.

eacf Computes and displays the extended autocorrelation function of a time

series.

garch.sim Simulates a GARCH process.

gBox Performs a goodness-of-fit test for fitted GARCH models.

harmonic Creates a matrix of the first m pairs of harmonic functions for

fitting a harmonic trend (cosine-sine trend, Fourier regression model with a time

series response.