model-building strategybrill/stat153/chap5.pdfmodel-building strategy finding appropriate models for...
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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.