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estimation of missing values
in an ARIMA process, 287
in an AR(p) process, 288
in a state-space model, 286
estimation of the white noise variance
least squares, 161
maximum likelihood, 160
using Burg’s algorithm, 148
using the Hannan-Rissanen algorithm,
157
using the innovations algorithm, 155
using the Yule-Walker equations, 142
expectation, 373
exponential distribution, 370
exponential family models, 301–302
exponential smoothing, 27–28, 322
F
filter (see linear filter)
Fisher information matrix, 387
forecasting, 63–77, 167–169 (see also
prediction)
forecasting ARIMA processes, 198–203
forecast function, 200–203
h-step predictor, 199
mean square error of, 200
forecast density, 293
forward prediction errors, 147
Fourier frequencies, 122
Fourier indices, 13
fractionally integrated ARMA process,
361
estimation of, 363
spectral density of, 363
Whittle likelihood approximation, 363
fractionally integrated white noise, 362
autocovariance of, 362
variance of, 362
frequency domain, 111
G
gamma distribution, 371
gamma function, 371
GARCH(p, q) process, 352–357
ARMA model with GARCH noise, 356
fitting GARCH models, 353–356
Gaussian-driven, 354
generalizations, 356
regression with GARCH noise, 356
Index 431
t-driven, 355
Gaussian likelihood
in time series context, 387
of a CAR(1) process, 359
of a multivariate AR process, 246
of an ARMA(p, q) process, 160
with missing observations, 284–285,
290
of GARCH model, 354
of regression with ARMA errors, 213
Gaussian linear process, 344
Gaussian time series, 380
Gauss-Markov theorem, 385
generalized distribution function, 115
generalized least squares (GLS)
estimation, 212, 386
generalized inverse, 272, 312
generalized state-space models
Bayesian, 292
filtering, 293
forecast density, 293
observation-driven, 299–311
parameter-driven, 292–299
prediction, 293
Gibbs phenomenon, 131
goals scored by England against
Scotland, 306–311
goodness of fit (see also tests of
randomness) based on ACF, 21
H
Hannan-Rissanen algorithm, 156
harmonic regression, 12–13
Hessian matrix, 161, 214
hidden process, 293
Holt-Winters algorithm, 322–326
seasonal, 326–328
hypothesis testing, 389–391
large-sample tests based on confidence
regions, 390–391
uniformly most powerful test, 390
I
independent random variables, 375
identification techniques, 187–193
for ARMA processes, 161, 169–174,
189
for AR(p) processes, 141
for MA(q) processes, 152
for seasonal ARIMA processes, 206
iid noise, 8, 16
sample ACF of, 61
multivariate, 232
innovations, 82, 273
innovations algorithm, 73–75, 150–151
fitted innovations MA(m) model, 151
multivariate, 246
input, 51
intervention analysis, 340–343
invertible
ARMA process, 86
multivariate ARMA process, 243
Itô integral, 358
ITSM, 31, 32, 43, 44, 81, 87, 95, 188,
333, 337–339, 395–421
J
joint distributions of a time series, 7
joint distribution of a random vector, 374
K
Kalman recursions
filtering, 271, 276
prediction, 271, 273
h-step, 274
smoothing, 271, 277
Kullback-Leibler discrepancy, 171
Kullback-Leibler index, 172
L
Lake Huron (LAKE.TSM), 10–11,
21–23, 63, 149–150, 155, 157, 163,
174, 193, 215–217, 291
latent process, 293
large-sample tests based on confidence
regions, 390–391
least squares estimation
for ARMA processes, 161
for regression model, 383–386
for transfer function models, 333–335
of trend, 10
likelihood function, 386 (see also
Gaussian likelihood)
linear combination of sinusoids, 116
linear difference equations, 201
linear filter, 26, 42, 51
input, 51