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432 Index
linear filter (cont.)
low-pass, 26, 130
moving-average, 31, 42
output, 51
simple moving-average, 129
linear process, 51, 232
ACVF of, 52
Gaussian, 344
multivariate, 232
linear regression (see regression)
local level model, 264
local linear trend model, 266
logistic equation, 345
long memory, 318, 362
long-memory model, 361–365
M
MA(1) process, 17
ACF of, 17, 48
estimation of missing values, 82
moment estimation, 145
noninvertible, 97
order selection, 152
PACF of, 110
sample ACF of, 61
spectral density of, 120
state-space representation of, 312
MA(q) (see moving average process)
MA(∞), 51
multivariate, 233
martingale difference sequence, 343
maximum likelihood estimation,
158–161, 386–387
ARMA processes, 160
large-sample distribution of, 162
confidence regions for, 161
mean
of a multivariate time series, 224
estimation of, 234
of a random variable, 373
of a random vector, 376
estimation of, 58
sample, 57
large-sample properties of, 58
mean square convergence, 393–394
properties of, 394
measurement error, 98
memory shortening, 318
method of moments estimation, 96, 140
minimum AICC AR model, 167
mink trappings (APPH.TSM), 257
missing values in ARMA processes
estimation of, 286
likelihood calculation with, 284
mixture distribution, 372
Monte Carlo EM algorithm (MCEM),
298
moving average (MA(q)) process, 50
ACF of, 89
sample, 94
ACVF of, 89
estimation
confidence intervals, 152
Hannan-Rissanen, 156
innovations, 150–151
maximum likelihood, 160, 162
order selection, 151, 152
partial autocorrelation of, 96
unit roots in, 196–198
multivariate AR process
estimation, 247–249
Burg’s algorithm, 248
maximum likelihood, 246–247
Whittle’s algorithm, 247
forecasting, 250–254
error covariance matrix of prediction,
251
multivariate ARMA process, 241–244
causal, 242
covariance matrix function of, 244
estimation
maximum likelihood, 246–247
invertible, 243
prediction, 244–246
error covariance matrix of prediction,
252
multivariate innovations algorithm, 246
multivariate normal distribution, 378
bivariate, 379–380
conditional distribution, 380
conditional expectation, 380
density function, 378
definition, 378
singular, 378
standardized, 378
multivariate time series, 223
covariance matrices of, 229, 230
mean vectors of, 229, 230
second-order properties of, 229–234
stationary, 230
multivariate white noise, 232
muskrat trappings (APPI.TSM), 257
negative binomial distribution, 372, 381
NILE.TSM, 363–365
NOISE.TSM, 334, 343
nonlinear models, 343–357
nonnegative definite matrix, 376
nonnegative definite function, 47
normal distribution, 370, 373
normal equations, 384
null hypothesis, 389
N
O
observation equation, 260
of CARMA(p, q) model, 359
ordinary least squares (OLS) estimators,
211, 383–385
one-step predictors, 71, 273
order selection, 141, 161, 169–174
AIC, 171
AICC, 141, 161, 173, 191, 247, 407
BIC, 173, 408
consistent, 173
efficient, 173
FPE, 170–171
orthogonal increment process, 117
orthonormal set, 123
overdifferencing, 196
overdispersed, 306
overshorts (OSHORTS.TSM), 96–99,
167, 197, 215
structural model for, 98
partial autocorrelation function (PACF),
71, 94–96
estimation of, 95
of an AR(p) process, 95
of an MA(1) process, 96
sample, 95
periodogram, 123–127
approximate distribution of, 124
point estimate, 388
Poisson distribution, 371, 374
model, 302
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