1 introduction .1
1.1 examples of time series 1
1.2 objectives of time series analysis 6
1.3 some simple time series models 7
1.3.1 some zero-mean models 8
1.3.2 models with trend and seasonality 9
1.3.3 a general approach to time series modeling 14
1.4 stationary models and the autocorrelation function 15
1.4.1 the sample autocorrelation function 18
1.4.2 a model for the lake huron data 21
1.5 estimation and elimination of trend and seasonal components 23
1.5.1 estimation and elimination of trend in the absence of seasonality 24
1.5.2 estimation and elimination of both trend and seasonality 31
1.6 testing the estimated noise sequence 35
problems 40
2 stationary processes 45
2.1 basic properties 45
2.2 linear processes 51
2.3 introduction to arma processes 55
2.4 properties of the sample mean and autocorrelation function 57
.2.4.1 estimation of/z 58
2.4.2 estimation of y(.) and p(.) 59
2.5 forecasting stationary time series 63
2.5.1 the durbin-levinson algorithm 69
2.5.2 the innovations algorithm 71
2.5.3 prediction of a stationary process in terms of infinitely many past values 75
2.6 the wold decomposition 77
problems 78
3 arma models 83
3.1 arma(p, q) processes 83
3.2 the acf and pacf of an arma(p, q) process 88
3.2.1 calculation of the acvf 88
3.2.2 the autocorrelation function 94
3.2.3 the partial autocorrelation function 94
3.2.4 examples 96
3.3 forecasting arma processes 100
problems 108
4 spectral analysis 111
4.1 spectral densities 112
4.2 the periodogram 121
4.3 time-invariant linear filters 127
4.4 the spectral density of an arma process 132
problems 134
5 modeling and forecasting with arma processes 137
5.1 preliminary estimation 138
5.1.1 yule-walker estimation 139
5.1.2 burg's algorithm 147
5.1.3 the innovations algorithm 150
5.1.4 the hannan-rissanen algorithm 156
5.2 maximum likelihood estimation 158
5.3 diagnostic checking 164
5.3.1 the graph of {rt,t=1, ...,n} 165
5.3.2 the sample acf of the residuals 166
5.3.3 tests for randomness of the residuals 166
5.4 forecasting167
5.5 order selection 169
5.5.1 the fpe criterion 170
5.5.2 the aicc criterion 171
problems 174
6 nonstationary and seasonal time series models 179
6.1 arima models for nonstationary time series 180
6.2 identification techniques 187
6.3 unit roots in time series models 193
6.3.1 unit roots in autoregressions 194
6.3.2 unit roots in moving averages 196
6.4 forecasting arima models 198
6.4.1 the forecast function 200
6.5 seasonal arima models 203
6.5.1 forecasting sarima processes 208
6.6 regression with arma errors 210
6.6.1 ols and gls estimation 210
6.6.2 ml estimation 213
problems 219
7 multivariate time series 223
7.1 examples 224
7.2 second-order properties of multivariate time series 229
7.3 estimation of the mean and covariance function 234
7.3.1 estimation of μ 234
7.3.2 estimation of г(h) 235
7.3.3 testing for independence of two stationary time series 237
7.3.4 bartlett's formula 238
7.4 multivariate arma processes 241
7.4.1 the covariance matrix function of a causal arma process 244
7.5 best linear predictors of second-order random vectors .. 244
7.6 modeling and forecasting with multivariate ar processes 246
7.6.1 estimation for autoregressive processes using whittle's algorithm 247
7.6.2 forecasting multivariate autoregressive processes 250
7.7 cointegration 254
problems 256
8 state-space models 259
8.1 state-space representations 260
8.2 the basic structural model 263
8.3 state-space representation of arima models 267
8.4 the kalman recursions 271
8.5 estimation for state-space models 277
8.6 state-space models with~issing observations 283
8.7 the em algorithm 289
8.8 generalized state-space models 292
8.8.1 parameter driven models 292
8.8.2 observation-driven models 299
problems 311
9 forecasting techniques 317
9.1 the arar algorithm 318
9.1.1 memory shortening 318
9.1.2 fitting a subset autoregression 319
9.1.3 forecasting 320
9.1.4 application of the arar algorithm 321
9.2 the holt-winters algorithm 322
9.2.1 the algorithm 322
9.2.2 holt-winters and arima forecasting 324
9.3 the holt-winters seasonal algorithm 326
9.3.1 the algorithm 326
9.3.2 holt-winters seasonal and arima forecasting 328
9.4 choosing a forecasting algorithm 328
problems 330
10 further topics 331
10.1 transfer function models 331
10.1.1 prediction based on a transfer function model 337
10.2 intervention analysis 340
10.3 nonlinear models 343
10.3.1 deviations from linearity 344
10.3.2 chaotic deterministic sequences 345
10.3.3 distinguishing between white noise and iid sequences 347
10.3.4 three useful classes of nonlinear models 348
10.3.5 modeling volatility 349
10.4 continuous-time models 357
10.5 long-memory models 361
problems 365
a random variables and probability distributions 369
a 1 distribution functions and expectation 369
a.2 random vectors 374
a.3 the multivariate normal distribution 377
problems 381
b statistical complements 383
b.1 least squares estimation 383
b.1.1 the gauss-markov theorem 385
b.1.2 generalized least squares 386
b.2 maximum likelihood estimation 386
b.2.1 properties of maximum likelihood estimators 387
b.3 confidence intervals 388
b.3.1 large-sample confidence regions 388
b.4 hypothesis testing 389
b.4.1 error probabilities 390
b.4.2 large-sample tests based on confidence regions 390
c mean square convergence 393
c.1 the cauchy criterion 393
d an itsm tutorial 395
d.1 getting started 396
d 1.1 running itsm 396
d.2 preparing your data for modeling 396
d.2.1 entering data 397
d.2.2 information 397
d.2.3 filing data 397
d.2.4 plotting data 398
d.2.5 transforming data 398
d.3 finding a model for your data 403
d.3.1 autofit 403
d.3.2 the sample acf and pacf 403
d.3.3 entering a model 404
d.3.4 preliminary estimation 406
d.3.5 the aicc statistic 408
d.3.6 changing your model 408
d.3.7 maximum likelihood estimation 409
d.3.8 optimization results 410
d.4 testing your model 411
d.4.1 plotting the residuals 412
d.4.2 acf/pacf of the residuals 412
d.4.3 testing for randomness of the residuals 414
d.5 prediction 415
d.5.1 forecast criteria 415
d.5.2 forecast results 415
d.6 model properties 416
d.6.1 arma models 417
d.6.2 model ace pacf 418
d.6.3 model representations 419
d.6.4 generating realizations of a random series 420
d.6.5 spectral properties 421
d.7 multivariate time series 421
references 423
index ... 429
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