Absolutely summable, , 12 , 302 , 304 , 305
Adjoint matrix, 12 , 29
Aggregation
Air pollution, 120 –137
Amplitude spectrum, 307
ARCH models, 203
Autoregressive representation of vector processes, –9
Bandwidth
Bartlett’s approximation, 241 , 242
Cauchy–Schwarz inequality,
Cholesky decomposition, 315 , 317 , 318 , 333 , 334 , 337
Clustering
hierarchical clustering, 460 , 463
model‐based cluster approach, 443
nonhierarchical clustering, 460 , 463
partitioning around mediods (PAM), 463 , 465
total within sum of square (TWSS), 461 , 463
Cointegration, 25
Cointegration in vector time series
error‐correction representation, 25
the trace test, 25
Conditional covariance matrix, 204 , 207
Correlation matrix function
partial, –7, 24
sample, 24 , 32
Co‐spectrum, 306 , 311
Covariance matrix function, 12 , 23 , 304 , 320
Cross‐spectrum, 306 , 312
Data
body weight of rats (WW7a), 238 , 509
daily log‐returns of DJIA 30 stocks from 7/1/2006 to 6/30/2010 (WW6c), 225 , 509
daily log‐returns of SP500, BAC, JPM, and C from 1/4/1994 to 12/30/2005 (WW6b), 222 , 509
daily mortality and air pollutants, ozone and carbon monoxide, from five California cities from 4/13/1999 to 9/14/1999 (WW3b), 120 , 509
daily stock returns from the first set of ten stocks from 8/2/2016 to 12/30/2016 (WW4a), 145 , 509
daily stock returns from the second set of ten stocks from 8/2/2016 to 12/30/2016 (WW5), 166 , 173 , 509
monthly total of vehicular theft data of Philadelphia from 1/2006 to 12/2014 (WW8a), 274 , 509
monthly unemployment rates for ten states in U. S. from 1/1976 to 8/2010 (WW2c), 58 , 509
new classified U.S. monthly retail sales from 6/2009 to 11/2016 (WW2b), 36 , 320 , 325 , 327 , 509
the oral condition of neck cancer patients in 1999 from Mid‐Michigan Medical Center (WW7b), 255 , 509
U.S. macroeconomic indicators from quarter 1 of 1959 to quarter 4 of 2007 (WW10), 452 , 510
U.S. monthly consumer price index (CPI) from five different sectors, energy, apparel, commodities, housing, and gas from 1/1986 to 12/2014 in Greater New York City (WW4b), 152 , 509
U.S. monthly retail sales from 6/2009 to 11/2016 (WW2a), 32 , 509
U.S. non‐seasonally adjusted labor force count for the 48 contiguous states and Washington D.C. from 1976 to 2014 (WW8b), 279 , 509
U.S. weekly interest over time on six exercise items from 4/15/2012 to 4/7/2017 (WW6a), 215 , 509
U.S. yearly retail sales and GDP, JIA, and CPI from 1992 to 2015 (WW3a), 109 ‐120, 509
U.S. yearly sexually transmitted disease morbidity rates from 1984 to 2014 (WW8c), 183 , 281 , 457 , 510
weekly log returns of DJIA, NASDQ, and S&P500 from 4/1990 to 12/2011 (WW9), 338 , 510
Dendrogram, 460 , 463 –465
Diagnostic checking
of vector time series models, 24 –25
Differencing, 21 , 36 , 47 , 145 , 152 , 265 , 274 , 329 , 452
Dimension reduction
contemporal aggregation, 437
factor analysis, 443 –444
hierarchical vector autoregressive (HVAR) method, 440 –441
in high dimensional multivariate time series, 437 –505
lag‐weighted Lasso method, 440
lasso method, 439 –440
model‐based cluster method, 443
multivariate time series, 437
principal component analysis, 209
regularization methods, 439 –441
space‐time AR (STAR) model, 442 –443
Discrete Prolate Spheroidal Sequences (DPSS), 313 , 314
Distance
Canberra distance, 460
Euclidian distance, 460
Kullback–Leibler (KL) distance, 460
Minkowski distance, 460
Eigenvalue, 13 , 17 , 140 , 141 , 143 –145, 147 , 149 , 150 , 153 , 154 , 165 , 168 , 205 , 207 , 210 , 213 –215, 314
Eigenvector, 140 , 141 , 143 –145, 147 , 150 , 153 , 154 , 165 , 168 , 210 , 213 , 214
Empirical example, , , 11 , 24 , 32 , 109 –129, 145 , 163 , 166 –169, 173 –175, 177 , 179 –181, 183 –193, 215 –229, 237 , 241 –243, 250 –252, 255 –257, 273 –289, 320 –329, 437 , 438 , 444 , 452 –459, 466 , 467
EViews, 25 , 169 , 176
Extended cross‐correlation matrix function, 24 , 39 , 45
Factor analysis of multivariate time series
Bartlett‐corrected test statistic, 172
common factors, 163
estimation, 165 –175
ith common factor, 171 , 173 , 177
ith communality, 165 , 169
loading matrix, 164
maximum likelihood method, 169 –173
orthogonal factor model, 163 –165
principal component method, 165 –166
specific factors, 163
Factor models with observable factors
dynamic factor models, 183
forecast equation, 183
Factor rotation
oblique rotation, 176 –177
orthogonal rotation, 176
varimax, 177
Factor scores
forecast, 181 , 183
with observable factors, 181 –183
Forecast
limits of, 124 , 129
step‐ahead, 25 , 43 , 44 , 46 , 47 , 183 , 189 , 284 , 444 , 445 , 475 , 486
Fourier frequencies, 302 , 303 , 309 , 312 , 320 , 336 , 337
Gain function, 307 , 311 , 312
GARCH models, , 203 –234
Generalized least squares (GLS) estimate, 106 , 108 –109, 114
Generalized space‐time autoregressive integrated moving average (GSTARIMA) models, 272
GO‐GARCH models
estimation, 210 –213
two step estimation, 210 –211
weighted scatter estimation, 211 –213
Granger causality, 18
GSTARIMA models, 272
Hermitian matrix, 306 , 318
Heteroscedasticity, 203
Hierarchical clustering
average linkage, 461
complete linkage, 461
median linkage, 461
single linkage, 461
High dimensional problem
contemporal aggregation, 437 , 466
factor analysis, 443 –444
hierarchical vector autoregressive (HVAR) method, 440 –441
lag‐weighted lasso method, 440
lasso method, 439 –440
model‐based cluster method, 443
multivariate time series, 437 , 438
regularization methods, 439 –441
space‐time AR (STAR) model, 442 –443
Identification of vector time series models, 21 –22
Information criterion
Invertible, , 11 –14, 16 , 19 , 20 , 27 , 209 , 210 , 225
Least squares estimation, 114 , 116 , 122
Log‐likelihood function, 24
Longitudinal data, 238 , 256
MATLAB, , 25 , 70 , 114 , 169 , 176 , 324 , 349 , 362
MINITAB, 169 , 176
Model‐based cluster approach, 443
Moving average representation of vector processes, , 11 , 15 , 17 , 23
Multivariate analysis of variance (MANOVA), 239 –243
Multivariate GARCH models
BEKK models, 207 –208
constant conditional correlation (CCC) models, 206
DVEC models, 204 –206
dynamic conditional correlation (DCC) models, 218
estimation, 210 –213
factor models, 208 –209
GO‐GARCH models, 209
O‐GARCH model, 209
representations, 204 –209
two step estimation method, 210 –211
VEC models, 204 –206
weighted scatter estimation (WSE) method, 211 –213
Multivariate multiple time series regression model
estimation, 108 –114
forecast, 115 , 116
generalized least squares estimator, 106
GLS procedure, 108 –109
VARX models, 114 , 115
Multivariate normal distribution, , 106 , 108 , 113 , 169 , 182
Multivariate spectral analysis of time series
spectral representations, 304 –309
Multivariate time series analysis
Multivariate time series outlier
multivariate additive outlier (MAO), 28
multivariate innovational outlier (MIO), 28
multivariate level shift (MLS), 28
multivariate temporary change (MTC), 28
Nonhierarchical clustering
Nonstationary vector autoregressive moving average processes, 21
O‐GARCH model, 209 –213, 225
Ordinary least squares (OLS), 109 , 111 –113
Orthogonal factor model, 163 –165
Outliers detection
likelihood ratio test (LRT), 31
Partial autoregression matrix,
Partial correlation matrix function
Partial lag autocorrelation matrix, 36
PCA (principal component analysis), 139 –142, 145 , 147 –157, 168 –170, 176 , 183 –188, 209 –212
Phase spectrum, 306 , 307 , 311
Positive definite matrix, , 12 , 265 , 314 , 442
Positive semidefinite matrix, , , 306 , 315
Principal component analysis (PCA) of multivariate time series
based on sample correlation matrix, 154 –157
based on sample covariance matrix, 153 –154
implications, 141 –142
population principal component analysis, 139 –141
sample principal components, 142 –145
Projection pursuit, 27 , 29 –32, 58 , 59
Projects, , 29 , 100 –101, 137 , 160 –161, 200 –201, 234 , 258 , 298 , 434 –435, 505
References, , 101 , 137 , 161 , 201 , 234 , 258 , 298 , 435 , 506
Regularization methods
hierarchical vector autoregressive (HVAR) method, 440 –441
lag‐weighted lasso method, 440
lasso method, 439 –440
Repeated measurements
AR(1) structure, 249
ARMA(1 ,1) structure, 250
fixed effects model, 243 –247
generalization, 254 –255
heterogeneous AR(1) structure, 250
heterogeneous Toeplitz structure, 249
identical and independent structure, 247 –248
independent but non‐identical structure, 248
Interaction, 238 , 243 , 245 , 246 , 250 , 252 , 256
multivariate analysis of variance (MANOVA), 239 –243
nested random effects model, 253 –254
random effects and mixed effects models, 252 –253
repeated measure analysis of variance, 239 –243
some common variance‐covariance structures, 247 –250
structure of common symmetry, 248
structure of heterogeneous common symmetry, 248 –249
Toeplitz structure, 249
Treatment group, 237 , 241 –244, 250
Treatment sum of squares, 239 , 240
Representation of vector processes
autoregressive representation, –8
moving average representation,
Sample moments of a vector time series
sample correlation matrix function, 23 –24
sample covariance matrix, 22 –23
sample mean vector, 22
SAS, , 25 , 43 , 44 , 47 , 58 , 113 , 115 , 116 , 118 , 122 , 129 , 169 , 176 , 218 , 242 , 243 , 251 , 252 , 255 , 256
Scree plot, 149 , 169
Seasonal GSTARIMA model, 272
Seasonal STARIMA model, 262 –271
Seasonal vector time series model
nonstationary seasonal vector time series, 26
Similarity measures
average linkage, 461
complete linkage, 461
Kullback–Leibler (KL) distance, 460
median linkage, 461
single linkage, 461
Slepian sequences or tapers, 313
Smoothed spectrum, 304 , 309 –313
Software code
MATLAB code, 70 –100
R code, 65 –69, 129 –130, 157 –160, 194 –200, 289 –298, 490 –505
SAS code, 67 , 69 , 130 –137, 229 –233, 257 , 258
Space‐time series models
generalized space‐time autoregressive integrated moving average (GSTARIMA) models, 272
space‐time autoregressive integrated moving average (STARIMA) models, 262
STARMA models, 266
Spatial series, 261 , 262
Spatial weighting matrix, 262 –264
Spectral density matrix estimation
Bayesian method, 316 –317, 325 –326
multitaper smoothing, 313 –314
penalized Whittle likelihood, 317 –318
sample spectrum, 320 –325
smoothed spectrum matrix, 309 –313
smoothing spline, 315 –316
VARMA spectral estimation, 318 –320
Spectral density matrix function, 305 , 306
Spectral distribution function, 302 , 305
Spectral representations
of covariance matrix function, 304 , 305
of multivariate time series processes, 304 –309
Spectral window
Bartlett window, 304
Blackman‐Tukey window, 304
Daniell, 320
Parzen window, 304
Rectangular window, 304
Spectrum analysis of nonstationary vector time series, 329 –337
Spectrum matrix
cospectrum, 306 , 311
cross‐amplitude spectrum, 306 , 311
gain function, 307 , 311
phase spectrum, 306 , 307 , 311
quadrature spectrum, 306 , 311
squared coherency function, 307 , 311
Spectrum representation of a nonstationary multivariate process
Bayesian methods, 336 –337
piecewise vector autoregressive model, 334 –336
smoothing spline ANOVA model, 333 –334
time‐varying autoregressive model, 332 –333
SPSS, 25 , 169 , 176
Spurious regression, 109
Squared coherency, 307 , 311
Squared Mahalanobis distance, 212
Square summable, 15 , 319
ST‐ACF and ST‐PACF, 267 –271, 273 , 279 , 281 , 284
STARIMA models
GSTARIMA models, 272
iterative model building of, 273
seasonal GSTARIMA model, 272
seasonal STARIMA model, 265 –266
STARMA models, 266 –267, 273 , 279 , 442 , 443 , 445
STAR model, 266 , 268 , 272 , 438 , 442 –443, 445 , 457 , 484 –486
STMA model
Systematic sampling, 466 , 467
Temporal aggregation, 437 , 466 , 467
Temporal disaggregation, 467
Time series regression, , 105 –137, 181
Toeplitz form, 255
Variance–covariance matrix, , 108 , 109 , 112 , 113 , 116 , 124 , 127 , 142 –144, 147 , 153 , 239 , 243 , 247 , 253 , 254 , 271 , 306 , 310 , 327 , 438
VARMA model, 25 , 27 –29, 216 –218, 225 , 267 , 273 , 318 , 437 , 438 , 443
VARMAX procedure, 116 , 118 , 122
VAR model, 25 , 26 , 114 –115, 327 , 329 , 333 –335, 439 –441, 443 –446
VARX model, 105 , 114 –121, 124 , 127
Vector autoregressive moving average (VARMA) processes
nonstationary, 21
stationary, 19 , 21
VARMA(1,1) model, 19
VARMA(p ,q ) model, 18 , 24 , 26 , 319 , 332 , 445
Vector autoregressive (VAR) processes
VAR(1) model, 17
VAR(p ) model, 14 , 15 , 18 , 114 , 261 , 319 , 438 , 439 , 444 , 445 , 488 , 489
Vector autoregressive representation,
Vector moving average (VMA) processes
Vector moving average representation, , 17 , 23
Vector time series model building
diagnostic checking, 24 –25
extended cross‐correlation matrices, 24
forecast, 24 –25
identification, 21 –22
parameter estimation, 24 –25
sample correlation matrix function, 23 –24
sample moments of a vector time series, 22 –24
Vector time series models
vector autoregressive moving (VARMA) processes, 18 –20
vector autoregressive (VAR) processes, 14 –18
vector autoregressive representation of,
vector moving average (VMA) processes, 11 –14
vector moving average representation of, 17 , 23
Vector time series regression models, 105 , 114 –120
Vector white noise (VWN) process, , , 11 , 14 , 19 , 114 , 115 , 118 , 265 , 318 , 442
Volatility, 203 –205, 208 , 210 , 223 , 225
Volatility model, 204 , 205 , 210
Weighted scatter estimation (WSE) method
linkage matrix, 211 , 212
weighting function, 212
Weighting function, 211 , 212 , 214 –215, 223 , 282 , 303 , 304
Wilk’s lambda, 240 , 242
Wishart distribution, 310 , 311
Yule–Walker matrix equations, 16
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