Index
A
accumulated effects 86–87
ACE (autocorrelations) 5, 37–38, 100–101
ACF (residual autocorrelation function) 71
ADF (Augmented Dickey-Fuller) test 31, 32–33
Akaike information criteria (AIC) 41, 61, 66–67
αβT matrix 118–119
α parameters
estimated under restrictions 123–124, 144–145
restrictions on 127–128
TEST statement for hypotheses on 126
testing hypotheses by VARMAX procedure 124–125
ARCH (Autoregressive Conditional Heteroscedasticity) effects 49, 101, 176
ARCH-effect testing 73
ARIMA models
See Autoregressive Integrated Moving Average (ARIMA) models
ARIMA procedure
Dickey-Fuller test and 35
estimating univariate ARIMA models 42
ARMA models
See Autoregressive Moving Average (ARMA) models
AR(p) models 38–39, 47–48, 49–50, 51–53, 74, 75, 104, 157
Augmented Dickey-Fuller (ADF) test 31, 32–33
autocorrelated errors, regression analysis with 13–18
autocorrelations (ACE) 5, 37–38, 100–101
AUTOREG procedure
Cochrane-Orcutt Estimation using 15–16
correction of standard errors with 13–14
Dickey-Fuller test and 35
GARCH models and 149, 158
inclusion of lagged dependent variable in regression 27
reverted regression 23–24
simultaneous estimation using 16–18
Autoregressive Conditional Heteroscedasticity (ARCH) effects 49, 101, 176
Autoregressive Integrated Moving Average (ARIMA) models
about 33, 37, 40, 43
infinite-order representations 40–41
multiplicative seasonal 41
autoregressive models 38–39
Autoregressive Moving Average (ARMA) models
about 33, 37, 40
infinite-order representations 40–41
autoregressive parameter matrices, prior covariance of 103–105
autoregressive terms, in models 120
B
Bayesian Vector Autoregressive (BVAR(p)) models
about 103
application of 108
for egg market 108–110
prior covariance of autoregressive parameter matrices 103–105
VARMAX procedure 105–106
BEKK parameterization 167–168, 172
β parameters
estimation with restrictions on 144–145
RESTRICT statement for 126–127
restrictions on 127–128
testing hypotheses on 120–124
testing hypotheses on using VARMAX procedure 122–124
tests for two restrictions on 123
β values, estimates for 135
bivariate case, tests for cointegration relation in 132
BOUND statement 77, 163
Box-Jenkins procedure 33, 35, 37–38
Brocklebank, J.C. 41, 42
BVAR(p) models
See Bayesian Vector Autoregressive (BVAR(p)) models
C
CAUSAL statement 95, 109
causality tests
for Danish egg market 91–101
estimation of final causality model 99–100
of production series 96–97
that use extended information sets 97–98
of total market series 94–95
CCC (Constant Conditional Correlation) parameterization 165–166, 169–170, 172–173, 176–177
Cochrane-Orcutt Estimation 10–12, 15–16
COINTEG statement 125, 135–136
ECTREND option 116, 122, 126
NORMALIZE=OHIO option 117–118, 122
cointegration
about 131–132
rank 4 model for five series specified with restrictions 141–145
Stock-Watson test for common trends for five series 139–141
using RESTRICT statement to determine form of models 138–139
cointegration rank 132
cointegration relations 131–132
cointegration tests
in five-dimensional series 133–134
using VARMAX procedure for two price series 132–133
COINTEST=(JOHANSEN) option, MODEL statement 133
conditional variance series 157–158
Constant Conditional Correlation (CCC) parameterization 165–166, 169–170, 172–173, 176–177
CORRCONSTANT=EXPECT option, GARCH statement 169
correlation matrix
of error terms 78
at lag 0 59–60
COWEST=NEWEYWEST option, MODEL statement 14
cross-correlation significance 70
D
DATALABEL=YEAR option 115
DCC (Dynamic Conditional Correlation) parameterization 166–167, 170–171, 178, 180–183
DFTEST option 46
diagonal elements, prior distribution for 104
Dickey, D.A. 41, 42
Dickey-Fuller tests
about 133–134
applying VARMAX procedure to wage series 46
for differenced series 66
simple applications of 32
for stationarity 63
for unit roots 30–32
in VARMAX procedure 46
vector error correction models and 116–117
DIF option 64, 92
differenced series
applying VARMAX procedure to 46–47
Dickey-Fuller tests for 66
regression models for 19–28
differencing
seasonal 35
time series 29–35
distribution, of residuals in VARMA(2,0) model 71–72
Durbin-Watson test 8–10, 49, 73, 116
DWPROB option, MODEL statement 9
Dynamic Conditional Correlation (DCC) parameterization 166–167, 170–171, 178, 180–183
E
ECM option, MODEL statement 116, 122
ECTREND option, COINTEG statement 116, 122, 126
effects
accumulated 86–87
of orthogonal shocks 88–89
EGARCH model 162–164
Engle, R.F. 96
error terms
correlation matrix of 78
lag 0 correlation of 83–84
estimated models, properties of 119
estimation
of error correction models with VARMAX procedure 116
of model parameters by RESTRICT statement 143–144
with restrictions on α and β parameters 144–145
for β values 135
estimation algorithm 178–180
F
fit
of final model 100–101
of fourth-order autoregressive model 67–70
fitted model 78–79, 151–153
fitted second-order autoregressive model, roots of 81–82
five series
rank 4 model for 141–145
Stock-Watson test for common trends for 139–141
five-dimensional series, cointegration tests in 133–134
forecasts 82–83
FORM option 172
FORM=CCC option 151, 158
fourth-order autoregressive model, fit of 67–70
G
Gammelgaard, S. 43
GARCH models
about 30
forms of 158–164
for univariate financial time series 149–155
GARCH statement
CORRCONSTANT=EXPECT option 169
OUTHT=CONDITIONAL option 151–153, 182
SUBFORM option 158, 172–173
Gaussian residuals, test for hypothesis of 49
Granger causality tests 63, 95–96
“gray zone” 9
H
HAC (heteroscedasticity and autocorrelation consistent) 14
Hendry, D.F. 96
heteroscedasticity and autocorrelation consistent (HAC) 14
hypotheses
null 33
TEST statement for on α parameters 126
testing on α parameters by VARMAX procedure 124–125
testing on β parameters 120–124
testing on β parameters using VARMAX procedure 122–124
I
IAC (inverse autocorrelations) 71
IACF (inverse autocorrelations) 100–101
ID statement 48
IGARCH model, using VARMAX procedure to fit 153–155
impulse response, plots of 85–86
independent variables, two lags of 25–26
infinite-order representations 59, 84–89
information criteria 41–42
INITIAL statement 127, 171
INTERVAL option 48
inverse autocorrelations (IAC) 71
inverse autocorrelations (IACF) 100–101
J
Jarque-Bera test 63, 73
Johansen, S. 112, 132
JOHANSEN option 134
Johansen rank tests 63
Juselieus, K. 112, 132
K
Koyck lag 28
KPSS unit root tests
about 33
application of 34
kth-order autocorrelation 38
Kwiatkowski, Phillips, Schmidt, and Shin (KPSS) 33
L
lag 0, correlation matrix at 59–60
lag correlation, of error terms 83–84
LAG function 11
lagged dependent variable
inclusion of in regression 27
interpreting models with 28
lagged independent variable
inclusion of 22, 24–25
two lags 25–26
LAGMAX=25 option, MODEL statement 93
Litterman, R.B. 103
Ljung-Box test 38, 49
long-run relation 113–114
Lütkepohl, H. 4, 57
M
MA(q) model 39–40, 50, 51, 53–54, 54–56, 74, 77
matrix formulation, of vector error correction model 113
METHOD=ML option 16, 47, 67
Milhøj, A. 4
MINIC option, MODEL statement 66
minus sign (−) 105
model fit 78–79, 94
MODEL statement 89, 92, 122
COINTEST=(JOHANSEN) option 133
COWEST=NEWEYWEST option 14
DWPROB option 9
ECM option 116, 122
LAGMAX=25 option 93
MINIC option 66
NOINT option 151
NSEASON=4 option 53
NSEASON=12 option 93, 108
PRIOR option 105, 106–108
SW option 140
models
See also specific types
autoregressive terms in 120
interpreting with lagged dependent variables 28
moving average 39–40
multiplicative seasonal ARIMA 41
for multivariate time series 57–61
with rank 2 135–137
selecting 66–67
for univariate time series 37–42
using RESTRICT statement to determine form of 138–139
VARMAX 58–59, 60–61
Morgan, D.P. 4
moving average models 39–40
multiplicative seasonal ARIMA models 41
multivariate GARCH models
about 165
BEKK parameterization 167–168, 172
bivariate example using two quotations for Danish stocks 168–173
CCC (Constant Conditional Correlation) parameterization 165–166, 169–170, 172–173, 176–177
DCC (Dynamic Conditional Correlation) parameterization 166–167, 170–171, 178, 180–183
multivariate series, modeling with VARMAX procedure 63–79
multivariate time series
about 57–58
modeling with VARMAX procedure 64–67
models for 57–61
multivariate VARMA-GARCH models
about 175–176
estimation algorithm 178–180
for residuals 176–177, 178, 180–183
wage-price time series 176
N
Newey-West method, adjusting standard deviations with 14–15
NLAG=1 option 16
NLOPTIONS statement, PALL option 178–179
NOINT option, MODEL statement 151
NORMALIZE option 133–134
NORMALIZE=OHIO option, COINTEG statement 117–118, 122, 126
NSEASON=4 option, MODEL statement 53
NSEASON=12 option, MODEL statement 93, 108
null hypothesis 33
O
ODS (SAS Output Delivery System) 1–2
off-diagonal elements, prior distribution for 104–105
options
See specific options
ordinary least squares (OLS) 8
ordinary regression models 1–2
orthogonal shocks, effects of 88–89
OUTHT=CONDITIONAL option, GARCH statement 151–153, 182
outliers, identification of 72–74
output
about 81
forecasts 82–83
infinite-order representations 84–89
lag 0 correlation of error terms 83–84
roots of fitted second-order autoregressive model 81–82
P
PACF (partial autocorrelations) 100–101
PALL option, NLOPTIONS statement 178–179
parameterized models, for time series 4–5
parameters
See also specific types
estimated for vector error correction models 117–120
estimating 67–68
estimation of by RESTRICT statement 143–144
in prior distribution 106–108
restriction of insignificant model 68–70
partial autocorrelations (PACF) 100–101
periods (.) 105
PGARCH model 161–162
plots, of impulse response 85–86
PLOTS=ALL option 46, 64, 70
plus sign (+) 105
portmanteau tests 61, 70–71, 94
price series
applying VARMAX procedure to 50–51
cointegration test for two using VARMAX procedure 132–133
PRINTALL option 46, 47, 64, 70, 156
PRINT=(DIAGNOSE) option 156
prior covariance, of autoregressive parameter matrices 103–105
prior distribution
for diagonal elements 104
for off-diagonal elements 104–105
parameters in 106–108
PRIOR option, MODEL statement 105, 106–108
PROC statement 13
procedures
See specific procedures
production series, causality tests of 96–97
properties
of estimated model 119
of final model 128–129
of fitted model 79
p-test 31
p-value 9
Q
QGARCH model 158–159
R
rank 2 model 135–137
rank 4 model 141–145
REG procedure 1–2, 8–10, 19–20, 22, 23–24, 27, 30–32, 116
regression, inclusion of lagged dependent variable in 27
regression analysis
with autocorrelated errors 13–18
reverted 23–24
for time series data 7–12
regression models
for differenced series 19–28
ordinary 1–2
in time series analysis 2–3
residual autocorrelation, in VARMA(2,0) model 70–71
residual autocorrelation function (ACF) 71
residuals
distribution of in VARMA(2,0) model 71–72
multivariate VARMA-GARCH models for 176–177, 178, 180–183
RESTRICT statement 68, 76, 126–127, 138–139, 143–144, 154, 164, 172, 176–177, 180
restrictions
alternative form of 142
estimated α parameters under 123–124
estimation with on α and β parameters 144–145
of insignificant model parameters 68–70
rank 4 model for five series specified with 141–145
tests for two on β parameters 123
on α and β parameters 127–128
reverted regression 23–24
Richard, J.F. 96
roots, of fitted second-order autoregressive model 81–82
S
SAS Output Delivery System (ODS) 1–2
Schwarz Bayesian criterion (SBC) 41, 61
seasonal differencing 35
SGPLOT procedure 19–20, 44, 92, 114, 115, 153
shrinkage, toward zero 107
simple regression 115–116
simultaneous estimation, using AUTOREG procedure 16–18
standard deviations, adjusting with Newey-West method 14–15
standard errors, correction of with AUTOREG procedure 13–14
STANDARD procedure 75
standardized series, analysis of 75–77
statements
See specific statements
stationarity 5, 29–30
STATIONARITY=(ADF) option 32
Stock-Watson test, for common trends for five series 139–141
SUBFORM option, GARCH statement 158, 172–173
SW option, MODEL statement 140
T
TEST statement 10, 68, 76, 99, 126, 153
tests
See also specific tests
for cointegration relation in bivariate case 132
for differencing time series 29–35
for two restrictions on β parameters 123
TGARCH model 159–161, 172–173
time series
about 3–4
differencing 29–35
model features 4
parameterized models for 4–5
regression analysis for data 7–12
regression models in analysis of 2–3
wage-price 43–45
total market series, causality tests of 94–95
U
unit roots
about 30
Dickey-Fuller tests for 30–32
KPSS unit root tests 33
univariate ARIMA models, estimating 42
univariate financial time series, GARCH models for 149–155
univariate GARCH models
about 147–149
wage series 155–158
univariate series, modeling with VARMAX procedure 43–56
univariate time series, models for 37–42
V
VARMA model
See Vector Autoregressive Moving Average (VARMA) model
VARMAX models
about 58–59, 60
building 60–61
VARMAX procedure
See also Bayesian Vector Autoregressive (BVAR(p)) models; causality tests; output; vector error correction models
about 4, 57, 63
AICc and 42
applying to differenced wage series 46–47
applying to number of cows series 51–53
applying to price series 50–51
applying to series of milk production 53–54
applying to wage series 46
Bayesian Vector Autoregressive (BVAR(p)) models and 105–106
BEKK parameterization and 167, 172
CCC models and 166
cointegration test for two price series using 132–133
cointegration tests in five-dimensional series 133–134
DCC models and 167
Dickey-Fuller tests and 35, 46, 66
estimates for β values 135
estimating AR(2) model 47–48
estimating parameters 68
estimating univariate ARIMA models 42
estimating vector error correction models with 116
GARCH models and 149, 158–161
Granger causality tests in 95–96
modeling multivariate series with 63–79
modeling multivariate time series with 64–67
modeling univariate series with 43–56
multiplicative seasonal ARIMA models and 41
Stock-Watson test for common trends 140
testing hypotheses on α parameters by 124–125
testing hypotheses on β parameters using 122–124
using to fit AR(2)-GARCH(1,1) models 157
using to fit GARCH(1,1) model 150–151
using to fit IGARCH model 153–155
using VARMA model for milk production and number of cows 74–79
wage series 155–158
Vector Autoregressive Moving Average (VARMA) model
about 58–59
for Danish egg market 92–94
Danish egg market and 91
distribution of residuals in 71–72
residual autocorrelation in 70–71
using for milk production and number of cows 74–79
vector error correction models
about 111–113
Dickey-Fuller tests and 116–117
estimated parameters 117–120
estimating with VARMAX procedure 116
example 114–117
matrix formulation of 113
properties of final model 128–129
RESTRICT statement for β parameters 126–127
restrictions on α and β parameters 127–128
TEST statement for hypotheses on α parameters 126
testing hypotheses on α parameters by VARMAX procedure 124–125
testing hypotheses on β parameters 120–122
testing hypotheses on β parameters using VARMAX procedure 122–124
W
wage series 46, 155–158
wage-price time series 43–45, 176
Wiener processes 132
X
X12 procedure 4
XLAG=3 option 93
Z
zero, shrinkage toward 107