ECON547 Time Series LAB2
Jon Duan
2016-07-21
VAR advantage and shortcoming
Vector autoregressions[https://www.otexts.org/fpp/9/2]
- Information criteria are commonly used to select the number of lags to be included.
- They are not built on some economic theory that imposes a theoretical structure to the equations. Every variable is assumed to influence every other variable in the system, which makes direct interpretation of the estimated coefficients very difficult.
VARs are useful in several contexts:
- forecasting a collection of related variables where no explicit interpretation is required;
- testing whether one variable is useful in forecasting another (the basis of Granger causality tests);
- impulse response analysis, where the response of one variable to a sudden but temporary change in another variable is analysed;
- forecast error variance decomposition, where the proportion of the forecast variance of one variable is attributed to the effect of other variables.html_document
Load data in R
#setwd("H:/Dropbox/book/uvic_econometrics/547-2016/lab/lab2")
setwd("/media/snowdj/documents/Dropbox/book/uvic_econometrics/547-2016/lab/lab2")
suppressMessages(library(vars))
suppressMessages(library(fpp))
#suppressMessages(library(readxl))
suppressMessages(library(ggfortify))
mydata <- read.csv("econ547lab02data.csv", stringsAsFactors = F)
mydata$X_date_ <- as.Date(mydata$X_date_, "%Y-%m-%d")
mydata <- read.zoo(mydata, format = "%Y-%m-%d")
mydata <- ts(mydata, frequency = 4, start = c(1975,4))
str(mydata)## Time-Series [1:89, 1:4] from 1976 to 1998: 9.62 9.6 9.57 9.55 9.53 ...
## - attr(*, "dimnames")=List of 2
## ..$ : NULL
## ..$ : chr [1:4] "lhc" "lrc" "lrm" "lrw"
## - attr(*, "index")= Date[1:89], format: "1975-10-01" "1976-01-01" ...head(mydata)## lhc lrc lrm lrw
## [1,] 9.623192 11.13184 12.07941 13.68748
## [2,] 9.599204 11.13272 12.05642 13.69605
## [3,] 9.568289 11.11106 12.06075 13.69202
## [4,] 9.547288 11.10917 12.07369 13.72911
## [5,] 9.526518 11.12232 12.10099 13.75605
## [6,] 9.476097 11.14304 12.11345 13.76211autoplot(mydata)
Difference data
diffdata <- diff(mydata)
head(diffdata)## lhc lrc lrm lrw
## [1,] -0.02398806 0.0008779375 -0.022994152 0.008564588
## [2,] -0.03091565 -0.0216597887 0.004330447 -0.004025515
## [3,] -0.02100037 -0.0018849865 0.012942662 0.037091429
## [4,] -0.02076992 0.0131504506 0.027296340 0.026933004
## [5,] -0.05042167 0.0207106696 0.012466571 0.006062178
## [6,] -0.01434714 0.0236890449 0.009957350 0.028722831Graph for differenced data
autoplot(diffdata)
Model selection
VARselect(diffdata, lag.max=8, type="const")$selection## AIC(n) HQ(n) SC(n) FPE(n)
## 6 1 1 6Estimate R tutorial
#detach("package:MTS", unload=TRUE)
var <- VAR(diffdata, p=1, type="const")Summary
summary(var)##
## VAR Estimation Results:
## =========================
## Endogenous variables: lhc, lrc, lrm, lrw
## Deterministic variables: const
## Sample size: 87
## Log Likelihood: 1013.209
## Roots of the characteristic polynomial:
## 0.7658 0.466 0.3178 0.03153
## Call:
## VAR(y = diffdata, p = 1, type = "const")
##
##
## Estimation results for equation lhc:
## ====================================
## lhc = lhc.l1 + lrc.l1 + lrm.l1 + lrw.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## lhc.l1 0.743965 0.075655 9.834 1.59e-15 ***
## lrc.l1 0.039250 0.124593 0.315 0.754
## lrm.l1 0.083690 0.165449 0.506 0.614
## lrw.l1 -0.007215 0.053451 -0.135 0.893
## const 0.003635 0.002298 1.582 0.118
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.01402 on 82 degrees of freedom
## Multiple R-Squared: 0.5994, Adjusted R-squared: 0.5798
## F-statistic: 30.67 on 4 and 82 DF, p-value: 1.316e-15
##
##
## Estimation results for equation lrc:
## ====================================
## lrc = lhc.l1 + lrc.l1 + lrm.l1 + lrw.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## lhc.l1 -0.0664483 0.0544663 -1.220 0.226
## lrc.l1 -0.1355711 0.0896984 -1.511 0.135
## lrm.l1 0.6876535 0.1191119 5.773 1.34e-07 ***
## lrw.l1 0.0562783 0.0384808 1.463 0.147
## const 0.0005973 0.0016542 0.361 0.719
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.01009 on 82 degrees of freedom
## Multiple R-Squared: 0.3708, Adjusted R-squared: 0.3401
## F-statistic: 12.08 on 4 and 82 DF, p-value: 9.14e-08
##
##
## Estimation results for equation lrm:
## ====================================
## lrm = lhc.l1 + lrc.l1 + lrm.l1 + lrw.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## lhc.l1 0.092855 0.046852 1.982 0.050843 .
## lrc.l1 0.132531 0.077159 1.718 0.089637 .
## lrm.l1 0.269603 0.102461 2.631 0.010160 *
## lrw.l1 0.017162 0.033101 0.518 0.605528
## const 0.005477 0.001423 3.849 0.000234 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.008682 on 82 degrees of freedom
## Multiple R-Squared: 0.2399, Adjusted R-squared: 0.2028
## F-statistic: 6.47 on 4 and 82 DF, p-value: 0.0001414
##
##
## Estimation results for equation lrw:
## ====================================
## lrw = lhc.l1 + lrc.l1 + lrm.l1 + lrw.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## lhc.l1 0.042254 0.164771 0.256 0.7983
## lrc.l1 0.413140 0.271355 1.523 0.1317
## lrm.l1 0.655201 0.360337 1.818 0.0727 .
## lrw.l1 0.004466 0.116412 0.038 0.9695
## const 0.004814 0.005004 0.962 0.3389
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.03053 on 82 degrees of freedom
## Multiple R-Squared: 0.08648, Adjusted R-squared: 0.04192
## F-statistic: 1.941 on 4 and 82 DF, p-value: 0.1114
##
##
##
## Covariance matrix of residuals:
## lhc lrc lrm lrw
## lhc 1.966e-04 2.095e-05 1.044e-05 8.142e-05
## lrc 2.095e-05 1.019e-04 -1.924e-05 6.151e-06
## lrm 1.044e-05 -1.924e-05 7.538e-05 7.721e-05
## lrw 8.142e-05 6.151e-06 7.721e-05 9.323e-04
##
## Correlation matrix of residuals:
## lhc lrc lrm lrw
## lhc 1.00000 0.14809 0.08574 0.19019
## lrc 0.14809 1.00000 -0.21961 0.01996
## lrm 0.08574 -0.21961 1.00000 0.29124
## lrw 0.19019 0.01996 0.29124 1.00000Test R tutorial
The null hypothesis of no serial correlation in the residuals
serial.test(var, lags.pt=10, type="PT.asymptotic")##
## Portmanteau Test (asymptotic)
##
## data: Residuals of VAR object var
## Chi-squared = 160.44, df = 144, p-value = 0.1653Var summary
summary(var)##
## VAR Estimation Results:
## =========================
## Endogenous variables: lhc, lrc, lrm, lrw
## Deterministic variables: const
## Sample size: 87
## Log Likelihood: 1013.209
## Roots of the characteristic polynomial:
## 0.7658 0.466 0.3178 0.03153
## Call:
## VAR(y = diffdata, p = 1, type = "const")
##
##
## Estimation results for equation lhc:
## ====================================
## lhc = lhc.l1 + lrc.l1 + lrm.l1 + lrw.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## lhc.l1 0.743965 0.075655 9.834 1.59e-15 ***
## lrc.l1 0.039250 0.124593 0.315 0.754
## lrm.l1 0.083690 0.165449 0.506 0.614
## lrw.l1 -0.007215 0.053451 -0.135 0.893
## const 0.003635 0.002298 1.582 0.118
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.01402 on 82 degrees of freedom
## Multiple R-Squared: 0.5994, Adjusted R-squared: 0.5798
## F-statistic: 30.67 on 4 and 82 DF, p-value: 1.316e-15
##
##
## Estimation results for equation lrc:
## ====================================
## lrc = lhc.l1 + lrc.l1 + lrm.l1 + lrw.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## lhc.l1 -0.0664483 0.0544663 -1.220 0.226
## lrc.l1 -0.1355711 0.0896984 -1.511 0.135
## lrm.l1 0.6876535 0.1191119 5.773 1.34e-07 ***
## lrw.l1 0.0562783 0.0384808 1.463 0.147
## const 0.0005973 0.0016542 0.361 0.719
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.01009 on 82 degrees of freedom
## Multiple R-Squared: 0.3708, Adjusted R-squared: 0.3401
## F-statistic: 12.08 on 4 and 82 DF, p-value: 9.14e-08
##
##
## Estimation results for equation lrm:
## ====================================
## lrm = lhc.l1 + lrc.l1 + lrm.l1 + lrw.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## lhc.l1 0.092855 0.046852 1.982 0.050843 .
## lrc.l1 0.132531 0.077159 1.718 0.089637 .
## lrm.l1 0.269603 0.102461 2.631 0.010160 *
## lrw.l1 0.017162 0.033101 0.518 0.605528
## const 0.005477 0.001423 3.849 0.000234 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.008682 on 82 degrees of freedom
## Multiple R-Squared: 0.2399, Adjusted R-squared: 0.2028
## F-statistic: 6.47 on 4 and 82 DF, p-value: 0.0001414
##
##
## Estimation results for equation lrw:
## ====================================
## lrw = lhc.l1 + lrc.l1 + lrm.l1 + lrw.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## lhc.l1 0.042254 0.164771 0.256 0.7983
## lrc.l1 0.413140 0.271355 1.523 0.1317
## lrm.l1 0.655201 0.360337 1.818 0.0727 .
## lrw.l1 0.004466 0.116412 0.038 0.9695
## const 0.004814 0.005004 0.962 0.3389
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.03053 on 82 degrees of freedom
## Multiple R-Squared: 0.08648, Adjusted R-squared: 0.04192
## F-statistic: 1.941 on 4 and 82 DF, p-value: 0.1114
##
##
##
## Covariance matrix of residuals:
## lhc lrc lrm lrw
## lhc 1.966e-04 2.095e-05 1.044e-05 8.142e-05
## lrc 2.095e-05 1.019e-04 -1.924e-05 6.151e-06
## lrm 1.044e-05 -1.924e-05 7.538e-05 7.721e-05
## lrw 8.142e-05 6.151e-06 7.721e-05 9.323e-04
##
## Correlation matrix of residuals:
## lhc lrc lrm lrw
## lhc 1.00000 0.14809 0.08574 0.19019
## lrc 0.14809 1.00000 -0.21961 0.01996
## lrm 0.08574 -0.21961 1.00000 0.29124
## lrw 0.19019 0.01996 0.29124 1.00000Forecast in VAR
fcst <- forecast(var)
plot(fcst, xlab="Year")
Forecast
summary(fcst)##
## Forecast method: VAR(1)
##
## Model Information:
##
## VAR Estimation Results:
## =======================
##
## Estimated coefficients for equation lhc:
## ========================================
## Call:
## lhc = lhc.l1 + lrc.l1 + lrm.l1 + lrw.l1 + const
##
## lhc.l1 lrc.l1 lrm.l1 lrw.l1 const
## 0.743965451 0.039249548 0.083690164 -0.007214767 0.003634752
##
##
## Estimated coefficients for equation lrc:
## ========================================
## Call:
## lrc = lhc.l1 + lrc.l1 + lrm.l1 + lrw.l1 + const
##
## lhc.l1 lrc.l1 lrm.l1 lrw.l1 const
## -0.0664482853 -0.1355710865 0.6876535018 0.0562782847 0.0005972531
##
##
## Estimated coefficients for equation lrm:
## ========================================
## Call:
## lrm = lhc.l1 + lrc.l1 + lrm.l1 + lrw.l1 + const
##
## lhc.l1 lrc.l1 lrm.l1 lrw.l1 const
## 0.092855153 0.132531499 0.269603292 0.017162046 0.005477467
##
##
## Estimated coefficients for equation lrw:
## ========================================
## Call:
## lrw = lhc.l1 + lrc.l1 + lrm.l1 + lrw.l1 + const
##
## lhc.l1 lrc.l1 lrm.l1 lrw.l1 const
## 0.042254042 0.413140041 0.655200761 0.004466028 0.004813776
##
##
##
## Error measures:
## ME RMSE MAE MPE
## lhc Training set -7.017717e-20 0.01361081 0.010030318 43.01055
## lrc Training set 9.779823e-21 0.00979887 0.006555468 123.33580
## lrm Training set 2.260702e-20 0.00842904 0.006092447 5856.65380
## lrw Training set 3.290382e-19 0.02964349 0.022340626 58.50191
## MAPE MASE ACF1
## lhc Training set 78.97531 0.6299238 -0.182100610
## lrc Training set 186.57130 0.5332630 -0.129157745
## lrm Training set 5947.27258 0.6350939 -0.019645607
## lrw Training set 134.17153 0.7260785 0.002108128
##
## Forecasts:
## lhc
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1998 Q1 0.02851435 1.054748e-02 0.04648123 0.001036385 0.05599232
## 1998 Q2 0.02649173 4.018966e-03 0.04896449 -0.007877397 0.06086086
## 1998 Q3 0.02468346 -1.486049e-05 0.04938178 -0.013089362 0.06245628
## 1998 Q4 0.02331189 -2.595149e-03 0.04921893 -0.016309508 0.06293329
## 1999 Q1 0.02222694 -4.360555e-03 0.04881443 -0.018435123 0.06288900
## 1999 Q2 0.02139285 -5.584928e-03 0.04837063 -0.019866102 0.06265181
## 1999 Q3 0.02074852 -6.455260e-03 0.04795229 -0.020856068 0.06235310
## 1999 Q4 0.02025378 -7.081569e-03 0.04758913 -0.021552028 0.06205959
## 2000 Q1 0.01987389 -7.538289e-03 0.04728607 -0.022049422 0.06179721
## 2000 Q2 0.01958263 -7.874495e-03 0.04703976 -0.022409421 0.06157469
##
## lrc
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1998 Q1 0.014375754 0.001440803 0.02731071 -0.005406546 0.03415805
## 1998 Q2 0.007722572 -0.008051241 0.02349638 -0.016401394 0.03184654
## 1998 Q3 0.008794147 -0.007121622 0.02470992 -0.015546922 0.03313522
## 1998 Q4 0.007865051 -0.008137460 0.02386756 -0.016608677 0.03233878
## 1999 Q1 0.007810961 -0.008211994 0.02383392 -0.016694034 0.03231596
## 1999 Q2 0.007607762 -0.008426392 0.02364192 -0.016914360 0.03212988
## 1999 Q3 0.007525529 -0.008514417 0.02356548 -0.017005452 0.03205651
## 1999 Q4 0.007449475 -0.008593820 0.02349277 -0.017086628 0.03198558
## 2000 Q1 0.007400161 -0.008645086 0.02344541 -0.017138927 0.03193925
## 2000 Q2 0.007361782 -0.008684610 0.02340817 -0.017179057 0.03190262
##
## lrm
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1998 Q1 0.01446204 3.335332e-03 0.02558876 -0.002554794 0.03147888
## 1998 Q2 0.01424173 2.406712e-03 0.02607675 -0.003858368 0.03234183
## 1998 Q3 0.01316970 9.993917e-04 0.02534000 -0.005443179 0.03178257
## 1998 Q4 0.01280392 4.912515e-04 0.02511659 -0.006026682 0.03163452
## 1999 Q1 0.01244882 5.748194e-05 0.02484016 -0.006502096 0.03139974
## 1999 Q2 0.01223345 -2.024926e-04 0.02466939 -0.006785682 0.03125258
## 1999 Q3 0.01206579 -3.961473e-04 0.02452773 -0.006993098 0.03112468
## 1999 Q4 0.01194537 -5.317993e-04 0.02442254 -0.007136813 0.03102755
## 2000 Q1 0.01185393 -6.321782e-04 0.02434004 -0.007241924 0.03094978
## 2000 Q2 0.01178520 -7.061541e-04 0.02427656 -0.007318678 0.03088908
##
## lrw
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## 1998 Q1 0.01819724 -0.02093351 0.05732798 -0.04164808 0.07804255
## 1998 Q2 0.02151463 -0.01847782 0.06150709 -0.03964855 0.08267782
## 1998 Q3 0.01855094 -0.02187958 0.05898146 -0.04328220 0.08038408
## 1998 Q4 0.01820161 -0.02233377 0.05873699 -0.04379191 0.08019513
## 1999 Q1 0.01751859 -0.02307054 0.05810772 -0.04455713 0.07959431
## 1999 Q2 0.01721469 -0.02340158 0.05783096 -0.04490253 0.07933191
## 1999 Q3 0.01695303 -0.02367876 0.05758482 -0.04518793 0.07909399
## 1999 Q4 0.01678081 -0.02385999 0.05742161 -0.04537393 0.07893555
## 2000 Q1 0.01664881 -0.02399727 0.05729490 -0.04551400 0.07881163
## 2000 Q2 0.01655189 -0.02409730 0.05720107 -0.04561567 0.07871945VARMA in R/ not available in EViews
http://faculty.chicagobooth.edu/ruey.tsay/teaching/mtsbk/ http://www.rinfinance.com/agenda/2013/talk/RueyTsay.pdf
library(MTS)##
## Attaching package: 'MTS'## The following object is masked from 'package:vars':
##
## VARmod <- VARMA(mydata)## Number of parameters: 20
## initial estimates: 0.1779 1.3911 0.1351 -0.5974 1.0007 0.0813 -0.2298 0.1245 -0.1163 0.4135 0.3242 0.1699 0.0446 0.1023 0.8151 0.0394 0.0722 0.325 -0.2076 0.9138
## Par. lower-bounds: -0.5772 0.9432 -0.2833 -2.1842 0.9389 -0.1761 -0.4063 0.0464 -0.1529 0.2608 0.2195 0.1236 0.0104 -0.0404 0.7173 -0.0038 -0.0575 -0.216 -0.5785 0.7497
## Par. upper-bounds: 0.9331 1.8391 0.5535 0.9895 1.0624 0.3388 -0.0533 0.2026 -0.0797 0.5662 0.4289 0.2163 0.0788 0.2449 0.9129 0.0827 0.2019 0.866 0.1633 1.0779
## Final Estimates: 0.07927819 1.379054 0.2791564 -0.5643976 0.9885154 0.04649285 -0.2451457 0.1808956 -0.1029267 0.4412872 0.3014631 0.1589724 0.01637567 0.008136092 0.884023 0.06429161 0.03478935 0.239377 -0.1318689 0.9403475
##
## Coefficient(s):
## Estimate Std. Error t value Pr(>|t|)
## lhc 0.079278 0.428347 0.185 0.853167
## lrc 1.379054 0.104232 13.231 < 2e-16 ***
## lrm 0.279156 0.183389 1.522 0.127956
## lrw -0.564398 0.642472 -0.878 0.379684
## lhc 0.988515 0.025323 39.037 < 2e-16 ***
## lrc 0.046493 0.127474 0.365 0.715318
## lrm -0.245146 0.083285 -2.943 0.003246 **
## lrw 0.180896 0.038933 4.646 3.38e-06 ***
## lhc -0.102927 0.013052 -7.886 3.11e-15 ***
## lrc 0.441287 0.029220 15.102 < 2e-16 ***
## lrm 0.301463 0.034254 8.801 < 2e-16 ***
## lrw 0.158972 0.014500 10.964 < 2e-16 ***
## lhc 0.016376 0.008886 1.843 0.065361 .
## lrc 0.008136 0.033356 0.244 0.807298
## lrm 0.884023 0.002454 360.303 < 2e-16 ***
## lrw 0.064292 0.018391 3.496 0.000473 ***
## lhc 0.034789 0.041377 0.841 0.400469
## lrc 0.239377 0.175999 1.360 0.173797
## lrm -0.131869 0.131821 -1.000 0.317135
## lrw 0.940347 0.040416 23.267 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## ---
## Estimates in matrix form:
## Constant term:
## Estimates: 0.07927819 1.379054 0.2791564 -0.5643976
## AR coefficient matrix
## AR( 1 )-matrix
## [,1] [,2] [,3] [,4]
## [1,] 0.9885 0.04649 -0.245 0.1809
## [2,] -0.1029 0.44129 0.301 0.1590
## [3,] 0.0164 0.00814 0.884 0.0643
## [4,] 0.0348 0.23938 -0.132 0.9403
##
## Residuals cov-matrix:
## [,1] [,2] [,3] [,4]
## [1,] 3.271263e-04 2.132811e-05 3.016422e-05 1.628138e-04
## [2,] 2.132811e-05 8.040960e-05 -4.119182e-06 7.018316e-05
## [3,] 3.016422e-05 -4.119182e-06 8.126028e-05 1.182583e-04
## [4,] 1.628138e-04 7.018316e-05 1.182583e-04 9.386931e-04
## ----
## aic= -33.7979
## bic= -33.23866summary(mod)## Length Class Mode
## data 356 mts numeric
## ARorder 1 -none- numeric
## MAorder 1 -none- numeric
## cnst 1 -none- logical
## coef 20 -none- numeric
## secoef 20 -none- numeric
## residuals 352 -none- numeric
## Sigma 16 -none- numeric
## aic 1 -none- numeric
## bic 1 -none- numeric
## Phi 16 -none- numeric
## Theta 0 -none- NULL
## Ph0 4 -none- numericPicking p and q
If you want some help picking p and q:
Eccm(mydata)## p-values table of Extended Cross-correlation Matrices:
## Column: MA order
## Row : AR order
## 0 1 2 3 4 5 6
## 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
## 1 0.0000 0.0024 0.0151 0.0197 0.3613 0.4949 0.7091
## 2 0.3249 0.9138 0.2013 0.9747 0.9999 0.9995 0.9201
## 3 0.6588 0.2265 0.6549 0.7650 0.9905 0.9644 0.7366
## 4 0.9477 0.6140 0.8744 0.7971 0.8377 0.8719 0.8336
## 5 0.9999 0.8348 0.9772 0.9934 0.9994 0.9980 0.9690VARMA(1,2)
mod <- VARMA(mydata, p=1, q =2)## Number of parameters: 52
## initial estimates: 0.0351 0.9792 -0.0913 -0.7615 1.0037 0.1328 -0.2706 0.1269 -0.0756 0.6036 0.2028 0.1245 0.0407 0.1644 0.7894 0.0311 0.0763 0.4109 -0.3048 0.9389 0.3589 -0.1944 0.585 -0.128 0.0816 0.1388 0.6159 -0.1287 0.0056 -0.3794 0.3935 0.0225 0.1442 -0.009 0.178 0.0506 0.3158 -0.0781 0.1414 -0.0552 0.1809 0.036 0.5513 0.0254 0.1031 1.7411 2.0088 -0.5688 0.2133 0.2831 1.7282 -0.3501
## Par. lower-bounds: -0.8679 0.5317 -0.5164 -2.5528 0.9254 -0.2054 -0.4939 0.0309 -0.1145 0.436 0.0921 0.0769 0.0038 0.0052 0.6843 -0.0141 -0.0792 -0.2599 -0.7477 0.7485 -0.3211 -1.2119 -0.4676 -0.3537 -0.5744 -0.8516 -0.4362 -0.3535 -0.3313 -0.8836 -0.1281 -0.0894 -0.1809 -0.4998 -0.3433 -0.0608 -0.0043 -0.5571 -0.3541 -0.1614 -0.1279 -0.4302 0.0561 -0.0804 -1.2457 -0.2773 -0.0792 -1.0167 -1.088 -1.6814 -0.3587 -0.7961
## Par. upper-bounds: 0.9382 1.4266 0.3338 1.0298 1.0821 0.471 -0.0472 0.2229 -0.0368 0.7711 0.3134 0.1721 0.0776 0.3236 0.8945 0.0763 0.2318 1.0817 0.1382 1.1294 1.0389 0.8232 1.6377 0.0978 0.7376 1.1292 1.668 0.0961 0.3426 0.1248 0.9151 0.1344 0.4693 0.4817 0.6993 0.162 0.6359 0.4008 0.6369 0.0511 0.4897 0.5022 1.0466 0.1312 1.4519 3.7594 4.0968 -0.121 1.5145 2.2476 3.8151 0.0958
## Final Estimates: 0.03413497 0.9926826 -0.0788096 -0.7647428 1.009944 0.1253087 -0.2777156 0.1250925 -0.0817825 0.5996175 0.1809903 0.1455225 0.03863587 0.159784 0.7903303 0.02686827 0.06486433 0.4016138 -0.3148048 0.9265616 0.403499 -0.1679248 0.6076047 -0.09878171 0.01619433 0.06390291 0.5136661 -0.0757035 -0.008321967 -0.2541014 0.4071056 0.004559816 0.1447705 0.06112168 0.1930929 -0.06076389 0.2817123 0.01480061 0.2697276 -0.03735633 0.06610726 0.005250734 0.4674704 -0.07938088 0.1528256 1.680832 1.970627 -0.2407804 0.2136075 0.2298716 1.694647 -0.2742968## Warning in sqrt(diag(solve(Hessian))): NaNs produced##
## Coefficient(s):
## Estimate Std. Error t value Pr(>|t|)
## lhc 0.034135 1.220725 0.028 0.977692
## lrc 0.992683 0.522335 1.900 0.057372 .
## lrm -0.078810 0.976916 -0.081 0.935703
## lrw -0.764743 4.788171 -0.160 0.873106
## lhc 1.009944 0.039345 25.669 < 2e-16 ***
## lrc 0.125309 0.095141 1.317 0.187812
## lrm -0.277716 0.063316 -4.386 1.15e-05 ***
## lrw 0.125093 0.034723 3.603 0.000315 ***
## lhc -0.081782 0.020005 -4.088 4.35e-05 ***
## lrc 0.599617 NA NA NA
## lrm 0.180990 0.043314 4.179 2.93e-05 ***
## lrw 0.145522 0.024955 5.831 5.50e-09 ***
## lhc 0.038636 0.029305 1.318 0.187364
## lrc 0.159784 0.052254 3.058 0.002229 **
## lrm 0.790330 0.004553 173.574 < 2e-16 ***
## lrw 0.026868 0.017598 1.527 0.126824
## lhc 0.064864 0.149083 0.435 0.663498
## lrc 0.401614 0.356015 1.128 0.259285
## lrm -0.314805 0.261479 -1.204 0.228613
## lrw 0.926562 0.103033 8.993 < 2e-16 ***
## 0.403499 0.439418 0.918 0.358484
## -0.167925 0.107430 -1.563 0.118027
## 0.607605 NA NA NA
## -0.098782 0.080996 -1.220 0.222621
## 0.016194 0.199868 0.081 0.935422
## 0.063903 NA NA NA
## 0.513666 NA NA NA
## -0.075703 0.039706 -1.907 0.056574 .
## -0.008322 0.237307 -0.035 0.972025
## -0.254101 NA NA NA
## 0.407106 NA NA NA
## 0.004560 0.047305 0.096 0.923209
## 0.144771 0.170937 0.847 0.397038
## 0.061122 NA NA NA
## 0.193093 NA NA NA
## -0.060764 0.025233 -2.408 0.016036 *
## 0.281712 0.400267 0.704 0.481551
## 0.014801 NA NA NA
## 0.269728 NA NA NA
## -0.037356 0.071990 -0.519 0.603822
## 0.066107 0.313322 0.211 0.832897
## 0.005251 0.100873 0.052 0.958486
## 0.467470 NA NA NA
## -0.079381 0.054986 -1.444 0.148834
## 0.152826 1.910975 0.080 0.936259
## 1.680832 NA NA NA
## 1.970627 NA NA NA
## -0.240780 0.344503 -0.699 0.484602
## 0.213608 1.002587 0.213 0.831283
## 0.229872 0.840785 0.273 0.784545
## 1.694647 NA NA NA
## -0.274297 0.218073 -1.258 0.208456
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## ---
## Estimates in matrix form:
## Constant term:
## Estimates: 0.03413497 0.9926826 -0.0788096 -0.7647428
## AR coefficient matrix
## AR( 1 )-matrix
## [,1] [,2] [,3] [,4]
## [1,] 1.0099 0.125 -0.278 0.1251
## [2,] -0.0818 0.600 0.181 0.1455
## [3,] 0.0386 0.160 0.790 0.0269
## [4,] 0.0649 0.402 -0.315 0.9266
## MA coefficient matrix
## MA( 1 )-matrix
## [,1] [,2] [,3] [,4]
## [1,] -0.40350 0.1679 -0.608 0.09878
## [2,] 0.00832 0.2541 -0.407 -0.00456
## [3,] -0.28171 -0.0148 -0.270 0.03736
## [4,] -0.15283 -1.6808 -1.971 0.24078
## MA( 2 )-matrix
## [,1] [,2] [,3] [,4]
## [1,] -0.0162 -0.06390 -0.514 0.0757
## [2,] -0.1448 -0.06112 -0.193 0.0608
## [3,] -0.0661 -0.00525 -0.467 0.0794
## [4,] -0.2136 -0.22987 -1.695 0.2743
##
## Residuals cov-matrix:
## [,1] [,2] [,3] [,4]
## [1,] 0.18095381 0.08603523 0.14528222 0.7241540
## [2,] 0.08603523 0.04103382 0.06915376 0.3447155
## [3,] 0.14528222 0.06915376 0.11686436 0.5821019
## [4,] 0.72415400 0.34471554 0.58210191 2.9016109
## ----
## aic= -24.93088
## bic= -23.47685summary(mod)## Length Class Mode
## data 356 mts numeric
## ARorder 1 -none- numeric
## MAorder 1 -none- numeric
## cnst 1 -none- logical
## coef 52 -none- numeric
## secoef 52 -none- numeric
## residuals 348 -none- numeric
## Sigma 16 -none- numeric
## aic 1 -none- numeric
## bic 1 -none- numeric
## Phi 16 -none- numeric
## Theta 32 -none- numeric
## Ph0 4 -none- numeric