#### HW#5, June 2, 2017
#### Problem 1 ####
> Kronspec(kdx=c(1,1,2))
Kronecker indices: 1 1 2
Dimension: 3
Notation:
0: fixed to 0
1: fixed to 1
2: estimation
AR coefficient matrices:
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,] 1 0 0 2 2 2 0 0 0
[2,] 0 1 0 2 2 2 0 0 0
[3,] 0 0 1 0 0 2 2 2 2
MA coefficient matrices:
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
[1,] 1 0 0 2 2 2 0 0 0
[2,] 0 1 0 2 2 2 0 0 0
[3,] 0 0 1 2 2 2 2 2 2
>
#### Problem 2
> setwd("C:/Users/rst/teaching/mts/sp2017")
> source("C:/Users/rst/HDSA/HTS/ProcMD.R")
> da <- read.csv("C:/Users/rst/HDSA/HTS/current.csv",header=T)
> m1 <- ProcMD(da)
> names(m1)
[1] "data" "dmtx"
> X <- m1$dmtx
> dim(X)
[1] 687 120
> name <- colnames(da)
> name[1]
[1] "sasdate"
> name <- name[-1]
> length(name)
[1] 120
> name
[1] "RPI" "W875RX1" "DPCERA3M086SBEA" "CMRMTSPLx"
[5] "RETAILx" "INDPRO" "IPFPNSS" "IPFINAL"
[9] "IPCONGD" "IPDCONGD" "IPNCONGD" "IPBUSEQ"
[13] "IPMAT" "IPDMAT" "IPNMAT" "IPMANSICS"
[17] "IPB51222S" "IPFUELS" "CUMFNS" "CLF16OV"
[21] "CE16OV" "UNRATE" "UEMPMEAN" "UEMPLT5"
[25] "UEMP5TO14" "UEMP15OV" "UEMP15T26" "UEMP27OV"
[29] "CLAIMSx" "PAYEMS" "USGOOD" "CES1021000001"
[33] "USCONS" "MANEMP" "DMANEMP" "NDMANEMP"
[37] "SRVPRD" "USTPU" "USWTRADE" "USTRADE"
[41] "USFIRE" "USGOVT" "CES0600000007" "AWOTMAN"
[45] "AWHMAN" "HOUST" "HOUSTNE" "HOUSTMW"
[49] "HOUSTS" "HOUSTW" "PERMIT" "PERMITNE"
[53] "PERMITMW" "PERMITS" "PERMITW" "AMDMNOx"
[57] "AMDMUOx" "BUSINVx" "ISRATIOx" "M1SL"
[61] "M2SL" "M2REAL" "AMBSL" "TOTRESNS"
[65] "NONBORRES" "BUSLOANS" "REALLN" "NONREVSL"
[69] "CONSPI" "S.P.500" "S.P..indust" "S.P.div.yield"
[73] "FEDFUNDS" "CP3Mx" "TB3MS" "TB6MS"
[77] "GS1" "GS5" "GS10" "AAA"
[81] "BAA" "COMPAPFFx" "TB3SMFFM" "TB6SMFFM"
[85] "T1YFFM" "T5YFFM" "T10YFFM" "AAAFFM"
[89] "BAAFFM" "EXSZUSx" "EXJPUSx" "EXUSUKx"
[93] "EXCAUSx" "WPSFD49207" "WPSFD49502" "WPSID61"
[97] "WPSID62" "OILPRICEx" "PPICMM" "CPIAUCSL"
[101] "CPIAPPSL" "CPITRNSL" "CPIMEDSL" "CUSR0000SAC"
[105] "CUSR0000SAD" "CUSR0000SAS" "CPIULFSL" "CUSR0000SA0L2"
[109] "CUSR0000SA0L5" "PCEPI" "DDURRG3M086SBEA" "DNDGRG3M086SBEA"
[113] "DSERRG3M086SBEA" "CES0600000008" "CES2000000008" "CES3000000008"
[117] "MZMSL" "DTCOLNVHFNM" "DTCTHFNM" "INVEST"
> require(MTS)
Loading required package: MTS
> dim(X)
[1] 687 120
> 687-36
[1] 651
> y <- X[2:687,6]
> X1 <- X[1:686,]
> m1 <- SWfore(y,X1,650,10)
MSE of out-of-sample forecasts: 2.805348e-05
> names(m1)
[1] "coef" "yhat" "MSE" "loadings" "DFindex"
> sqrt(m1$MSE)
[1] 0.005296554
> m2 <- SWfore(y,X1,650,30)
MSE of out-of-sample forecasts: 2.477952e-05
> sqrt(m2$MSE)
[1] 0.004977903
> m3 <- SWfore(y,X1,650,50)
MSE of out-of-sample forecasts: 4.120186e-05
> sqrt(m3$MSE)
[1] 0.006418868
>
### Problem 3
> y <- X[2:687,22]
> m4 <- SWfore(y,X1,650,10)
MSE of out-of-sample forecasts: 0.02042371
> sqrt(m4$MSE)
[1] 0.1429115
> m5 <- SWfore(y,X1,650,30)
MSE of out-of-sample forecasts: 0.02657126
> sqrt(m5$MSE)
[1] 0.1630069
> m6 <- SWfore(y,X1,650,50)
MSE of out-of-sample forecasts: 0.0267818
> sqrt(m6$MSE)
[1] 0.1636515
>
> y1 <- X[,22]
> pacf(y1)
> n1 <- arima(y1,order=c(2,0,1),seasonal=list(order=c(1,0,1),period=12))
> n1
Call:
arima(x = y1, order = c(2, 0, 1), seasonal = list(order = c(1, 0, 1), period = 12))
Coefficients:
ar1 ar2 ma1 sar1 sma1 intercept
0.6960 0.2032 -0.6952 0.5405 -0.7987 -0.0002
s.e. 0.0574 0.0421 0.0478 0.0726 0.0513 0.0084
sigma^2 estimated as 0.02546: log likelihood = 284.6, aic = -555.2
> tsdiag(n1,gof=24)
> yf <- y1[652:687]
> n1 <- arima(y1[1:651],order=c(2,0,1),seasonal=list(order=c(1,0,1),period=12),include.mean=F)
> p1 <- predict(n1,36)
> err <- yf-p1$pred
> sqrt(mean(err^2))
[1] 0.1501588
>
#### Problem 4 ###
> X <- cbind(UNP,IP,PMI,TCU)
> require(MTS)
> MTSplot(X)
> dX <- diffM(X)
> colnames(X)
[1] "UNP" "IP" "PMI" "TCU"
> zt <- dX[,1:2]; xt <- dX[,3:4]
> VARorder(zt)
selected order: aic = 6
selected order: bic = 3
selected order: hq = 5
Summary table:
p AIC BIC HQ M(p) p-value
[1,] 0 -4.9852 -4.9852 -4.9852 0.0000 0.0000
[2,] 1 -5.1900 -5.1607 -5.1786 127.0760 0.0000
[3,] 2 -5.2502 -5.1915 -5.2273 42.6817 0.0000
[4,] 3 -5.2857 -5.1976 -5.2514 28.2548 0.0000
[5,] 4 -5.3059 -5.1885 -5.2602 19.3748 0.0007
[6,] 5 -5.3209 -5.1741 -5.2637 16.2423 0.0027
[7,] 6 -5.3209 -5.1448 -5.2523 7.6620 0.1048
[8,] 7 -5.3167 -5.1112 -5.2367 5.2267 0.2648
[9,] 8 -5.3113 -5.0765 -5.2199 4.5298 0.3390
[10,] 9 -5.3052 -5.0410 -5.2024 4.1135 0.3909
[11,] 10 -5.2997 -5.0062 -5.1855 4.4571 0.3477
[12,] 11 -5.2921 -4.9693 -5.1665 3.2369 0.5190
[13,] 12 -5.3140 -4.9618 -5.1768 19.7158 0.0006
[14,] 13 -5.3082 -4.9266 -5.1596 4.2164 0.3775
> m1 <- VAR(zt,5)
Constant term:
Estimates: 0.01966672 0.05670193
Std.Error: 0.007383392 0.0211047
AR coefficient matrix
AR( 1 )-matrix
[,1] [,2]
[1,] -0.062 -0.0854
[2,] -0.506 0.0814
standard error
[,1] [,2]
[1,] 0.043 0.015
[2,] 0.123 0.043
AR( 2 )-matrix
[,1] [,2]
[1,] 0.0631 -0.0378
[2,] -0.2135 0.1062
standard error
[,1] [,2]
[1,] 0.0433 0.0153
[2,] 0.1239 0.0437
AR( 3 )-matrix
[,1] [,2]
[1,] 0.0884 -0.0175
[2,] -0.1540 0.1682
standard error
[,1] [,2]
[1,] 0.0432 0.0153
[2,] 0.1234 0.0437
AR( 4 )-matrix
[,1] [,2]
[1,] 0.0928 -0.0102
[2,] 0.2344 0.1457
standard error
[,1] [,2]
[1,] 0.0429 0.0155
[2,] 0.1227 0.0442
AR( 5 )-matrix
[,1] [,2]
[1,] 0.0687 -0.0154
[2,] 0.3259 0.0124
standard error
[,1] [,2]
[1,] 0.0422 0.0153
[2,] 0.1208 0.0437
Residuals cov-mtx:
[,1] [,2]
[1,] 0.02454520 -0.02039505
[2,] -0.02039505 0.20054556
det(SSE) = 0.004506472
AIC = -5.335463
BIC = -5.188709
HQ = -5.27833
> m1a <- refVAR(m1,thres=1)
Constant term:
Estimates: 0.01738008 0.05777494
Std.Error: 0.00710883 0.02074842
AR coefficient matrix
AR( 1 )-matrix
[,1] [,2]
[1,] -0.0613 -0.0885
[2,] -0.5061 0.0828
standard error
[,1] [,2]
[1,] 0.0429 0.0148
[2,] 0.1227 0.0427
AR( 2 )-matrix
[,1] [,2]
[1,] 0.0659 -0.0401
[2,] -0.2141 0.1076
standard error
[,1] [,2]
[1,] 0.0432 0.0152
[2,] 0.1238 0.0434
AR( 3 )-matrix
[,1] [,2]
[1,] 0.0982 -0.0182
[2,] -0.1571 0.1687
standard error
[,1] [,2]
[1,] 0.042 0.0153
[2,] 0.123 0.0437
AR( 4 )-matrix
[,1] [,2]
[1,] 0.110 0.000
[2,] 0.227 0.146
standard error
[,1] [,2]
[1,] 0.0402 0.0000
[2,] 0.1195 0.0441
AR( 5 )-matrix
[,1] [,2]
[1,] 0.0853 0
[2,] 0.3161 0
standard error
[,1] [,2]
[1,] 0.0399 0
[2,] 0.1157 0
Residuals cov-mtx:
[,1] [,2]
[1,] 0.02460541 -0.0204294
[2,] -0.02042940 0.2005734
det(SSE) = 0.00451783
AIC = -5.342962
BIC = -5.218222
HQ = -5.2944
> m1b <- refVAR(m1a,thres=1.2)
Constant term:
Estimates: 0.01542687 0.05777494
Std.Error: 0.006918662 0.02074842
AR coefficient matrix
AR( 1 )-matrix
[,1] [,2]
[1,] -0.0568 -0.0898
[2,] -0.5061 0.0828
standard error
[,1] [,2]
[1,] 0.0428 0.0147
[2,] 0.1227 0.0427
AR( 2 )-matrix
[,1] [,2]
[1,] 0.0769 -0.0403
[2,] -0.2141 0.1076
standard error
[,1] [,2]
[1,] 0.0422 0.0152
[2,] 0.1238 0.0434
AR( 3 )-matrix
[,1] [,2]
[1,] 0.112 0.000
[2,] -0.157 0.169
standard error
[,1] [,2]
[1,] 0.0403 0.0000
[2,] 0.1228 0.0437
AR( 4 )-matrix
[,1] [,2]
[1,] 0.118 0.000
[2,] 0.227 0.146
standard error
[,1] [,2]
[1,] 0.0396 0.0000
[2,] 0.1195 0.0441
AR( 5 )-matrix
[,1] [,2]
[1,] 0.0878 0
[2,] 0.3161 0
standard error
[,1] [,2]
[1,] 0.0398 0
[2,] 0.1157 0
Residuals cov-mtx:
[,1] [,2]
[1,] 0.02466481 -0.0204294
[2,] -0.02042940 0.2005734
det(SSE) = 0.004529745
AIC = -5.343667
BIC = -5.226265
HQ = -5.297961
> MTSdiag(m1b,gof=24)
[1] "Covariance matrix:"
UNP IP
UNP 0.0247 -0.0205
IP -0.0205 0.2009
CCM at lag: 0
[,1] [,2]
[1,] 1.00 -0.29
[2,] -0.29 1.00
Simplified matrix:
CCM at lag: 1
. .
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CCM at lag: 2
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CCM at lag: 3
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CCM at lag: 4
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CCM at lag: 5
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CCM at lag: 6
+ .
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CCM at lag: 7
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CCM at lag: 8
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CCM at lag: 9
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. +
CCM at lag: 10
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CCM at lag: 11
+ .
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CCM at lag: 12
- .
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CCM at lag: 13
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CCM at lag: 14
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CCM at lag: 15
. +
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CCM at lag: 16
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CCM at lag: 17
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CCM at lag: 18
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- .
CCM at lag: 19
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CCM at lag: 20
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CCM at lag: 21
. .
. .
CCM at lag: 22
. .
+ .
CCM at lag: 23
. .
. .
CCM at lag: 24
- .
. -
Hit Enter for p-value plot of individual ccm:
Hit Enter to compute MQ-statistics:
Ljung-Box Statistics:
m Q(m) df p-value
[1,] 1.000 0.103 4.000 1.00
[2,] 2.000 0.643 8.000 1.00
[3,] 3.000 3.161 12.000 0.99
[4,] 4.000 4.192 16.000 1.00
[5,] 5.000 4.646 20.000 1.00
[6,] 6.000 12.877 24.000 0.97
[7,] 7.000 16.039 28.000 0.97
[8,] 8.000 22.671 32.000 0.89
[9,] 9.000 28.502 36.000 0.81
[10,] 10.000 31.339 40.000 0.83
[11,] 11.000 38.214 44.000 0.72
[12,] 12.000 57.672 48.000 0.16
[13,] 13.000 60.359 52.000 0.20
[14,] 14.000 64.459 56.000 0.20
[15,] 15.000 77.452 60.000 0.06
[16,] 16.000 78.128 64.000 0.11
[17,] 17.000 83.000 68.000 0.10
[18,] 18.000 90.558 72.000 0.07
[19,] 19.000 100.377 76.000 0.03
[20,] 20.000 101.698 80.000 0.05
[21,] 21.000 106.778 84.000 0.05
[22,] 22.000 119.222 88.000 0.01
[23,] 23.000 125.628 92.000 0.01
[24,] 24.000 145.757 96.000 0.00
Hit Enter to obtain residual plots:
>
>
> VARXorder(zt,xt,maxp=11,maxm=6)
selected order(p,s): aic = 7 4
selected order(p,s): bic = 4 4
selected order(p,s): hq = 5 4
> ?VARX
> m2 <- VARX(zt,5,xt,4)
constant term:
est: 0.0118 0.0272
se: 0.0075 0.0088
AR( 1 ) matrix
UNP IP
UNP -0.176 0.048
IP -0.007 0.168
standard errors
[,1] [,2]
[1,] 0.041 0.035
[2,] 0.048 0.041
AR( 2 ) matrix
UNP IP
UNP 0.008 -0.051
IP 0.026 0.165
standard errors
[,1] [,2]
[1,] 0.040 0.035
[2,] 0.047 0.041
AR( 3 ) matrix
UNP IP
UNP 0.080 -0.039
IP -0.015 0.195
standard errors
[,1] [,2]
[1,] 0.040 0.034
[2,] 0.047 0.040
AR( 4 ) matrix
UNP IP
UNP 0.127 -0.029
IP 0.023 0.230
standard errors
[,1] [,2]
[1,] 0.040 0.034
[2,] 0.047 0.040
AR( 5 ) matrix
UNP IP
UNP 0.123 -0.039
IP 0.021 0.029
standard errors
[,1] [,2]
[1,] 0.039 0.014
[2,] 0.046 0.017
Coefficients of exogenous
lag- 0 coefficient matrix
PMI TCU
UNP -0.010 -0.067
IP -0.005 0.812
standard errors
[,1] [,2]
[1,] 0.003 0.012
[2,] 0.003 0.015
lag- 1 coefficient matrix
PMI TCU
UNP -0.011 -0.099
IP -0.003 -0.180
standard errors
[,1] [,2]
[1,] 0.003 0.031
[2,] 0.003 0.037
lag- 2 coefficient matrix
PMI TCU
UNP -0.008 0.027
IP 0.002 -0.124
standard errors
[,1] [,2]
[1,] 0.003 0.032
[2,] 0.004 0.037
lag- 3 coefficient matrix
PMI TCU
UNP 0.005 0.017
IP 0.001 -0.129
standard errors
[,1] [,2]
[1,] 0.003 0.031
[2,] 0.003 0.037
lag- 4 coefficient matrix
PMI TCU
UNP -0.003 0.004
IP 0.002 -0.176
standard errors
[,1] [,2]
[1,] 0.003 0.030
[2,] 0.003 0.036
Residual Covariance Matrix
UNP IP
UNP 0.01966 0.00068
IP 0.00068 0.02696
===========
Information criteria:
AIC: -7.401912
BIC: -7.091729
> m2a <- refVARX(m2,thres=1.0)
constant term:
est: 0.0073 0.0283
se: 0.007 0.0087
AR( 1 ) matrix
[,1] [,2]
[1,] -0.182 0.017
[2,] 0.000 0.171
standard errors
[,1] [,2]
[1,] 0.04 0.029
[2,] 1.00 0.041
AR( 2 ) matrix
[,1] [,2]
[1,] 0 -0.031
[2,] 0 0.167
standard errors
[,1] [,2]
[1,] 1 0.014
[2,] 1 0.041
AR( 3 ) matrix
[,1] [,2]
[1,] 0.087 -0.023
[2,] 0.000 0.192
standard errors
[,1] [,2]
[1,] 0.039 0.014
[2,] 1.000 0.040
AR( 4 ) matrix
[,1] [,2]
[1,] 0.142 0.000
[2,] 0.000 0.225
standard errors
[,1] [,2]
[1,] 0.038 1.00
[2,] 1.000 0.04
AR( 5 ) matrix
[,1] [,2]
[1,] 0.128 -0.040
[2,] 0.000 0.025
standard errors
[,1] [,2]
[1,] 0.038 0.014
[2,] 1.000 0.015
Coefficients of exogenous
lag- 0 coefficient matrix
[,1] [,2]
[1,] -0.010 -0.072
[2,] -0.005 0.812
standard errors
[,1] [,2]
[1,] 0.003 0.012
[2,] 0.003 0.013
lag- 1 coefficient matrix
[,1] [,2]
[1,] -0.011 -0.076
[2,] 0.000 -0.183
standard errors
[,1] [,2]
[1,] 0.003 0.027
[2,] 1.000 0.036
lag- 2 coefficient matrix
[,1] [,2]
[1,] -0.007 0.000
[2,] 0.000 -0.126
standard errors
[,1] [,2]
[1,] 0.003 1.000
[2,] 1.000 0.036
lag- 3 coefficient matrix
[,1] [,2]
[1,] 0.006 0.000
[2,] 0.000 -0.124
standard errors
[,1] [,2]
[1,] 0.003 1.000
[2,] 1.000 0.036
lag- 4 coefficient matrix
[,1] [,2]
[1,] 0 0.000
[2,] 0 -0.174
standard errors
[,1] [,2]
[1,] 1 1.000
[2,] 1 0.034
Residual Covariance Matrix
UNP IP
UNP 0.01986 0.00064
IP 0.00064 0.02710
===========
Information criteria:
AIC: -7.436783
BIC: -7.23738
> MTSdiag(m2a,gof=24)
[1] "Covariance matrix:"
UNP IP
UNP 0.019898 0.000642
IP 0.000642 0.027149
CCM at lag: 0
[,1] [,2]
[1,] 1.0000 0.0276
[2,] 0.0276 1.0000
Simplified matrix:
CCM at lag: 1
. .
. .
CCM at lag: 2
. .
. .
CCM at lag: 3
. .
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CCM at lag: 4
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CCM at lag: 5
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CCM at lag: 6
+ .
. .
CCM at lag: 7
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CCM at lag: 8
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CCM at lag: 9
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. +
CCM at lag: 10
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CCM at lag: 11
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CCM at lag: 12
- .
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CCM at lag: 13
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CCM at lag: 14
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CCM at lag: 15
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CCM at lag: 16
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CCM at lag: 17
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CCM at lag: 18
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CCM at lag: 19
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CCM at lag: 20
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CCM at lag: 21
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CCM at lag: 22
. .
+ .
CCM at lag: 23
. .
. .
CCM at lag: 24
- .
. -
Hit Enter for p-value plot of individual ccm:
Hit Enter to compute MQ-statistics:
Ljung-Box Statistics:
m Q(m) df p-value
[1,] 1.000 0.862 4.000 0.93
[2,] 2.000 1.829 8.000 0.99
[3,] 3.000 3.930 12.000 0.98
[4,] 4.000 6.322 16.000 0.98
[5,] 5.000 7.379 20.000 1.00
[6,] 6.000 14.780 24.000 0.93
[7,] 7.000 19.426 28.000 0.88
[8,] 8.000 22.747 32.000 0.89
[9,] 9.000 32.956 36.000 0.61
[10,] 10.000 33.671 40.000 0.75
[11,] 11.000 35.706 44.000 0.81
[12,] 12.000 47.127 48.000 0.51
[13,] 13.000 49.074 52.000 0.59
[14,] 14.000 52.140 56.000 0.62
[15,] 15.000 56.669 60.000 0.60
[16,] 16.000 59.787 64.000 0.63
[17,] 17.000 62.030 68.000 0.68
[18,] 18.000 67.244 72.000 0.64
[19,] 19.000 70.982 76.000 0.64
[20,] 20.000 75.728 80.000 0.61
[21,] 21.000 80.526 84.000 0.59
[22,] 22.000 91.045 88.000 0.39
[23,] 23.000 98.102 92.000 0.31
[24,] 24.000 117.203 96.000 0.07
Hit Enter to obtain residual plots:
>
> m3 <- VARX(Zt,5,Xt,0)
constant term:
est: 0.0085 0.0541
se: 0.0071 0.0089
AR( 1 ) matrix
UNP IP
UNP -0.180 0.031
IP 0.007 0.362
standard errors
[,1] [,2]
[1,] 0.04 0.030
[2,] 0.05 0.038
AR( 2 ) matrix
UNP IP
UNP 0.002 -0.030
IP 0.080 0.057
standard errors
[,1] [,2]
[1,] 0.04 0.014
[2,] 0.05 0.017
AR( 3 ) matrix
UNP IP
UNP 0.062 -0.022
IP 0.067 0.070
standard errors
[,1] [,2]
[1,] 0.04 0.014
[2,] 0.05 0.017
AR( 4 ) matrix
UNP IP
UNP 0.114 -0.023
IP 0.095 0.057
standard errors
[,1] [,2]
[1,] 0.039 0.014
[2,] 0.049 0.018
AR( 5 ) matrix
UNP IP
UNP 0.108 -0.043
IP 0.061 0.043
standard errors
[,1] [,2]
[1,] 0.038 0.014
[2,] 0.048 0.018
Coefficients of exogenous
lag- 0 coefficient matrix
pmi tcu pmi1 tcu1
UNP -0.011 -0.075 -0.010 -0.087
IP -0.002 0.810 -0.002 -0.350
standard errors
[,1] [,2] [,3] [,4]
[1,] 0.003 0.012 0.003 0.027
[2,] 0.004 0.015 0.004 0.033
Residual Covariance Matrix
UNP IP
UNP 0.02011 0.00020
IP 0.00020 0.03118
===========
Information criteria:
AIC: -7.27329
BIC: -7.051442
> m3a <- refVARX(m3,thres=1.0)
constant term:
est: 0.0086 0.0537
se: 0.0071 0.0089
AR( 1 ) matrix
[,1] [,2]
[1,] -0.18 0.031
[2,] 0.00 0.363
standard errors
[,1] [,2]
[1,] 0.04 0.030
[2,] 1.00 0.038
AR( 2 ) matrix
[,1] [,2]
[1,] 0.000 -0.030
[2,] 0.076 0.056
standard errors
[,1] [,2]
[1,] 1.000 0.013
[2,] 0.049 0.017
AR( 3 ) matrix
[,1] [,2]
[1,] 0.061 -0.023
[2,] 0.062 0.071
standard errors
[,1] [,2]
[1,] 0.040 0.014
[2,] 0.049 0.017
AR( 4 ) matrix
[,1] [,2]
[1,] 0.114 -0.023
[2,] 0.091 0.059
standard errors
[,1] [,2]
[1,] 0.039 0.014
[2,] 0.048 0.018
AR( 5 ) matrix
[,1] [,2]
[1,] 0.108 -0.043
[2,] 0.060 0.045
standard errors
[,1] [,2]
[1,] 0.038 0.014
[2,] 0.047 0.017
Coefficients of exogenous
lag- 0 coefficient matrix
[,1] [,2] [,3] [,4]
[1,] -0.011 -0.075 -0.01 -0.087
[2,] 0.000 0.806 0.00 -0.354
standard errors
[,1] [,2] [,3] [,4]
[1,] 0.003 0.012 0.003 0.027
[2,] 1.000 0.014 1.000 0.033
Residual Covariance Matrix
UNP IP
UNP 0.02011 0.00020
IP 0.00020 0.03121
===========
Information criteria:
AIC: -7.285949
BIC: -7.093681
>
> m3 <- VARX(Zt,6,Xt,0)
constant term:
est: 0.0074 0.0527
se: 0.0071 0.0089
AR( 1 ) matrix
UNP IP
UNP -0.193 0.027
IP 0.013 0.360
standard errors
[,1] [,2]
[1,] 0.04 0.031
[2,] 0.05 0.038
AR( 2 ) matrix
UNP IP
UNP -0.017 -0.034
IP 0.086 0.053
standard errors
[,1] [,2]
[1,] 0.04 0.014
[2,] 0.05 0.017
AR( 3 ) matrix
UNP IP
UNP 0.049 -0.021
IP 0.069 0.065
standard errors
[,1] [,2]
[1,] 0.04 0.014
[2,] 0.05 0.017
AR( 4 ) matrix
UNP IP
UNP 0.106 -0.021
IP 0.101 0.057
standard errors
[,1] [,2]
[1,] 0.039 0.014
[2,] 0.049 0.018
AR( 5 ) matrix
UNP IP
UNP 0.116 -0.035
IP 0.078 0.043
standard errors
[,1] [,2]
[1,] 0.039 0.014
[2,] 0.048 0.018
AR( 6 ) matrix
UNP IP
UNP 0.114 0.006
IP 0.011 0.032
standard errors
[,1] [,2]
[1,] 0.038 0.014
[2,] 0.048 0.017
Coefficients of exogenous
lag- 0 coefficient matrix
pmi tcu pmi1 tcu1
UNP -0.011 -0.074 -0.011 -0.087
IP -0.002 0.813 0.000 -0.350
standard errors
[,1] [,2] [,3] [,4]
[1,] 0.003 0.012 0.003 0.027
[2,] 0.004 0.015 0.004 0.034
Residual Covariance Matrix
UNP IP
UNP 0.01984 0.00025
IP 0.00025 0.03060
===========
Information criteria:
AIC: -7.292062
BIC: -7.040307
> m3a <- refVARX(m3,thres=1.0)
constant term:
est: 0.0101 0.0527
se: 0.0065 0.0088
AR( 1 ) matrix
[,1] [,2]
[1,] -0.192 0.00
[2,] 0.000 0.36
standard errors
[,1] [,2]
[1,] 0.04 1.000
[2,] 1.00 0.038
AR( 2 ) matrix
[,1] [,2]
[1,] 0.000 -0.031
[2,] 0.085 0.053
standard errors
[,1] [,2]
[1,] 1.000 0.013
[2,] 0.049 0.017
AR( 3 ) matrix
[,1] [,2]
[1,] 0.055 -0.017
[2,] 0.067 0.065
standard errors
[,1] [,2]
[1,] 0.039 0.013
[2,] 0.049 0.017
AR( 4 ) matrix
[,1] [,2]
[1,] 0.107 -0.018
[2,] 0.101 0.058
standard errors
[,1] [,2]
[1,] 0.039 0.014
[2,] 0.048 0.017
AR( 5 ) matrix
[,1] [,2]
[1,] 0.113 -0.033
[2,] 0.077 0.043
standard errors
[,1] [,2]
[1,] 0.038 0.014
[2,] 0.048 0.017
AR( 6 ) matrix
[,1] [,2]
[1,] 0.107 0.000
[2,] 0.000 0.031
standard errors
[,1] [,2]
[1,] 0.036 1.000
[2,] 1.000 0.017
Coefficients of exogenous
lag- 0 coefficient matrix
[,1] [,2] [,3] [,4]
[1,] -0.011 -0.075 -0.011 -0.065
[2,] 0.000 0.810 0.000 -0.353
standard errors
[,1] [,2] [,3] [,4]
[1,] 0.003 0.012 0.003 0.012
[2,] 1.000 0.014 1.000 0.033
Residual Covariance Matrix
UNP IP
UNP 0.01988 0.00025
IP 0.00025 0.03062
===========
Information criteria:
AIC: -7.313026
BIC: -7.113102
> MTSdiag(m3a,gof=24)
[1] "Covariance matrix:"
UNP IP
UNP 0.019914 0.000252
IP 0.000252 0.030671
CCM at lag: 0
[,1] [,2]
[1,] 1.0000 0.0102
[2,] 0.0102 1.0000
Simplified matrix:
CCM at lag: 1
. .
. .
CCM at lag: 2
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. +
CCM at lag: 3
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. +
CCM at lag: 4
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. +
CCM at lag: 5
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CCM at lag: 6
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. +
CCM at lag: 7
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CCM at lag: 8
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CCM at lag: 9
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. +
CCM at lag: 10
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CCM at lag: 11
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CCM at lag: 12
- .
. .
CCM at lag: 13
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. +
CCM at lag: 14
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CCM at lag: 15
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. +
CCM at lag: 16
. .
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CCM at lag: 17
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CCM at lag: 18
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CCM at lag: 19
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CCM at lag: 20
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CCM at lag: 21
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CCM at lag: 22
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CCM at lag: 23
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CCM at lag: 24
- .
. -
Hit Enter for p-value plot of individual ccm:
Hit Enter to compute MQ-statistics:
Ljung-Box Statistics:
m Q(m) df p-value
[1,] 1.00 5.91 4.00 0.21
[2,] 2.00 13.39 8.00 0.10
[3,] 3.00 25.19 12.00 0.01
[4,] 4.00 50.51 16.00 0.00
[5,] 5.00 54.79 20.00 0.00
[6,] 6.00 62.19 24.00 0.00
[7,] 7.00 69.65 28.00 0.00
[8,] 8.00 74.69 32.00 0.00
[9,] 9.00 89.17 36.00 0.00
[10,] 10.00 91.06 40.00 0.00
[11,] 11.00 97.14 44.00 0.00
[12,] 12.00 109.25 48.00 0.00
[13,] 13.00 116.48 52.00 0.00
[14,] 14.00 118.23 56.00 0.00
[15,] 15.00 126.85 60.00 0.00
[16,] 16.00 130.73 64.00 0.00
[17,] 17.00 133.29 68.00 0.00
[18,] 18.00 139.73 72.00 0.00
[19,] 19.00 142.53 76.00 0.00
[20,] 20.00 147.75 80.00 0.00
[21,] 21.00 152.14 84.00 0.00
[22,] 22.00 158.81 88.00 0.00
[23,] 23.00 166.45 92.00 0.00
[24,] 24.00 185.07 96.00 0.00
Hit Enter to obtain residual plots: