chap8_2
qaj
2023-12-23
例子 2.1
- Cholesky 分解
ldl.decomp <- function(A){
R <- chol(A)
D <- diag(diag(R)) # 第一格diag()函数返回的是向量，第二个diag()函数把向量变成矩阵。
Linv <- solve(t(solve(D, R))) # 第一个solve求解L，第二个solve求解L的逆矩阵。
G <- D^2
list(Linv=Linv, G=G)
}Sigma <- rbind(c(2,1), c(1,1)); Sigma## [,1] [,2]
## [1,] 2 1
## [2,] 1 1ldl.decomp(Sigma)## $Linv
## [,1] [,2]
## [1,] 1.0 0
## [2,] -0.5 1
##
## $G
## [,1] [,2]
## [1,] 2 0.0
## [2,] 0 0.5Sigma <- rbind(c(1,1), c(1,2)); Sigma## [,1] [,2]
## [1,] 1 1
## [2,] 1 2ldl.decomp(Sigma)## $Linv
## [,1] [,2]
## [1,] 1 0
## [2,] -1 1
##
## $G
## [,1] [,2]
## [1,] 1 0
## [2,] 0 1例 2.2
- 考虑二元VAR(1)模型 \[ \begin{aligned} \left(\begin{array}{c} r_{1t} \\ r_{2t} \end{array}\right) = & \left(\begin{array}{c} 5 \\ 3 \end{array}\right) + \left(\begin{array}{cc} 0.2 & 0.3 \\ -0.6 & 1.1 \end{array}\right) \left(\begin{array}{c} r_{1,t-1} \\ r_{2,t-1} \end{array}\right) + \left(\begin{array}{c} a_{1t} \\ a_{2t} \end{array}\right) \\ \boldsymbol\Sigma =& \left(\begin{array}{cc} 1 & 0.8 \\ 0.8 & 2 \end{array}\right) \end{aligned} \]
计算\(\boldsymbol\Phi\)的特征值的绝对值：
abs(eigen(rbind(c(0.2, 0.3), c(-0.6, 1.1)))$value)## [1] 0.8 0.5英、加、美GDP的VAR建模
- 读入数据，计算对数增长率:
library(readr)
library(zoo)##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericlibrary(xts)
library(lubridate)##
## Attaching package: 'lubridate'## The following objects are masked from 'package:base':
##
## date, intersect, setdiff, unionda <- read_table("D:/齐安静 教学/时间序列分析/北大/ftsdata/q-gdp-ukcaus.txt")##
## ── Column specification ────────────────────────────────────────────────────────
## cols(
## year = col_double(),
## mon = col_double(),
## uk = col_double(),
## ca = col_double(),
## us = col_double()
## )ts.gdp3 <- ts(as.matrix(da[,c("uk", "ca", "us")]),
start=c(1980,1), frequency=4)
ts.gdp3r <- diff(log(ts.gdp3))*100- 三个国家的季度GDP对数增长率的时间序列图：
plot(as.xts(ts.gdp3r), type="l",
multi.panel=TRUE, theme="white",
main="英国、加拿大、美国GDP的季度增长率(%)",
major.ticks="years",
grid.ticks.on = "years")
- 利用MTS包的VAR()函数估计VAR(1)模型：
library(MTS, quietly = TRUE)
Z <- coredata(as.xts(ts.gdp3r))
m1.gdp3r <- MTS::VAR(Z, 1)## Constant term:
## Estimates: 0.1713324 0.1182869 0.2785892
## Std.Error: 0.06790162 0.07193106 0.07877173
## AR coefficient matrix
## AR( 1 )-matrix
## [,1] [,2] [,3]
## [1,] 0.434 0.189 0.0373
## [2,] 0.185 0.245 0.3917
## [3,] 0.322 0.182 0.1674
## standard error
## [,1] [,2] [,3]
## [1,] 0.0811 0.0827 0.0872
## [2,] 0.0859 0.0877 0.0923
## [3,] 0.0940 0.0960 0.1011
##
## Residuals cov-mtx:
## [,1] [,2] [,3]
## [1,] 0.28933472 0.01965508 0.06619853
## [2,] 0.01965508 0.32469319 0.16862723
## [3,] 0.06619853 0.16862723 0.38938665
##
## det(SSE) = 0.02721916
## AIC = -3.459834
## BIC = -3.256196
## HQ = -3.377107- 估计VAR(2)模型：
Z <- coredata(as.xts(ts.gdp3r))
m2.gdp3r <- MTS::VAR(Z, 2)## Constant term:
## Estimates: 0.1258163 0.1231581 0.2895581
## Std.Error: 0.07266338 0.07382941 0.0816888
## AR coefficient matrix
## AR( 1 )-matrix
## [,1] [,2] [,3]
## [1,] 0.393 0.103 0.0521
## [2,] 0.351 0.338 0.4691
## [3,] 0.491 0.240 0.2356
## standard error
## [,1] [,2] [,3]
## [1,] 0.0934 0.0984 0.0911
## [2,] 0.0949 0.1000 0.0926
## [3,] 0.1050 0.1106 0.1024
## AR( 2 )-matrix
## [,1] [,2] [,3]
## [1,] 0.0566 0.106 0.01889
## [2,] -0.1914 -0.175 -0.00868
## [3,] -0.3120 -0.131 0.08531
## standard error
## [,1] [,2] [,3]
## [1,] 0.0924 0.0876 0.0938
## [2,] 0.0939 0.0890 0.0953
## [3,] 0.1038 0.0984 0.1055
##
## Residuals cov-mtx:
## [,1] [,2] [,3]
## [1,] 0.28244420 0.02654091 0.07435286
## [2,] 0.02654091 0.29158166 0.13948786
## [3,] 0.07435286 0.13948786 0.35696571
##
## det(SSE) = 0.02258974
## AIC = -3.502259
## BIC = -3.094982
## HQ = -3.336804利用MTS包的VARorder函数可以计算VAR定阶的统计量检验和各种信息准则
library(MTS, quietly = TRUE)
Z <- coredata(as.xts(ts.gdp3r))
m3.gdp3r <- VARorder(Z/100)## selected order: aic = 2
## selected order: bic = 1
## selected order: hq = 1
## Summary table:
## p AIC BIC HQ M(p) p-value
## [1,] 0 -30.9560 -30.9560 -30.9560 0.0000 0.0000
## [2,] 1 -31.8830 -31.6794 -31.8003 115.1329 0.0000
## [3,] 2 -31.9643 -31.5570 -31.7988 23.5389 0.0051
## [4,] 3 -31.9236 -31.3127 -31.6754 10.4864 0.3126
## [5,] 4 -31.8971 -31.0826 -31.5662 11.5767 0.2382
## [6,] 5 -31.7818 -30.7636 -31.3682 2.7406 0.9737
## [7,] 6 -31.7112 -30.4893 -31.2148 6.7822 0.6598
## [8,] 7 -31.6180 -30.1925 -31.0389 4.5469 0.8719
## [9,] 8 -31.7570 -30.1279 -31.0952 24.4833 0.0036
## [10,] 9 -31.6897 -29.8569 -30.9451 6.4007 0.6992
## [11,] 10 -31.5994 -29.5630 -30.7721 4.3226 0.8889
## [12,] 11 -31.6036 -29.3636 -30.6936 11.4922 0.2435
## [13,] 12 -31.6183 -29.1746 -30.6255 11.8168 0.2238
## [14,] 13 -31.6718 -29.0245 -30.5964 14.1266 0.1179- 还可以用vars包的VAR()函数估计。 估计VAR(1)：
da1 <- coredata(as.xts(ts.gdp3r))
var1 <- vars::VAR(da1, p=1)
summary(var1)##
## VAR Estimation Results:
## =========================
## Endogenous variables: uk, ca, us
## Deterministic variables: const
## Sample size: 124
## Log Likelihood: -304.407
## Roots of the characteristic polynomial:
## 0.7091 0.08735 0.05004
## Call:
## vars::VAR(y = da1, p = 1)
##
##
## Estimation results for equation uk:
## ===================================
## uk = uk.l1 + ca.l1 + us.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## uk.l1 0.43435 0.08106 5.358 4.12e-07 ***
## ca.l1 0.18888 0.08275 2.282 0.0242 *
## us.l1 0.03727 0.08716 0.428 0.6697
## const 0.17133 0.06790 2.523 0.0129 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.5468 on 120 degrees of freedom
## Multiple R-Squared: 0.3687, Adjusted R-squared: 0.3529
## F-statistic: 23.36 on 3 and 120 DF, p-value: 5.596e-12
##
##
## Estimation results for equation ca:
## ===================================
## ca = uk.l1 + ca.l1 + us.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## uk.l1 0.18499 0.08587 2.154 0.0332 *
## ca.l1 0.24475 0.08766 2.792 0.0061 **
## us.l1 0.39166 0.09233 4.242 4.38e-05 ***
## const 0.11829 0.07193 1.644 0.1027
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.5792 on 120 degrees of freedom
## Multiple R-Squared: 0.4685, Adjusted R-squared: 0.4552
## F-statistic: 35.26 on 3 and 120 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation us:
## ===================================
## us = uk.l1 + ca.l1 + us.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## uk.l1 0.32153 0.09404 3.419 0.000859 ***
## ca.l1 0.18196 0.09600 1.895 0.060438 .
## us.l1 0.16740 0.10111 1.656 0.100410
## const 0.27859 0.07877 3.537 0.000577 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.6343 on 120 degrees of freedom
## Multiple R-Squared: 0.3044, Adjusted R-squared: 0.287
## F-statistic: 17.5 on 3 and 120 DF, p-value: 1.725e-09
##
##
##
## Covariance matrix of residuals:
## uk ca us
## uk 0.29898 0.02031 0.06841
## ca 0.02031 0.33552 0.17425
## us 0.06841 0.17425 0.40237
##
## Correlation matrix of residuals:
## uk ca us
## uk 1.00000 0.06413 0.1972
## ca 0.06413 1.00000 0.4742
## us 0.19722 0.47424 1.0000- 利用MTS包的VARorder函数可以计算VAR定阶的统计量检验和各种信息准则：
library(MTS, quietly = TRUE)
Z <- coredata(as.xts(ts.gdp3r))
m3.gdp3r <- VARorder(Z/100)## selected order: aic = 2
## selected order: bic = 1
## selected order: hq = 1
## Summary table:
## p AIC BIC HQ M(p) p-value
## [1,] 0 -30.9560 -30.9560 -30.9560 0.0000 0.0000
## [2,] 1 -31.8830 -31.6794 -31.8003 115.1329 0.0000
## [3,] 2 -31.9643 -31.5570 -31.7988 23.5389 0.0051
## [4,] 3 -31.9236 -31.3127 -31.6754 10.4864 0.3126
## [5,] 4 -31.8971 -31.0826 -31.5662 11.5767 0.2382
## [6,] 5 -31.7818 -30.7636 -31.3682 2.7406 0.9737
## [7,] 6 -31.7112 -30.4893 -31.2148 6.7822 0.6598
## [8,] 7 -31.6180 -30.1925 -31.0389 4.5469 0.8719
## [9,] 8 -31.7570 -30.1279 -31.0952 24.4833 0.0036
## [10,] 9 -31.6897 -29.8569 -30.9451 6.4007 0.6992
## [11,] 10 -31.5994 -29.5630 -30.7721 4.3226 0.8889
## [12,] 11 -31.6036 -29.3636 -30.6936 11.4922 0.2435
## [13,] 12 -31.6183 -29.1746 -30.6255 11.8168 0.2238
## [14,] 13 -31.6718 -29.0245 -30.5964 14.1266 0.1179- 还可以用vars包的VAR()函数估计。 估计VAR(1)
da1 <- coredata(as.xts(ts.gdp3r))
var1 <- vars::VAR(da1, p=1)
summary(var1)##
## VAR Estimation Results:
## =========================
## Endogenous variables: uk, ca, us
## Deterministic variables: const
## Sample size: 124
## Log Likelihood: -304.407
## Roots of the characteristic polynomial:
## 0.7091 0.08735 0.05004
## Call:
## vars::VAR(y = da1, p = 1)
##
##
## Estimation results for equation uk:
## ===================================
## uk = uk.l1 + ca.l1 + us.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## uk.l1 0.43435 0.08106 5.358 4.12e-07 ***
## ca.l1 0.18888 0.08275 2.282 0.0242 *
## us.l1 0.03727 0.08716 0.428 0.6697
## const 0.17133 0.06790 2.523 0.0129 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.5468 on 120 degrees of freedom
## Multiple R-Squared: 0.3687, Adjusted R-squared: 0.3529
## F-statistic: 23.36 on 3 and 120 DF, p-value: 5.596e-12
##
##
## Estimation results for equation ca:
## ===================================
## ca = uk.l1 + ca.l1 + us.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## uk.l1 0.18499 0.08587 2.154 0.0332 *
## ca.l1 0.24475 0.08766 2.792 0.0061 **
## us.l1 0.39166 0.09233 4.242 4.38e-05 ***
## const 0.11829 0.07193 1.644 0.1027
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.5792 on 120 degrees of freedom
## Multiple R-Squared: 0.4685, Adjusted R-squared: 0.4552
## F-statistic: 35.26 on 3 and 120 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation us:
## ===================================
## us = uk.l1 + ca.l1 + us.l1 + const
##
## Estimate Std. Error t value Pr(>|t|)
## uk.l1 0.32153 0.09404 3.419 0.000859 ***
## ca.l1 0.18196 0.09600 1.895 0.060438 .
## us.l1 0.16740 0.10111 1.656 0.100410
## const 0.27859 0.07877 3.537 0.000577 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.6343 on 120 degrees of freedom
## Multiple R-Squared: 0.3044, Adjusted R-squared: 0.287
## F-statistic: 17.5 on 3 and 120 DF, p-value: 1.725e-09
##
##
##
## Covariance matrix of residuals:
## uk ca us
## uk 0.29898 0.02031 0.06841
## ca 0.02031 0.33552 0.17425
## us 0.06841 0.17425 0.40237
##
## Correlation matrix of residuals:
## uk ca us
## uk 1.00000 0.06413 0.1972
## ca 0.06413 1.00000 0.4742
## us 0.19722 0.47424 1.0000- 用AIC自动选择阶，指定ic=“AIC”和lag.max=5:
da1 <- coredata(as.xts(ts.gdp3r))
var2 <- vars::VAR(da1, ic="AIC", lag.max=5)
summary(var2)##
## VAR Estimation Results:
## =========================
## Endogenous variables: uk, ca, us
## Deterministic variables: const
## Sample size: 121
## Log Likelihood: -252.647
## Roots of the characteristic polynomial:
## 0.785 0.7516 0.7516 0.7336 0.7336 0.6144 0.6144 0.5679 0.5679 0.5165 0.5122 0.5122
## Call:
## vars::VAR(y = da1, lag.max = 5, ic = "AIC")
##
##
## Estimation results for equation uk:
## ===================================
## uk = uk.l1 + ca.l1 + us.l1 + uk.l2 + ca.l2 + us.l2 + uk.l3 + ca.l3 + us.l3 + uk.l4 + ca.l4 + us.l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## uk.l1 0.515685 0.095298 5.411 3.8e-07 ***
## ca.l1 0.071863 0.094459 0.761 0.44844
## us.l1 0.063904 0.088668 0.721 0.47264
## uk.l2 -0.050370 0.101228 -0.498 0.61979
## ca.l2 0.160043 0.098620 1.623 0.10754
## us.l2 -0.001984 0.095495 -0.021 0.98346
## uk.l3 0.052363 0.102949 0.509 0.61205
## ca.l3 -0.278830 0.098341 -2.835 0.00547 **
## us.l3 0.141145 0.093728 1.506 0.13501
## uk.l4 0.040130 0.091050 0.441 0.66028
## ca.l4 0.261731 0.086947 3.010 0.00325 **
## us.l4 -0.246485 0.088234 -2.794 0.00617 **
## const 0.147957 0.074786 1.978 0.05043 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.5013 on 108 degrees of freedom
## Multiple R-Squared: 0.478, Adjusted R-squared: 0.42
## F-statistic: 8.242 on 12 and 108 DF, p-value: 7.84e-11
##
##
## Estimation results for equation ca:
## ===================================
## ca = uk.l1 + ca.l1 + us.l1 + uk.l2 + ca.l2 + us.l2 + uk.l3 + ca.l3 + us.l3 + uk.l4 + ca.l4 + us.l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## uk.l1 0.37782 0.10199 3.705 0.000336 ***
## ca.l1 0.31603 0.10109 3.126 0.002276 **
## us.l1 0.40963 0.09489 4.317 3.52e-05 ***
## uk.l2 -0.17399 0.10834 -1.606 0.111193
## ca.l2 -0.25407 0.10554 -2.407 0.017771 *
## us.l2 0.06295 0.10220 0.616 0.539247
## uk.l3 0.09615 0.11018 0.873 0.384757
## ca.l3 0.12035 0.10525 1.143 0.255362
## us.l3 0.01366 0.10031 0.136 0.891925
## uk.l4 0.07470 0.09744 0.767 0.445006
## ca.l4 -0.09031 0.09305 -0.971 0.333959
## us.l4 -0.09781 0.09443 -1.036 0.302622
## const 0.07757 0.08004 0.969 0.334593
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.5365 on 108 degrees of freedom
## Multiple R-Squared: 0.5661, Adjusted R-squared: 0.5179
## F-statistic: 11.74 on 12 and 108 DF, p-value: 8.189e-15
##
##
## Estimation results for equation us:
## ===================================
## us = uk.l1 + ca.l1 + us.l1 + uk.l2 + ca.l2 + us.l2 + uk.l3 + ca.l3 + us.l3 + uk.l4 + ca.l4 + us.l4 + const
##
## Estimate Std. Error t value Pr(>|t|)
## uk.l1 0.51905 0.10938 4.745 6.41e-06 ***
## ca.l1 0.17304 0.10841 1.596 0.1134
## us.l1 0.15039 0.10177 1.478 0.1424
## uk.l2 -0.21784 0.11618 -1.875 0.0635 .
## ca.l2 -0.15931 0.11319 -1.407 0.1622
## us.l2 0.22561 0.10960 2.058 0.0420 *
## uk.l3 0.04783 0.11816 0.405 0.6864
## ca.l3 -0.07856 0.11287 -0.696 0.4879
## us.l3 0.07384 0.10758 0.686 0.4939
## uk.l4 0.15409 0.10450 1.475 0.1433
## ca.l4 -0.15182 0.09979 -1.521 0.1311
## us.l4 -0.05350 0.10127 -0.528 0.5984
## const 0.23868 0.08584 2.781 0.0064 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.5754 on 108 degrees of freedom
## Multiple R-Squared: 0.4531, Adjusted R-squared: 0.3923
## F-statistic: 7.457 on 12 and 108 DF, p-value: 7.522e-10
##
##
##
## Covariance matrix of residuals:
## uk ca us
## uk 0.25130 0.04912 0.1001
## ca 0.04912 0.28783 0.1084
## us 0.10009 0.10840 0.3310
##
## Correlation matrix of residuals:
## uk ca us
## uk 1.0000 0.1826 0.3470
## ca 0.1826 1.0000 0.3512
## us 0.3470 0.3512 1.0000- 对例3.1的三个国家的GDP季度增速建模， VAR(2)模型的残差的多元混成检验程序如下：
resi <- m2.gdp3r$residuals
MTS::mq(resi, adj=3^2 * 2+3)## Ljung-Box Statistics:
## m Q(m) df p-value
## [1,] 1.000 0.816 -12.000 1.00
## [2,] 2.000 3.978 -3.000 1.00
## [3,] 3.000 16.665 6.000 1.00
## [4,] 4.000 35.122 15.000 0.00
## [5,] 5.000 38.189 24.000 0.03
## [6,] 6.000 41.239 33.000 0.15
## [7,] 7.000 47.621 42.000 0.25
## [8,] 8.000 61.677 51.000 0.15
## [9,] 9.000 67.366 60.000 0.24
## [10,] 10.000 76.930 69.000 0.24
## [11,] 11.000 81.567 78.000 0.37
## [12,] 12.000 93.112 87.000 0.31
## [13,] 13.000 105.327 96.000 0.24
## [14,] 14.000 116.279 105.000 0.21
## [15,] 15.000 128.974 114.000 0.16
## [16,] 16.000 134.704 123.000 0.22
## [17,] 17.000 138.552 132.000 0.33
## [18,] 18.000 146.256 141.000 0.36
## [19,] 19.000 162.418 150.000 0.23
## [20,] 20.000 171.948 159.000 0.23
## [21,] 21.000 174.913 168.000 0.34
## [22,] 22.000 182.056 177.000 0.38
## [23,] 23.000 190.276 186.000 0.40
## [24,] 24.000 202.141 195.000 0.35
- 例3.3 对三个国家的GDP对数增长率的VAR(2)模型进行化简。
library(MTS, quietly = TRUE)
Z <- coredata(as.xts(ts.gdp3r))
cat("================ Full model:\n")## ================ Full model:mods1.gdp3r <- VAR(Z, 2)## Constant term:
## Estimates: 0.1258163 0.1231581 0.2895581
## Std.Error: 0.07266338 0.07382941 0.0816888
## AR coefficient matrix
## AR( 1 )-matrix
## [,1] [,2] [,3]
## [1,] 0.393 0.103 0.0521
## [2,] 0.351 0.338 0.4691
## [3,] 0.491 0.240 0.2356
## standard error
## [,1] [,2] [,3]
## [1,] 0.0934 0.0984 0.0911
## [2,] 0.0949 0.1000 0.0926
## [3,] 0.1050 0.1106 0.1024
## AR( 2 )-matrix
## [,1] [,2] [,3]
## [1,] 0.0566 0.106 0.01889
## [2,] -0.1914 -0.175 -0.00868
## [3,] -0.3120 -0.131 0.08531
## standard error
## [,1] [,2] [,3]
## [1,] 0.0924 0.0876 0.0938
## [2,] 0.0939 0.0890 0.0953
## [3,] 0.1038 0.0984 0.1055
##
## Residuals cov-mtx:
## [,1] [,2] [,3]
## [1,] 0.28244420 0.02654091 0.07435286
## [2,] 0.02654091 0.29158166 0.13948786
## [3,] 0.07435286 0.13948786 0.35696571
##
## det(SSE) = 0.02258974
## AIC = -3.502259
## BIC = -3.094982
## HQ = -3.336804cat("\n\n================ Restricted model:\n")##
##
## ================ Restricted model:mods2.gdp3r <- refVAR(mods1.gdp3r, thres=1.96)## Constant term:
## Estimates: 0.1628247 0 0.2827525
## Std.Error: 0.06814101 0 0.07972864
## AR coefficient matrix
## AR( 1 )-matrix
## [,1] [,2] [,3]
## [1,] 0.467 0.207 0.000
## [2,] 0.334 0.270 0.496
## [3,] 0.468 0.225 0.232
## standard error
## [,1] [,2] [,3]
## [1,] 0.0790 0.0686 0.0000
## [2,] 0.0921 0.0875 0.0913
## [3,] 0.1027 0.0963 0.1023
## AR( 2 )-matrix
## [,1] [,2] [,3]
## [1,] 0.000 0 0
## [2,] -0.197 0 0
## [3,] -0.301 0 0
## standard error
## [,1] [,2] [,3]
## [1,] 0.0000 0 0
## [2,] 0.0921 0 0
## [3,] 0.1008 0 0
##
## Residuals cov-mtx:
## [,1] [,2] [,3]
## [1,] 0.29003669 0.01803456 0.07055856
## [2,] 0.01803456 0.30802503 0.14598345
## [3,] 0.07055856 0.14598345 0.36268779
##
## det(SSE) = 0.02494104
## AIC = -3.531241
## BIC = -3.304976
## HQ = -3.439321- 对这个简化的模型的检验， 可以提取残差后用MTS::mq()函数检验， 这时自由度缩减个数由原来的\(3^2 \times 2+3=21\)个， 减少到12个， 因为模型中约束了9个参数等于零。 MTS包还提供了一个MTSdiag()函数， 输入模型结果和adj=自由度缩减个数， 作残差的CCM估计表、图和残差的多元混成检验：
library(MTS, quietly = TRUE)
MTSdiag(mods2.gdp3r, adj=12)## [1] "Covariance matrix:"
## uk ca us
## uk 0.2924 0.0182 0.0711
## ca 0.0182 0.3084 0.1472
## us 0.0711 0.1472 0.3657
## CCM at lag: 0
## [,1] [,2] [,3]
## [1,] 1.0000 0.0605 0.218
## [2,] 0.0605 1.0000 0.438
## [3,] 0.2175 0.4382 1.000
## Simplified matrix:
## CCM at lag: 1
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## 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 1.78 -3.00 1.00
## [2,] 2.00 12.41 6.00 1.00
## [3,] 3.00 22.60 15.00 0.09
## [4,] 4.00 37.71 24.00 0.04
## [5,] 5.00 41.65 33.00 0.14
## [6,] 6.00 44.95 42.00 0.35
## [7,] 7.00 51.50 51.00 0.45
## [8,] 8.00 64.87 60.00 0.31
## [9,] 9.00 72.50 69.00 0.36
## [10,] 10.00 81.58 78.00 0.37
## [11,] 11.00 86.12 87.00 0.51
## [12,] 12.00 98.08 96.00 0.42
## [13,] 13.00 112.31 105.00 0.30
## [14,] 14.00 121.89 114.00 0.29
## [15,] 15.00 134.58 123.00 0.22
## [16,] 16.00 139.16 132.00 0.32
## [17,] 17.00 145.85 141.00 0.37
## [18,] 18.00 152.56 150.00 0.43
## [19,] 19.00 165.91 159.00 0.34
## [20,] 20.00 175.22 168.00 0.34
## [21,] 21.00 180.56 177.00 0.41
## [22,] 22.00 187.40 186.00 0.46
## [23,] 23.00 193.78 195.00 0.51
## [24,] 24.00 204.65 204.00 0.47
## Hit Enter to obtain residual plots:
- 残差的相关阵
cor(mods2.gdp3r$residuals)## [,1] [,2] [,3]
## [1,] 1.00000000 0.06054285 0.2175489
## [2,] 0.06054285 1.00000000 0.4382489
## [3,] 0.21754885 0.43824891 1.0000000例3.4
- 检验美国的GDP季度增长率是不是英国和加拿大的增长率的单向格兰杰原因。
GrangerTest(Z, p=2, locInput=3)## Number of targeted zero parameters: 4
## Chi-square test for Granger Causality and p-value: 27.2262 1.789152e-05- 对加拿大进行检验：
GrangerTest(Z, p=2, locInput=2)## Number of targeted zero parameters: 4
## Chi-square test for Granger Causality and p-value: 48.83871 6.309173e-10- 英国进行检验
GrangerTest(Z, p=2, locInput=1)## Number of targeted zero parameters: 4
## Chi-square test for Granger Causality and p-value: 8.948851 0.06239076
## Constant term:
## Estimates: 0.2104448 0.1231581 0.2895581
## Std.Error: 0.06685632 0.07382941 0.0816888
## AR coefficient matrix
## AR( 1 )-matrix
## [,1] [,2] [,3]
## [1,] 0.473 0.000 0.000
## [2,] 0.351 0.338 0.469
## [3,] 0.491 0.240 0.236
## standard error
## [,1] [,2] [,3]
## [1,] 0.0899 0.000 0.0000
## [2,] 0.0949 0.100 0.0926
## [3,] 0.1050 0.111 0.1024
## AR( 2 )-matrix
## [,1] [,2] [,3]
## [1,] 0.151 0.000 0.00000
## [2,] -0.191 -0.175 -0.00868
## [3,] -0.312 -0.131 0.08531
## standard error
## [,1] [,2] [,3]
## [1,] 0.0859 0.0000 0.0000
## [2,] 0.0939 0.0890 0.0953
## [3,] 0.1038 0.0984 0.1055
##
## Residuals cov-mtx:
## [,1] [,2] [,3]
## [1,] 0.30423344 0.02654091 0.07435286
## [2,] 0.02654091 0.29158166 0.13948786
## [3,] 0.07435286 0.13948786 0.35696571
##
## det(SSE) = 0.02443371
## AIC = -3.487791
## BIC = -3.17102
## HQ = -3.359104- 例3.5
例3.1中建立的三个国家的GDP增速的VAR(2)模型是基于1980年第二季度到2011年第二季度的数据， 用建立的模型进行超前1到8步预测。 第一个预测对应2011年第三季度， 最后一个预测对应2013年第二季度。
library(MTS, quietly = TRUE)
VARpred(m2.gdp3r, 8)## orig 125
## Forecasts at origin: 125
## uk ca us
## [1,] 0.3129 0.05166 0.1660
## [2,] 0.2647 0.31687 0.4889
## [3,] 0.3143 0.48231 0.5205
## [4,] 0.3839 0.53053 0.5998
## [5,] 0.4412 0.56978 0.6297
## [6,] 0.4799 0.59478 0.6530
## [7,] 0.5068 0.60967 0.6630
## [8,] 0.5247 0.61689 0.6688
## Standard Errors of predictions:
## [,1] [,2] [,3]
## [1,] 0.5315 0.5400 0.5975
## [2,] 0.5804 0.7165 0.7077
## [3,] 0.6202 0.7672 0.7345
## [4,] 0.6484 0.7785 0.7442
## [5,] 0.6629 0.7824 0.7475
## [6,] 0.6692 0.7838 0.7484
## [7,] 0.6719 0.7842 0.7486
## [8,] 0.6729 0.7843 0.7487
## Root mean square errors of predictions:
## [,1] [,2] [,3]
## [1,] 0.5461 0.5549 0.6140
## [2,] 0.6001 0.7799 0.7499
## [3,] 0.6365 0.7879 0.7456
## [4,] 0.6601 0.7832 0.7484
## [5,] 0.6689 0.7841 0.7488
## [6,] 0.6719 0.7844 0.7488
## [7,] 0.6730 0.7844 0.7487
## [8,] 0.6734 0.7844 0.7487- 多步预测会趋近到序列均值，计算序列均值：
Z <- coredata(as.xts(ts.gdp3r))
colMeans(Z)## uk ca us
## 0.5223092 0.6153672 0.6473996可以看出在超前8步时的点预测值很接近于序列的均值。
不管是不考虑参数估计误差的标准误差(Standard errors of predictions) 还是考虑参数估计误差的标准误差(Root mean squared errors of predictions)， 超前\(l\)步预报当\(l\to\infty\)时都应该趋于序列的标准差。 计算序列的样本标准差：
Z <- coredata(as.xts(ts.gdp3r))
apply(Z, 2, sd)## uk ca us
## 0.7086442 0.7851955 0.7872912例3.6
- 对例3.2中建立的简化的VAR(2)模型作脉冲响应函数图和累积响应图。 三个分量两两的图形共\(3 \times 3 \times 2\)幅。
library(MTS, quietly = TRUE)
VARMAirf(Phi = mods2.gdp3r$Phi,
Sigma = mods2.gdp3r$Sig,
orth=FALSE)
## Press return to continue
例3.7
- 对例3.2中建立的简化的VAR(2)模型作正交化的脉冲响应函数图和累积响应图。
library(MTS, quietly = TRUE)
VARMAirf(Phi = mods2.gdp3r$Phi,
Sigma = mods2.gdp3r$Sig,
orth=TRUE)
## Press return to continue