MFE5209 Vector AutoRegressive
Wu Fuheng
Wednesday, October 08, 2014
Problem 1
a)
library(MTS)
C=matrix(c(0.8,-0.3,0.4,0.6), nrow = 2)
S=matrix(c(2,0.5,0.5,1.0), nrow = 2)
m1 = VARMAsim(300, arlags = c(1), phi = C, sigma = S)
yt = m1$series
plot(yt)
b)
ccm(yt, lags = 5)## [1] "Covariance matrix:"
## [,1] [,2]
## [1,] 5.8253 -0.0456
## [2,] -0.0456 2.7397
## CCM at lag: 0
## [,1] [,2]
## [1,] 1.0000 -0.0114
## [2,] -0.0114 1.0000
## Simplified matrix:
## CCM at lag: 1
## + +
## - +
## CCM at lag: 2
## + +
## - +
## CCM at lag: 3
## + +
## - +
## CCM at lag: 4
## . +
## - .
## CCM at lag: 5
## - +
## - -
## Hit Enter for p-value plot of individual ccm:
c)
mq(yt, lag=10)## Ljung-Box Statistics:
## m Q(m) df p-value
## [1,] 1 381 4 0
## [2,] 2 652 8 0
## [3,] 3 856 12 0
## [4,] 4 1005 16 0
## [5,] 5 1114 20 0
## [6,] 6 1188 24 0
## [7,] 7 1248 28 0
## [8,] 8 1292 32 0
## [9,] 9 1328 36 0
## [10,] 10 1357 40 0
We can see all p-value is 0, so we can reject \(H_0\) of no cross-correlations with 5% significance level.
d)
m2 = VARMAsim(200,malags = c(1),theta = C,sigma = S);
zt = m2$series
ccm(zt, lags = 2)## [1] "Covariance matrix:"
## [,1] [,2]
## [1,] 4.430 0.233
## [2,] 0.233 1.332
## CCM at lag: 0
## [,1] [,2]
## [1,] 1.0000 0.0961
## [2,] 0.0961 1.0000
## Simplified matrix:
## CCM at lag: 1
## - .
## . -
## CCM at lag: 2
## + .
## . .
## Hit Enter for p-value plot of individual ccm:
mq(zt, lag=10)## Ljung-Box Statistics:
## m Q(m) df p-value
## [1,] 1 109 4 0
## [2,] 2 120 8 0
## [3,] 3 126 12 0
## [4,] 4 132 16 0
## [5,] 5 133 20 0
## [6,] 6 136 24 0
## [7,] 7 137 28 0
## [8,] 8 139 32 0
## [9,] 9 143 36 0
## [10,] 10 152 40 0
Problem 2
The file q-fdebt.txt contains the U.S. quarterly federal debts held by (a)foreign and international investors, (b) federal reserve banks, and (c) the public.The data are from the Federal Reserve Bank of St. Louis, from 1970 to 2012 for 171 observations, and not seasonally adjusted. The debts are in billions of dollars. Take the log transformation and the first difference for each time series. Let zt be the differenced log series.
a) Plot the time series \(z_t\).
library(MTS)
da=read.table("q-fdebt.txt",header=T)
debt=log(da[,3:5])
tdx=da[,1]+da[,2]/12
#MTSplot(debt,tdx);
zt=diffM(debt);
MTSplot(zt,tdx[-1])
b) Obtain the first five lags of sample CCMs of \(z_t\).
ccm(zt, lags = 5)## [1] "Covariance matrix:"
## hbfin hbfrbn hbpun
## hbfin 0.002745 -0.000175 0.000350
## hbfrbn -0.000175 0.002709 0.000104
## hbpun 0.000350 0.000104 0.000488
## CCM at lag: 0
## [,1] [,2] [,3]
## [1,] 1.0000 -0.0642 0.3027
## [2,] -0.0642 1.0000 0.0906
## [3,] 0.3027 0.0906 1.0000
## Simplified matrix:
## CCM at lag: 1
## + . .
## . + .
## + . +
## CCM at lag: 2
## + . .
## . . +
## . . +
## CCM at lag: 3
## + . .
## . . +
## . . +
## CCM at lag: 4
## + . .
## . . .
## . . +
## CCM at lag: 5
## . . .
## . - .
## . . +
## Hit Enter for p-value plot of individual ccm:
c) Use a VAR model to fit the data \(z_t\), with an appropriate order selection. Justify your order choice. Interpret the fitted model.
VARorder(zt)## selected order: aic = 6
## selected order: bic = 1
## selected order: hq = 5
## Summary table:
## p AIC BIC HQ M(p) p-value
## [1,] 0 -20.2022 -20.2022 -20.2022 0.0000 0.0000
## [2,] 1 -20.6589 -20.4929 -20.5915 85.7893 0.0000
## [3,] 2 -20.6950 -20.3630 -20.5603 21.2235 0.0117
## [4,] 3 -20.7588 -20.2607 -20.5567 24.8554 0.0031
## [5,] 4 -21.0911 -20.4271 -20.8216 62.8856 0.0000
## [6,] 5 -21.2401 -20.4100 -20.9033 35.8101 0.0000
## [7,] 6 -21.2942 -20.2981 -20.8900 21.9979 0.0089
## [8,] 7 -21.2443 -20.0822 -20.7727 7.5246 0.5827
## [9,] 8 -21.2225 -19.8944 -20.6836 11.0615 0.2715
## [10,] 9 -21.2600 -19.7659 -20.6537 18.4248 0.0306
## [11,] 10 -21.1983 -19.5382 -20.5246 5.5442 0.7845
## [12,] 11 -21.2105 -19.3844 -20.4695 14.4641 0.1067
## [13,] 12 -21.2087 -19.2166 -20.4003 12.4426 0.1895
## [14,] 13 -21.2025 -19.0443 -20.3267 11.6032 0.2366v=VAR(zt,p=6)## Constant term:
## Estimates: 0.0128167 -0.002779114 0.006766811
## Std.Error: 0.005855736 0.00658421 0.002145951
## AR coefficient matrix
## AR( 1 )-matrix
## [,1] [,2] [,3]
## [1,] 0.1108 -0.0809 0.2687
## [2,] 0.0518 0.4170 0.0199
## [3,] 0.0251 -0.0849 0.3841
## standard error
## [,1] [,2] [,3]
## [1,] 0.0863 0.0709 0.2459
## [2,] 0.0970 0.0797 0.2765
## [3,] 0.0316 0.0260 0.0901
## AR( 2 )-matrix
## [,1] [,2] [,3]
## [1,] 0.119139 0.0637 -0.3936
## [2,] 0.056577 0.0576 0.2447
## [3,] -0.000517 -0.0512 0.0929
## standard error
## [,1] [,2] [,3]
## [1,] 0.0835 0.0790 0.2477
## [2,] 0.0939 0.0888 0.2785
## [3,] 0.0306 0.0289 0.0908
## AR( 3 )-matrix
## [,1] [,2] [,3]
## [1,] 0.04859 -0.0830 0.0328
## [2,] 0.03783 -0.0854 0.5337
## [3,] 0.00947 0.0431 0.0409
## standard error
## [,1] [,2] [,3]
## [1,] 0.0815 0.0771 0.1913
## [2,] 0.0916 0.0867 0.2151
## [3,] 0.0299 0.0283 0.0701
## AR( 4 )-matrix
## [,1] [,2] [,3]
## [1,] 0.1351 -0.0978 0.3302
## [2,] -0.2165 -0.0268 -0.0845
## [3,] -0.0124 -0.0465 0.7027
## standard error
## [,1] [,2] [,3]
## [1,] 0.0809 0.0765 0.1945
## [2,] 0.0909 0.0860 0.2187
## [3,] 0.0296 0.0280 0.0713
## AR( 5 )-matrix
## [,1] [,2] [,3]
## [1,] -0.1806 0.0837 0.0546
## [2,] -0.0330 -0.2288 -0.1699
## [3,] -0.0825 0.0151 -0.2283
## standard error
## [,1] [,2] [,3]
## [1,] 0.0799 0.0751 0.2531
## [2,] 0.0899 0.0845 0.2846
## [3,] 0.0293 0.0275 0.0927
## AR( 6 )-matrix
## [,1] [,2] [,3]
## [1,] 0.0558 0.000863 0.1647
## [2,] 0.1469 0.321765 0.0126
## [3,] 0.0221 0.003439 -0.1248
## standard error
## [,1] [,2] [,3]
## [1,] 0.0780 0.0724 0.240
## [2,] 0.0877 0.0814 0.270
## [3,] 0.0286 0.0265 0.088
##
## Residuals cov-mtx:
## [,1] [,2] [,3]
## [1,] 0.0013575684 -0.0002676928 0.0001986357
## [2,] -0.0002676928 0.0017163512 0.0000364333
## [3,] 0.0001986357 0.0000364333 0.0001823216
##
## det(SSE) = 3.383589e-10
## AIC = -21.17162
## BIC = -20.17554
## HQ = -20.76742d) Is the fitted model adequate? Draw the conclusion using the 5% significance level.
resi=v$residuals
mq(resi,adj=18)## Ljung-Box Statistics:
## m Q(m) df p-value
## [1,] 1.000 0.662 -9.000 1.00
## [2,] 2.000 1.803 0.000 1.00
## [3,] 3.000 5.426 9.000 0.80
## [4,] 4.000 9.182 18.000 0.96
## [5,] 5.000 13.157 27.000 0.99
## [6,] 6.000 16.077 36.000 1.00
## [7,] 7.000 22.219 45.000 1.00
## [8,] 8.000 31.981 54.000 0.99
## [9,] 9.000 36.715 63.000 1.00
## [10,] 10.000 40.591 72.000 1.00
## [11,] 11.000 53.223 81.000 0.99
## [12,] 12.000 62.088 90.000 0.99
## [13,] 13.000 73.361 99.000 0.97
## [14,] 14.000 77.671 108.000 0.99
## [15,] 15.000 82.331 117.000 0.99
## [16,] 16.000 97.883 126.000 0.97
## [17,] 17.000 105.248 135.000 0.97
## [18,] 18.000 110.913 144.000 0.98
## [19,] 19.000 115.581 153.000 0.99
## [20,] 20.000 121.451 162.000 0.99
## [21,] 21.000 129.525 171.000 0.99
## [22,] 22.000 131.996 180.000 1.00
## [23,] 23.000 137.719 189.000 1.00
## [24,] 24.000 141.346 198.000 1.00