ECON5513 Assignment 2
Cormac Gallagher
27 April 2017
Question 1
Excersie 1
A three-dimensional VAR(2)-process where the third variable does not Granger-cause the first variable was generated using the following model:
\[\left[ \begin{array}{c} y_{1t} \\ y_{2t} \\ y_{3t} \end{array} \right] = \left[ \begin{array}{c} 5 \\ 10 \\ 15 \end{array} \right] + \left[ \begin{array}{ccc} -0.5 & -0.1 & 0 \\ 0.2 & 0.5 & 0.2 \\ -0.4 & -0.6 & 0.1 \end{array} \right] \left[ \begin{array}{c} y_{1t-1} \\ y_{2t-1} \\ y_{3t-1} \end{array} \right] + \left[ \begin{array}{ccc} 0.3 & 0.1 & 0 \\ 0.1 & -0.3 & -0.1 \\ 0.3 & -0.2 & 0.3 \end{array} \right] \left[ \begin{array}{c} y_{1t-2} \\ y_{2t-2} \\ y_{3t-2} \end{array} \right] + \left[ \begin{array}{c} u_{1t} \\ u_{2t} \\ u_{3t} \end{array} \right]\]
In a three-equation model with 2 lags \(\{y_{3t}\}\) does not Granger-cause \(\{y_{1t}\}\), if and only if all of the coefficients of \(A_{13}(L)\) are equal to zero.
\[A_{13}^{(1)}=A_{13}^{(2)}=0\]
Setting lag polynomial A(L)
Apoly <- array(c(1.0, -0.5, 0.3, 0.0, 0.2, 0.1, 0.0, -0.4, 0.3,
0.0, -0.1, 0.1, 1.0, 0.5, -0.3, 0.0, -0.6, -0.2,
0.0, 0.0, 0.0, 0.0, 0.2, -0.1, 1.0, 0.1, 0.3),
c(3, 3, 3))Setting the corvariance to identity matrix
B <- diag(3)Setting constant terms to 5, 10, and 15
TRD <- c(5, 10, 15)Generating the VAR(2) model
var2 <- ARMA(A = Apoly, B = B, TREND = TRD)
var2## TREND= 5 10 15
## A(L) =
## 1-0.5L1+0.3L2 0-0.1L1+0.1L2 0
## 0+0.2L1+0.1L2 1+0.5L1-0.3L2 0+0.2L1-0.1L2
## 0-0.4L1+0.3L2 0-0.6L1-0.2L2 1+0.1L1+0.3L2
##
## B(L) =
## 1 0 0
## 0 1 0
## 0 0 1Excersie 2
Simulating 250 observations of the VAR(2) model:
varsim <- simulate(var2, sampleT = 250, noise = list(w = matrix(rnorm(750), nrow = 250, ncol = 3)),
rng = list(seed = c(123456))) Obtaining the generated series
vardat <- matrix(varsim$output, nrow = 250, ncol = 3)
colnames(vardat) <- c("y1", "y2", "y3") plot.ts(vardat, main = "VAR(2) Model Simulation", xlab = "Sample") 
Determining an appropriate lag order
VARselect(vardat, lag.max = 3, type = "const") ## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 2 2 2 2
##
## $criteria
## 1 2 3
## AIC(n) 0.5168466 -0.18138552 -0.14248269
## HQ(n) 0.5854899 -0.06125981 0.02912548
## SC(n) 0.6873432 0.11698353 0.28375881
## FPE(n) 1.6767462 0.83415172 0.86731778VARselect(vardat, lag.max = 3, type = NULL)## $selection
## AIC(n) HQ(n) SC(n) FPE(n)
## 2 2 2 2
##
## $criteria
## 1 2 3
## AIC(n) 0.5168466 -0.18138552 -0.14248269
## HQ(n) 0.5854899 -0.06125981 0.02912548
## SC(n) 0.6873432 0.11698353 0.28375881
## FPE(n) 1.6767462 0.83415172 0.86731778Estimating the model
varsimest <- VAR(vardat, p = 2, type = "const", season = NULL, exogen = NULL)
varsimest <- VAR(vardat, type = "const", lag.max = 3, ic = "SC") # Selection according to AICsummary(varsimest)##
## VAR Estimation Results:
## =========================
## Endogenous variables: y1, y2, y3
## Deterministic variables: const
## Sample size: 248
## Log Likelihood: -1014.579
## Roots of the characteristic polynomial:
## 0.8447 0.6935 0.6935 0.5674 0.5674 0.3909
## Call:
## VAR(y = vardat, type = "const", lag.max = 3, ic = "SC")
##
##
## Estimation results for equation y1:
## ===================================
## y1 = y1.l1 + y2.l1 + y3.l1 + y1.l2 + y2.l2 + y3.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## y1.l1 0.51441 0.06206 8.289 8.02e-15 ***
## y2.l1 0.15761 0.05978 2.636 0.00892 **
## y3.l1 0.03098 0.04884 0.634 0.52648
## y1.l2 -0.27445 0.06406 -4.285 2.65e-05 ***
## y2.l2 -0.01467 0.06798 -0.216 0.82929
## y3.l2 0.04417 0.04700 0.940 0.34830
## const 2.85013 1.25360 2.274 0.02387 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.9733 on 241 degrees of freedom
## Multiple R-Squared: 0.266, Adjusted R-squared: 0.2478
## F-statistic: 14.56 on 6 and 241 DF, p-value: 3.586e-14
##
##
## Estimation results for equation y2:
## ===================================
## y2 = y1.l1 + y2.l1 + y3.l1 + y1.l2 + y2.l2 + y3.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## y1.l1 -0.24622 0.06096 -4.039 7.23e-05 ***
## y2.l1 -0.49199 0.05873 -8.377 4.51e-15 ***
## y3.l1 -0.31946 0.04798 -6.659 1.85e-10 ***
## y1.l2 -0.04724 0.06293 -0.751 0.4536
## y2.l2 0.39694 0.06679 5.944 9.71e-09 ***
## y3.l2 0.10852 0.04617 2.350 0.0196 *
## const 10.98080 1.23153 8.916 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.9561 on 241 degrees of freedom
## Multiple R-Squared: 0.588, Adjusted R-squared: 0.5777
## F-statistic: 57.33 on 6 and 241 DF, p-value: < 2.2e-16
##
##
## Estimation results for equation y3:
## ===================================
## y3 = y1.l1 + y2.l1 + y3.l1 + y1.l2 + y2.l2 + y3.l2 + const
##
## Estimate Std. Error t value Pr(>|t|)
## y1.l1 0.37339 0.06099 6.123 3.71e-09 ***
## y2.l1 0.48029 0.05875 8.175 1.69e-14 ***
## y3.l1 -0.13324 0.04799 -2.776 0.00593 **
## y1.l2 -0.35776 0.06295 -5.683 3.80e-08 ***
## y2.l2 0.12328 0.06681 1.845 0.06621 .
## y3.l2 -0.37265 0.04619 -8.068 3.37e-14 ***
## const 18.11876 1.23195 14.707 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
##
## Residual standard error: 0.9565 on 241 degrees of freedom
## Multiple R-Squared: 0.5585, Adjusted R-squared: 0.5475
## F-statistic: 50.82 on 6 and 241 DF, p-value: < 2.2e-16
##
##
##
## Covariance matrix of residuals:
## y1 y2 y3
## y1 0.94726 0.09110 0.02705
## y2 0.09110 0.91420 -0.04062
## y3 0.02705 -0.04062 0.91482
##
## Correlation matrix of residuals:
## y1 y2 y3
## y1 1.00000 0.09789 0.02906
## y2 0.09789 1.00000 -0.04442
## y3 0.02906 -0.04442 1.00000# Checking the roots
roots <- roots(varsimest)
roots## [1] 0.8447123 0.6934515 0.6934515 0.5673669 0.5673669 0.3908880
