Specifcations

A VAR(p) system

$\mathbf y_t = \boldsymbol{\mu} + \mathbf{\Phi}_1 \mathbf{y}_{t-1} + \cdots + \mathbf{\Phi}_p y_{t-p} + \boldsymbol{\epsilon}_t$

$\mathbf{y}_t$ and $\boldsymbol{\mu}$ are $K$ -dimensional vectors
$\boldsymbol{\epsilon}_t \sim \mathbf{iid} (\mathbf{0}, \boldsymbol{\Omega})$ , where $\boldsymbol{\Omega}$ is a $K\times K$ covariance matrix
$\mathbf{\Phi}_1, \ldots, \mathbf{\Phi}_p$ are $K\times K$ parameter matrices
Example: bivariate VAR(1)

$\begin{align} y_{1t} & = \mu_1 + \phi_{11} y_{1,t-1} + \phi_{12} y_{2,t-1} + \epsilon_{1t} \\ y_{2t} & = \mu_2 + \phi_{21} y_{1,t-1} + \phi_{22} y_{2,t-1} + \epsilon_{2t} \end{align}$

Estimation

OLS each equation separately to obtain $\hat {\mathbf{\Phi}}_1, \ldots, \hat{\mathbf{\Phi}}_p$
Compute residual vector $\hat{\boldsymbol{\epsilon}}_t = (\hat{\epsilon}_1, \ldots, \hat{\epsilon}_K)'$
Estimate variance structure $\hat{\boldsymbol{\Omega}} = T^{-1} \sum_{t=1}^T \hat{\boldsymbol{\epsilon}_t } \hat{\boldsymbol{\epsilon}_t }'$

Example

Bierens, H. and Martins, L. (2010)
Monthly: 1973:Jan – 1999:Dec
- cpiUSA: US Consumer Price Index
- cpiCAN: Canada Consumer Price Index

library(tsDyn) 
data(barry) 
barry <- barry[, c("cpiUSA", "cpiCAN")]
plot(barry)

Data processing and VAR regression

dbarry <- diff(barry)
plot(dbarry)

mreg <- lineVar(dbarry, lag = 3, model = "VAR")
print(mreg)

##                 Intercept cpiUSA -1   cpiCAN -1    cpiUSA -2  cpiCAN -2
## Equation cpiUSA 0.1089088 0.4678089 -0.04648345 -0.028101665 0.07909299
## Equation cpiCAN 0.0987480 0.2705002 -0.02301329 -0.004867705 0.19486643
##                   cpiUSA -3 cpiCAN -3
## Equation cpiUSA 0.007891732 0.0939838
## Equation cpiCAN 0.010717830 0.1536533

Information criteria

Akaika: $AIC = \log |\hat{\boldsymbol{\Omega}} | + \frac{2}{T} p K^2$
Bayesian (Schwarz) $BIC = \log |\hat{\boldsymbol{\Omega}} | + \frac{\log T}{T}p K^2$
Hannan: $HIC= \log |\hat{\boldsymbol{\Omega}} | + \frac{2\log \log T}{T} p K^2$

Impulse Response

A tool that tracks the dynamics and transmission of a shock in the VAR system
Remind a bivariate system $\begin{align} y_{1t} & = \mu_1 + \phi_{11} y_{1,t-1} + \phi_{12} y_{2,t-1} + \epsilon_{1t} \\ y_{2t} & = \mu_2 + \phi_{21} y_{1,t-1} + \phi_{22} y_{2,t-1} + \epsilon_{2t} \end{align}$
Decompose $\epsilon_{2t} = \rho \epsilon_{1t} + v_{2t}$ , where $\rho$ is the population regression coefficient
By design, $\epsilon_{1t}$ is uncorrelated with $v_{2t}$
Rewrite the VAR into a structural VAR (SVAR)

$\begin{align} y_{1t} & = \mu_1 + \phi_{11} y_{1,t-1} + \phi_{12} y_{2,t-1} + \epsilon_{1t} \\ y_{2t} & = \mu_2 + \phi_{21} y_{1,t-1} + \phi_{22} y_{2,t-1} + (\rho \epsilon_{1t} + v_{2t}) \\ & = \tilde{\mu}_2 + \rho y_{1t} + \tilde{\phi}_{21} y_{1,t-1} + \tilde{\phi}_{22} y_{2,t-1} + v_{2t} \end{align}$

SVAR is an alternative mathematical representation of the plain VAR.
It doesn’t imply causality by itself, but it reflects the researcher’s perspective.

Path of impact

A shock of $\epsilon_{1t}$ at time $t$ (today) (Usually consider the magnitude of one standard deviation) directly affects $y_{1t}$ in

$y_{1t} = \mu_1 + \phi_{11} y_{1,t-1} + \phi_{12} y_{2,t-1} + \epsilon_{1t}.$

Next, $\epsilon_{1t}$ affects $y_{2t}$ via $\rho y_{1t}$ as well:

$\begin{align} \\ y_{2t} & = \tilde{\mu}_2 +\rho y_{1t} + \tilde{\phi}_{21} y_{1,t-1} + \tilde{\phi}_{22} y_{2,t-1} + v_{2t} \end{align}$

Although the error terms are independent over time, the effect of a shock from $\epsilon_{1t}$ will continue tomorrow ( $t+1$ ) as its has shifted $y_{1t}$ and $y_{2t}$ .
- The channel for $y_{1,t+1} = \mu_1 + \phi_{11} y_{1t} + \phi_{12} y_{2t} + \epsilon_{1,t+1}$ is via $(y_{1t}, y_{2t})$
- The channel for $y_{2,t+1} = \tilde{\mu}_2 + \rho y_{1,t+1} + \tilde{\phi}_{21} y_{1t} + \tilde{\phi}_{22} y_{2t} + v_{2,t+1}$ is via $(y_{1,t+1}, y_{1t}, y_{2t})$
Keep going to $t+2, t+3, \ldots$ to obtain its path of impact over time

Example

irf_var = irf(mreg, impulse = "cpiUSA", response = c("cpiUSA", "cpiCAN"), boot = FALSE)
print(irf_var)

## 
## Impulse response coefficients
## $cpiUSA
##            cpiUSA      cpiCAN
##  [1,] 0.154994135 0.060646708
##  [2,] 0.069688566 0.040530272
##  [3,] 0.031158080 0.028981581
##  [4,] 0.021399138 0.026299843
##  [5,] 0.014563998 0.017653617
##  [6,] 0.010441039 0.013341153
##  [7,] 0.007891908 0.010156895
##  [8,] 0.005755647 0.007318570
##  [9,] 0.004270164 0.005491120
## [10,] 0.003176343 0.004072063
## [11,] 0.002344197 0.003000950

plot(irf_var)

irf_var = irf(mreg, impulse = "cpiCAN", response = c("cpiUSA", "cpiCAN"), boot = FALSE)
print(irf_var)

## 
## Impulse response coefficients
## $cpiCAN
##             cpiUSA       cpiCAN
##  [1,]  0.000000000  0.227148412
##  [2,] -0.010558642 -0.005227431
##  [3,]  0.013269409  0.041527784
##  [4,]  0.025508725  0.036568544
##  [5,]  0.012570420  0.013169955
##  [6,]  0.011451505  0.016622132
##  [7,]  0.008911022  0.011112575
##  [8,]  0.005981957  0.007496385
##  [9,]  0.004731050  0.006244477
## [10,]  0.003462496  0.004370711
## [11,]  0.002529312  0.003245789

plot(irf_var)

Variance Decomposition

Let $E_t[ \cdot]$ be the conditional expectation given all information contained from the start of the time up to the time $t$ .

In the structural equation where the shock is originated (the first equation in our two-equation example), the forecast error

$e_{1,T+1}:= y_{1,T+1} - E_T[y_{1,T+1}].$

In this SVAR system, $e_{1,T+1} = \epsilon_{1,T+1}$ . The variance of the forecast error is

$var[e_{1,T+1}] = var[\epsilon_{1,T+1}] = E_T[\epsilon^2_{1,T+1}]$

In the second equation, to which $\epsilon_{1,T+1}$ spread,

$\begin{align} e_{2,T+1} & = & y_{2,T+1} - E_T[y_{2,T+1}] \\ & = & \tilde{\mu}_2 + \rho y_{1,T+1} + \tilde{\phi}_{21} y_{1T} + \tilde{\phi}_{22} y_{2T} + v_{2,T+1} \\ & & - (\tilde{\mu}_2 + \rho E_T[ y_{1,T+1} ] + \tilde{\phi}_{21} y_{2T} + \tilde{\phi}_{22} y_{2T}) \\ & = & \rho ( y_{1,T+1} - E_T[y_{1,T+1}] ) + v_{2,T+1} \\ & = & \rho \epsilon_{1,T+1} + v_{2,T+1} \end{align}$

Thus $var[e_{2,T+1}] = \rho^2 E_T[\epsilon_{1,T+1}^2] + E_T[ v_{2,T+1}^2].$

Alternatively, we look at the decomposition $\epsilon_{2,T+1} = \rho \epsilon_{1,T+1} + v_{2,T+1}$ and reach the same conclusion.
Consider a longer horizon for $T+h$ with $h>1$ . The estimation error spreads and transmits in the SVAR system as well.

Granger Causality

In the VAR(p) system, does it make statistical significant difference if we completely shut down $y_2$ from the regression equation for $y_1$ ?
Unrestricted model: y_1 ~ Lags(y_1, 1:p) + Lags(y_2, 1:p) + Lags(y_3, 1:p)... + Lags(y_K, 1:p)
Restricted model: y_1 ~ Lags(y_1, 1:p) + Lags(y_3, 1:p)... + Lags(y_K, 1:p)
The null hypothesis restricts all coefficients associated with $y_2$ as 0.
Under the null, the Wald statistic asymptotically follows $\chi^2(p)$ .
Caution: Granger causality is not “causal”. Instead, it is merely a statistical predictive relationship.

Example

Thurman W.N. & Fisher M.E. (1988), Chickens, Eggs, and Causality, or Which Came First?, American Journal of Agricultural Economics, 237-238
1930–1983 US chicken population and egg production

library(lmtest)
data(ChickEgg)

# based on F-test
grangertest(egg ~ chicken, order = 3, data = ChickEgg)

## Granger causality test
## 
## Model 1: egg ~ Lags(egg, 1:3) + Lags(chicken, 1:3)
## Model 2: egg ~ Lags(egg, 1:3)
##   Res.Df Df      F Pr(>F)
## 1     44                 
## 2     47 -3 0.5916 0.6238

grangertest(chicken ~ egg, order = 3, data = ChickEgg)

## Granger causality test
## 
## Model 1: chicken ~ Lags(chicken, 1:3) + Lags(egg, 1:3)
## Model 2: chicken ~ Lags(chicken, 1:3)
##   Res.Df Df     F   Pr(>F)   
## 1     44                     
## 2     47 -3 5.405 0.002966 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Example (cont.)

Based on the Wald test

# based on Chi-square test
grangertest(egg ~ chicken, order = 3, data = ChickEgg, test = "Chisq")

## Granger causality test
## 
## Model 1: egg ~ Lags(egg, 1:3) + Lags(chicken, 1:3)
## Model 2: egg ~ Lags(egg, 1:3)
##   Res.Df Df  Chisq Pr(>Chisq)
## 1     44                     
## 2     47 -3 1.7748     0.6204

grangertest(chicken ~ egg, order = 3, data = ChickEgg, test = "Chisq")

## Granger causality test
## 
## Model 1: chicken ~ Lags(chicken, 1:3) + Lags(egg, 1:3)
## Model 2: chicken ~ Lags(chicken, 1:3)
##   Res.Df Df  Chisq Pr(>Chisq)   
## 1     44                        
## 2     47 -3 16.215   0.001025 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Vector Autoregression

VAR

Specifcations

Estimation

Example

Data processing and VAR regression

Information criteria

Impulse Response

Path of impact

Example

Variance Decomposition

Granger Causality

Example

Example (cont.)