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.)