Time Series_2_Cointegration and VAR

1. Cointegration

n×1 vector y_t of time series is cointegrated if each series individually integrated in the order d (d = 1, nonstationary unit-root processes, i.e., random walks; d = 0, stationary process).

1.1 Simulation for Cointegration, based on Hamilton (1994)

   

#generate the two time series of length 1000
set.seed(20200229)		#fix the random seed
N <- 100			#define length of simulation
x <- cumsum(rnorm(N))		#simulate a normal random walk
gamma <- 0.7			#set an initial parameter value
y <- gamma * x + rnorm(N)	#simulate the cointegrating series
plot(x, col="black", type='l', lwd=2, ylab='simulated values')	 
lines(y,col="skyblue", lwd=2)				

Both series are individually random walks.

In real world we don't know gamma, so need estimation based on raw data, running linear regression of one series on the other, i.e., Engle-Granger method of testing cointegration.

1.2 Statistical Tests (Augmented Dickey Fuller test)

install.packages('urca')
library('urca')
summary(ur.df(x,type="none"))	
summary(ur.df(y,type="none"))	

NULL: Unit root exists in the process

Reject NULL if test-statistic is smaller than critical value. Result shows test-statistic is larger than critical value at three usual sig.levels. We can't reject NULL.

Conclusion: Both series are individually unit root process.

1.3 Linear Combination

z <- y - gamma * x			
plot(z,type='l')				
summary(ur.df(z,type="none"))

 z_t seems to be a white noise process and rejection of unit root if confirmed by ADF tests:

1.4 Estimate the Cointegrating Relationship using Engle-Granger method

Step 1: Run linear regression y_t on x_t (simple OLS estimation);

Step 2: Test residuals for the presence of a unit root.

coin <- lm(y ~ x -1)		#regression without constant
coin$resid			#obtain the residuals
summary(ur.df(coin$resid))	#ADF test of residuals

Reject NULL.

Pair trading (statistical arbitrage) exploits such cointegrating relationship, aiming to set up strategy based on the spread between two time series. If two series are cointegrated, we expect their stationary linear combination to revert to 0. By selling relatively expensive and buying cheaper one, wait for the reversion to make profit.

Intuition: Linear combination of time series (which form cointegrating relationship) is determined by underlying (long-term) economic forces, e.g., same industry companies may grow similarly; spot and forward price of financial product are bound together by no-arbitrage principle; FX rates of interlinked countries tend to move together.

2. Vector Autoregressive Model

2.1 Theory Bgs

 A_i are n×n coefficient matrices for all i = 1...p;

u_t is a vector white noise process (assumes lack of auto correlation, while allow contemporaneous (同时发生的) correlation between components, i.e., u_t has non-diagonal covariance matrix, meaning that contemporaneous dependencies aren't modeled) with positive definite covariance matrix. Thus we can use OLS to estimate equation by equation. (叫做 reduced form VAR models)

另一种叫做 structural VAR form model, SVAR models the contemporaneous effects among variables: 

 Where: 

Disturbance terms are defined to be uncorrelated, thus are referred to as structural shocks.

2.2 Three-component VAR model (equity return, stock index, US Treasury bond interest rates)

2.2.1 Task: Forecast stock market index by using additional variables and identify impluse responses.

         Assume: There exists a hidden long-term relationship between these three components.

2.2.2 Process:

rm(list = ls()) #clear the whole workspace
install.packages('xts');library(xts)
install.packages('vars');library('vars')
install.packages('quantmod');library('quantmod')
getSymbols('SNP', from='2012-01-02', to='2020-02-29')		
getSymbols('MSFT', from='2012-01-02', to='2020-02-29')
getSymbols('DTB3', src='FRED')	#3-month T-Bill interest rates
chartSeries(MSFT, theme=chartTheme('white'))   	

Cl(MSFT) 	#closing prices
Op(MSFT)	#open prices
Hi(MSFT)	#daily highest price
Lo(MSFT)	#daily lowest price
ClCl(MSFT)	#close-to-close daily return
Ad(MSFT)	#daily adjusted closing price

Subset to obtain period of interests ,while the downloaded prices are supposed to be nonstationary series and should be transformed into stationary series:

#indexing time series data
DTB3.sub <- DTB3['2012-01-02/2020-02-29']
#Calculate returns from Adjusted log series
SNP.ret  <- diff(log(Ad(SNP)))
MSFT.ret <- diff(log(Ad(MSFT)))
#replace NA values
DTB3.sub[is.na(DTB3.sub)] <- 0
DTB3.sub <- na.omit(DTB3.sub) # return with incomplete removed
# merge the three databases to get the same length
# innerjoin returns only rows in which one set have matching keys in the other
dataDaily <- na.omit(merge(SNP.ret,MSFT.ret,DTB3.sub), join='inner')

VAR modeling usually deals with lower frequency data, so transform data to monthly (or quarterly) frequency.

#obtain monthly data, package xts
SNP.M  <- to.monthly(SNP.ret)$SNP.ret.Close
MSFT.M <- to.monthly(MSFT.ret)$MSFT.ret.Close
DTB3.M <- to.monthly(DTB3.sub)$DTB3.sub.Close

# allow a max of 4 lags
# choose the model with best(lowest Akaike Information Criterion value)
var1 <- VAR(dataDaily, lag.max=4, ic="AIC")

Or see multiple information criteria:

VARselect(dataDaily,lag.max=4)

summary(var1)

coef(var1)	#concise summary of the estimated variables
residuals(var1)	#list of residuals (of the corresponding ~lm)
fitted(var1)	#list of fitted values
Phi(var1)	#coefficient matrices of VMA representation
plot(var1, plot.type='multiple') #Diagram of fit and residuals for each variables

#confidence interval for bootstrapped error bands
var.irf <- irf(var1, ci=0.9)
plot(var.irf)

 Number of required restrictions for SVAR model is K(K-1)/2, so in our case is 3.