VAR model with zero coefficients

Question

ll,

I'm working with a bivariate time series $(X_{t},Y_{t})$. Looking at the two time series separately, $X_{t}$ appears to be white noise. This is supported by looking at the empirical ACF and PACF of $X_{t}$ and by performing the Ljung-Box-Test for various lags (very high p-values). My model for $X_{t}$ would therefore be:

$$X_{t} = \mu + \epsilon_{t}$$ with $\epsilon_{t} \sim WN(\mu, \sigma_{\epsilon})$.

Looking at the empirical ACF and PACF of the other time series $Y_{t}$ I get significant values of both ACF and PACF even for relatively high lags. This could suggest an ARMA model but using AIC and a maximum model order of $p,q = 5$ points to an AR(1) model, like so:

$$Y_{t} = \alpha_{1}Y_{t-1}+\epsilon^{y}_{t}$$

Finally, I had a look at the empirical cross-correlation function of $Y_{t}$ and $X_{t}$. It shows a significant value for a lag of $h = -1$, that is a significant correlation between $Y_{t}$ and $X_{t+1}$. It would therefore seem appropriate to account for this by letting $X_{t}$ depend on $Y_{t-1}$, yielding something like

$$X_{t} = \mu + \beta_{1}Y_{t-1}+ \epsilon_{t}$$

I'm wondering how I would go about fitting such a model (preferably using R). Would I have to use a package for VAR models and specify the $X_{t-1}$ coefficient for $Y_{t}$ as zero?

Thanks!

Answer 1

The VAR() function in the R package "vars" will fit a VAR model.

I suspect it will fit a coefficient for X_t-1, but will not find the estimated coefficient to be significantly different from zero. Here's an example:

library(vars)
base.url <- "http://staff.elena.aut.ac.nz/Paul-Cowpertwait/ts/"
www <- paste0(base.url, "stockmarket.dat")
stock <- read.table(www, header = TRUE)
nyl <- cbind(stock$London, stock$NewYork)
colnames(nyl) <- c("London", "NewYork")
stock.var <- VAR(nyl, type = "const") 
summary(stock.var)

For New York, both London_t-1 and NY_t-1 are fit, but only NY_t-1 is significantly different from zero.