Our Model

(This section is under construction...)

The bivariate return process is specified as follows:

where

• X_t = (X_1t, X_2t)' is the vector of returns for week t (X_1t: percentage change of WTI crude oil price, X_2t: percentage return on DJIA),
• µ is the vector of expected returns,
• (ν_t) is bivariate Gaussian white noise with the unity matrix as covariance matrix,
• the conditional covariance matrix H_t is similar to the BEKK model, but augmented by an additional term which allows for asymmetry.

Our current model is fitted using weekly return data on WTI crude oil and the DJIA from 1991-01-08 through 2007-09-25 (873 weeks). We are particularly interested in the news impact on the variance of weekly returns on the DJIA, that is: How do this week's returns on WTI price and the DJIA impact next week's variance of returns on the DJIA? The following images show this news impact function.

Figure 4: News impact on the variance of returns on DJIA

This shows: If the WTI price increases this week and the DJIA loses, the next week's DJIA volatility will increase the most. Vice versa, if WTI crude oil becomes cheaper and the DJIA increases, there is no impact on volatility. This shows clearly that there is an asymmetric link between the crude oil market and the stock market. This phenomenon cannot be seen by simply computing the correlation between weekly returns on WTI and on the DJIA.