Via : Journal of Financial Economics (May 1996)

Usually the first multivariate time series model that one comes across is a VAR model. It is a logical progression from modeling a univariate ARMA process. Most of the textbooks that introduce VAR start off with the Standard VAR and then go at length in to procedures such as estimating the parameters, hypothesis testing for the number of lags to consider, innovation accounting topics such as Impulse Response Decomposition, Forecast error variance decomposition. When one wants to apply VAR to any real world situation, one inevitably starts with Structural VAR. One can easily transform a Structural VAR to Standard VAR and use the standard innovation accounting tools.

Once you want to do any sort of analysis linking back to original Structural VAR, there is a BIG problem, i.e. one cannot identify all the parameters of a Structural VAR from a Standard VAR. To be more precise, if it is a n dimensional vector that is being modeled, (n*n-n)/2 restrictions must be imposed on the Structural VAR. These things look very abstract from a pure math sense. That is where papers like these help one understand the importance of imposing restrictions on a standard model to infer things from real life data. Intuitively I could understand that one must impose constraints and one of the standard ways to do it is via causal ordering. My understanding was shallow until I read this paper.

What’s this paper about ?

Order types influence the stock returns and vice versa. The paper by Hasbrouck analyzes the price impact of index arb program trades, non index arb program trades, non program trades.

This relationship is captured by building a VAR model. The vector of variables considered are

  1. Stock index futures basis adjusted for carrying cost

  2. Log difference on the stock index futures contract

  3. Total Signed order flow

  4. Signed program order flow

  5. Signed index-arbitrage order flow

  6. Square root of the cumulative signed trade volume

  7. Log quote midpoint return

The variables considered are in the above order for a specific reason. The author chooses the causal ordering in the same order. A Standard VAR is built. However to infer whether there is a price impact from index-arb program , non index-arb program and non program trades, impulse response functions are used. This is where my fundas about imposing additional restrictions on Structural VAR became clear. The author imposes constraints for contemporaneous structural innovations and computes the various impulse response functions to show that “Program orders in general, and index-arb program orders in particular are found to have statistically and economically significant price impacts”. Forecast error variance decomposition is also used to analyze the impact of various components on the long term variance of the stock price.

The author’s intent of analyzing the contemporaneous relationship between program orders on stock returns using VAR model, makes this an ideal read for someone looking to understand practical aspects of VAR modeling.