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Interpreting the White Noise Assumption in a VAR

Article Quant Q&A · Author: Nord1
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Summary

The document asks what the time indices mean in the multivariate white noise condition for a vector autoregression. Here, t and s label two time periods: when t equals s, the expression is the residual covariance at the same time; when they differ, it is the covariance across different times. The index s is simply a second time index used to state that condition. For white noise, residuals have zero mean, a stable same-period covariance matrix, and zero covariance across distinct periods.

The supplied answer incorrectly associates the unequal-time condition with exogenous variables. Zero covariance across different dates is a serial uncorrelatedness condition; it does not indicate that the model contains an exogenous variable. The discussion is brief and does not cover stronger assumptions sometimes used in VARs, such as independence or distributional requirements, nor does it explain estimation or diagnostics.

Key ideas

  • The indices t and s represent time periods for residual vectors.
  • When t equals s, the expression gives same-period residual covariance.
  • When t differs from s, white noise requires zero covariance across periods.
  • The unequal-time condition does not imply the presence of exogenous variables.

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# Understand the white noise condition in Vector Autoregression


# Understand the white noise condition in Vector Autoregression












In the following vector autoregression model with lag polynomial representation:

$$\Phi (L) y_t= \epsilon_t$$

where $Y$ is the vector of endogenous variables, $\Phi$ is the parameters matrix, $\epsilon$ is the error term, and $L$ is the lag polynomial factor.

The basic assumption in the above model is that the residual follow a multivariate white noise, i.e.

$$E(\epsilon_t )=0$$

and $E(\epsilon_t \epsilon_s^{‚})$ equals either $0$ if $t \neq s$, or $\sum{\epsilon}$ if $t=s$.

My question is, what $t=s$ and $t \neq s$ really mean in the above condition and from where we get $s$?

## Answer by Nord1 (score 1, accepted)

https://quant.stackexchange.com/a/41794

I think the mistake is how to define $\ Y_t$. It is supposed to contain endogenous and exogenous variables. Hence, the multivariate white noise in the VAR analysis should full fill the following conditions:

$E(\epsilon_t )=0$ and $E(\epsilon_t \epsilon_s^{‚})$ equals either $0$ if $t \neq s$, or $\sum{\epsilon}$ if $t=s$.

In the case of $t=s$, this refer to multivariate covariance stationary condition for the endogenous variable. In the case of $t \neq s$, this refer to the exogenous condition (if the model have an exogenous variables). In this case, $E(\epsilon_t \epsilon_s^{‚})$ should equal zero (ie. $t \neq s$ refer to the existence of an exogenous variable).

reference: "Lecture note from Christopher F Baum"

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.