4.8.3 Autoregressive Processes

          

4.8.3  Autoregressive Processes

An n-dimensional autoregressive process of order p, AR(p), has form

[4.55]

where a is an n-dimensional vector, the bk are n×n matrices, and W is n-dimensional white noise. The name “autoregressive” indicates that [4.52] defines a regression of tX on its own past values. In applications, AR(1) and AR(2) processes are common.

Exhibit 4.12 indicates a realization of the univariate AR(2) process

[4.56]

where W is variance 1 Gaussian white noise.

           

Exhibit 4.12:  A realization of AR(2) process [4.56].

2 Responses to 4.8.3 Autoregressive Processes

  1. Aktar at #

    If the coefficients and one step prediction error of an AR model are given, how to find the variance of the model and also how to compute the covariance matrix? Would you please let me know ASAP

    Reply
  2. gholton at #

    In assessing the correlation structure, you would also want to consider autocorrelations. A standard reference for AR ad related models is Hamilton, James D. (1994). Time Series Analysis , Princeton: Princeton University Press. Hamilton’s focus is univariate time series, but he does address multivariate time series as well.

    MA, AR and ARMA models have little relevance for value-at-risk. I mention them briefly in this book for two reasons: 1) they provide some simple examples of time series models to help readers understand the theory, and 2) they are a useful foundation to have before learning about other models, especially ARCH-GARCH models.

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