Estimation of Vector Autoregressive Processes

In this chapter, it is assumed that a

K

-dimensional multiple time series

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$$y_1 , \ldots ,y_T \;with\;y_t = \left( {y_{1t} , \ldots ,y_{Kt} } \right)^\prime $$

is available that is known to be generated by a stationary, stable VAR(

p

) process

3.1.1

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$$ y_t = v + A_{1yt - 1} + \ldots + A_p y_{t - p} + u_t . $$

All symbols have their usual meanings, that is, ν = (ν

1

,…, ν

K

)′ is a (

K

× 1) vector of intercept terms, the

A

i

are (

K × K

) coefficient matrices and

u

t

is white noise with nonsingular covariance matrix Σ

u

. In contrast to the assumptions of the previous chapter, the coefficients ν,

A

1

,…,

A

p

, and Σ

u

are assumed to be unknown in the following. The time series data will be used to estimate the coefficients. Note that notationwise we do not distinguish between the stochastic process and a time series as a realization of a stochastic process. The particular meaning of a symbol should be obvious from the context.