Description Usage Arguments Details Value Note Author(s) References See Also Examples

This function computes univariate and multivariate ARCH-LM tests for a VAR(p).

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`x` |
Object of class ‘ |

`lags.single` |
An integer specifying the lags to be used for the univariate ARCH statistics. |

`lags.multi` |
An integer specifying the lags to be used for the multivariate ARCH statistic. |

`multivariate.only` |
If |

The multivariate ARCH-LM test is based on the following regression (the univariate test can be considered as special case of the exhibtion below and is skipped):

*
vech(\bold{\hat{u}}_t \bold{\hat{u}}_t') = \bold{β}_0 + B_1
vech(\bold{\hat{u}}_{t-1} \bold{\hat{u}}_{t-1}') + … + B_q
vech(\bold{\hat{u}}_{t-q} \bold{\hat{u}}_{t-q}' + \bold{v}_t)
*

whereby *\bold{v}_t* assigns a spherical error process and
*vech* is the column-stacking operator for symmetric matrices
that stacks the columns from the main diagonal on downwards. The
dimension of *\bold{β}_0* is *\frac{1}{2}K(K +1)* and for
the coefficient matrices *B_i* with *i=1, …, q*,
*\frac{1}{2}K(K +1) \times \frac{1}{2}K(K +1)*. The null
hypothesis is: *H_0 := B_1 = B_2 = … = B_q = 0* and the
alternative is: *H_1: B_1 \neq 0 or B_2 \neq 0 or … B_q \neq
0*.
The test statistic is:

*
VARCH_{LM}(q) = \frac{1}{2}T K (K + 1)R_m^2 \quad ,
*

with

*
R_m^2 = 1 - \frac{2}{K(K+1)}tr(\hat{Ω} \hat{Ω}_0^{-1})
\quad ,
*

and *\hat{Ω}* assigns the covariance matrix of the above
defined regression model. This test statistic is distributed as
*χ^2(qK^2(K+1)^2/4)*.

A list with class attribute ‘`varcheck`

’ holding the
following elements:

`resid` |
A matrix with the residuals of the VAR. |

`arch.uni` |
A list with objects of class ‘ |

`arch.mul` |
An object with class attribute ‘ |

This function was named `arch`

in earlier versions of package
vars; it is now deprecated. See `vars-deprecated`

too.

Bernhard Pfaff

Doornik, J. A. and D. F. Hendry (1997), *Modelling Dynamic
Systems Using PcFiml 9.0 for Windows*, International Thomson
Business Press, London.

Engle, R. F. (1982), Autoregressive conditional heteroscedasticity
with estimates of the variance of United Kingdom inflation,
*Econometrica*, **50**: 987-1007.

Hamilton, J. (1994), *Time Series Analysis*, Princeton
University Press, Princeton.

LÃ¼tkepohl, H. (2006), *New Introduction to Multiple Time Series
Analysis*, Springer, New York.

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