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

This function computes the multivariate Portmanteau- and Breusch-Godfrey test for serially correlated errors.

1 2 | ```
serial.test(x, lags.pt = 16, lags.bg = 5, type = c("PT.asymptotic",
"PT.adjusted", "BG", "ES") )
``` |

`x` |
Object of class ‘ |

`lags.pt` |
An integer specifying the lags to be used for the Portmanteau statistic. |

`lags.bg` |
An integer specifying the lags to be used for the Breusch-Godfrey statistic. |

`type` |
Character, the type of test. The default is an asymptotic Portmanteau test. |

The Portmanteau statistic for testing the absence of up to the order *h*
serially correlated disturbances in a stable VAR(p) is defined as:

*
Q_h = T ∑_{j = 1}^h
tr(\hat{C}_j'\hat{C}_0^{-1}\hat{C}_j\hat{C}_0^{-1}) \quad ,
*

where *\hat{C}_i = \frac{1}{T}∑_{t = i + 1}^T \bold{\hat{u}}_t
\bold{\hat{u}}_{t - i}'*. The test statistic is approximately
distributed as *χ^2(K^2(h - p))*. This test statistic is
choosen by setting `type = "PT.asymptotic"`

. For smaller sample sizes
and/or values of *h* that are not sufficiently large, a corrected
test statistic is computed as:

*
Q_h^* = T^2 ∑_{j = 1}^h
\frac{1}{T - j}tr(\hat{C}_j'\hat{C}_0^{-1}\hat{C}_j\hat{C}_0^{-1}) \quad ,
*

This test statistic can be accessed, if `type = "PT.adjusted"`

is
set.

The Breusch-Godfrey LM-statistic is based upon the following auxiliary regressions:

*
\bold{\hat{u}}_t = A_1 \bold{y}_{t-1} + … + A_p\bold{y}_{t-p} +
CD_t + B_1\bold{\hat{u}}_{t-1} + … + B_h\bold{\hat{u}}_{t-h} +
\bold{\varepsilon}_t
*

The null hypothesis is: *H_0: B_1 = … = B_h = 0* and
correspondingly the alternative hypothesis is of the form *H_1:
\exists \; B_i \ne 0* for *i = 1, 2, …, h*. The test statistic
is defined as:

*
LM_h = T(K - tr(\tilde{Σ}_R^{-1}\tilde{Σ}_e)) \quad ,
*

where *\tilde{Σ}_R* and *\tilde{Σ}_e* assign the
residual covariance matrix of the restricted and unrestricted
model, respectively. The test statistic *LM_h* is distributed as
*χ^2(hK^2)*. This test statistic is calculated if ```
type =
"BG"
```

is used.

Edgerton and Shukur (1999) proposed a small sample correction, which is defined as:

*
LMF_h = \frac{1 - (1 - R_r^2)^{1/r}}{(1 - R_r^2)^{1/r}} \frac{Nr -
q}{K m} \quad ,
*

with *R_r^2 = 1 - |\tilde{Σ}_e | / |\tilde{Σ}_R|*,
*r = ((K^2m^2 - 4)/(K^2 + m^2 - 5))^{1/2}*, *q = 1/2 K m - 1*
and *N = T - K - m - 1/2(K - m + 1)*, whereby *n* is the
number of regressors in the original system and *m = Kh*. The
modified test statistic is distributed as *F(hK^2, int(Nr -
q))*. This modified statistic will be returned, if ```
type =
"ES"
```

is provided in the call to `serial()`

.

A list with class attribute ‘`varcheck`

’ holding the
following elements:

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

`pt.mul` |
A list with objects of class attribute ‘ |

`LMh` |
An object with class attribute ‘ |

`LMFh` |
An object with class attribute ‘ |

This function was named `serial`

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

too.

Bernhard Pfaff

Breusch, T . S. (1978), Testing for autocorrelation in dynamic linear
models, *Australian Economic Papers*, **17**: 334-355.

Edgerton, D. and Shukur, G. (1999), Testing autocorrelation in a
system perspective, *Econometric Reviews*, **18**: 43-386.

Godfrey, L. G. (1978), Testing for higher order serial correlation in
regression equations when the regressors include lagged dependent
variables, *Econometrica*, **46**: 1303-1313.

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

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

1 2 3 |

```
Loading required package: MASS
Loading required package: strucchange
Loading required package: zoo
Attaching package: 'zoo'
The following objects are masked from 'package:base':
as.Date, as.Date.numeric
Loading required package: sandwich
Loading required package: urca
Loading required package: lmtest
Portmanteau Test (adjusted)
data: Residuals of VAR object var.2c
Chi-squared = 231.59, df = 224, p-value = 0.3497
```

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