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

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

x

Object of class ‘`varest`

’; generated by
`VAR()`

, or an object of class ‘`vec2var`

’;
generated by `vec2var()`

.

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.

A list with class attribute ‘`varcheck`

’ holding the
following elements:

A matrix with the residuals of the VAR.

A list with objects of class attribute ‘`htest`

’
containing the multivariate Portmanteau-statistic (asymptotic and
adjusted.

An object with class attribute ‘`htest`

’
containing the Breusch-Godfrey LM-statistic.

An object with class attribute ‘`htest`

’
containing the Edgerton-Shukur F-statistic.

The Portmanteau statistic for testing the absence of up to the order \(h\)
serially correlated disturbances in a stable VAR(p) is defined as:
$$
Q_h = T \sum_{j = 1}^h
tr(\hat{C}_j'\hat{C}_0^{-1}\hat{C}_j\hat{C}_0^{-1}) \quad ,
$$
where \(\hat{C}_i = \frac{1}{T}\sum_{t = i + 1}^T \bold{\hat{u}}_t
\bold{\hat{u}}_{t - i}'\). The test statistic is approximately
distributed as \(\chi^2(K^2(h - p))\). 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:
$$
Q_h^* = T^2 \sum_{j = 1}^h
\frac{1}{T - j}tr(\hat{C}_j'\hat{C}_0^{-1}\hat{C}_j\hat{C}_0^{-1}) \quad ,
$$
This test statistic can be accessed, if `type = "PT.adjusted"`

is
set.

The Breusch-Godfrey LM-statistic is based upon the following auxiliary regressions: $$ \bold{\hat{u}}_t = A_1 \bold{y}_{t-1} + \ldots + A_p\bold{y}_{t-p} + CD_t + B_1\bold{\hat{u}}_{t-1} + \ldots + B_h\bold{\hat{u}}_{t-h} + \bold{\varepsilon}_t $$ The null hypothesis is: \(H_0: B_1 = \ldots = B_h = 0\) and correspondingly the alternative hypothesis is of the form \(H_1: \exists \; B_i \ne 0\) for \(i = 1, 2, \ldots, h\). The test statistic is defined as:

$$
LM_h = T(K - tr(\tilde{\Sigma}_R^{-1}\tilde{\Sigma}_e)) \quad ,
$$
where \(\tilde{\Sigma}_R\) and \(\tilde{\Sigma}_e\) assign the
residual covariance matrix of the restricted and unrestricted
model, respectively. The test statistic \(LM_h\) is distributed as
\(\chi^2(hK^2)\). 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 \(R_r^2 = 1 - |\tilde{\Sigma}_e | / |\tilde{\Sigma}_R|\),
\(r = ((K^2m^2 - 4)/(K^2 + m^2 - 5))^{1/2}\), \(q = 1/2 K m - 1\)
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()`

.

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.

L<U+34AE5C2F>hl, H. (2006), *New Introduction to Multiple Time Series
Analysis*, Springer, New York.

```
# NOT RUN {
data(Canada)
var.2c <- VAR(Canada, p = 2, type = "const")
serial.test(var.2c, lags.pt = 16, type = "PT.adjusted")
# }
```

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