critMatrix | R Documentation |

An information criterion is calculated for different orders of an autoregressive-moving-average (ARMA) model.

```
critMatrix(
X,
p.max = 5,
q.max = 5,
criterion = c("bic", "aic"),
include.mean = TRUE
)
```

`X` |
a numeric vector that contains the observed time series ordered from past to present; the series is assumed to follow an ARMA process. |

`p.max` |
an integer value |

`q.max` |
an integer value |

`criterion` |
a character value that defines the information criterion
that will be calculated; the Bayesian Information Criterion ( |

`include.mean` |
a logical value; this argument regulates whether to
estimate the mean of the series ( |

This function is part of the `smoots`

package and was implemented under
version 1.1.0. The series passed to `X`

is assumed to follow an
ARMA(`p,q`

) model. A `p.max + 1`

by `q.max + 1`

matrix is
calculated for this series. More precisely, the criterion chosen via the
argument `criterion`

is calculated for all combinations of orders
`p = 0, 1, ..., p_{max}`

and
`q = 0, 1, ..., q_{max}`

.

Within the function, two information criteria are supported: the Bayesian Information Criterion (BIC) and Akaike's Information Criterion (AIC). The AIC is given by

`AIC_{p,q} := \ln(\hat{\sigma}_{p,q}^{2}) + \frac{2(p+q)}{n},`

where `\hat{sigma}_{p,q}^{2}`

is the estimated
innovation variance, `p`

and `q`

are the ARMA orders and `n`

is
the number of observations.

The BIC, on the other hand, is defined by

`BIC_{p,q} := k \ln(n) - 2\ln(\hat{L})`

with `k`

being the number of estimated parameters and
`\hat{L}`

being the estimated Log-Likelihood. Since the parameter
`k`

only differs with respect to the orders `p`

and `q`

for all
estimated models, the term `k \ln(n)`

is reduced to
`(p + q) \ln(n)`

within the function. Exemplarily,
if the mean of the series is estimated as well, it is usually considered
within the parameter `k`

when calculating the BIC.
However, since the mean is estimated for all models, not considering this
estimated parameter within the calculation of the BIC will reduce all BIC
values by the same amount of `\ln(n)`

. Therefore, the selection
via this simplified criterion is still valid, if the number of the estimated
parameters only differs with respect to `p`

and `q`

between the
models that the BIC is obtained for.

The optimal orders are considered to be the ones which minimize either the BIC or the AIC. The use of the BIC is however recommended, because the BIC is consistent, whereas the AIC is not.

NOTE:

Within this function, the `arima`

function of the
`stats`

package with its method `"CSS-ML"`

is used throughout for
the estimation of ARMA models.

The function returns a `p.max + 1`

by `q.max + 1`

matrix, where the
rows represent the AR orders from `p = 0`

to `p = p_{max}`

and the columns represent the MA orders from `q = 0`

to
`q = q_{max}`

. The values within the matrix are the values of
the previously selected information criterion for the different combinations
of `p`

and `q`

.

Dominik Schulz (Research Assistant) (Department of Economics, Paderborn University),

Package Creator and Maintainer

```
## Not run:
# Simulate an ARMA(2,1) process
set.seed(23)
X.sim <- stats::arima.sim(model = list(ar = c(1.2, -0.71), ma = 0.46),
n = 1000) + 13.1
# Application of the function
critMatrix(X.sim)
# Result: Via the BIC, the orders p.opt = 2 and q.opt = 1 are selected.
## End(Not run)
```

smoots documentation built on Sept. 11, 2023, 9:07 a.m.

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