# Analytical Mean, Variance and Autocorrelation of an INGARCH Process

### Description

Functions to calculate the analytical mean, variance and autocorrelation / partial autocorrelation / autocovariance function of an integer-valued generalised autoregressive conditional heteroscedasticity (INGARCH) process.

### Usage

1 2 3 4 | ```
ingarch.mean(intercept, past_obs=NULL, past_mean=NULL)
ingarch.var(intercept, past_obs=NULL, past_mean=NULL)
ingarch.acf(intercept, past_obs=NULL, past_mean=NULL, lag.max=10,
type=c("acf", "pacf", "acvf"), plot=TRUE, ...)
``` |

### Arguments

`intercept` |
numeric positive value for the intercept |

`past_obs` |
numeric non-negative vector containing the coefficients |

`past_mean` |
numeric non-negative vector containing the coefficients |

`lag.max` |
integer value indicating how many lags of the (partial) autocorrelation / autocovariance function should be calculated. |

`type` |
character. If |

`plot` |
logical. If |

`...` |
additional arguments to be passed to function |

### Details

The INGARCH model of order *p* and *q* used here follows the definition

*
Z[t]|F[t-1] ~ Poi(κ[t]),*

where *F[t-1]* is the history of the process up to time *t-1* and *Poi* is the Poisson distribution parametrised by its mean (cf. Ferland et al., 2006).
The conditional mean *κ[t]* is given by

*
κ[t] = β[0] + β[1] Z[t-1] + … + β[p] Z[t-p]
+ α[1] κ[t-1] + … + α[q] κ[t-q].*

The function `ingarch.acf`

depends on the function `tacvfARMA`

from package `ltsa`

, which needs to be installed.

### Author(s)

Tobias Liboschik

### References

Ferland, R., Latour, A. and Oraichi, D. (2006) Integer-valued GARCH process. *Journal of Time Series Analysis* **27(6)**, 923–942, http://dx.doi.org/10.1111/j.1467-9892.2006.00496.x.

### See Also

`tsglm`

for fitting a more genereal GLM for time series of counts of which this INGARCH model is a special case. `tsglm.sim`

for simulation from such a model.

### Examples

1 2 3 4 5 | ```
ingarch.mean(0.3, c(0.1,0.1), 0.1)
## Not run:
ingarch.var(0.3, c(0.1,0.1), 0.1)
ingarch.acf(0.3, c(0.1,0.1,0.1), 0.1, type="acf", lag.max=15)
## End(Not run)
``` |

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