pgam.likelihood: Likelihood function to be maximized

Description

This is the log-likelihood function that is passed to optim for likelihood maximization.

Usage

pgam.likelihood(par, y, x, offset, fperiod, env = parent.frame())

Arguments

par

vector of parameters to be optimized

observed time series which is the response variable of the model

observed explanatory variables for parametric fit

offset

model offset. Just like in GLM

fperiod

vector of seasonal factors to be passed to pgam.par2psi

env

the caller environment for log-likelihood value to be stored

Value

List containing log-likelihood value, optimum linear predictor and the gamma parameters vectors.

Details

Log-likelihood function of hyperparameters $\omega$ and $\beta$ is given by $$\log L\left(\omega,\beta\right)=\sum_{t=\tau+1}^{n}{\log \Gamma\left(a_{t|t-1}+y_{t}\right)-\log y_{t}!-\cr \log \Gamma\left(a_{t|t-1}\right)+a_{t|t-1}\log b_{t|t-1}-\left(a_{t|t-1}+y_{t}\right)\log \left(1+b_{t|t-1}\right)}$$ where $a_{t|t-1}$ and $b_{t|t-1}$ are estimated as it is shown in pgam.filter.

References

Harvey, A. C., Fernandes, C. (1989) Time series models for count data or qualitative observations. Journal of Business and Economic Statistics, 7(4):407--417

Harvey, A. C. (1990) Forecasting, structural time series models and the Kalman Filter. Cambridge, New York

Campos, E. L., De Leon, A. C. M. P., Fernandes, C. A. C. (2003) Modelo Poisson-Gama para Séries Temporais de Dados de Contagem - Teoria e Aplicações. 10a ESTE - Escola de Séries Temporais e Econometria

Junger, W. L. (2004) Modelo Poisson-Gama Semi-Paramétrico: Uma Abordagem de Penalização por Rugosidade. MSc Thesis. Rio de Janeiro, PUC-Rio, Departamento de Engenharia Elétrica