log_lin_estimnarpq {PNAR}R Documentation

Estimation of the log-linear Poisson NAR(p) model with p lags and q covariates (log-PNAR(p))

Description

Estimation of the log-linear Poisson Network Autoregressive model of order pp with qq covariates (log-PNAR(pp)).

Usage

log_lin_estimnarpq(y, W, p, Z = NULL, uncons = FALSE, init = NULL,
xtol_rel = 1e-8, maxeval = 100)

Arguments

y

A TTTT x NN time series object or a TTTT x NN numerical matrix with the NN multivariate count time series over TTTT time periods.

W

The NN x NN row-normalized non-negative adjacency matrix describing the network. The main diagonal entries of the matrix should be zeros, all the other entries should be non-negative and the maximum sum of elements over the rows should equal one. The function row-normalizes the matrix if a non-normalized adjacency matrix is provided.

p

The number of lags in the model.

Z

An NN x qq matrix of covariates (one for each column), where qq is the number of covariates in the model.

uncons

logical, if TRUE an unconstrained optimization is performed (default is FALSE).

init

A vector of starting values for the optimization algorithm. If this is NULL, the function computes them internally.

xtol_rel

The stopping tolerance of the optimization algorithm.

maxeval

The maximum number of evalutions the optimization algorithm will perform.

Details

This function performs a constrained estimation of the linear Poisson NAR(pp) model with qq non-negative valued covariates, for each node of the network i=1,...,Ni=1,...,N over the time sample t=1,...,TTt=1,...,TT, defined as

νi,t=β0+h=1p(β1hXi,th+β2hYi,th)+l=1qδlZi,l, \nu_{i,t}=\beta_{0}+\sum_{h=1}^{p}(\beta_{1h}X_{i,t-h}+\beta_{2h}Y_{i,t-h})+\sum_{l=1}^{q}\delta_{l}Z_{i,l},

where Xi,t=j=1NWijYj,tX_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node ii connections, with the weights of the mean being WijW_{ij}, the single element of the network matrix WW. The sequence νi,t\nu_{i,t} is the log of the expectation of Yi,tY_{i,t}, conditional to its past values. The parameter β0\beta_{0} is the intercept of the model, β1h\beta_{1h} are the network coefficients, β2h\beta_{2h} are the autoregressive parameters, and δl\delta_{l} are the coefficients assocciated to the covariates Zi,lZ_{i,l}.

The estimation of the parameters of the model is performed by Quasi Maximum Likelihood Estimation (QMLE), maximizing the following quasi log-likelihood

l(θ)=t=1TTi=1N[Yi,tνi,t(θ)eνi,t(θ)] l(\theta)=\sum_{t=1}^{TT}\sum_{i=1}^{N}\left[Y_{i,t}\nu_{i,t}(\theta)-e^{\nu_{i,t}(\theta)}\right]

with respect to the vector of unknown parameters θ\theta described above.

By default, the optimization is constrained in the stationary region where h=1p(β1h+β2h)<1\sum_{h=1}^{p}(|\beta_{1h}|+|\beta_{2h}|)<1; this can be removed by setting uncons = TRUE. However, the model estimates might be inconsistent if the estimated parameters lie outside the stationary region.

The ordinary least squares estimates are employed as starting values of the optimization procedure. Robust standard errors and z-tests are also returned.

Value

A list with attribute class "PNAR" including:

coefs

A matrix with the estimated QMLE coefficients, their standard errors, their Z-test statistics and the relevant p-values computed via the standard normal approximation.

score

The value of the quasi score function at the optimization point. It should be close to 0 if the optimization is successful.

loglik

The value of the maximized quasi log-likelihood.

ic

A vector with the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and the Quasi information criterion (QIC).

Alternatively, these can be printed via the function summary.PNAR.

Author(s)

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

References

Armillotta, M. and K. Fokianos (2022a). Poisson network autoregression. https://arxiv.org/abs/2104.06296

Armillotta, M., Tsagris, M. and Fokianos, K. (2022c). The R-package PNAR for modelling count network time series. https://arxiv.org/abs/2211.02582

See Also

lin_estimnarpq

Examples

data(crime)
data(crime_W)
mod1 <- log_lin_estimnarpq(crime, crime_W, p = 2)
summary(mod1)

[Package PNAR version 1.6 Index]