score_test_nonlinpq_h0 {PNAR}

Linearity test against non-linear ID-PNAR(p) model

Description

Quasi score test for testing linearity of Poisson Network Autoregressive model of order p against the non-linear Intercep Drift (ID) version (ID-PNAR(p)).

Usage

score_test_nonlinpq_h0(b, y, W, p, d, Z = NULL)

Arguments

b

The estimated parameters from the linear PNAR model, in the following order: (intercept, network parameters, autoregressive parameters, covariates). The dimension of the vector should be 2p + 1 + q, where q denotes the number of covariates.

y

A TT \times N time series object or a TT \times N numerical matrix with the N multivariate count time series over TT time periods.

W

The N \times N 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.

d

The lag parameter of non-linear variable (should be between 1 and p).

Z

An N \times q matrix of covariates (one for each column), where q is the number of covariates in the model. Note that they must be non-negative.

Details

The function computes the quasi score test for testing linearity of Poisson Network Autoregressive model of order p against the following ID-PNAR(p) model. For each node of the network i=1,...,N over the time sample t=1,...,TT

\lambda_{i,t}=\frac{\beta_{0}}{(1+X_{i,t-d})^{\gamma}}+\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 X_{i,t}=\sum_{j=1}^{N}W_{ij}Y_{j,t} is the network effect, i.e. the weighted average impact of node i connections, with the weights of the mean being W_{ij}, the single element of the network matrix W. The sequence \lambda_{i,t} is the expectation of Y_{i,t} conditional to its past values.

The null hypothesis of the test is defined as H_{0}: \gamma=0, versus the alternative H_{1}: \gamma >0. The test statistic has the form

LM=S^{'}(\hat{\theta})\Sigma^{-1}(\hat{\theta})S(\hat{\theta}),

where

S(\hat{\theta})=\sum_{t=1}^{TT}\sum_{i=1}^{N}\left(\frac{Y_{i,t}}{\lambda_{i,t}(\hat{\theta})}-1\right)\frac{\partial\lambda_{i,t}(\hat{\theta})}{\partial\gamma}

is the partition of the quasi score related to the non-linear parameter \gamma, evaluated at the estimated parameters \hat{\theta} under the null assumption H_{0} (linear model), and \Sigma(\hat{\theta}) is the variance of S(\hat{\theta}). Under H_{0}, the test asymptotically follows the \chi^2 distribution with 1 degree of freedom. For details see Armillotta and Fokianos (2022b, Sec. 4).

Value

A list with attribute class "htest" including:

statistic

The value of the \chi^2 test statistic.

parameter

The degrees of freedom of the \chi^2 distribution. This is always 1.

p.value

The p-value of the \chi^2 test statistic.

null.value

The value of the \gamma parameter, which is equal to 0 under the null hypothesis.

alternative

The alternative hypothesis, \gamma has to be greater than 0.

method

The name of the test.

data.name

Information on the arguments used.

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

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. and K. Fokianos (2022b). Testing linearity for network autoregressive models. https://arxiv.org/abs/2202.03852

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

score_test_stnarpq_j, score_test_tnarpq_j, lin_estimnarpq

Examples

data(crime)
data(crime_W)
mod1 <- lin_estimnarpq(crime, crime_W, p = 2)
ca <- mod1$coefs[, 1]
score_test_nonlinpq_h0(ca, crime, crime_W, p = 2, d = 1)