score_test_stnarpq_DV {PNAR}

Bound p-value for testing for smooth transition effects on PNAR(p) model

Description

Computation of Davies bound p-value for the sup-type test for testing linearity of Poisson Network Autoregressive model of order p (PNAR(p)) versus the non-linear Smooth Transition alternative (ST-PNAR(p)).

Usage

score_test_stnarpq_DV(b, y, W, p, d, Z = NULL, gama_L = NULL,
gama_U = NULL, len = 100)

Arguments

b

The estimated parameters from the linear 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.

gama_L

The lower value of the nuisance parameter \gamma to consider. Use NULL if there is not information about its value. See the details for default computation.

gama_U

The upper value of the nuisance parameter \gamma to consider. Use NULL if there is not information about its value. See the details for default computation.

len

The length of the grid of values of \gamma values to consider.

Details

The function computes an upper-bound for the p-value of the sup-type test for testing linearity of Poisson Network Autoregressive model of order p (PNAR(p)) versus the following Smooth Transition alternative (ST-PNAR(p)). For each node of the network i=1,...,N over the time sample t=1,...,TT

\lambda_{i,t}=\beta_{0}+\sum_{h=1}^{p}(\beta_{1h}X_{i,t-h}+\beta_{2h}Y_{i,t-h}+\alpha_{h}e^{-\gamma X_{i,t-d}^{2}}X_{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}: \alpha_{1}=...=\alpha_{p}=0, versus the alternative that at least one among \alpha_{h} is not 0. The test statistic has the form

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

where

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

is the partition of the quasi score related to the vector of non-linear parameters \alpha=(\alpha_{1},...,\alpha_{p}), evaluated at the estimated parameters \hat{\theta} under the null assumption H_{0} (linear model), and \Sigma(\hat{\theta},\gamma) is the variance of S(\hat{\theta},\gamma). Since the test statistic depends on an unknown nuisance parameter (\gamma), the supremum of the statistic is considered in the test, \sup_{\gamma}LM(\gamma). The function computes the bound of the p-value, suggested by Davies (1987), for the test statistic \sup_{\gamma}LM(\gamma), with scalar nuisance parameter \gamma, as follows.

P(\chi^{2}_{k} \geq M)+V M^{1/2(k-1)}\frac{e^{-M/2}2^{-k/2}}{\Gamma(k/2)}

where M is the maximum of the test statistic LM(\gamma), computed by the available sample, over a grid of values for the nuisance parameter \gamma_{F}=(\gamma_{L},\gamma_{1},...,\gamma_{l},\gamma_{U}); k is the number of non-linear parameters tested. So the first summand of the bound is just the p-value of a chi-square test with k degrees of freedom. The second summand is a correction term depending on V, which is the approximated total variation computed as

V=|LM^{1/2}(\gamma_{1})-LM^{1/2}(\gamma_{L})|+|LM^{1/2}(\gamma_{2})-LM^{1/2}(\gamma_{1})|+...+|LM^{1/2}(\gamma_{U})-LM^{1/2}(\gamma_{l})|.

The feasible bound allows to approximate the p-values of the sup-type test in a straightforward way, by adding to the tail probability of a chi-square distribution a correction term which depends on the total variation of the process. For details see Armillotta and Fokianos (2022b, Sec. 5).

The values of gama_L and gama_U are computed internally as gama_L =-\log(0.9)/X^{2} and gama_U =-\log(0.1)/X^{2}, where X is the overall mean of X_{i,t} over the nodes i=1,...,N and times t=1,...,TT. Since the non-linear function e^{-\gamma X_{i,t-d}^{2}} ranges between 0 and 1, by considering X to be a representative value for the network mean, gama_U and gama_L would be the values of \gamma leading the non-linear switching function to be 0.1 and 0.9, respectively, so that in the optimization procedure the extremes of the function domain are excluded. Alternatively, their values can be supplied by the user.

Value

A list including:

DV

The Davies bound of p-values for sup test.

supLM

The value of the sup test statistic in the sample y.

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

Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 33-43.

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, global_optimise_LM_stnarpq

Examples

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