score_test_stnarpq_j {PNAR}

Bootstrap test for smooth transition effects on PNAR(p) model

Description

Computation of bootstrap 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_j(supLM, b, y, W, p, d, Z = NULL, J = 499,
gama_L = NULL, gama_U = NULL, tol = 1e-9, ncores = 1, seed = NULL)

Arguments

supLM

The optimized value of the test statistic. See the function global_optimise_LM_stnarpq.

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.

J

The number of bootstrap samples to draw.

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.

tol

Tolerance level for the optimizer.

ncores

Number of cores to use for parallel computing. By default the number of cores is set to 1 (no parallel computing). Note: If for some reason the parallel does not work then load the doParallel package yourseleves.

seed

To replicate the results use a seed for the generator, an integer number.

Details

The function computes a bootstrap p-value for 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). This value can be computed for the available sample by using the function global_optimise_LM_stnarpq and should be supplied here as an input supLM.

The function performs the bootstrap resampling of the test statistic \sup_{\gamma}LM(\gamma) by employing Gaussian perturbations of the score S(\hat{\theta},\gamma). 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 value can be supplied by the user.

Note: For large datasets the function may require few minutes to run. Parallel computing is suggested to speed up the computations.

Value

A list including:

pJ

The bootstrap p-value of the sup test.

cpJ

The adjusted version of bootstrap p-value of the sup test.

gamaj

The optimal values of the \gamma parameter for score test boostrap replications.

supLMj

The values of perturbed test statistic at the optimum point gamaj.

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_DV, global_optimise_LM_stnarpq, score_test_tnarpq_j

Examples

# load data
data(crime)
data(crime_W)
#estimate linear PNAR model
mod1 <- lin_estimnarpq(crime, crime_W, p = 2)
b <- mod1$coefs[, 1]

g <- global_optimise_LM_stnarpq(b = b, y = crime, W = crime_W, p = 2, d = 1)
supg <- g$supLM
score_test_stnarpq_j(supLM = supg, b = b, y = crime, W = crime_W, p = 2, d = 1, J = 5)