score_test_nonlinpq_h0: 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)

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.

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.

Author

Mirko Armillotta, Michail Tsagris and Konstantinos Fokianos.

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 (2023, Sec. 4).

References

Armillotta, M. and K. Fokianos (2023). Nonlinear network autoregression. Annals of Statistics, 51(6): 2526--2552.

Armillotta, M. and K. Fokianos (2024). Count network autoregression. Journal of Time Series Analysis, 45(4): 584--612.

Armillotta, M., Tsagris, M. and Fokianos, K. (2024). Inference for Network Count Time Series with the R Package PNAR. The R Journal, 15/4: 255--269.

Examples

Run this code

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)

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