# Wald_test: Perform Wald test In uGMAR: Estimate Univariate Gaussian and Student's t Mixture Autoregressive Models

## Description

`Wald_test` performs a Wald test for a GMAR, StMAR, or G-StMAR model.

## Usage

 `1` ```Wald_test(gsmar, A, c, h = 6e-06) ```

## Arguments

 `gsmar` a class 'gsmar' object, typically generated by `fitGSMAR` or `GSMAR`. `A` a size (k x n_params) matrix with full row rank specifying a part of the null hypothesis, where n_params is the number of parameters in the (unconstrained) model. See details for more information. `c` a length k vector specifying a part of the null hypothesis. See details for more information. `h` the difference used to approximate the derivatives.

## Details

Denoting the true parameter value by θ_{0}, we test the null hypothesis Aθ_{0}=c. Under the null, the test statistic is asymptotically χ^2-distributed with k (`=nrow(A)`) degrees of freedom. The parameter θ_{0} is assumed to have the same form as in the model supplied in the argument `gsmar` and it is presented in the documentation of the argument `params` in the function `GSMAR` (see `?GSMAR`).

Note that this function does not check whether the specified constraints are feasible (e.g., whether the implied constrained model would be stationary or have positive definite error term covariance matrices).

## Value

A list with class "htest" containing the following components:

 `statistic` the value of the Wald statistics. `parameter` the degrees of freedom of the Wald statistic. `p.value` the p-value of the test. `alternative` a character string describing the alternative hypothesis. `method` a character string indicating the type of the test (Wald test). `data.name` a character string giving the names of the supplied model, constraint matrix A, and vector c. `gsmar` the supplied argument gsmar. `A` the supplied argument A. `c` the supplied argument c. `h` the supplied argument h.

## References

• Kalliovirta L., Meitz M. and Saikkonen P. 2015. Gaussian Mixture Autoregressive model for univariate time series. Journal of Time Series Analysis, 36, 247-266.

• Meitz M., Preve D., Saikkonen P. 2021. A mixture autoregressive model based on Student's t-distribution. Communications in Statistics - Theory and Methods, doi: 10.1080/03610926.2021.1916531

• Virolainen S. 2021. A mixture autoregressive model based on Gaussian and Student's t-distributions. Studies in Nonlinear Dynamics & Econometrics, doi: 10.1515/snde-2020-0060

`LR_test`, `fitGSMAR`, `GSMAR`, `diagnostic_plot`, `profile_logliks`, `quantile_residual_tests`, `cond_moment_plot`
 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```# GMAR p=1, M=2 model: fit12 <- fitGSMAR(simudata, p=1, M=2, model="GMAR", ncalls=1, seeds=1) # Test with Wald test whether the AR coefficients are the same in both # regimes: # There are 7 parameters in the model and the AR coefficient of the # first regime is the 2nd element, whereas the AR coefficient of the second # regime is in the 5th element. A <- matrix(c(0, 1, 0, 0, -1, 0, 0), nrow=1, ncol=7) c <- 0 Wald_test(fit12, A=A, c=c) ```