Fit the First-Order Discrete-Time Vector Autoregressive Model

The function uses a multigroup model to fit the first-order discrete-time vector autoregressive model.

Usage

FitDTVARMx(
  data,
  observed,
  id,
  alpha_fixed = FALSE,
  alpha_values = NULL,
  alpha_free = NULL,
  alpha_lbound = NULL,
  alpha_ubound = NULL,
  beta_values = NULL,
  beta_free = NULL,
  beta_lbound = NULL,
  beta_ubound = NULL,
  psi_diag = FALSE,
  psi_l_values = NULL,
  psi_l_free = NULL,
  psi_l_lbound = NULL,
  psi_l_ubound = NULL,
  theta_fixed = FALSE,
  theta_l_values = NULL,
  theta_l_free = NULL,
  theta_l_lbound = NULL,
  theta_l_ubound = NULL,
  mu0_fixed = FALSE,
  mu0_func = FALSE,
  mu0_values = NULL,
  mu0_free = NULL,
  mu0_lbound = NULL,
  mu0_ubound = NULL,
  sigma0_fixed = FALSE,
  sigma0_func = FALSE,
  sigma0_diag = FALSE,
  sigma0_l_values = NULL,
  sigma0_l_free = NULL,
  sigma0_l_lbound = NULL,
  sigma0_l_ubound = NULL,
  try = 1000,
  ncores = NULL,
  ...
)

Arguments

data: Data frame. A data frame object of data for potentially multiple subjects that contain a column of subject ID numbers (i.e., an ID variable), and at least one column of observed values.
observed: Character vector. A vector of character strings of the names of the observed variables in the data.
id: Character string. A character string of the name of the ID variable in the data.
alpha_fixed: Logical. If alpha_fixed = TRUE, the dynamic model intercept vector alpha is fixed. If alpha_fixed = FALSE, the dynamic model intercept vector alpha is estimated.
alpha_values: Optional starting values for alpha. If alpha_fixed = TRUE, alpha_values will be used as fixed values. If alpha_fixed = FALSE, alpha_values will be used as starting values. If alpha_values = NULL, a vector of zeroes is used.
alpha_free: Optional logical vector representing free parameters for alpha.
alpha_lbound: Optional lower bound for alpha. Ignored if alpha_fixed = TRUE.
alpha_ubound: Optional upper bound for alpha. Ignored if alpha_fixed = TRUE.
beta_values: Numeric matrix. Optional starting values for beta.
beta_free: Optional logical matrix representing free parameters for beta.
beta_lbound: Numeric matrix. Optional lower bound for beta.
beta_ubound: Numeric matrix. Optional upper bound for beta.
psi_diag: Logical. If psi_diag = TRUE, psi is a diagonal matrix.
psi_l_values: Numeric matrix. Optional starting values for psi.
psi_l_free: Optional logical matrix representing free parameters for psi.
psi_l_lbound: Numeric matrix. Optional lower bound for psi.
psi_l_ubound: Optional upper bound for psi.
theta_fixed: Logical. If theta_fixed = TRUE, the measurement error matrix theta is fixed to zero. If theta_fixed = FALSE, estimate the diagonal measurement error matrix theta.
theta_l_values: Optional starting values for the Cholseky decomposition of theta. Ignored if theta_fixed = TRUE.
theta_l_free: Optional logical matrix representing free parameters for the Cholseky decomposition of theta.
theta_l_lbound: Optional lower bound for the Cholseky decomposition of theta. Ignored if theta_fixed = TRUE.
theta_l_ubound: Optional upper bound for the Cholseky decomposition of theta. Ignored if theta_fixed = TRUE.
mu0_fixed: Logical. If mu0_fixed = TRUE, initial mean vector mu0 is fixed. If mu0_fixed = FALSE, initial mean vector mu0 is estimated.
mu0_func: Logical. If mu0_func = TRUE, initial mean vector mu0 is fixed to $ \left( \matfbf{I} - \boldsymbol{\beta} \right)^{-1} \boldsymbol{\alpha} $. The argument is only evaluated when mu0_fixed = TRUE.
mu0_values: Optional starting values for mu0. If mu0_fixed = TRUE, mu0_values will be used as fixed values. If mu0_fixed = FALSE, mu0_values will be used as starting values. If mu0_values = NULL, a vector of zeroes is used.
mu0_free: Optional logical vector representing free parameters for mu0.
mu0_lbound: Optional lower bound for mu0. Ignored if mu0_fixed = TRUE.
mu0_ubound: Optional upper bound for mu0. Ignored if mu0_fixed = TRUE.
sigma0_fixed: Logical. If sigma0_fixed = TRUE, initial mean vector sigma0 is fixed. If sigma0_fixed = FALSE, initial mean vector sigma0 is estimated.
sigma0_func: Logical. If sigma0_func = TRUE, initial covariance matrix sigma0 is fixed to $ \left( \matfbf{J} - \boldsymbol{\beta} \otimes \boldsymbol{\beta} \right)^{-1} \mathrm{Vec} \left( \boldsymbol{\Psi} \right) $. The argument is only evaluated when sigma0_fixed = TRUE.
sigma0_diag: Logical. If sigma0_diag = TRUE, sigma0 is a diagonal matrix.
sigma0_l_values: Optional starting values for the Cholesky decomposition of sigma0. If sigma0_fixed = TRUE, sigma0_l_values %*% t(sigma0_l_values) will be used as fixed values for sigma0. If sigma0_fixed = FALSE, sigma0_l_values will be used as starting values for the Cholesky decomposition of sigma0. If sigma0_l_values = NULL, an identity matrix is used.
sigma0_l_free: Optional logical matrix representing free parameters for the Cholesky decomposition of sigma0.
sigma0_l_lbound: Optional lower bound for the Cholesky decomposition of sigma0. Ignored if sigma0_fixed = TRUE.
sigma0_l_ubound: Optional upper bound for the Cholesky decomposition of sigma0. Ignored if sigma0_fixed = TRUE.
try: Positive integer. Number of extra optimization tries.
ncores: Positive integer. Number of cores to use.
...: Additional optional arguments to pass to mxTryHardctsem.

Value

Returns an object of class dtvarmx which is a list with the following elements:

call: Function call.
args: List of function arguments.
fun: Function used ("FitDTVARMx").
output: A fitted OpenMx model.

Details

The measurement model is given by $$ \mathbf{y}_{i, t} = \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta} \right) $$ where $\mathbf{y}_{i, t}$, $\boldsymbol{\eta}_{i, t}$, and $\boldsymbol{\varepsilon}_{i, t}$ are random variables and $\boldsymbol{\Lambda}$, and $\boldsymbol{\Theta}$ are model parameters. $\mathbf{y}_{i, t}$ represents a vector of observed random variables, $\boldsymbol{\eta}_{i, t}$ a vector of latent random variables, and $\boldsymbol{\varepsilon}_{i, t}$ a vector of random measurement errors, at time $t$ and individual $i$. $\boldsymbol{\Lambda}$ denotes a matrix of factor loadings, and $\boldsymbol{\Theta}$ the covariance matrix of $\boldsymbol{\varepsilon}$. In this model, $\boldsymbol{\Lambda}$ is an identity matrix and $\boldsymbol{\Theta}$ is a diagonal matrix.

The dynamic structure is given by $$ \boldsymbol{\eta}_{i, t} = \boldsymbol{\alpha} + \boldsymbol{\beta} \boldsymbol{\eta}_{i, t - 1} + \boldsymbol{\zeta}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\zeta}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Psi} \right) $$ where $\boldsymbol{\eta}_{i, t}$, $\boldsymbol{\eta}_{i, t - 1}$, and $\boldsymbol{\zeta}_{i, t}$ are random variables, and $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$, and $\boldsymbol{\Psi}$ are model parameters. Here, $\boldsymbol{\eta}_{i, t}$ is a vector of latent variables at time $t$ and individual $i$, $\boldsymbol{\eta}_{i, t - 1}$ represents a vector of latent variables at time $t - 1$ and individual $i$, and $\boldsymbol{\zeta}_{i, t}$ represents a vector of dynamic noise at time $t$ and individual $i$. $\boldsymbol{\alpha}$ denotes a vector of intercepts, $\boldsymbol{\beta}$ a matrix of autoregression and cross regression coefficients, and $\boldsymbol{\Psi}$ the covariance matrix of $\boldsymbol{\zeta}_{i, t}$.

References

Hunter, M. D. (2017). State space modeling in an open source, modular, structural equation modeling environment. Structural Equation Modeling: A Multidisciplinary Journal, 25(2), 307–324. doi:10.1080/10705511.2017.1369354

Neale, M. C., Hunter, M. D., Pritikin, J. N., Zahery, M., Brick, T. R., Kirkpatrick, R. M., Estabrook, R., Bates, T. C., Maes, H. H., & Boker, S. M. (2015). OpenMx 2.0: Extended structural equation and statistical modeling. Psychometrika, 81(2), 535–549. doi:10.1007/s11336-014-9435-8

Author

Ivan Jacob Agaloos Pesigan

Examples

if (FALSE) { # \dontrun{
# Generate data using the simStateSpace package------------------------------
set.seed(42)
sim <- simStateSpace::SimSSMVARFixed(
  n = 5,
  time = 100,
  mu0 = rep(x = 0, times = 3),
  sigma0_l = t(chol(diag(3))),
  alpha = rep(x = 0, times = 3),
  beta = matrix(
    data = c(
      0.7, 0.5, -0.1,
      0.0, 0.6, 0.4,
      0, 0, 0.5
    ),
    nrow = 3
  ),
  psi_l = t(chol(diag(3)))
)
data <- as.data.frame(sim)

# Fit the model--------------------------------------------------------------
library(fitDTVARMx)
fit <- FitDTVARMx(
  data = data,
  observed = c("y1", "y2", "y3"),
  id = "id"
)
print(fit)
summary(fit)
coef(fit)
vcov(fit)
} # }