Fit the First-Order Vector Autoregressive Model by ID

The function fits the first-order vector autoregressive model for each unit ID.

Usage

FitVARMxID(
  data,
  observed,
  id,
  time = NULL,
  ct = FALSE,
  center = TRUE,
  mu_fixed = FALSE,
  mu_free = NULL,
  mu_values = NULL,
  mu_lbound = NULL,
  mu_ubound = NULL,
  alpha_fixed = FALSE,
  alpha_free = NULL,
  alpha_values = NULL,
  alpha_lbound = NULL,
  alpha_ubound = NULL,
  beta_fixed = FALSE,
  beta_free = NULL,
  beta_values = NULL,
  beta_lbound = NULL,
  beta_ubound = NULL,
  psi_diag = FALSE,
  psi_fixed = FALSE,
  psi_d_free = NULL,
  psi_d_values = NULL,
  psi_d_lbound = NULL,
  psi_d_ubound = NULL,
  psi_d_equal = FALSE,
  psi_l_free = NULL,
  psi_l_values = NULL,
  psi_l_lbound = NULL,
  psi_l_ubound = NULL,
  nu_fixed = TRUE,
  nu_free = NULL,
  nu_values = NULL,
  nu_lbound = NULL,
  nu_ubound = NULL,
  theta_diag = TRUE,
  theta_fixed = TRUE,
  theta_d_free = NULL,
  theta_d_values = NULL,
  theta_d_lbound = NULL,
  theta_d_ubound = NULL,
  theta_d_equal = FALSE,
  theta_l_free = NULL,
  theta_l_values = NULL,
  theta_l_lbound = NULL,
  theta_l_ubound = NULL,
  mu0_fixed = TRUE,
  mu0_func = TRUE,
  mu0_free = NULL,
  mu0_values = NULL,
  mu0_lbound = NULL,
  mu0_ubound = NULL,
  sigma0_fixed = TRUE,
  sigma0_func = TRUE,
  sigma0_diag = FALSE,
  sigma0_d_free = NULL,
  sigma0_d_values = NULL,
  sigma0_d_lbound = NULL,
  sigma0_d_ubound = NULL,
  sigma0_d_equal = FALSE,
  sigma0_l_free = NULL,
  sigma0_l_values = NULL,
  sigma0_l_lbound = NULL,
  sigma0_l_ubound = NULL,
  robust = FALSE,
  seed = NULL,
  tries_explore = 100,
  tries_local = 100,
  max_attempts = 10,
  silent = FALSE,
  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.
time: Character string. A character string of the name of the TIME variable in the data. Used when ct = TRUE.
ct: Logical. If TRUE, fit a continuous-time vector autoregressive model. If FALSE, fit a discrete-time vector autoregressive model.
center: Logical. If TRUE, use the mean-centered (mean-reverting) state equation. When center = TRUE, alpha is implied and the set-point mu is estimated. When center = FALSE, alpha is estimated and mu is implied.
mu_fixed: Logical. If TRUE, the set-point mean vector mu is fixed to mu_values. If mu_fixed = TRUE and mu_values = NULL, mu is fixed to a zero vector. If FALSE, mu is estimated.
mu_free: Logical vector indicating which elements of mu are freely estimated. If NULL, all elements are free. Ignored if mu_fixed = TRUE.
mu_values: Numeric vector of values for mu. If mu_fixed = TRUE, these are fixed values. If mu_fixed = FALSE, these are starting values. If NULL, defaults to a vector of zeros.
mu_lbound: Numeric vector of lower bounds for mu. Ignored if mu_fixed = TRUE.
mu_ubound: Numeric vector of upper bounds for mu. Ignored if mu_fixed = TRUE.
alpha_fixed: Logical. If TRUE, the dynamic model intercept vector alpha is fixed to alpha_values. If FALSE, alpha is estimated.
alpha_free: Logical vector indicating which elements of alpha are freely estimated. If NULL, all elements are free. Ignored if alpha_fixed = TRUE.
alpha_values: Numeric vector of values for alpha. If alpha_fixed = TRUE, these are fixed values. If alpha_fixed = FALSE, these are starting values. If NULL, defaults to a vector of zeros.
alpha_lbound: Numeric vector of lower bounds for alpha. Ignored if alpha_fixed = TRUE.
alpha_ubound: Numeric vector of upper bounds for alpha. Ignored if alpha_fixed = TRUE.
beta_fixed: Logical. If TRUE, the dynamic model coefficient matrix beta is fixed. If FALSE, beta is estimated.
beta_free: Logical matrix indicating which elements of beta are freely estimated. If NULL, all elements are free. Ignored if beta_fixed = TRUE.
beta_values: Numeric matrix. Values for beta. If beta_fixed = TRUE, these are fixed values; if beta_fixed = FALSE, these are starting values. If NULL, defaults to a diagonal matrix with -0.001 when ct = TRUE and 0.001 when ct = FALSE.
beta_lbound: Numeric matrix. Lower bounds for beta. Ignored if beta_fixed = TRUE. If NULL and ct = FALSE, defaults to -2.5. If NULL and ct = TRUE, defaults to NA.
beta_ubound: Numeric matrix. Upper bounds for beta. Ignored if beta_fixed = TRUE. If NULL and ct = FALSE, defaults to +2.5. If NULL and ct = TRUE, diagonal upper bounds are set to -1e-05 and the off-diagonals are set to NA.
psi_diag: Logical. If TRUE, psi is diagonal. If FALSE, psi is symmetric.
psi_fixed: Logical. If TRUE, the process noise covariance matrix psi is fixed using psi_d_values and psi_l_values. If psi_d_values is NULL it is fixed to a zero matrix. If FALSE, psi is estimated.
psi_d_free: Logical vector indicating free/fixed status of the elements of psi_d. If NULL, all element of psi_d are free.
psi_d_values: Numeric vector with starting values for psi_d. If psi_fixed = TRUE, these are fixed values. If psi_fixed = FALSE, these are starting values.
psi_d_lbound: Numeric vector with lower bounds for psi_d.
psi_d_ubound: Numeric vector with upper bounds for psi_d.
psi_d_equal: Logical. When TRUE, all free diagonal elements of psi_d are constrained to be equal and estimated as a single shared parameter. Ignored if no diagonal elements are free.
psi_l_free: Logical matrix indicating which strictly-lower-triangular elements of psi_l are free. If NULL, all element of psi_l are free. Ignored if psi_diag = TRUE.
psi_l_values: Numeric matrix of starting values for the strictly-lower-triangular elements of psi_l.
psi_l_lbound: Numeric matrix with lower bounds for psi_l.
psi_l_ubound: Numeric matrix with upper bounds for psi_l.
nu_fixed: Logical. If TRUE, the measurement model intercept vector nu is fixed to nu_values. If FALSE, nu is estimated.
nu_free: Logical vector indicating which elements of nu are freely estimated. If NULL, all elements are free. Ignored if nu_fixed = TRUE.
nu_values: Numeric vector of values for nu. If nu_fixed = TRUE, these are fixed values. If nu_fixed = FALSE, these are starting values. If NULL, defaults to a vector of zeros.
nu_lbound: Numeric vector of lower bounds for nu. Ignored if nu_fixed = TRUE.
nu_ubound: Numeric vector of upper bounds for nu. Ignored if nu_fixed = TRUE.
theta_diag: Logical. If TRUE, theta is diagonal. If FALSE, theta is symmetric.
theta_fixed: Logical. If TRUE, the measurement error covariance matrix theta is fixed using theta_d_values and theta_l_values. If theta_d_values is NULL it is fixed to a zero matrix. If FALSE, theta is estimated.
theta_d_free: Logical vector indicating free/fixed status of the diagonal parameters theta_d. If NULL, all element of theta_d are free.
theta_d_values: Numeric vector with starting values for theta_d. If theta_fixed = TRUE, these are fixed values. If theta_fixed = FALSE, these are starting values.
theta_d_lbound: Numeric vector with lower bounds for theta_d.
theta_d_ubound: Numeric vector with upper bounds for theta_d.
theta_d_equal: Logical. When TRUE, all free diagonal elements of theta_d are constrained to be equal and estimated as a single shared parameter. Ignored if no diagonal elements are free.
theta_l_free: Logical matrix indicating which strictly-lower-triangular elements of theta_l are free. If NULL, all element of theta_l are free. Ignored if theta_diag = TRUE.
theta_l_values: Numeric matrix of starting values for the strictly-lower-triangular elements of theta_l.
theta_l_lbound: Numeric matrix with lower bounds for theta_l.
theta_l_ubound: Numeric matrix with upper bounds for theta_l.
mu0_fixed: Logical. If TRUE, the initial mean vector mu0 is fixed. If mu0_fixed = TRUE and mu0_func = TRUE, mu0 is fixed to the implied stable mean vector. If mu0_fixed = TRUE and mu0_values = NULL, mu0 is fixed to a zero vector. If FALSE, mu0 is estimated.
mu0_func: Logical. If TRUE and mu0_fixed = TRUE, mu0 is fixed to the implied stable mean vector.
mu0_free: Logical vector indicating which elements of mu0 are free. Ignored if mu0_fixed = TRUE.
mu0_values: Numeric vector of values for mu0. If mu0_fixed = TRUE, these are fixed values. If mu0_fixed = FALSE, these are starting values. If NULL, defaults to a vector of zeros. Ignored if mu0_fixed = TRUE and mu0_func = TRUE.
mu0_lbound: Numeric vector of lower bounds for mu0. Ignored if mu0_fixed = TRUE.
mu0_ubound: Numeric vector of upper bounds for mu0. Ignored if mu0_fixed = TRUE.
sigma0_fixed: Logical. If TRUE, the initial condition covariance matrix sigma0 is fixed using sigma0_d_values and sigma0_l_values. If sigma0_fixed = TRUE and sigma0_func = TRUE, sigma0 is fixed to the implied stable covariance matrix. If sigma0_fixed = TRUE and sigma0_d_values = NULL, sigma0 is fixed to a diffused matrix.
sigma0_func: Logical. If TRUE and sigma0_fixed = TRUE, sigma0 is fixed to the implied stable covariance matrix.
sigma0_diag: Logical. If TRUE, sigma0 is diagonal. If FALSE, sigma0 is symmetric.
sigma0_d_free: Logical vector indicating free/fixed status of the elements of sigma0_d. If NULL, all element of sigma0_d are free.
sigma0_d_values: Numeric vector with starting values for sigma0_d. If sigma0_fixed = TRUE, these are fixed values. If sigma0_fixed = FALSE, these are starting values.
sigma0_d_lbound: Numeric vector with lower bounds for sigma0_d.
sigma0_d_ubound: Numeric vector with upper bounds for sigma0_d.
sigma0_d_equal: Logical. When TRUE, all free diagonal elements of sigma0_d are constrained to be equal and estimated as a single shared parameter. Ignored if no diagonal elements are free.
sigma0_l_free: Logical matrix indicating which strictly-lower-triangular elements of sigma0_l are free. If NULL, all element of sigma0_l are free. Ignored if sigma0_diag = TRUE.
sigma0_l_values: Numeric matrix of starting values for the strictly-lower-triangular elements of sigma0_l.
sigma0_l_lbound: Numeric matrix with lower bounds for sigma0_l.
sigma0_l_ubound: Numeric matrix with upper bounds for sigma0_l.
robust: Logical. If TRUE, calculate robust (sandwich) sampling variance-covariance matrix.
seed: Random seed for reproducibility.
tries_explore: Integer. Number of extra tries for the wide exploration phase.
tries_local: Integer. Number of extra tries for local polishing.
max_attempts: Integer. Maximum number of remediation attempts after the first Hessian computation fails the criteria.
silent: Logical. If TRUE, suppresses messages during the model fitting stage.
ncores: Positive integer. Number of cores to use.

Value

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

call: Function call.
args: List of function arguments.
fun: Function used ("FitVARMxID").
model: A list of generated OpenMx models.
output: A list of fitted OpenMx models.
converged: A logical vector indicating converged cases.
robust: A list of output from OpenMx::imxRobustSE() with argument details = TRUE for each id if robust = TRUE.

Details

Measurement Model

By default, the measurement model is given by $$ \mathbf{y}_{i, t} = \boldsymbol{\eta}_{i, t} . $$ However, the full measurement model can be parameterized as follows $$ \mathbf{y}_{i, t} = \boldsymbol{\nu}_{i} + \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta}_{i} \right) $$ where $\mathbf{y}_{i, t}$, $\boldsymbol{\eta}_{i, t}$, and $\boldsymbol{\varepsilon}_{i, t}$ are random variables and $\boldsymbol{\nu}_{i}$, $\boldsymbol{\Lambda}$, and $\boldsymbol{\Theta}_{i}$ 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 $i$. $\boldsymbol{\nu}_{i}$, denotes a vector of intercepts (fixed to a null vector by default), $\boldsymbol{\Lambda}$ a matrix of factor loadings, and $\boldsymbol{\Theta}_{i}$ the covariance matrix of $\boldsymbol{\varepsilon}$. In this model, $\boldsymbol{\Lambda}$ is an identity matrix and $\boldsymbol{\Theta}_{i}$ is a diagonal matrix.

Discrete-Time Dynamic Structure

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

If center = TRUE, the dynamic structure is parameterized as follows $$ \boldsymbol{\eta}_{i, t} = \boldsymbol{\mu}_{i} + \boldsymbol{\beta}_{i} \left( \boldsymbol{\eta}_{i, t - 1} - \boldsymbol{\mu}_{i} \right) + \boldsymbol{\zeta}_{i, t} $$ where $\boldsymbol{\mu}_{i}$ is equilibrium level of the latent state toward which the system is pulled over time.

Continuous-Time Dynamic Structure

The continuous-time parameterization, when ct = TRUE, for the dynamic structure is given by $$ \mathrm{d} \boldsymbol{\eta}_{i, t} = \left( \boldsymbol{\alpha}_{i} + \boldsymbol{\beta}_{i} \boldsymbol{\eta}_{i, t} \right) \mathrm{d} t + \boldsymbol{\Psi}_{i}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t} $$ note that $\mathrm{d}\boldsymbol{W}$ is a Wiener process or Brownian motion, which represents random fluctuations.

If center = TRUE, the dynamic structure is parameterized as follows $$ \mathrm{d} \boldsymbol{\eta}_{i, t} = \boldsymbol{\beta}_{i} \left( \boldsymbol{\eta}_{i, t} - \boldsymbol{\mu}_{i} \right) \mathrm{d} t + \boldsymbol{\Psi}_{i}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t} $$

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

# \donttest{
# Generate data using the simStateSpace package-------------------------
library(simStateSpace)
set.seed(42)
n <- 5
time <- 100
p <- 2
alpha <- rep(x = 0, times = p)
beta <- 0.50 * diag(p)
psi <- 0.001 * diag(p)
psi_l <- t(chol(psi))
mu0 <- SSMMeanEta(
  beta = beta,
  alpha = alpha
)
sigma0 <- SSMCovEta(
  beta = beta,
  psi = psi
)
sigma0_l <- t(chol(sigma0))
sim <- SimSSMVARFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l
)
data <- as.data.frame(sim)

# Fit the model---------------------------------------------------------
# center = TRUE
library(fitVARMxID)
fit <- FitVARMxID(
  data = data,
  observed = paste0("y", seq_len(p)),
  id = "id",
  center = TRUE
)
#> Running DTVAR_ID1 with 9 parameters
#> 
#> Beginning initial fit attempt
#> Running DTVAR_ID1 with 9 parameters
#> 
#>  Lowest minimum so far:  -822.060942313193
#> 
#> Solution found
#> 


#> 
#>  Solution found!  Final fit=-822.06094 (started at 367.82482)  (1 attempt(s): 1 valid, 0 errors)
#>  Start values from best fit:
#> 0.331612745243841,-0.0197631400065759,0.0270181292624815,0.539260086092713,0.0057946951525851,0.007106199232002,0.0623713422440293,-6.91195717901486,-6.98806630902827
#> Running DTVAR_ID2 with 9 parameters
#> 
#> Beginning initial fit attempt
#> Running DTVAR_ID2 with 9 parameters
#> 
#>  Lowest minimum so far:  -834.035953636412
#> 
#> Solution found
#> 


#> 
#>  Solution found!  Final fit=-834.03595 (started at 367.84346)  (1 attempt(s): 1 valid, 0 errors)
#>  Start values from best fit:
#> 0.546679030340841,-0.0201020488638611,-0.0487574618599386,0.297450290600703,-0.00284153166188121,-0.0199122101192943,0.176134255014816,-7.14253861726531,-6.87719884724769
#> Running DTVAR_ID3 with 9 parameters
#> 
#> Beginning initial fit attempt
#> Running DTVAR_ID3 with 9 parameters
#> 
#>  Lowest minimum so far:  -808.550007472268
#> 
#> Solution found
#> 


#> 
#>  Solution found!  Final fit=-808.55001 (started at 367.87017)  (1 attempt(s): 1 valid, 0 errors)
#>  Start values from best fit:
#> 0.528105561142997,-0.332501283495961,0.0108536161301484,0.361419705300276,-0.00426229999671594,-0.00173173484843314,-0.21387789357269,-6.9657202697529,-6.80024208411447
#> Running DTVAR_ID4 with 9 parameters
#> 
#> Beginning initial fit attempt
#> Running DTVAR_ID4 with 9 parameters
#> 
#>  Lowest minimum so far:  -845.339407101868
#> 
#> Solution found
#> 


#> 
#>  Solution found!  Final fit=-845.33941 (started at 367.80199)  (1 attempt(s): 1 valid, 0 errors)
#>  Start values from best fit:
#> 0.160844162100514,-0.121824766994102,-0.0298919550660853,0.481815653357755,-0.0130955259827441,0.00326100822229185,0.00601848006477796,-7.23151794331632,-6.89984913049538
#> Running DTVAR_ID5 with 9 parameters
#> 
#> Beginning initial fit attempt
#> Running DTVAR_ID5 with 9 parameters
#> 
#>  Lowest minimum so far:  -832.354548096018
#> 
#> Solution found
#> 


#> 
#>  Solution found!  Final fit=-832.35455 (started at 367.7976)  (1 attempt(s): 1 valid, 0 errors)
#>  Start values from best fit:
#> 0.299055985152669,0.0348621727088368,-0.0801687055865619,0.465890390856868,-0.00111115045695215,0.00331563195705935,-0.166419668597626,-6.91180267781828,-7.08999323366805
print(fit)
#> Call:
#> FitVARMxID(data = data, observed = paste0("y", seq_len(p)), id = "id", 
#>     center = TRUE)
#> 
#> Convergence:
#> 100.0%
#> 
#> Estimated paramaters per individual.
#>   mu[1,1] mu[2,1] beta[1,1] beta[2,1] beta[1,2] beta[2,2] psi[1,1] psi[2,1]
#> 1  0.0058  0.0071    0.3316   -0.0198    0.0270    0.5393    1e-03    1e-04
#> 2 -0.0028 -0.0199    0.5467   -0.0201   -0.0488    0.2975    8e-04    1e-04
#> 3 -0.0043 -0.0017    0.5281   -0.3325    0.0109    0.3614    9e-04   -2e-04
#> 4 -0.0131  0.0033    0.1608   -0.1218   -0.0299    0.4818    7e-04    0e+00
#> 5 -0.0011  0.0033    0.2991    0.0349   -0.0802    0.4659    1e-03   -2e-04
#>   psi[2,2]
#> 1   0.0009
#> 2   0.0011
#> 3   0.0012
#> 4   0.0010
#> 5   0.0009
summary(fit)
#> Call:
#> FitVARMxID(data = data, observed = paste0("y", seq_len(p)), id = "id", 
#>     center = TRUE)
#> 
#> Convergence:
#> 100.0%
#> 
#> Estimated paramaters per individual.
#>   mu[1,1] mu[2,1] beta[1,1] beta[2,1] beta[1,2] beta[2,2] psi[1,1] psi[2,1]
#> 1  0.0058  0.0071    0.3316   -0.0198    0.0270    0.5393    1e-03    1e-04
#> 2 -0.0028 -0.0199    0.5467   -0.0201   -0.0488    0.2975    8e-04    1e-04
#> 3 -0.0043 -0.0017    0.5281   -0.3325    0.0109    0.3614    9e-04   -2e-04
#> 4 -0.0131  0.0033    0.1608   -0.1218   -0.0299    0.4818    7e-04    0e+00
#> 5 -0.0011  0.0033    0.2991    0.0349   -0.0802    0.4659    1e-03   -2e-04
#>   psi[2,2]
#> 1   0.0009
#> 2   0.0011
#> 3   0.0012
#> 4   0.0010
#> 5   0.0009
coef(fit)
#> $`1`
#>       mu[1,1]       mu[2,1]     beta[1,1]     beta[2,1]     beta[1,2] 
#>  5.794695e-03  7.106199e-03  3.316127e-01 -1.976314e-02  2.701813e-02 
#>     beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2] 
#>  5.392601e-01  9.953214e-04  6.207953e-05  9.262857e-04 
#> 
#> $`2`
#>       mu[1,1]       mu[2,1]     beta[1,1]     beta[2,1]     beta[1,2] 
#> -0.0028415317 -0.0199122101  0.5466790303 -0.0201020489 -0.0487574619 
#>     beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2] 
#>  0.2974502906  0.0007904397  0.0001392235  0.0010550290 
#> 
#> $`3`
#>       mu[1,1]       mu[2,1]     beta[1,1]     beta[2,1]     beta[1,2] 
#> -0.0042623000 -0.0017317348  0.5281055611 -0.3325012835  0.0108536161 
#>     beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2] 
#>  0.3614197053  0.0009432480 -0.0002017399  0.0011560438 
#> 
#> $`4`
#>       mu[1,1]       mu[2,1]     beta[1,1]     beta[2,1]     beta[1,2] 
#> -1.309553e-02  3.261008e-03  1.608442e-01 -1.218248e-01 -2.989196e-02 
#>     beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2] 
#>  4.818157e-01  7.231704e-04  4.352386e-06  1.007466e-03 
#> 
#> $`5`
#>       mu[1,1]       mu[2,1]     beta[1,1]     beta[2,1]     beta[1,2] 
#> -0.0011111505  0.0033156320  0.2990559852  0.0348621727 -0.0801687056 
#>     beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2] 
#>  0.4658903909  0.0009954751 -0.0001656666  0.0008606361 
#> 
vcov(fit)
#> $`1`
#>                 mu[1,1]       mu[2,1]     beta[1,1]     beta[2,1]     beta[1,2]
#> mu[1,1]    2.224531e-05  2.743979e-06 -5.546138e-06  5.372819e-06  1.426060e-05
#> mu[2,1]    2.743979e-06  4.246265e-05 -8.513834e-08  2.692920e-07 -6.084754e-06
#> beta[1,1] -5.546138e-06 -8.513834e-08  8.856671e-03  5.015831e-04 -6.327507e-04
#> beta[2,1]  5.372819e-06  2.692920e-07  5.015831e-04  8.222799e-03 -5.447017e-05
#> beta[1,2]  1.426060e-05 -6.084754e-06 -6.327507e-04 -5.447017e-05  7.838173e-03
#> beta[2,2]  1.561657e-06  2.505703e-05 -3.578235e-05 -5.710681e-04  4.375766e-04
#> psi[1,1]  -6.356703e-11  4.508286e-10 -6.465520e-08 -2.604991e-09 -2.201980e-09
#> psi[2,1]   1.052388e-09  1.585577e-09 -1.335929e-08 -4.590393e-08 -4.219508e-08
#> psi[2,2]   2.048862e-10 -5.688192e-10 -2.195562e-09  1.320718e-09 -6.345910e-09
#>               beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2]
#> mu[1,1]    1.561657e-06 -6.356703e-11  1.052388e-09  2.048862e-10
#> mu[2,1]    2.505703e-05  4.508286e-10  1.585577e-09 -5.688192e-10
#> beta[1,1] -3.578235e-05 -6.465520e-08 -1.335929e-08 -2.195562e-09
#> beta[2,1] -5.710681e-04 -2.604991e-09 -4.590393e-08  1.320718e-09
#> beta[1,2]  4.375766e-04 -2.201980e-09 -4.219508e-08 -6.345910e-09
#> beta[2,2]  7.234812e-03  3.359103e-09  1.100977e-08 -1.024578e-07
#> psi[1,1]   3.359103e-09  1.981853e-08  1.237250e-09  7.717316e-11
#> psi[2,1]   1.100977e-08  1.237250e-09  9.257098e-09  1.150936e-09
#> psi[2,2]  -1.024578e-07  7.717316e-11  1.150936e-09  1.716564e-08
#> 
#> $`2`
#>                 mu[1,1]       mu[2,1]     beta[1,1]     beta[2,1]     beta[1,2]
#> mu[1,1]    3.711430e-05  1.028041e-06 -1.259632e-06 -1.486527e-06 -6.233540e-06
#> mu[2,1]    1.028041e-06  2.111878e-05 -5.442272e-07 -5.152523e-06  3.401863e-06
#> beta[1,1] -1.259632e-06 -5.442272e-07  7.020210e-03  1.240322e-03 -7.462724e-04
#> beta[2,1] -1.486527e-06 -5.152523e-06  1.240322e-03  9.418442e-03 -1.528294e-04
#> beta[1,2] -6.233540e-06  3.401863e-06 -7.462724e-04 -1.528294e-04  6.909651e-03
#> beta[2,2] -5.125566e-07 -6.251956e-06 -1.319441e-04 -9.994834e-04  1.212534e-03
#> psi[1,1]   1.736505e-10  4.490027e-10 -8.316323e-08 -4.757987e-09 -5.289942e-10
#> psi[2,1]   4.315405e-10  1.152683e-09 -1.582423e-09 -3.001699e-08 -4.177727e-08
#> psi[2,2]  -6.295115e-11 -1.484608e-10 -2.478939e-10 -5.296051e-10 -4.988704e-09
#>               beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2]
#> mu[1,1]   -5.125566e-07  1.736505e-10  4.315405e-10 -6.295115e-11
#> mu[2,1]   -6.251956e-06  4.490027e-10  1.152683e-09 -1.484608e-10
#> beta[1,1] -1.319441e-04 -8.316323e-08 -1.582423e-09 -2.478939e-10
#> beta[2,1] -9.994834e-04 -4.757987e-09 -3.001699e-08 -5.296051e-10
#> beta[1,2]  1.212534e-03 -5.289942e-10 -4.177727e-08 -4.988704e-09
#> beta[2,2]  9.204832e-03 -6.900777e-10 -4.464222e-09 -6.176501e-08
#> psi[1,1]  -6.900777e-10  1.249914e-08  2.199571e-09  3.880986e-10
#> psi[2,1]  -4.464222e-09  2.199571e-09  8.528443e-09  2.938846e-09
#> psi[2,2]  -6.176501e-08  3.880986e-10  2.938846e-09  2.226704e-08
#> 
#> $`3`
#>                 mu[1,1]       mu[2,1]     beta[1,1]     beta[2,1]     beta[1,2]
#> mu[1,1]    4.014794e-05 -2.654775e-05 -4.975392e-06  3.545267e-08  6.952597e-06
#> mu[2,1]   -2.654775e-05  4.483621e-05 -7.485950e-06  1.159674e-05 -2.039629e-05
#> beta[1,1] -4.975392e-06 -7.485950e-06  8.624703e-03 -1.819206e-03  2.783507e-03
#> beta[2,1]  3.545267e-08  1.159674e-05 -1.819206e-03  1.025332e-02 -5.999974e-04
#> beta[1,2]  6.952597e-06 -2.039629e-05  2.783507e-03 -5.999974e-04  6.799241e-03
#> beta[2,2] -1.948947e-06  7.272828e-06 -6.216471e-04  3.312132e-03 -1.494444e-03
#> psi[1,1]  -1.436036e-09  1.121628e-09 -8.827167e-08  6.928421e-08 -1.229378e-08
#> psi[2,1]  -5.397337e-09 -2.352972e-09  1.362150e-07 -6.786872e-08  2.827157e-08
#> psi[2,2]   4.203122e-09 -3.178531e-10 -8.092158e-08  6.993384e-09 -5.252178e-09
#>               beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2]
#> mu[1,1]   -1.948947e-06 -1.436036e-09 -5.397337e-09  4.203122e-09
#> mu[2,1]    7.272828e-06  1.121628e-09 -2.352972e-09 -3.178531e-10
#> beta[1,1] -6.216471e-04 -8.827167e-08  1.362150e-07 -8.092158e-08
#> beta[2,1]  3.312132e-03  6.928421e-08 -6.786872e-08  6.993384e-09
#> beta[1,2] -1.494444e-03 -1.229378e-08  2.827157e-08 -5.252178e-09
#> beta[2,2]  8.033171e-03  3.692584e-09  4.371743e-09 -7.603266e-08
#> psi[1,1]   3.692584e-09  1.776845e-08 -3.803148e-09  8.297955e-10
#> psi[2,1]   4.371743e-09 -3.803148e-09  1.136282e-08 -4.703032e-09
#> psi[2,2]  -7.603266e-08  8.297955e-10 -4.703032e-09  2.676450e-08
#> 
#> $`4`
#>                 mu[1,1]       mu[2,1]     beta[1,1]     beta[2,1]     beta[1,2]
#> mu[1,1]    1.043712e-05 -3.604418e-06  4.814324e-06 -1.628211e-06 -1.951168e-07
#> mu[2,1]   -3.604418e-06  3.796609e-05  3.531995e-07 -3.004506e-06  7.858673e-06
#> beta[1,1]  4.814324e-06  3.531995e-07  9.865757e-03  3.424691e-05  2.406062e-04
#> beta[2,1] -1.628211e-06 -3.004506e-06  3.424691e-05  1.342206e-02 -2.793637e-05
#> beta[1,2] -1.951168e-07  7.858673e-06  2.406062e-04 -2.793637e-05  5.470098e-03
#> beta[2,2]  7.280477e-08 -1.813611e-06 -3.572421e-06  3.342110e-04  1.482257e-05
#> psi[1,1]   1.858414e-10  6.042985e-10 -1.927989e-08  1.700900e-08  1.075401e-08
#> psi[2,1]   1.409379e-10 -3.190850e-09  6.911712e-09 -4.795484e-08 -6.116108e-08
#> psi[2,2]  -2.788222e-10 -8.534671e-10 -6.757041e-09  2.822853e-10 -7.525012e-09
#>               beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2]
#> mu[1,1]    7.280477e-08  1.858414e-10  1.409379e-10 -2.788222e-10
#> mu[2,1]   -1.813611e-06  6.042985e-10 -3.190850e-09 -8.534671e-10
#> beta[1,1] -3.572421e-06 -1.927989e-08  6.911712e-09 -6.757041e-09
#> beta[2,1]  3.342110e-04  1.700900e-08 -4.795484e-08  2.822853e-10
#> beta[1,2]  1.482257e-05  1.075401e-08 -6.116108e-08 -7.525012e-09
#> beta[2,2]  7.419211e-03 -1.769062e-09  1.850297e-08 -9.165652e-08
#> psi[1,1]  -1.769062e-09  1.046050e-08  6.372424e-11  1.141598e-12
#> psi[2,1]   1.850297e-08  6.372424e-11  7.297033e-09  9.078742e-11
#> psi[2,2]  -9.165652e-08  1.141598e-12  9.078742e-11  2.030675e-08
#> 
#> $`5`
#>                 mu[1,1]       mu[2,1]     beta[1,1]     beta[2,1]     beta[1,2]
#> mu[1,1]    2.115776e-05 -6.312505e-06 -9.365997e-07 -7.441133e-06  2.767397e-07
#> mu[2,1]   -6.312505e-06  2.876505e-05  3.661845e-08  8.679316e-08 -8.396412e-07
#> beta[1,1] -9.365997e-07  3.661845e-08  9.209205e-03 -1.527788e-03  1.910733e-03
#> beta[2,1] -7.441133e-06  8.679316e-08 -1.527788e-03  8.067790e-03 -3.233289e-04
#> beta[1,2]  2.767397e-07 -8.396412e-07  1.910733e-03 -3.233289e-04  9.374810e-03
#> beta[2,2] -6.090495e-08 -7.854492e-06 -3.163124e-04  1.665073e-03 -1.546388e-03
#> psi[1,1]  -1.799574e-10 -2.759494e-11 -6.116907e-08  1.073268e-08 -2.432120e-09
#> psi[2,1]  -1.077773e-09 -4.450186e-10  1.327054e-08 -2.513345e-08 -2.731715e-08
#> psi[2,2]   5.203866e-10  2.457130e-10 -6.550906e-10 -7.755098e-09  2.272470e-08
#>               beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2]
#> mu[1,1]   -6.090495e-08 -1.799574e-10 -1.077773e-09  5.203866e-10
#> mu[2,1]   -7.854492e-06 -2.759494e-11 -4.450186e-10  2.457130e-10
#> beta[1,1] -3.163124e-04 -6.116907e-08  1.327054e-08 -6.550906e-10
#> beta[2,1]  1.665073e-03  1.073268e-08 -2.513345e-08 -7.755098e-09
#> beta[1,2] -1.546388e-03 -2.432120e-09 -2.731715e-08  2.272470e-08
#> beta[2,2]  8.178308e-03  1.594600e-09  1.057114e-08 -8.224086e-08
#> psi[1,1]   1.594600e-09  1.982445e-08 -3.299155e-09  5.501745e-10
#> psi[2,1]   1.057114e-08 -3.299155e-09  8.842776e-09 -2.854392e-09
#> psi[2,2]  -8.224086e-08  5.501745e-10 -2.854392e-09  1.481772e-08
#> 
converged(fit)
#>    1    2    3    4    5 
#> TRUE TRUE TRUE TRUE TRUE 
confint(fit)
#> $`1`
#>                     est           se          z            p          2.5%
#> mu[1,1]    5.794695e-03 4.716494e-03  1.2286023 2.192209e-01 -0.0034494626
#> mu[2,1]    7.106199e-03 6.516337e-03  1.0905206 2.754839e-01 -0.0056655862
#> beta[1,1]  3.316127e-01 9.410989e-02  3.5236760 4.256043e-04  0.1471607586
#> beta[2,1] -1.976314e-02 9.067965e-02 -0.2179446 8.274723e-01 -0.1974919839
#> beta[1,2]  2.701813e-02 8.853346e-02  0.3051742 7.602335e-01 -0.1465042608
#> beta[2,2]  5.392601e-01 8.505770e-02  6.3399328 2.298653e-10  0.3725500634
#> psi[1,1]   9.953214e-04 1.407783e-04  7.0701331 1.547852e-12  0.0007194010
#> psi[2,1]   6.207953e-05 9.621382e-05  0.6452247 5.187816e-01 -0.0001264961
#> psi[2,2]   9.262857e-04 1.310177e-04  7.0699276 1.550145e-12  0.0006694957
#>                  97.5%
#> mu[1,1]   0.0150388529
#> mu[2,1]   0.0198779847
#> beta[1,1] 0.5160647319
#> beta[2,1] 0.1579657039
#> beta[1,2] 0.2005405193
#> beta[2,2] 0.7059701088
#> psi[1,1]  0.0012712419
#> psi[2,1]  0.0002506551
#> psi[2,2]  0.0011830757
#> 
#> $`2`
#>                     est           se          z            p          2.5%
#> mu[1,1]   -0.0028415317 6.092150e-03 -0.4664251 6.409113e-01 -1.478193e-02
#> mu[2,1]   -0.0199122101 4.595517e-03 -4.3329637 1.471154e-05 -2.891926e-02
#> beta[1,1]  0.5466790303 8.378669e-02  6.5246521 6.815955e-11  3.824601e-01
#> beta[2,1] -0.0201020489 9.704866e-02 -0.2071337 8.359054e-01 -2.103139e-01
#> beta[1,2] -0.0487574619 8.312431e-02 -0.5865608 5.574987e-01 -2.116781e-01
#> beta[2,2]  0.2974502906 9.594182e-02  3.1003196 1.933119e-03  1.094078e-01
#> psi[1,1]   0.0007904397 1.117995e-04  7.0701511 1.547651e-12  5.713166e-04
#> psi[2,1]   0.0001392235 9.234957e-05  1.5075707 1.316644e-01 -4.177832e-05
#> psi[2,2]   0.0010550290 1.492214e-04  7.0702241 1.546837e-12  7.625603e-04
#>                   97.5%
#> mu[1,1]    0.0090988639
#> mu[2,1]   -0.0109051615
#> beta[1,1]  0.7108979333
#> beta[2,1]  0.1701098267
#> beta[1,2]  0.1141631945
#> beta[2,2]  0.4854927942
#> psi[1,1]   0.0010095628
#> psi[2,1]   0.0003202253
#> psi[2,2]   0.0013474976
#> 
#> $`3`
#>                     est           se          z            p          2.5%
#> mu[1,1]   -0.0042623000 0.0063362402 -0.6726860 5.011471e-01 -0.0166811025
#> mu[2,1]   -0.0017317348 0.0066959846 -0.2586229 7.959262e-01 -0.0148556235
#> beta[1,1]  0.5281055611 0.0928692778  5.6865475 1.296333e-08  0.3460851215
#> beta[2,1] -0.3325012835 0.1012586808 -3.2836818 1.024605e-03 -0.5309646510
#> beta[1,2]  0.0108536161 0.0824575110  0.1316268 8.952795e-01 -0.1507601357
#> beta[2,2]  0.3614197053 0.0896279579  4.0324438 5.519981e-05  0.1857521358
#> psi[1,1]   0.0009432480 0.0001332983  7.0762167 1.481432e-12  0.0006819880
#> psi[2,1]  -0.0002017399 0.0001065965 -1.8925564 5.841689e-02 -0.0004106652
#> psi[2,2]   0.0011560438 0.0001635986  7.0663433 1.590696e-12  0.0008353964
#>                   97.5%
#> mu[1,1]    8.156503e-03
#> mu[2,1]    1.139215e-02
#> beta[1,1]  7.101260e-01
#> beta[2,1] -1.340379e-01
#> beta[1,2]  1.724674e-01
#> beta[2,2]  5.370873e-01
#> psi[1,1]   1.204508e-03
#> psi[2,1]   7.185418e-06
#> psi[2,2]   1.476691e-03
#> 
#> $`4`
#>                     est           se           z            p          2.5%
#> mu[1,1]   -1.309553e-02 3.230653e-03 -4.05352239 5.045217e-05 -0.0194274903
#> mu[2,1]    3.261008e-03 6.161663e-03  0.52924156 5.966379e-01 -0.0088156295
#> beta[1,1]  1.608442e-01 9.932652e-02  1.61934768 1.053725e-01 -0.0338322305
#> beta[2,1] -1.218248e-01 1.158536e-01 -1.05154040 2.930105e-01 -0.3488936994
#> beta[1,2] -2.989196e-02 7.396011e-02 -0.40416319 6.860927e-01 -0.1748511120
#> beta[2,2]  4.818157e-01 8.613484e-02  5.59373709 2.222334e-08  0.3129944678
#> psi[1,1]   7.231704e-04 1.022766e-04  7.07073379 1.541165e-12  0.0005227120
#> psi[2,1]   4.352386e-06 8.542267e-05  0.05095119 9.593644e-01 -0.0001630730
#> psi[2,2]   1.007466e-03 1.425018e-04  7.06984995 1.551013e-12  0.0007281677
#>                   97.5%
#> mu[1,1]   -0.0067635616
#> mu[2,1]    0.0153376460
#> beta[1,1]  0.3555205547
#> beta[2,1]  0.1052441654
#> beta[1,2]  0.1150672019
#> beta[2,2]  0.6506368389
#> psi[1,1]   0.0009236288
#> psi[2,1]   0.0001717777
#> psi[2,2]   0.0012867644
#> 
#> $`5`
#>                     est           se          z            p          2.5%
#> mu[1,1]   -0.0011111505 4.599757e-03 -0.2415672 8.091155e-01 -0.0101265080
#> mu[2,1]    0.0033156320 5.363306e-03  0.6182068 5.364391e-01 -0.0071962544
#> beta[1,1]  0.2990559852 9.596461e-02  3.1163155 1.831262e-03  0.1109688153
#> beta[2,1]  0.0348621727 8.982088e-02  0.3881300 6.979199e-01 -0.1411835079
#> beta[1,2] -0.0801687056 9.682360e-02 -0.8279872 4.076777e-01 -0.2699394786
#> beta[2,2]  0.4658903909 9.043400e-02  5.1517173 2.581119e-07  0.2886430126
#> psi[1,1]   0.0009954751 1.407993e-04  7.0701700 1.547440e-12  0.0007195135
#> psi[2,1]  -0.0001656666 9.403604e-05 -1.7617357 7.811397e-02 -0.0003499739
#> psi[2,2]   0.0008606361 1.217281e-04  7.0701532 1.547627e-12  0.0006220535
#>                  97.5%
#> mu[1,1]   0.0079042071
#> mu[2,1]   0.0138275183
#> beta[1,1] 0.4871431550
#> beta[2,1] 0.2109078533
#> beta[1,2] 0.1096020674
#> beta[2,2] 0.6431377691
#> psi[1,1]  0.0012714367
#> psi[2,1]  0.0000186406
#> psi[2,2]  0.0010992187
#> 
plot(fit)










# Fit the model---------------------------------------------------------
# center = FALSE
library(fitVARMxID)
fit <- FitVARMxID(
  data = data,
  observed = paste0("y", seq_len(p)),
  id = "id",
  center = FALSE
)
#> Running DTVAR_ID1 with 9 parameters
#> 
#> Beginning initial fit attempt
#> Running DTVAR_ID1 with 9 parameters
#> 
#>  Lowest minimum so far:  -822.060942313204
#> 
#> Solution found
#> 


#> 
#>  Solution found!  Final fit=-822.06094 (started at 367.82482)  (1 attempt(s): 1 valid, 0 errors)
#>  Start values from best fit:
#> 0.331612810420896,-0.0197631306798523,0.027018148758157,0.539260047061468,0.00368111266356549,0.00338863296810361,0.0623714148367602,-6.9119567442846,-6.98806613376272
#> Running DTVAR_ID2 with 9 parameters
#> 
#> Beginning initial fit attempt
#> Running DTVAR_ID2 with 9 parameters
#> 
#>  Lowest minimum so far:  -834.035953636483
#> 
#> Solution found
#> 


#> 
#>  Solution found!  Final fit=-834.03595 (started at 367.84346)  (1 attempt(s): 1 valid, 0 errors)
#>  Start values from best fit:
#> 0.546679083679444,-0.0201029544964559,-0.0487573560553898,0.29745045841592,-0.0022590347266253,-0.0140464530832986,0.176135728656938,-7.14253888427364,-6.87719774329376
#> Running DTVAR_ID3 with 9 parameters
#> 
#> Beginning initial fit attempt
#> Running DTVAR_ID3 with 9 parameters
#> 
#>  Lowest minimum so far:  -808.550007472284
#> 
#> Solution found
#> 


#> 
#>  Solution found!  Final fit=-808.55001 (started at 367.87017)  (1 attempt(s): 1 valid, 0 errors)
#>  Start values from best fit:
#> 0.52810552384264,-0.332501184944442,0.0108536588742205,0.361419710176755,-0.00199256396304628,-0.00252307444345536,-0.213878020138358,-6.9657199385558,-6.80024241140314
#> Running DTVAR_ID4 with 9 parameters
#> 
#> Beginning initial fit attempt
#> Running DTVAR_ID4 with 9 parameters
#> 
#>  Lowest minimum so far:  -845.339407101152
#> 
#> Solution found
#> 


#> 
#>  Solution found!  Final fit=-845.33941 (started at 367.80199)  (1 attempt(s): 1 valid, 0 errors)
#>  Start values from best fit:
#> 0.160844825671735,-0.121822547746862,-0.0298910806237062,0.481816258706521,-0.0108917261651341,9.44597699330245e-05,0.00601786289000614,-7.23151852202068,-6.89984834600342
#> Running DTVAR_ID5 with 9 parameters
#> 
#> Beginning initial fit attempt
#> Running DTVAR_ID5 with 9 parameters
#> 
#>  Lowest minimum so far:  -832.354548096109
#> 
#> Solution found
#> 


#> 
#>  Solution found!  Final fit=-832.35455 (started at 367.7976)  (1 attempt(s): 1 valid, 0 errors)
#>  Start values from best fit:
#> 0.299056714950255,0.0348620015095441,-0.080168388399385,0.465889962011985,-0.0005130370000419,0.00180964949230418,-0.166419692672795,-6.91180279386919,-7.08999354610295
print(fit)
#> Call:
#> FitVARMxID(data = data, observed = paste0("y", seq_len(p)), id = "id", 
#>     center = FALSE)
#> 
#> Convergence:
#> 100.0%
#> 
#> Estimated paramaters per individual.
#>   alpha[1,1] alpha[2,1] beta[1,1] beta[2,1] beta[1,2] beta[2,2] psi[1,1]
#> 1     0.0037     0.0034    0.3316   -0.0198    0.0270    0.5393    1e-03
#> 2    -0.0023    -0.0140    0.5467   -0.0201   -0.0488    0.2975    8e-04
#> 3    -0.0020    -0.0025    0.5281   -0.3325    0.0109    0.3614    9e-04
#> 4    -0.0109     0.0001    0.1608   -0.1218   -0.0299    0.4818    7e-04
#> 5    -0.0005     0.0018    0.2991    0.0349   -0.0802    0.4659    1e-03
#>   psi[2,1] psi[2,2]
#> 1    1e-04   0.0009
#> 2    1e-04   0.0011
#> 3   -2e-04   0.0012
#> 4    0e+00   0.0010
#> 5   -2e-04   0.0009
summary(fit)
#> Call:
#> FitVARMxID(data = data, observed = paste0("y", seq_len(p)), id = "id", 
#>     center = FALSE)
#> 
#> Convergence:
#> 100.0%
#> 
#> Estimated paramaters per individual.
#>   alpha[1,1] alpha[2,1] beta[1,1] beta[2,1] beta[1,2] beta[2,2] psi[1,1]
#> 1     0.0037     0.0034    0.3316   -0.0198    0.0270    0.5393    1e-03
#> 2    -0.0023    -0.0140    0.5467   -0.0201   -0.0488    0.2975    8e-04
#> 3    -0.0020    -0.0025    0.5281   -0.3325    0.0109    0.3614    9e-04
#> 4    -0.0109     0.0001    0.1608   -0.1218   -0.0299    0.4818    7e-04
#> 5    -0.0005     0.0018    0.2991    0.0349   -0.0802    0.4659    1e-03
#>   psi[2,1] psi[2,2]
#> 1    1e-04   0.0009
#> 2    1e-04   0.0011
#> 3   -2e-04   0.0012
#> 4    0e+00   0.0010
#> 5   -2e-04   0.0009
coef(fit)
#> $`1`
#>    alpha[1,1]    alpha[2,1]     beta[1,1]     beta[2,1]     beta[1,2] 
#>  3.681113e-03  3.388633e-03  3.316128e-01 -1.976313e-02  2.701815e-02 
#>     beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2] 
#>  5.392600e-01  9.953219e-04  6.207963e-05  9.262859e-04 
#> 
#> $`2`
#>    alpha[1,1]    alpha[2,1]     beta[1,1]     beta[2,1]     beta[1,2] 
#> -0.0022590347 -0.0140464531  0.5466790837 -0.0201029545 -0.0487573561 
#>     beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2] 
#>  0.2974504584  0.0007904395  0.0001392246  0.0010550305 
#> 
#> $`3`
#>    alpha[1,1]    alpha[2,1]     beta[1,1]     beta[2,1]     beta[1,2] 
#> -0.0019925640 -0.0025230744  0.5281055238 -0.3325011849  0.0108536589 
#>     beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2] 
#>  0.3614197102  0.0009432483 -0.0002017401  0.0011560435 
#> 
#> $`4`
#>    alpha[1,1]    alpha[2,1]     beta[1,1]     beta[2,1]     beta[1,2] 
#> -1.089173e-02  9.445977e-05  1.608448e-01 -1.218225e-01 -2.989108e-02 
#>     beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2] 
#>  4.818163e-01  7.231700e-04  4.351938e-06  1.007467e-03 
#> 
#> $`5`
#>    alpha[1,1]    alpha[2,1]     beta[1,1]     beta[2,1]     beta[1,2] 
#> -0.0005130370  0.0018096495  0.2990567150  0.0348620015 -0.0801683884 
#>     beta[2,2]      psi[1,1]      psi[2,1]      psi[2,2] 
#>  0.4658899620  0.0009954750 -0.0001656666  0.0008606358 
#> 
vcov(fit)
#> $`1`
#>               alpha[1,1]    alpha[2,1]     beta[1,1]     beta[2,1]
#> alpha[1,1]  1.041603e-05  6.394386e-07 -5.052813e-05  1.065411e-06
#> alpha[2,1]  6.394386e-07  9.499927e-06 -2.801871e-06 -4.336068e-05
#> beta[1,1]  -5.052813e-05 -2.801871e-06  8.856580e-03  5.015575e-04
#> beta[2,1]   1.065411e-06 -4.336068e-05  5.015575e-04  8.222857e-03
#> beta[1,2]  -4.233847e-05 -5.314284e-06 -6.329150e-04 -5.463914e-05
#> beta[2,2]  -2.535518e-06 -3.652708e-05 -3.563574e-05 -5.710657e-04
#> psi[1,1]    3.362529e-10  1.975913e-10 -6.466855e-08 -2.573362e-09
#> psi[2,1]    1.039750e-09  9.398095e-10 -1.334551e-08 -4.606274e-08
#> psi[2,2]    2.105697e-10  4.622880e-10 -2.154239e-09  1.310655e-09
#>                beta[1,2]     beta[2,2]      psi[1,1]      psi[2,1]
#> alpha[1,1] -4.233847e-05 -2.535518e-06  3.362529e-10  1.039750e-09
#> alpha[2,1] -5.314284e-06 -3.652708e-05  1.975913e-10  9.398095e-10
#> beta[1,1]  -6.329150e-04 -3.563574e-05 -6.466855e-08 -1.334551e-08
#> beta[2,1]  -5.463914e-05 -5.710657e-04 -2.573362e-09 -4.606274e-08
#> beta[1,2]   7.838493e-03  4.374482e-04 -2.271557e-09 -4.256298e-08
#> beta[2,2]   4.374482e-04  7.234741e-03  3.353554e-09  1.101489e-08
#> psi[1,1]   -2.271557e-09  3.353554e-09  1.981856e-08  1.237235e-09
#> psi[2,1]   -4.256298e-08  1.101489e-08  1.237235e-09  9.257335e-09
#> psi[2,2]   -6.485253e-09 -1.024456e-07  7.716120e-11  1.150882e-09
#>                 psi[2,2]
#> alpha[1,1]  2.105697e-10
#> alpha[2,1]  4.622880e-10
#> beta[1,1]  -2.154239e-09
#> beta[2,1]   1.310655e-09
#> beta[1,2]  -6.485253e-09
#> beta[2,2]  -1.024456e-07
#> psi[1,1]    7.716120e-11
#> psi[2,1]    1.150882e-09
#> psi[2,2]    1.716566e-08
#> 
#> $`2`
#>               alpha[1,1]    alpha[2,1]     beta[1,1]     beta[2,1]
#> alpha[1,1]  1.032520e-05  1.895363e-06  4.491604e-06 -4.439589e-07
#> alpha[2,1]  1.895363e-06  1.388419e-05  4.954290e-07  3.218983e-06
#> beta[1,1]   4.491604e-06  4.954290e-07  7.020108e-03  1.240217e-03
#> beta[2,1]  -4.439589e-07  3.218983e-06  1.240217e-03  9.418247e-03
#> beta[1,2]   1.328065e-04  2.597512e-05 -7.461932e-04 -1.528260e-04
#> beta[2,2]   2.323419e-05  1.760379e-04 -1.316317e-04 -9.990625e-04
#> psi[1,1]   -1.467923e-10  2.928268e-10 -8.315922e-08 -4.779164e-09
#> psi[2,1]   -5.798542e-10  6.511350e-10 -1.399162e-09 -2.990968e-08
#> psi[2,2]   -1.336522e-10 -1.333107e-09 -1.481766e-10 -4.862756e-10
#>                beta[1,2]     beta[2,2]      psi[1,1]      psi[2,1]
#> alpha[1,1]  1.328065e-04  2.323419e-05 -1.467923e-10 -5.798542e-10
#> alpha[2,1]  2.597512e-05  1.760379e-04  2.928268e-10  6.511350e-10
#> beta[1,1]  -7.461932e-04 -1.316317e-04 -8.315922e-08 -1.399162e-09
#> beta[2,1]  -1.528260e-04 -9.990625e-04 -4.779164e-09 -2.990968e-08
#> beta[1,2]   6.909674e-03  1.212584e-03 -5.267598e-10 -4.155880e-08
#> beta[2,2]   1.212584e-03  9.204365e-03 -6.247447e-10 -4.119408e-09
#> psi[1,1]   -5.267598e-10 -6.247447e-10  1.249911e-08  2.199620e-09
#> psi[2,1]   -4.155880e-08 -4.119408e-09  2.199620e-09  8.528063e-09
#> psi[2,2]   -4.896433e-09 -6.155999e-08  3.881269e-10  2.938828e-09
#>                 psi[2,2]
#> alpha[1,1] -1.336522e-10
#> alpha[2,1] -1.333107e-09
#> beta[1,1]  -1.481766e-10
#> beta[2,1]  -4.862756e-10
#> beta[1,2]  -4.896433e-09
#> beta[2,2]  -6.155999e-08
#> psi[1,1]    3.881269e-10
#> psi[2,1]    2.938828e-09
#> psi[2,2]    2.226711e-08
#> 
#> $`3`
#>               alpha[1,1]    alpha[2,1]     beta[1,1]     beta[2,1]
#> alpha[1,1]  9.428506e-06 -2.010191e-06  3.931285e-05 -8.901042e-06
#> alpha[2,1] -2.010191e-06  1.178478e-05 -1.526270e-05  5.685430e-05
#> beta[1,1]   3.931285e-05 -1.526270e-05  8.624447e-03 -1.818927e-03
#> beta[2,1]  -8.901042e-06  5.685430e-05 -1.818927e-03  1.025298e-02
#> beta[1,2]   2.713936e-05 -1.585628e-05  2.783340e-03 -5.998268e-04
#> beta[2,2]  -6.234640e-06  3.202370e-05 -6.213590e-04  3.311956e-03
#> psi[1,1]   -1.087323e-09  5.406398e-10 -8.823624e-08  6.928668e-08
#> psi[2,1]   -1.890360e-09 -3.577258e-09  1.364448e-07 -6.764901e-08
#> psi[2,2]    1.632746e-09  1.092380e-09 -8.091549e-08  6.958967e-09
#>                beta[1,2]     beta[2,2]      psi[1,1]      psi[2,1]
#> alpha[1,1]  2.713936e-05 -6.234640e-06 -1.087323e-09 -1.890360e-09
#> alpha[2,1] -1.585628e-05  3.202370e-05  5.406398e-10 -3.577258e-09
#> beta[1,1]   2.783340e-03 -6.213590e-04 -8.823624e-08  1.364448e-07
#> beta[2,1]  -5.998268e-04  3.311956e-03  6.928668e-08 -6.764901e-08
#> beta[1,2]   6.799025e-03 -1.494191e-03 -1.226517e-08  2.847862e-08
#> beta[2,2]  -1.494191e-03  8.032890e-03  3.724920e-09  4.538555e-09
#> psi[1,1]   -1.226517e-08  3.724920e-09  1.776846e-08 -3.803147e-09
#> psi[2,1]    2.847862e-08  4.538555e-09 -3.803147e-09  1.136256e-08
#> psi[2,2]   -5.258376e-09 -7.606566e-08  8.298031e-10 -4.702820e-09
#>                 psi[2,2]
#> alpha[1,1]  1.632746e-09
#> alpha[2,1]  1.092380e-09
#> beta[1,1]  -8.091549e-08
#> beta[2,1]   6.958967e-09
#> beta[1,2]  -5.258376e-09
#> beta[2,2]  -7.606566e-08
#> psi[1,1]    8.298031e-10
#> psi[2,1]   -4.702820e-09
#> psi[2,2]    2.676429e-08
#> 
#> $`4`
#>               alpha[1,1]    alpha[2,1]     beta[1,1]     beta[2,1]
#> alpha[1,1]  9.037944e-06  5.971385e-08  1.324679e-04 -9.195905e-07
#> alpha[2,1]  5.971385e-08  1.220647e-05  1.226437e-06  1.729198e-04
#> beta[1,1]   1.324679e-04  1.226437e-06  9.866100e-03  3.402501e-05
#> beta[2,1]  -9.195905e-07  1.729198e-04  3.402501e-05  1.342181e-02
#> beta[1,2]  -1.461605e-05  3.635078e-06  2.405932e-04 -2.785645e-05
#> beta[2,2]  -8.980359e-08 -2.074502e-05 -3.676591e-06  3.344239e-04
#> psi[1,1]   -1.140286e-10  5.653372e-10 -1.928848e-08  1.710657e-08
#> psi[2,1]    3.124039e-10 -2.321308e-09  6.868227e-09 -4.763686e-08
#> psi[2,2]   -3.243453e-10 -1.735596e-10 -6.826949e-09  3.194079e-10
#>                beta[1,2]     beta[2,2]      psi[1,1]      psi[2,1]
#> alpha[1,1] -1.461605e-05 -8.980359e-08 -1.140286e-10  3.124039e-10
#> alpha[2,1]  3.635078e-06 -2.074502e-05  5.653372e-10 -2.321308e-09
#> beta[1,1]   2.405932e-04 -3.676591e-06 -1.928848e-08  6.868227e-09
#> beta[2,1]  -2.785645e-05  3.344239e-04  1.710657e-08 -4.763686e-08
#> beta[1,2]   5.470059e-03  1.493185e-05  1.079037e-08 -6.126221e-08
#> beta[2,2]   1.493185e-05  7.419103e-03 -1.708817e-09  1.861242e-08
#> psi[1,1]    1.079037e-08 -1.708817e-09  1.046046e-08  6.374499e-11
#> psi[2,1]   -6.126221e-08  1.861242e-08  6.374499e-11  7.297141e-09
#> psi[2,2]   -7.529354e-09 -9.163166e-08  1.154333e-12  9.071773e-11
#>                 psi[2,2]
#> alpha[1,1] -3.243453e-10
#> alpha[2,1] -1.735596e-10
#> beta[1,1]  -6.826949e-09
#> beta[2,1]   3.194079e-10
#> beta[1,2]  -7.529354e-09
#> beta[2,2]  -9.163166e-08
#> psi[1,1]    1.154333e-12
#> psi[2,1]    9.071773e-11
#> psi[2,2]    2.030681e-08
#> 
#> $`5`
#>               alpha[1,1]    alpha[2,1]     beta[1,1]     beta[2,1]
#> alpha[1,1]  9.968760e-06 -1.649475e-06  3.243225e-06 -5.834209e-06
#> alpha[2,1] -1.649475e-06  8.582750e-06 -5.982085e-07  3.749137e-06
#> beta[1,1]   3.243225e-06 -5.982085e-07  9.208669e-03 -1.527171e-03
#> beta[2,1]  -5.834209e-06  3.749137e-06 -1.527171e-03  8.067261e-03
#> beta[1,2]  -2.883141e-05  4.306912e-06  1.910692e-03 -3.230196e-04
#> beta[2,2]   4.101270e-06 -2.945464e-05 -3.156739e-04  1.665135e-03
#> psi[1,1]   -1.883995e-10 -2.257554e-12 -6.106649e-08  1.085567e-08
#> psi[2,1]   -6.860031e-10 -2.637887e-10  1.350304e-08 -2.491736e-08
#> psi[2,2]    3.083962e-10  3.770891e-10 -6.765003e-10 -7.777452e-09
#>                beta[1,2]     beta[2,2]      psi[1,1]      psi[2,1]
#> alpha[1,1] -2.883141e-05  4.101270e-06 -1.883995e-10 -6.860031e-10
#> alpha[2,1]  4.306912e-06 -2.945464e-05 -2.257554e-12 -2.637887e-10
#> beta[1,1]   1.910692e-03 -3.156739e-04 -6.106649e-08  1.350304e-08
#> beta[2,1]  -3.230196e-04  1.665135e-03  1.085567e-08 -2.491736e-08
#> beta[1,2]   9.374112e-03 -1.545459e-03 -2.321767e-09 -2.714276e-08
#> beta[2,2]  -1.545459e-03  8.177319e-03  1.768581e-09  1.092935e-08
#> psi[1,1]   -2.321767e-09  1.768581e-09  1.982437e-08 -3.299070e-09
#> psi[2,1]   -2.714276e-08  1.092935e-08 -3.299070e-09  8.842287e-09
#> psi[2,2]    2.269641e-08 -8.223520e-08  5.501767e-10 -2.854173e-09
#>                 psi[2,2]
#> alpha[1,1]  3.083962e-10
#> alpha[2,1]  3.770891e-10
#> beta[1,1]  -6.765003e-10
#> beta[2,1]  -7.777452e-09
#> beta[1,2]   2.269641e-08
#> beta[2,2]  -8.223520e-08
#> psi[1,1]    5.501767e-10
#> psi[2,1]   -2.854173e-09
#> psi[2,2]    1.481758e-08
#> 
converged(fit)
#>    1    2    3    4    5 
#> TRUE TRUE TRUE TRUE TRUE 
confint(fit)
#> $`1`
#>                      est           se          z            p          2.5%
#> alpha[1,1]  3.681113e-03 3.227387e-03  1.1405860 2.540422e-01 -0.0026444498
#> alpha[2,1]  3.388633e-03 3.082195e-03  1.0994219 2.715841e-01 -0.0026523586
#> beta[1,1]   3.316128e-01 9.410940e-02  3.5236948 4.255741e-04  0.1471617720
#> beta[2,1]  -1.976313e-02 9.067997e-02 -0.2179437 8.274730e-01 -0.1974926059
#> beta[1,2]   2.701815e-02 8.853526e-02  0.3051682 7.602380e-01 -0.1465077784
#> beta[2,2]   5.392600e-01 8.505728e-02  6.3399635 2.298197e-10  0.3725508417
#> psi[1,1]    9.953219e-04 1.407784e-04  7.0701309 1.547877e-12  0.0007194012
#> psi[2,1]    6.207963e-05 9.621504e-05  0.6452175 5.187862e-01 -0.0001264984
#> psi[2,2]    9.262859e-04 1.310178e-04  7.0699254 1.550170e-12  0.0006694958
#>                   97.5%
#> alpha[1,1] 0.0100066751
#> alpha[2,1] 0.0094296245
#> beta[1,1]  0.5160638489
#> beta[2,1]  0.1579663445
#> beta[1,2]  0.2005440759
#> beta[2,2]  0.7059692524
#> psi[1,1]   0.0012712425
#> psi[2,1]   0.0002506577
#> psi[2,2]   0.0011830760
#> 
#> $`2`
#>                      est           se          z            p          2.5%
#> alpha[1,1] -0.0022590347 3.213285e-03 -0.7030297 4.820373e-01 -8.556958e-03
#> alpha[2,1] -0.0140464531 3.726150e-03 -3.7696965 1.634462e-04 -2.134957e-02
#> beta[1,1]   0.5466790837 8.378609e-02  6.5247002 6.813769e-11  3.824614e-01
#> beta[2,1]  -0.0201029545 9.704765e-02 -0.2071452 8.358965e-01 -2.103129e-01
#> beta[1,2]  -0.0487573561 8.312445e-02 -0.5865585 5.575002e-01 -2.116783e-01
#> beta[2,2]   0.2974504584 9.593938e-02  3.1004000 1.932594e-03  1.094127e-01
#> psi[1,1]    0.0007904395 1.117994e-04  7.0701559 1.547597e-12  5.713166e-04
#> psi[2,1]    0.0001392246 9.234751e-05  1.5076165 1.316527e-01 -4.177317e-05
#> psi[2,2]    0.0010550305 1.492217e-04  7.0702232 1.546847e-12  7.625614e-04
#>                    97.5%
#> alpha[1,1]  0.0040388883
#> alpha[2,1] -0.0067433341
#> beta[1,1]   0.7108967927
#> beta[2,1]   0.1701069507
#> beta[1,2]   0.1141635705
#> beta[2,2]   0.4854881895
#> psi[1,1]    0.0010095623
#> psi[2,1]    0.0003202224
#> psi[2,2]    0.0013474996
#> 
#> $`3`
#>                      est           se          z            p          2.5%
#> alpha[1,1] -0.0019925640 0.0030705872 -0.6489195 5.163904e-01 -0.0080108043
#> alpha[2,1] -0.0025230744 0.0034328963 -0.7349696 4.623580e-01 -0.0092514276
#> beta[1,1]   0.5281055238 0.0928679012  5.6866314 1.295697e-08  0.3460877822
#> beta[2,1]  -0.3325011849 0.1012569899 -3.2837356 1.024410e-03 -0.5309612384
#> beta[1,2]   0.0108536589 0.0824562004  0.1316294 8.952774e-01 -0.1507575243
#> beta[2,2]   0.3614197102 0.0896263936  4.0325143 5.518327e-05  0.1857552067
#> psi[1,1]    0.0009432483 0.0001332984  7.0762167 1.481432e-12  0.0006819883
#> psi[2,1]   -0.0002017401 0.0001065953 -1.8925797 5.841379e-02 -0.0004106630
#> psi[2,2]    0.0011560435 0.0001635980  7.0663682 1.590410e-12  0.0008353973
#>                    97.5%
#> alpha[1,1]  4.025676e-03
#> alpha[2,1]  4.205279e-03
#> beta[1,1]   7.101233e-01
#> beta[2,1]  -1.340411e-01
#> beta[1,2]   1.724648e-01
#> beta[2,2]   5.370842e-01
#> psi[1,1]    1.204508e-03
#> psi[2,1]    7.182852e-06
#> psi[2,2]    1.476690e-03
#> 
#> $`4`
#>                      est           se           z            p          2.5%
#> alpha[1,1] -1.089173e-02 3.006317e-03 -3.62294619 2.912664e-04 -0.0167840000
#> alpha[2,1]  9.445977e-05 3.493776e-03  0.02703659 9.784306e-01 -0.0067532148
#> beta[1,1]   1.608448e-01 9.932825e-02  1.61932616 1.053771e-01 -0.0338349576
#> beta[2,1]  -1.218225e-01 1.158526e-01 -1.05153098 2.930148e-01 -0.3488893766
#> beta[1,2]  -2.989108e-02 7.395985e-02 -0.40415282 6.861003e-01 -0.1748497150
#> beta[2,2]   4.818163e-01 8.613422e-02  5.59378461 2.221725e-08  0.3129962951
#> psi[1,1]    7.231700e-04 1.022764e-04  7.07074165 1.541078e-12  0.0005227119
#> psi[2,1]    4.351938e-06 8.542331e-05  0.05094555 9.593689e-01 -0.0001630747
#> psi[2,2]    1.007467e-03 1.425020e-04  7.06984484 1.551070e-12  0.0007281681
#>                    97.5%
#> alpha[1,1] -0.0049994523
#> alpha[2,1]  0.0069421344
#> beta[1,1]   0.3555246089
#> beta[2,1]   0.1052442811
#> beta[1,2]   0.1150675538
#> beta[2,2]   0.6506362223
#> psi[1,1]    0.0009236280
#> psi[2,1]    0.0001717785
#> psi[2,2]    0.0012867656
#> 
#> $`5`
#>                      est           se          z            p          2.5%
#> alpha[1,1] -0.0005130370 3.157334e-03 -0.1624906 8.709196e-01 -0.0067012986
#> alpha[2,1]  0.0018096495 2.929633e-03  0.6177052 5.367697e-01 -0.0039323259
#> beta[1,1]   0.2990567150 9.596181e-02  3.1164138 1.830652e-03  0.1109750209
#> beta[2,1]   0.0348620015 8.981793e-02  0.3881408 6.979119e-01 -0.1411779084
#> beta[1,2]  -0.0801683884 9.682000e-02 -0.8280148 4.076621e-01 -0.2699320934
#> beta[2,2]   0.4658899620 9.042853e-02  5.1520240 2.576900e-07  0.2886532997
#> psi[1,1]    0.0009954750 1.407990e-04  7.0701840 1.547283e-12  0.0007195140
#> psi[2,1]   -0.0001656666 9.403343e-05 -1.7617845 7.810571e-02 -0.0003499688
#> psi[2,2]    0.0008606358 1.217275e-04  7.0701858 1.547264e-12  0.0006220544
#>                   97.5%
#> alpha[1,1] 5.675225e-03
#> alpha[2,1] 7.551625e-03
#> beta[1,1]  4.871384e-01
#> beta[2,1]  2.109019e-01
#> beta[1,2]  1.095953e-01
#> beta[2,2]  6.431266e-01
#> psi[1,1]   1.271436e-03
#> psi[2,1]   1.863549e-05
#> psi[2,2]   1.099217e-03
#> 
plot(fit)









# }