Parametric Bootstrap for the State Space Model (Fixed Parameters)

This function simulates data from a state-space model and fits the model using the dynr package. The process is repeated R times. It assumes that the parameters remain constant across individuals and over time. At the moment, the function only supports type = 0.

Usage

PBSSMFixed(
  R,
  path,
  prefix,
  n,
  time,
  delta_t = 1,
  mu0,
  sigma0_l,
  alpha,
  beta,
  psi_l,
  nu,
  lambda,
  theta_l,
  type = 0,
  x = NULL,
  gamma = NULL,
  kappa = NULL,
  mu0_fixed = FALSE,
  sigma0_fixed = FALSE,
  alpha_level = 0.05,
  optimization_flag = TRUE,
  hessian_flag = FALSE,
  verbose = FALSE,
  weight_flag = FALSE,
  debug_flag = FALSE,
  perturb_flag = FALSE,
  xtol_rel = 1e-07,
  stopval = -9999,
  ftol_rel = -1,
  ftol_abs = -1,
  maxeval = as.integer(-1),
  maxtime = -1,
  ncores = NULL,
  seed = NULL
)

Arguments

R

Positive integer. Number of bootstrap samples.

path

Path to a directory to store bootstrap samples and estimates.

prefix

Character string. Prefix used for the file names for the bootstrap samples and estimates.

n

Positive integer. Number of individuals.

time

Positive integer. Number of time points.

delta_t

Numeric. Time interval. The default value is 1.0 with an option to use a numeric value for the discretized state space model parameterization of the linear stochastic differential equation model.

mu0

Numeric vector. Mean of initial latent variable values (\(\boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0}\)).

sigma0_l

Numeric matrix. Cholesky factorization (t(chol(sigma0))) of the covariance matrix of initial latent variable values (\(\boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0}\)).

alpha

Numeric vector. Vector of constant values for the dynamic model (\(\boldsymbol{\alpha}\)).

beta

Numeric matrix. Transition matrix relating the values of the latent variables at the previous to the current time point (\(\boldsymbol{\beta}\)).

psi_l

Numeric matrix. Cholesky factorization (t(chol(psi))) of the covariance matrix of the process noise (\(\boldsymbol{\Psi}\)).

nu

Numeric vector. Vector of intercept values for the measurement model (\(\boldsymbol{\nu}\)).

lambda

Numeric matrix. Factor loading matrix linking the latent variables to the observed variables (\(\boldsymbol{\Lambda}\)).

theta_l

Numeric matrix. Cholesky factorization (t(chol(theta))) of the covariance matrix of the measurement error (\(\boldsymbol{\Theta}\)).

type

Integer. State space model type. See Details for more information.

x

List. Each element of the list is a matrix of covariates for each individual i in n. The number of columns in each matrix should be equal to time.

gamma

Numeric matrix. Matrix linking the covariates to the latent variables at current time point (\(\boldsymbol{\Gamma}\)).

kappa

Numeric matrix. Matrix linking the covariates to the observed variables at current time point (\(\boldsymbol{\kappa}\)).

mu0_fixed

Logical. If mu0_fixed = TRUE, fix the initial mean vector to mu0. If mu0_fixed = FALSE, mu0 is estimated.

sigma0_fixed

Logical. If sigma0_fixed = TRUE, fix the initial covariance matrix to tcrossprod(sigma0_l). If sigma0_fixed = FALSE, sigma0 is estimated.

alpha_level

Numeric vector. Significance level \(\alpha\).

optimization_flag

a flag (TRUE/FALSE) indicating whether optimization is to be done.

hessian_flag

a flag (TRUE/FALSE) indicating whether the Hessian matrix is to be calculated.

verbose

a flag (TRUE/FALSE) indicating whether more detailed intermediate output during the estimation process should be printed

weight_flag

a flag (TRUE/FALSE) indicating whether the negative log likelihood function should be weighted by the length of the time series for each individual

debug_flag

a flag (TRUE/FALSE) indicating whether users want additional dynr output that can be used for diagnostic purposes

perturb_flag

a flag (TRUE/FLASE) indicating whether to perturb the latent states during estimation. Only useful for ensemble forecasting.

xtol_rel

Stopping criteria option for parameter optimization. See dynr::dynr.model() for more details.

stopval

Stopping criteria option for parameter optimization. See dynr::dynr.model() for more details.

ftol_rel

Stopping criteria option for parameter optimization. See dynr::dynr.model() for more details.

ftol_abs

Stopping criteria option for parameter optimization. See dynr::dynr.model() for more details.

maxeval

Stopping criteria option for parameter optimization. See dynr::dynr.model() for more details.

maxtime

Stopping criteria option for parameter optimization. See dynr::dynr.model() for more details.

ncores

Positive integer. Number of cores to use. If ncores = NULL, use a single core. Consider using multiple cores when number of bootstrap samples R is a large value.

seed

Random seed.

Value

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

call

Function call.

args

Function arguments.

thetahatstar

Sampling distribution of \(\boldsymbol{\hat{\theta}}\).

vcov

Sampling variance-covariance matrix of \(\boldsymbol{\hat{\theta}}\).

est

Vector of estimated \(\boldsymbol{\hat{\theta}}\).

fun

Function used ("PBSSMFixed").

method

Bootstrap method used ("parametric").

Details

Type 0

The measurement model is given by $$ \mathbf{y}_{i, t} = \boldsymbol{\nu} + \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta} \right) $$ where \(\mathbf{y}_{i, t}\), \(\boldsymbol{\eta}_{i, t}\), and \(\boldsymbol{\varepsilon}_{i, t}\) are random variables and \(\boldsymbol{\nu}\), \(\boldsymbol{\Lambda}\), 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 \(i\). \(\boldsymbol{\nu}\) denotes a vector of intercepts, \(\boldsymbol{\Lambda}\) a matrix of factor loadings, and \(\boldsymbol{\Theta}\) the covariance matrix of \(\boldsymbol{\varepsilon}\).

An alternative representation of the measurement error is given by $$ \boldsymbol{\varepsilon}_{i, t} = \boldsymbol{\Theta}^{\frac{1}{2}} \mathbf{z}_{i, t}, \quad \mathrm{with} \quad \mathbf{z}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \mathbf{I} \right) $$ where \(\mathbf{z}_{i, t}\) is a vector of independent standard normal random variables and \( \left( \boldsymbol{\Theta}^{\frac{1}{2}} \right) \left( \boldsymbol{\Theta}^{\frac{1}{2}} \right)^{\prime} = \boldsymbol{\Theta} . \)

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 \(\boldsymbol{\zeta}_{i, t}\).

An alternative representation of the dynamic noise is given by $$ \boldsymbol{\zeta}_{i, t} = \boldsymbol{\Psi}^{\frac{1}{2}} \mathbf{z}_{i, t}, \quad \mathrm{with} \quad \mathbf{z}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \mathbf{I} \right) $$ where \( \left( \boldsymbol{\Psi}^{\frac{1}{2}} \right) \left( \boldsymbol{\Psi}^{\frac{1}{2}} \right)^{\prime} = \boldsymbol{\Psi} . \)

Type 1

The measurement model is given by $$ \mathbf{y}_{i, t} = \boldsymbol{\nu} + \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta} \right) . $$

The dynamic structure is given by $$ \boldsymbol{\eta}_{i, t} = \boldsymbol{\alpha} + \boldsymbol{\beta} \boldsymbol{\eta}_{i, t - 1} + \boldsymbol{\Gamma} \mathbf{x}_{i, t} + \boldsymbol{\zeta}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\zeta}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Psi} \right) $$ where \(\mathbf{x}_{i, t}\) represents a vector of covariates at time \(t\) and individual \(i\), and \(\boldsymbol{\Gamma}\) the coefficient matrix linking the covariates to the latent variables.

Type 2

The measurement model is given by $$ \mathbf{y}_{i, t} = \boldsymbol{\nu} + \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\kappa} \mathbf{x}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta} \right) $$ where \(\boldsymbol{\kappa}\) represents the coefficient matrix linking the covariates to the observed variables.

The dynamic structure is given by $$ \boldsymbol{\eta}_{i, t} = \boldsymbol{\alpha} + \boldsymbol{\beta} \boldsymbol{\eta}_{i, t - 1} + \boldsymbol{\Gamma} \mathbf{x}_{i, t} + \boldsymbol{\zeta}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\zeta}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Psi} \right) . $$

References

Chow, S.-M., Ho, M. R., Hamaker, E. L., & Dolan, C. V. (2010). Equivalence and differences between structural equation modeling and state-space modeling techniques. Structural Equation Modeling: A Multidisciplinary Journal, 17(2), 303–332. doi:10.1080/10705511003661553

See also

Other Bootstrap for State Space Models Functions: PBSSMLinSDEFixed(), PBSSMOUFixed(), PBSSMVARFixed()

Author

Ivan Jacob Agaloos Pesigan

Examples

# \donttest{
# prepare parameters
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 50
delta_t <- 1
## dynamic structure
p <- 3
mu0 <- rep(x = 0, times = p)
sigma0 <- 0.001 * diag(p)
sigma0_l <- t(chol(sigma0))
alpha <- rep(x = 0, times = p)
beta <- 0.50 * diag(p)
psi <- 0.001 * diag(p)
psi_l <- t(chol(psi))
## measurement model
k <- 3
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.001 * diag(k)
theta_l <- t(chol(theta))

path <- tempdir()

pb <- PBSSMFixed(
  R = 10L, # use at least 1000 in actual research
  path = path,
  prefix = "ssm",
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 0,
  ncores = 1, # consider using multiple cores
  seed = 42
)
print(pb)
#> Call:
#> PBSSMFixed(R = 10L, path = path, prefix = "ssm", n = n, time = time, 
#>     delta_t = delta_t, mu0 = mu0, sigma0_l = sigma0_l, alpha = alpha, 
#>     beta = beta, psi_l = psi_l, nu = nu, lambda = lambda, theta_l = theta_l, 
#>     type = 0, ncores = 1, seed = 42)
#> 
#> Parametric bootstrap confidence intervals.
#> type = "pc"
#>              est     se  R    2.5%  97.5%
#> beta_1_1   0.500 0.2556 10  0.1627 0.9432
#> beta_2_1   0.000 0.1011 10 -0.1649 0.1286
#> beta_3_1   0.000 0.1535 10 -0.2590 0.2353
#> beta_1_2   0.000 0.1033 10 -0.2115 0.0959
#> beta_2_2   0.500 0.1432 10  0.3345 0.7640
#> beta_3_2   0.000 0.0693 10 -0.1540 0.0468
#> beta_1_3   0.000 0.1674 10 -0.4192 0.0627
#> beta_2_3   0.000 0.0980 10 -0.1156 0.1599
#> beta_3_3   0.500 0.2063 10  0.2255 0.7774
#> psi_1_1    0.001 0.0007 10  0.0001 0.0021
#> psi_2_2    0.001 0.0006 10  0.0002 0.0020
#> psi_3_3    0.001 0.0008 10  0.0002 0.0021
#> theta_1_1  0.001 0.0005 10  0.0000 0.0016
#> theta_2_2  0.001 0.0004 10  0.0001 0.0015
#> theta_3_3  0.001 0.0007 10  0.0000 0.0017
#> mu0_1_1    0.000 0.0215 10 -0.0357 0.0194
#> mu0_2_1    0.000 0.0158 10 -0.0217 0.0178
#> mu0_3_1    0.000 0.0195 10 -0.0176 0.0381
#> sigma0_1_1 0.001 0.0007 10  0.0000 0.0018
#> sigma0_2_2 0.001 0.0022 10  0.0000 0.0061
#> sigma0_3_3 0.001 0.0017 10  0.0000 0.0044
summary(pb)
#> Call:
#> PBSSMFixed(R = 10L, path = path, prefix = "ssm", n = n, time = time, 
#>     delta_t = delta_t, mu0 = mu0, sigma0_l = sigma0_l, alpha = alpha, 
#>     beta = beta, psi_l = psi_l, nu = nu, lambda = lambda, theta_l = theta_l, 
#>     type = 0, ncores = 1, seed = 42)
#>              est     se  R    2.5%  97.5%
#> beta_1_1   0.500 0.2556 10  0.1627 0.9432
#> beta_2_1   0.000 0.1011 10 -0.1649 0.1286
#> beta_3_1   0.000 0.1535 10 -0.2590 0.2353
#> beta_1_2   0.000 0.1033 10 -0.2115 0.0959
#> beta_2_2   0.500 0.1432 10  0.3345 0.7640
#> beta_3_2   0.000 0.0693 10 -0.1540 0.0468
#> beta_1_3   0.000 0.1674 10 -0.4192 0.0627
#> beta_2_3   0.000 0.0980 10 -0.1156 0.1599
#> beta_3_3   0.500 0.2063 10  0.2255 0.7774
#> psi_1_1    0.001 0.0007 10  0.0001 0.0021
#> psi_2_2    0.001 0.0006 10  0.0002 0.0020
#> psi_3_3    0.001 0.0008 10  0.0002 0.0021
#> theta_1_1  0.001 0.0005 10  0.0000 0.0016
#> theta_2_2  0.001 0.0004 10  0.0001 0.0015
#> theta_3_3  0.001 0.0007 10  0.0000 0.0017
#> mu0_1_1    0.000 0.0215 10 -0.0357 0.0194
#> mu0_2_1    0.000 0.0158 10 -0.0217 0.0178
#> mu0_3_1    0.000 0.0195 10 -0.0176 0.0381
#> sigma0_1_1 0.001 0.0007 10  0.0000 0.0018
#> sigma0_2_2 0.001 0.0022 10  0.0000 0.0061
#> sigma0_3_3 0.001 0.0017 10  0.0000 0.0044
confint(pb)
#>                    2.5 %      97.5 %
#> beta_1_1    1.626566e-01 0.943187625
#> beta_2_1   -1.649366e-01 0.128602513
#> beta_3_1   -2.590191e-01 0.235339527
#> beta_1_2   -2.115189e-01 0.095917751
#> beta_2_2    3.344581e-01 0.764011522
#> beta_3_2   -1.539633e-01 0.046832519
#> beta_1_3   -4.192253e-01 0.062713408
#> beta_2_3   -1.155608e-01 0.159944872
#> beta_3_3    2.254559e-01 0.777431680
#> psi_1_1     9.826323e-05 0.002141036
#> psi_2_2     2.209297e-04 0.002044133
#> psi_3_3     1.776410e-04 0.002123510
#> theta_1_1   3.124256e-12 0.001573813
#> theta_2_2   1.272953e-04 0.001472064
#> theta_3_3   2.516683e-12 0.001725308
#> mu0_1_1    -3.570665e-02 0.019411810
#> mu0_2_1    -2.165274e-02 0.017832927
#> mu0_3_1    -1.755279e-02 0.038134516
#> sigma0_1_1  2.884978e-13 0.001846362
#> sigma0_2_2  3.379446e-21 0.006122529
#> sigma0_3_3  6.439289e-14 0.004418048
vcov(pb)
#>                 beta_1_1      beta_2_1      beta_3_1      beta_1_2
#> beta_1_1    6.533223e-02 -4.892502e-03  4.434204e-04  2.584880e-03
#> beta_2_1   -4.892502e-03  1.022457e-02 -1.285054e-02 -2.874452e-03
#> beta_3_1    4.434204e-04 -1.285054e-02  2.357017e-02  3.462465e-03
#> beta_1_2    2.584880e-03 -2.874452e-03  3.462465e-03  1.068070e-02
#> beta_2_2   -1.917220e-03  6.318208e-03 -7.631873e-03 -1.063096e-02
#> beta_3_2   -9.238484e-04  9.573481e-04 -2.282590e-03 -9.384263e-04
#> beta_1_3   -1.631705e-02  6.876483e-03 -1.129819e-02  9.816551e-04
#> beta_2_3    1.380578e-02 -1.618225e-04 -1.499778e-03  2.305538e-03
#> beta_3_3    5.333069e-03 -3.993522e-03  3.349283e-03  7.611216e-03
#> psi_1_1    -1.695797e-04  6.795878e-06  5.856309e-06 -5.605841e-06
#> psi_2_2     1.064620e-05 -2.040384e-05  2.746069e-05  4.240714e-05
#> psi_3_3    -7.632148e-05  1.718862e-05 -1.434953e-05 -1.271601e-05
#> theta_1_1   1.321908e-04 -9.522526e-06 -5.320303e-06  2.583021e-06
#> theta_2_2  -1.787805e-05  1.247195e-05 -1.087854e-05 -2.869378e-05
#> theta_3_3   9.768817e-05 -1.801930e-05  1.100342e-05  1.457112e-05
#> mu0_1_1     3.074668e-03 -9.919380e-04  3.518734e-04  9.576441e-04
#> mu0_2_1    -2.537954e-03  2.203528e-04  5.306683e-05 -5.500862e-05
#> mu0_3_1    -2.678580e-03  5.422391e-04  1.229654e-05 -2.839870e-04
#> sigma0_1_1 -5.707850e-05  2.478245e-05 -5.533134e-06 -4.066630e-05
#> sigma0_2_2 -2.840790e-04 -3.243701e-07  2.506603e-05 -1.449269e-04
#> sigma0_3_3 -1.784925e-04 -6.746114e-05  5.641729e-05  3.581213e-05
#>                 beta_2_2      beta_3_2      beta_1_3      beta_2_3
#> beta_1_1   -1.917220e-03 -9.238484e-04 -1.631705e-02  1.380578e-02
#> beta_2_1    6.318208e-03  9.573481e-04  6.876483e-03 -1.618225e-04
#> beta_3_1   -7.631873e-03 -2.282590e-03 -1.129819e-02 -1.499778e-03
#> beta_1_2   -1.063096e-02 -9.384263e-04  9.816551e-04  2.305538e-03
#> beta_2_2    2.051400e-02  3.027703e-03  1.158091e-02 -4.152647e-03
#> beta_3_2    3.027703e-03  4.799100e-03  8.838937e-04  1.252695e-03
#> beta_1_3    1.158091e-02  8.838937e-04  2.801958e-02 -5.241715e-03
#> beta_2_3   -4.152647e-03  1.252695e-03 -5.241715e-03  9.600205e-03
#> beta_3_3   -2.080218e-02  9.603971e-04 -1.757538e-02  1.564136e-02
#> psi_1_1     1.144514e-06 -7.120163e-07  3.805336e-05 -3.420922e-05
#> psi_2_2    -7.150714e-05 -7.577085e-06 -4.660790e-05  2.162676e-05
#> psi_3_3     6.620606e-05 -1.789972e-06  8.938601e-05 -6.430443e-05
#> theta_1_1   2.703048e-06 -5.206255e-06 -2.263280e-05  2.019187e-05
#> theta_2_2   4.849494e-05  2.280593e-06  3.292302e-05 -2.020231e-05
#> theta_3_3  -5.853384e-05  3.885636e-07 -7.965013e-05  5.941341e-05
#> mu0_1_1    -1.726918e-03 -1.489813e-04 -1.293152e-03  1.005562e-03
#> mu0_2_1     1.433180e-05  5.677906e-04  1.301728e-04 -2.170207e-04
#> mu0_3_1     1.090520e-03  1.800958e-04  1.583044e-03 -2.040271e-04
#> sigma0_1_1  4.705749e-05 -3.094374e-07  9.229273e-06  5.862384e-06
#> sigma0_2_2  1.774061e-04  1.654498e-05  1.072799e-04 -5.391496e-05
#> sigma0_3_3 -8.726443e-05 -2.769356e-05  1.710784e-05 -1.031826e-04
#>                 beta_3_3       psi_1_1       psi_2_2       psi_3_3
#> beta_1_1    5.333069e-03 -1.695797e-04  1.064620e-05 -7.632148e-05
#> beta_2_1   -3.993522e-03  6.795878e-06 -2.040384e-05  1.718862e-05
#> beta_3_1    3.349283e-03  5.856309e-06  2.746069e-05 -1.434953e-05
#> beta_1_2    7.611216e-03 -5.605841e-06  4.240714e-05 -1.271601e-05
#> beta_2_2   -2.080218e-02  1.144514e-06 -7.150714e-05  6.620606e-05
#> beta_3_2    9.603971e-04 -7.120163e-07 -7.577085e-06 -1.789972e-06
#> beta_1_3   -1.757538e-02  3.805336e-05 -4.660790e-05  8.938601e-05
#> beta_2_3    1.564136e-02 -3.420922e-05  2.162676e-05 -6.430443e-05
#> beta_3_3    4.257860e-02 -7.173644e-06  8.382611e-05 -1.501124e-04
#> psi_1_1    -7.173644e-06  4.604387e-07  1.296426e-09  1.665908e-07
#> psi_2_2     8.382611e-05  1.296426e-09  3.109041e-07 -2.567570e-07
#> psi_3_3    -1.501124e-04  1.665908e-07 -2.567570e-07  6.310119e-07
#> theta_1_1  -8.598973e-06 -3.462260e-07 -1.215671e-08 -7.923548e-08
#> theta_2_2  -6.431497e-05  1.828432e-08 -2.187735e-07  2.179041e-07
#> theta_3_3   1.259605e-04 -2.304052e-07  2.214667e-07 -5.509069e-07
#> mu0_1_1     2.005304e-03 -8.472259e-06  5.501769e-06 -7.383944e-06
#> mu0_2_1     5.446704e-04  7.097892e-06  8.860079e-07 -6.600987e-07
#> mu0_3_1    -4.826952e-04  7.957369e-06 -2.552742e-06  3.129514e-06
#> sigma0_1_1  1.618549e-06  1.271673e-07 -1.354422e-07  3.695267e-08
#> sigma0_2_2 -1.032026e-04  7.445604e-07 -6.673824e-07  5.215311e-07
#> sigma0_3_3 -6.184073e-05  5.331454e-07  1.464423e-07  3.221839e-07
#>                theta_1_1     theta_2_2     theta_3_3       mu0_1_1
#> beta_1_1    1.321908e-04 -1.787805e-05  9.768817e-05  3.074668e-03
#> beta_2_1   -9.522526e-06  1.247195e-05 -1.801930e-05 -9.919380e-04
#> beta_3_1   -5.320303e-06 -1.087854e-05  1.100342e-05  3.518734e-04
#> beta_1_2    2.583021e-06 -2.869378e-05  1.457112e-05  9.576441e-04
#> beta_2_2    2.703048e-06  4.849494e-05 -5.853384e-05 -1.726918e-03
#> beta_3_2   -5.206255e-06  2.280593e-06  3.885636e-07 -1.489813e-04
#> beta_1_3   -2.263280e-05  3.292302e-05 -7.965013e-05 -1.293152e-03
#> beta_2_3    2.019187e-05 -2.020231e-05  5.941341e-05  1.005562e-03
#> beta_3_3   -8.598973e-06 -6.431497e-05  1.259605e-04  2.005304e-03
#> psi_1_1    -3.462260e-07  1.828432e-08 -2.304052e-07 -8.472259e-06
#> psi_2_2    -1.215671e-08 -2.187735e-07  2.214667e-07  5.501769e-06
#> psi_3_3    -7.923548e-08  2.179041e-07 -5.509069e-07 -7.383944e-06
#> theta_1_1   2.876069e-07 -8.989565e-09  1.403556e-07  6.688178e-06
#> theta_2_2  -8.989565e-09  1.698276e-07 -1.902346e-07 -3.628896e-06
#> theta_3_3   1.403556e-07 -1.902346e-07  5.010858e-07  8.485310e-06
#> mu0_1_1     6.688178e-06 -3.628896e-06  8.485310e-06  4.636515e-04
#> mu0_2_1    -6.406365e-06 -1.208342e-06 -1.196712e-06 -2.147625e-04
#> mu0_3_1    -6.377624e-06  1.406503e-06 -4.629364e-06 -3.445786e-04
#> sigma0_1_1 -1.246015e-07  1.368014e-07 -7.740760e-08 -4.132591e-06
#> sigma0_2_2 -5.198438e-07  5.460939e-07 -6.070267e-07 -1.697032e-05
#> sigma0_3_3 -2.871752e-07 -1.081965e-07 -3.207816e-07 -2.813377e-06
#>                  mu0_2_1       mu0_3_1    sigma0_1_1    sigma0_2_2
#> beta_1_1   -2.537954e-03 -2.678580e-03 -5.707850e-05 -2.840790e-04
#> beta_2_1    2.203528e-04  5.422391e-04  2.478245e-05 -3.243701e-07
#> beta_3_1    5.306683e-05  1.229654e-05 -5.533134e-06  2.506603e-05
#> beta_1_2   -5.500862e-05 -2.839870e-04 -4.066630e-05 -1.449269e-04
#> beta_2_2    1.433180e-05  1.090520e-03  4.705749e-05  1.774061e-04
#> beta_3_2    5.677906e-04  1.800958e-04 -3.094374e-07  1.654498e-05
#> beta_1_3    1.301728e-04  1.583044e-03  9.229273e-06  1.072799e-04
#> beta_2_3   -2.170207e-04 -2.040271e-04  5.862384e-06 -5.391496e-05
#> beta_3_3    5.446704e-04 -4.826952e-04  1.618549e-06 -1.032026e-04
#> psi_1_1     7.097892e-06  7.957369e-06  1.271673e-07  7.445604e-07
#> psi_2_2     8.860079e-07 -2.552742e-06 -1.354422e-07 -6.673824e-07
#> psi_3_3    -6.600987e-07  3.129514e-06  3.695267e-08  5.215311e-07
#> theta_1_1  -6.406365e-06 -6.377624e-06 -1.246015e-07 -5.198438e-07
#> theta_2_2  -1.208342e-06  1.406503e-06  1.368014e-07  5.460939e-07
#> theta_3_3  -1.196712e-06 -4.629364e-06 -7.740760e-08 -6.070267e-07
#> mu0_1_1    -2.147625e-04 -3.445786e-04 -4.132591e-06 -1.697032e-05
#> mu0_2_1     2.500344e-04  1.978200e-04 -5.841757e-07  4.433364e-06
#> mu0_3_1     1.978200e-04  3.792046e-04  4.564303e-06  1.815494e-05
#> sigma0_1_1 -5.841757e-07  4.564303e-06  5.591436e-07  1.204200e-06
#> sigma0_2_2  4.433364e-06  1.815494e-05  1.204200e-06  4.685236e-06
#> sigma0_3_3  7.156246e-06 -2.122177e-06 -8.071715e-07 -5.062892e-07
#>               sigma0_3_3
#> beta_1_1   -1.784925e-04
#> beta_2_1   -6.746114e-05
#> beta_3_1    5.641729e-05
#> beta_1_2    3.581213e-05
#> beta_2_2   -8.726443e-05
#> beta_3_2   -2.769356e-05
#> beta_1_3    1.710784e-05
#> beta_2_3   -1.031826e-04
#> beta_3_3   -6.184073e-05
#> psi_1_1     5.331454e-07
#> psi_2_2     1.464423e-07
#> psi_3_3     3.221839e-07
#> theta_1_1  -2.871752e-07
#> theta_2_2  -1.081965e-07
#> theta_3_3  -3.207816e-07
#> mu0_1_1    -2.813377e-06
#> mu0_2_1     7.156246e-06
#> mu0_3_1    -2.122177e-06
#> sigma0_1_1 -8.071715e-07
#> sigma0_2_2 -5.062892e-07
#> sigma0_3_3  2.937108e-06
coef(pb)
#>   beta_1_1   beta_2_1   beta_3_1   beta_1_2   beta_2_2   beta_3_2   beta_1_3 
#>      0.500      0.000      0.000      0.000      0.500      0.000      0.000 
#>   beta_2_3   beta_3_3    psi_1_1    psi_2_2    psi_3_3  theta_1_1  theta_2_2 
#>      0.000      0.500      0.001      0.001      0.001      0.001      0.001 
#>  theta_3_3    mu0_1_1    mu0_2_1    mu0_3_1 sigma0_1_1 sigma0_2_2 sigma0_3_3 
#>      0.001      0.000      0.000      0.000      0.001      0.001      0.001 
print(pb, type = "bc") # bias-corrected
#> Call:
#> PBSSMFixed(R = 10L, path = path, prefix = "ssm", n = n, time = time, 
#>     delta_t = delta_t, mu0 = mu0, sigma0_l = sigma0_l, alpha = alpha, 
#>     beta = beta, psi_l = psi_l, nu = nu, lambda = lambda, theta_l = theta_l, 
#>     type = 0, ncores = 1, seed = 42)
#> 
#> Parametric bootstrap confidence intervals.
#> type = "bc"
#>              est     se  R    2.5%  97.5%
#> beta_1_1   0.500 0.2556 10  0.2865 0.9969
#> beta_2_1   0.000 0.1011 10 -0.1682 0.1064
#> beta_3_1   0.000 0.1535 10 -0.2993 0.1344
#> beta_1_2   0.000 0.1033 10 -0.2115 0.0959
#> beta_2_2   0.500 0.1432 10  0.3230 0.7018
#> beta_3_2   0.000 0.0693 10 -0.0957 0.0532
#> beta_1_3   0.000 0.1674 10 -0.1853 0.0634
#> beta_2_3   0.000 0.0980 10 -0.1285 0.1437
#> beta_3_3   0.500 0.2063 10  0.2299 0.7827
#> psi_1_1    0.001 0.0007 10  0.0000 0.0015
#> psi_2_2    0.001 0.0006 10  0.0001 0.0019
#> psi_3_3    0.001 0.0008 10  0.0002 0.0021
#> theta_1_1  0.001 0.0005 10  0.0004 0.0017
#> theta_2_2  0.001 0.0004 10  0.0004 0.0016
#> theta_3_3  0.001 0.0007 10  0.0000 0.0018
#> mu0_1_1    0.000 0.0215 10 -0.0357 0.0194
#> mu0_2_1    0.000 0.0158 10 -0.0217 0.0178
#> mu0_3_1    0.000 0.0195 10 -0.0187 0.0283
#> sigma0_1_1 0.001 0.0007 10  0.0000 0.0019
#> sigma0_2_2 0.001 0.0022 10  0.0000 0.0068
#> sigma0_3_3 0.001 0.0017 10  0.0000 0.0042
summary(pb, type = "bc")
#> Call:
#> PBSSMFixed(R = 10L, path = path, prefix = "ssm", n = n, time = time, 
#>     delta_t = delta_t, mu0 = mu0, sigma0_l = sigma0_l, alpha = alpha, 
#>     beta = beta, psi_l = psi_l, nu = nu, lambda = lambda, theta_l = theta_l, 
#>     type = 0, ncores = 1, seed = 42)
#>              est     se  R    2.5%  97.5%
#> beta_1_1   0.500 0.2556 10  0.2865 0.9969
#> beta_2_1   0.000 0.1011 10 -0.1682 0.1064
#> beta_3_1   0.000 0.1535 10 -0.2993 0.1344
#> beta_1_2   0.000 0.1033 10 -0.2115 0.0959
#> beta_2_2   0.500 0.1432 10  0.3230 0.7018
#> beta_3_2   0.000 0.0693 10 -0.0957 0.0532
#> beta_1_3   0.000 0.1674 10 -0.1853 0.0634
#> beta_2_3   0.000 0.0980 10 -0.1285 0.1437
#> beta_3_3   0.500 0.2063 10  0.2299 0.7827
#> psi_1_1    0.001 0.0007 10  0.0000 0.0015
#> psi_2_2    0.001 0.0006 10  0.0001 0.0019
#> psi_3_3    0.001 0.0008 10  0.0002 0.0021
#> theta_1_1  0.001 0.0005 10  0.0004 0.0017
#> theta_2_2  0.001 0.0004 10  0.0004 0.0016
#> theta_3_3  0.001 0.0007 10  0.0000 0.0018
#> mu0_1_1    0.000 0.0215 10 -0.0357 0.0194
#> mu0_2_1    0.000 0.0158 10 -0.0217 0.0178
#> mu0_3_1    0.000 0.0195 10 -0.0187 0.0283
#> sigma0_1_1 0.001 0.0007 10  0.0000 0.0019
#> sigma0_2_2 0.001 0.0022 10  0.0000 0.0068
#> sigma0_3_3 0.001 0.0017 10  0.0000 0.0042
confint(pb, type = "bc")
#>                    2.5 %      97.5 %
#> beta_1_1    2.865140e-01 0.996913972
#> beta_2_1   -1.681965e-01 0.106357315
#> beta_3_1   -2.993036e-01 0.134425625
#> beta_1_2   -2.115189e-01 0.095917751
#> beta_2_2    3.229898e-01 0.701814986
#> beta_3_2   -9.572536e-02 0.053242497
#> beta_1_3   -1.853209e-01 0.063398680
#> beta_2_3   -1.284541e-01 0.143676250
#> beta_3_3    2.298664e-01 0.782746723
#> psi_1_1     5.292005e-07 0.001546135
#> psi_2_2     1.337401e-04 0.001858861
#> psi_3_3     1.540076e-04 0.002069710
#> theta_1_1   4.351684e-04 0.001661737
#> theta_2_2   3.720811e-04 0.001564831
#> theta_3_3   7.356208e-12 0.001796755
#> mu0_1_1    -3.570665e-02 0.019411810
#> mu0_2_1    -2.165274e-02 0.017832927
#> mu0_3_1    -1.867286e-02 0.028251855
#> sigma0_1_1  8.120570e-11 0.001893493
#> sigma0_2_2  9.878043e-21 0.006782330
#> sigma0_3_3  1.756386e-14 0.004185875
# }