Parametric Bootstrap (The Vector Autoregressive Model)

Model

The measurement model is given by $\begin{equation} \mathbf{y}_{i, t} = \boldsymbol{\eta}_{i, t} \end{equation}$ where $\mathbf{y}_{i, t}$ represents a vector of observed variables and $\boldsymbol{\eta}_{i, t}$ a vector of latent variables for individual $i$ and time $t$ . Since the observed and latent variables are equal, we only generate data from the dynamic structure.

The dynamic structure is given by $\begin{equation} \boldsymbol{\eta}_{i, t} = \boldsymbol{\alpha} + \boldsymbol{\beta} \boldsymbol{\eta}_{i, t - 1} + \boldsymbol{\zeta}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\zeta}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Psi} \right) \end{equation}$ 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 $\begin{equation} \boldsymbol{\zeta}_{i, t} = \boldsymbol{\Psi}^{\frac{1}{2}} \mathbf{z}_{i, t}, \quad \mathrm{with} \quad \mathbf{z}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \mathbf{I} \right) \end{equation}$ where $\left( \boldsymbol{\Psi}^{\frac{1}{2}} \right) \left( \boldsymbol{\Psi}^{\frac{1}{2}} \right)^{\prime} = \boldsymbol{\Psi}$ .

Parameters

Notation

Let $t = 100$ be the number of time points and $n = 5$ be the number of individuals.

Let the initial condition $\boldsymbol{\eta}_{0}$ be given by

$\begin{equation} \boldsymbol{\eta}_{0} \sim \mathcal{N} \left( \boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0}, \boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0} \right) \end{equation}$

$\begin{equation} \boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0} = \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ \end{array} \right) \end{equation}$

$\begin{equation} \boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0} = \left( \begin{array}{ccc} 1 & 0.2 & 0.2 \\ 0.2 & 1 & 0.2 \\ 0.2 & 0.2 & 1 \\ \end{array} \right) . \end{equation}$

Let the constant vector $\boldsymbol{\alpha}$ be given by

$\begin{equation} \boldsymbol{\alpha} = \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ \end{array} \right) . \end{equation}$

Let the transition matrix $\boldsymbol{\beta}$ be given by

$\begin{equation} \boldsymbol{\beta} = \left( \begin{array}{ccc} 0.7 & 0 & 0 \\ 0.5 & 0.6 & 0 \\ -0.1 & 0.4 & 0.5 \\ \end{array} \right) . \end{equation}$

Let the dynamic process noise $\boldsymbol{\Psi}$ be given by

$\begin{equation} \boldsymbol{\Psi} = \left( \begin{array}{ccc} 0.1 & 0 & 0 \\ 0 & 0.1 & 0 \\ 0 & 0 & 0.1 \\ \end{array} \right) . \end{equation}$

R Function Arguments

n
#> [1] 5
time
#> [1] 100
mu0
#> [1] 0 0 0
sigma0
#>      [,1] [,2] [,3]
#> [1,]  1.0  0.2  0.2
#> [2,]  0.2  1.0  0.2
#> [3,]  0.2  0.2  1.0
sigma0_l # sigma0_l <- t(chol(sigma0))
#>      [,1]      [,2]      [,3]
#> [1,]  1.0 0.0000000 0.0000000
#> [2,]  0.2 0.9797959 0.0000000
#> [3,]  0.2 0.1632993 0.9660918
alpha
#> [1] 0 0 0
beta
#>      [,1] [,2] [,3]
#> [1,]  0.7  0.0  0.0
#> [2,]  0.5  0.6  0.0
#> [3,] -0.1  0.4  0.5
psi
#>      [,1] [,2] [,3]
#> [1,]  0.1  0.0  0.0
#> [2,]  0.0  0.1  0.0
#> [3,]  0.0  0.0  0.1
psi_l # psi_l <- t(chol(psi))
#>           [,1]      [,2]      [,3]
#> [1,] 0.3162278 0.0000000 0.0000000
#> [2,] 0.0000000 0.3162278 0.0000000
#> [3,] 0.0000000 0.0000000 0.3162278

Parametric Bootstrap

R <- 5L # use at least 1000 in actual research
path <- getwd()
prefix <- "var"

We use the PBSSMVARFixed function from the bootStateSpace package to perform parametric bootstraping using the parameters described above. The argument R specifies the number of bootstrap replications. The generated data and model estimates are stored in path using the specified prefix for the file names. The ncores = parallel::detectCores() argument instructs the function to use all available CPU cores in the system.

NOTE: Fitting the VAR model multiple times is computationally intensive.

library(bootStateSpace)
pb <- PBSSMVARFixed(
  R = R,
  path = path,
  prefix = prefix,
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  ncores = parallel::detectCores(),
  seed = 42
)
summary(pb)
#> Call:
#> PBSSMVARFixed(R = R, path = path, prefix = prefix, n = n, time = time, 
#>     mu0 = mu0, sigma0_l = sigma0_l, alpha = alpha, beta = beta, 
#>     psi_l = psi_l, ncores = parallel::detectCores(), seed = 42)
#>             est     se R    2.5%   97.5%
#> beta_1_1    0.7 0.0280 5  0.6634  0.7330
#> beta_2_1    0.5 0.0261 5  0.4794  0.5376
#> beta_3_1   -0.1 0.0301 5 -0.1475 -0.0760
#> beta_1_2    0.0 0.0486 5 -0.0489  0.0495
#> beta_2_2    0.6 0.0148 5  0.5847  0.6166
#> beta_3_2    0.4 0.0201 5  0.3629  0.4124
#> beta_1_3    0.0 0.0367 5 -0.0568  0.0305
#> beta_2_3    0.0 0.0203 5 -0.0112  0.0385
#> beta_3_3    0.5 0.0227 5  0.4716  0.5200
#> psi_1_1     0.1 0.0065 5  0.0880  0.1030
#> psi_2_2     0.1 0.0040 5  0.0942  0.1035
#> psi_3_3     0.1 0.0047 5  0.0947  0.1057
#> mu0_1_1     0.0 0.7384 5 -0.9757  0.8926
#> mu0_2_1     0.0 0.4396 5 -0.5231  0.5924
#> mu0_3_1     0.0 0.3699 5 -0.6983  0.2269
#> sigma0_1_1  1.0 0.8489 5  0.3086  2.2269
#> sigma0_2_1  0.2 0.3379 5 -0.3639  0.4107
#> sigma0_3_1  0.2 0.3344 5 -0.1751  0.6238
#> sigma0_2_2  1.0 0.3817 5  0.2378  1.0539
#> sigma0_3_2  0.2 0.2974 5 -0.0067  0.7087
#> sigma0_3_3  1.0 0.7525 5  0.2928  2.0012
summary(pb, type = "bc")
#> Call:
#> PBSSMVARFixed(R = R, path = path, prefix = prefix, n = n, time = time, 
#>     mu0 = mu0, sigma0_l = sigma0_l, alpha = alpha, beta = beta, 
#>     psi_l = psi_l, ncores = parallel::detectCores(), seed = 42)
#>             est     se R    2.5%   97.5%
#> beta_1_1    0.7 0.0280 5  0.6677  0.7349
#> beta_2_1    0.5 0.0261 5  0.4848  0.5422
#> beta_3_1   -0.1 0.0301 5 -0.1435 -0.0750
#> beta_1_2    0.0 0.0486 5 -0.0496  0.0475
#> beta_2_2    0.6 0.0148 5  0.5847  0.6171
#> beta_3_2    0.4 0.0201 5  0.3673  0.4132
#> beta_1_3    0.0 0.0367 5 -0.0256  0.0347
#> beta_2_3    0.0 0.0203 5 -0.0120  0.0322
#> beta_3_3    0.5 0.0227 5  0.4723  0.5203
#> psi_1_1     0.1 0.0065 5  0.0893  0.1032
#> psi_2_2     0.1 0.0040 5  0.0950  0.1035
#> psi_3_3     0.1 0.0047 5  0.0944  0.1053
#> mu0_1_1     0.0 0.7384 5 -1.0545  0.7686
#> mu0_2_1     0.0 0.4396 5 -0.4268  0.6248
#> mu0_3_1     0.0 0.3699 5 -0.2398  0.2557
#> sigma0_1_1  1.0 0.8489 5  0.2877  2.1599
#> sigma0_2_1  0.2 0.3379 5 -0.3993  0.4029
#> sigma0_3_1  0.2 0.3344 5 -0.2298  0.4150
#> sigma0_2_2  1.0 0.3817 5  0.3415  1.0726
#> sigma0_3_2  0.2 0.2974 5  0.0217  0.7383
#> sigma0_3_3  1.0 0.7525 5  0.4639  2.1281

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. https://doi.org/10.1080/10705511003661553

Ou, L., Hunter, M. D., & Chow, S.-M. (2019). What’s for dynr: A package for linear and nonlinear dynamic modeling in R. The R Journal, 11(1), 91. https://doi.org/10.32614/rj-2019-012

R Core Team. (2024). R: A language and environment for statistical computing. R Foundation for Statistical Computing. https://www.R-project.org/