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 $\left( \boldsymbol{\Psi}^{\frac{1}{2}} \right) \left( \boldsymbol{\Psi}^{\frac{1}{2}} \right)^{\prime} = \boldsymbol{\Psi}$ .

Data Generation

Notation

Let $t = 500$ be the number of time points and $n = 100$ be the number of individuals.

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

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

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

$\begin{equation} \boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0} = \left( \begin{array}{ccc} 0.1960784 & 0.1183232 & 0.0298539 \\ 0.1183232 & 0.3437711 & 0.1381855 \\ 0.0298539 & 0.1381855 & 0.2663828 \\ \end{array} \right) . \end{equation}$

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] 100
time
#> [1] 500
mu0
#> [1] 0 0 0
sigma0
#>            [,1]      [,2]       [,3]
#> [1,] 0.19607843 0.1183232 0.02985385
#> [2,] 0.11832319 0.3437711 0.13818551
#> [3,] 0.02985385 0.1381855 0.26638284
sigma0_l # sigma0_l <- t(chol(sigma0))
#>            [,1]      [,2]     [,3]
#> [1,] 0.44280744 0.0000000 0.000000
#> [2,] 0.26721139 0.5218900 0.000000
#> [3,] 0.06741949 0.2302597 0.456966
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

Visualizing the Dynamics Without Process Noise (n = 5 with Different Initial Condition)

Using the `SimSSMVARFixed` Function from the `simStateSpace` Package to Simulate Data

library(simStateSpace)
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)
head(data)
#>   id time          y1          y2          y3
#> 1  1    0 -0.81728749  0.01319023  0.57975035
#> 2  1    1 -0.62264147 -0.94877859  0.54422379
#> 3  1    2 -0.06731464 -0.69530903 -0.09626706
#> 4  1    3 -0.09174342 -0.44804477 -0.27983813
#> 5  1    4 -0.04104160 -0.72769549 -0.19238403
#> 6  1    5  0.18907246 -0.81602072 -0.73831208
summary(data)
#>        id              time             y1                  y2           
#>  Min.   :  1.00   Min.   :  0.0   Min.   :-1.831372   Min.   :-2.644556  
#>  1st Qu.: 25.75   1st Qu.:124.8   1st Qu.:-0.297127   1st Qu.:-0.395276  
#>  Median : 50.50   Median :249.5   Median :-0.003762   Median :-0.003310  
#>  Mean   : 50.50   Mean   :249.5   Mean   :-0.002119   Mean   :-0.002361  
#>  3rd Qu.: 75.25   3rd Qu.:374.2   3rd Qu.: 0.290142   3rd Qu.: 0.390052  
#>  Max.   :100.00   Max.   :499.0   Max.   : 1.975304   Max.   : 2.318903  
#>        y3            
#>  Min.   :-2.1828250  
#>  1st Qu.:-0.3458073  
#>  Median :-0.0015880  
#>  Mean   :-0.0004972  
#>  3rd Qu.: 0.3443516  
#>  Max.   : 2.1134391
plot(sim)

Model Fitting

Prepare Data

dynr_data <- dynr::dynr.data(
  data = data,
  id = "id",
  time = "time",
  observed = c("y1", "y2", "y3")
)

Prepare Initial Condition

dynr_initial <- dynr::prep.initial(
  values.inistate = mu0,
  params.inistate = c("mu0_1_1", "mu0_2_1", "mu0_3_1"),
  values.inicov = sigma0,
  params.inicov = matrix(
    data = c(
      "sigma0_1_1", "sigma0_2_1", "sigma0_3_1",
      "sigma0_2_1", "sigma0_2_2", "sigma0_3_2",
      "sigma0_3_1", "sigma0_3_2", "sigma0_3_3"
    ),
    nrow = 3
  )
)

Prepare Measurement Model

dynr_measurement <- dynr::prep.measurement(
  values.load = diag(3),
  params.load = matrix(data = "fixed", nrow = 3, ncol = 3),
  state.names = c("eta_1", "eta_2", "eta_3"),
  obs.names = c("y1", "y2", "y3")
)

Prepare Dynamic Process

dynr_dynamics <- dynr::prep.formulaDynamics(
  formula = list(
    eta_1 ~ alpha_1_1 * 1 + beta_1_1 * eta_1 + beta_1_2 * eta_2 + beta_1_3 * eta_3,
    eta_2 ~ alpha_2_1 * 1 + beta_2_1 * eta_1 + beta_2_2 * eta_2 + beta_2_3 * eta_3,
    eta_3 ~ alpha_3_1 * 1 + beta_3_1 * eta_1 + beta_3_2 * eta_2 + beta_3_3 * eta_3
  ),
  startval = c(
    alpha_1_1 = alpha[1], alpha_2_1 = alpha[2], alpha_3_1 = alpha[3],
    beta_1_1 = beta[1, 1], beta_1_2 = beta[1, 2], beta_1_3 = beta[1, 3],
    beta_2_1 = beta[2, 1], beta_2_2 = beta[2, 2], beta_2_3 = beta[2, 3],
    beta_3_1 = beta[3, 1], beta_3_2 = beta[3, 2], beta_3_3 = beta[3, 3]
  ),
  isContinuousTime = FALSE
)

Prepare Process Noise

dynr_noise <- dynr::prep.noise(
  values.latent = psi,
  params.latent = matrix(
    data = c(
      "psi_1_1", "psi_2_1", "psi_3_1",
      "psi_2_1", "psi_2_2", "psi_3_2",
      "psi_3_1", "psi_3_2", "psi_3_3"
    ),
    nrow = 3
  ),
  values.observed = matrix(data = 0, nrow = 3, ncol = 3),
  params.observed = matrix(data = "fixed", nrow = 3, ncol = 3)
)

Prepare the Model

model <- dynr::dynr.model(
  data = dynr_data,
  initial = dynr_initial,
  measurement = dynr_measurement,
  dynamics = dynr_dynamics,
  noise = dynr_noise,
  outfile = "var.c"
)

Fit the Model

results <- dynr::dynr.cook(
  model,
  debug_flag = TRUE,
  verbose = FALSE
)
#> [1] "Get ready!!!!"
#> using C compiler: ‘gcc (Ubuntu 13.3.0-6ubuntu2~24.04) 13.3.0’
#> Optimization function called.
#> Starting Hessian calculation ...
#> Finished Hessian calculation.
#> Original exit flag:  3 
#> Modified exit flag:  3 
#> Optimization terminated successfully: ftol_rel or ftol_abs was reached. 
#> Original fitted parameters:  -0.0006607231 3.211261e-05 0.0003389175 0.6935267 
#> 0.0005250874 -0.0005643153 0.4984296 0.6011778 -0.002500733 -0.1056564 
#> 0.4056584 0.4971785 -2.308965 -0.007690335 0.004142653 -2.309886 -0.001043527 
#> -2.300854 -0.002408683 0.001315326 0.02286415 -1.674994 0.3533811 -0.060354 
#> -1.299978 0.4453573 -1.556499 
#> 
#> Transformed fitted parameters:  -0.0006607231 3.211261e-05 0.0003389175 
#> 0.6935267 0.0005250874 -0.0005643153 0.4984296 0.6011778 -0.002500733 
#> -0.1056564 0.4056584 0.4971785 0.09936408 -0.000764143 0.0004116308 0.09927844 
#> -0.0001067592 0.1001751 -0.002408683 0.001315326 0.02286415 0.1873094 
#> 0.06619159 -0.01130487 0.2959286 0.1173817 0.2656113 
#> 
#> Doing end processing
#> Successful trial
#> Total Time: 39.83028 
#> Backend Time: 39.8202

Summary

summary(results)
#> Coefficients:
#>              Estimate Std. Error t value   ci.lower   ci.upper Pr(>|t|)    
#> alpha_1_1  -6.607e-04  1.411e-03  -0.468 -3.426e-03  2.105e-03   0.3198    
#> alpha_2_1   3.211e-05  1.411e-03   0.023 -2.732e-03  2.797e-03   0.4909    
#> alpha_3_1   3.389e-04  1.417e-03   0.239 -2.438e-03  3.116e-03   0.4055    
#> beta_1_1    6.935e-01  3.613e-03 191.939  6.864e-01  7.006e-01   <2e-16 ***
#> beta_1_2    5.251e-04  3.035e-03   0.173 -5.424e-03  6.474e-03   0.4313    
#> beta_1_3   -5.643e-04  3.087e-03  -0.183 -6.615e-03  5.486e-03   0.4275    
#> beta_2_1    4.984e-01  3.612e-03 138.009  4.914e-01  5.055e-01   <2e-16 ***
#> beta_2_2    6.012e-01  3.034e-03 198.146  5.952e-01  6.071e-01   <2e-16 ***
#> beta_2_3   -2.501e-03  3.086e-03  -0.810 -8.549e-03  3.547e-03   0.2089    
#> beta_3_1   -1.057e-01  3.628e-03 -29.123 -1.128e-01 -9.855e-02   <2e-16 ***
#> beta_3_2    4.057e-01  3.048e-03 133.101  3.997e-01  4.116e-01   <2e-16 ***
#> beta_3_3    4.972e-01  3.100e-03 160.389  4.911e-01  5.033e-01   <2e-16 ***
#> psi_1_1     9.936e-02  6.290e-04 157.960  9.813e-02  1.006e-01   <2e-16 ***
#> psi_2_1    -7.641e-04  4.446e-04  -1.719 -1.636e-03  1.073e-04   0.0428 *  
#> psi_3_1     4.116e-04  4.466e-04   0.922 -4.637e-04  1.287e-03   0.1783    
#> psi_2_2     9.928e-02  6.285e-04 157.956  9.805e-02  1.005e-01   <2e-16 ***
#> psi_3_2    -1.068e-04  4.464e-04  -0.239 -9.817e-04  7.682e-04   0.4055    
#> psi_3_3     1.002e-01  6.342e-04 157.962  9.893e-02  1.014e-01   <2e-16 ***
#> mu0_1_1    -2.409e-03  4.327e-02  -0.056 -8.721e-02  8.239e-02   0.4778    
#> mu0_2_1     1.315e-03  5.439e-02   0.024 -1.053e-01  1.079e-01   0.4904    
#> mu0_3_1     2.286e-02  5.153e-02   0.444 -7.813e-02  1.239e-01   0.3286    
#> sigma0_1_1  1.873e-01  2.645e-02   7.083  1.355e-01  2.391e-01   <2e-16 ***
#> sigma0_2_1  6.619e-02  2.444e-02   2.709  1.829e-02  1.141e-01   0.0034 ** 
#> sigma0_3_1 -1.130e-02  2.223e-02  -0.509 -5.487e-02  3.226e-02   0.3055    
#> sigma0_2_2  2.959e-01  4.184e-02   7.073  2.139e-01  3.779e-01   <2e-16 ***
#> sigma0_3_2  1.174e-01  3.040e-02   3.861  5.779e-02  1.770e-01   0.0001 ***
#> sigma0_3_3  2.656e-01  3.754e-02   7.076  1.920e-01  3.392e-01   <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> -2 log-likelihood value at convergence = 79944.89
#> AIC = 79998.89
#> BIC = 80237.02

Parameter Estimates

alpha_hat
#> [1] -6.607231e-04  3.211261e-05  3.389175e-04
beta_hat
#>            [,1]         [,2]          [,3]
#> [1,]  0.6935267 0.0005250874 -0.0005643153
#> [2,]  0.4984296 0.6011777566 -0.0025007326
#> [3,] -0.1056564 0.4056584147  0.4971784809
psi_hat
#>               [,1]          [,2]          [,3]
#> [1,]  0.0993640778 -0.0007641430  0.0004116308
#> [2,] -0.0007641430  0.0992784439 -0.0001067592
#> [3,]  0.0004116308 -0.0001067592  0.1001751099
mu0_hat
#> [1] -0.002408683  0.001315326  0.022864146
sigma0_hat
#>             [,1]       [,2]        [,3]
#> [1,]  0.18730937 0.06619159 -0.01130487
#> [2,]  0.06619159 0.29592857  0.11738173
#> [3,] -0.01130487 0.11738173  0.26561129

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

Pesigan, I. J. A., Russell, M. A., & Chow, S.-M. (2025). Inferences and effect sizes for direct, indirect, and total effects in continuous-time mediation models. Psychological Methods. https://doi.org/10.1037/met0000779

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

Ivan Jacob Agaloos Pesigan

2025-10-09

Model

Data Generation

Notation

R Function Arguments

Visualizing the Dynamics Without Process Noise (n = 5 with Different Initial Condition)

Using the `SimSSMVARFixed` Function from the `simStateSpace` Package to Simulate Data

Model Fitting

Prepare Data

Prepare Initial Condition

Prepare Measurement Model

Prepare Dynamic Process

Prepare Process Noise

Prepare the Model

Fit the Model

Summary

Parameter Estimates

References

The Vector Autoregressive Model

Ivan Jacob Agaloos Pesigan

2025-10-09

Model

Data Generation

Notation

R Function Arguments

Visualizing the Dynamics Without Process Noise (n = 5 with Different Initial Condition)

Using the SimSSMVARFixed Function from the simStateSpace Package to Simulate Data

Model Fitting

Prepare Data

Prepare Initial Condition

Prepare Measurement Model

Prepare Dynamic Process

Prepare Process Noise

Prepare the Model

Fit the Model

Summary

Parameter Estimates

References

Using the `SimSSMVARFixed` Function from the `simStateSpace` Package to Simulate Data