Parametric Bootstrap (The Ornstein–Uhlenbeck Model)

Ivan Jacob Agaloos Pesigan

2025-02-14

Model

The measurement model is given by $$\begin{equation} \mathbf{y}_{i, t} = \boldsymbol{\nu} + \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta} \right) \end{equation}$$ 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 $$\begin{equation} \boldsymbol{\varepsilon}_{i, t} = \boldsymbol{\Theta}^{\frac{1}{2}} \mathbf{z}_{i, t}, \quad \mathrm{with} \quad \mathbf{z}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \mathbf{I} \right) \end{equation}$$ 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 $$\begin{equation} \mathrm{d} \boldsymbol{\eta}_{i, t} = \boldsymbol{\Phi} \left( \boldsymbol{\eta}_{i, t} - \boldsymbol{\mu} \right) \mathrm{d}t + \boldsymbol{\Sigma}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t} \end{equation}$$ where $\boldsymbol{\mu}$ is the long-term mean or equilibrium level, $\boldsymbol{\Phi}$ is the rate of mean reversion, determining how quickly the variable returns to its mean, $\boldsymbol{\Sigma}$ is the matrix of volatility or randomness in the process, and $\mathrm{d}\boldsymbol{W}$ is a Wiener process or Brownian motion, which represents random fluctuations.

Parameters

Notation

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

Let the measurement model intecept vector $\boldsymbol{\nu}$ be given by

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

Let the factor loadings matrix $\boldsymbol{\Lambda}$ be given by

$$\begin{equation} \boldsymbol{\Lambda} = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) . \end{equation}$$

Let the measurement error covariance matrix $\boldsymbol{\Theta}$ be given by

$$\begin{equation} \boldsymbol{\Theta} = \left( \begin{array}{ccc} 0.2 & 0 & 0 \\ 0 & 0.2 & 0 \\ 0 & 0 & 0.2 \\ \end{array} \right) . \end{equation}$$

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 long-term mean vector $\boldsymbol{\mu}$ be given by

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

Let the rate of mean reversion matrix $\boldsymbol{\Phi}$ be given by

$$\begin{equation} \boldsymbol{\Phi} = \left( \begin{array}{ccc} -0.357 & 0 & 0 \\ 0.771 & -0.511 & 0 \\ -0.45 & 0.729 & -0.693 \\ \end{array} \right) . \end{equation}$$

Let the dynamic process noise covariance matrix $\boldsymbol{\Sigma}$ be given by

$$\begin{equation} \boldsymbol{\Sigma} = \left( \begin{array}{ccc} 0.2445556 & 0.0220159 & -0.0500476 \\ 0.0220159 & 0.070678 & 0.0153946 \\ -0.0500476 & 0.0153946 & 0.0755306 \\ \end{array} \right) . \end{equation}$$

Let $\Delta t = 0.1$.

R Function Arguments

n
#> [1] 5
time
#> [1] 100
delta_t
#> [1] 0.1
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
mu
#> [1] 0 0 0
phi
#>        [,1]   [,2]   [,3]
#> [1,] -0.357  0.000  0.000
#> [2,]  0.771 -0.511  0.000
#> [3,] -0.450  0.729 -0.693
sigma
#>             [,1]       [,2]        [,3]
#> [1,]  0.24455556 0.02201587 -0.05004762
#> [2,]  0.02201587 0.07067800  0.01539456
#> [3,] -0.05004762 0.01539456  0.07553061
sigma_l # sigma_l <- t(chol(sigma))
#>             [,1]      [,2]     [,3]
#> [1,]  0.49452559 0.0000000 0.000000
#> [2,]  0.04451917 0.2620993 0.000000
#> [3,] -0.10120330 0.0759256 0.243975
nu
#> [1] 0 0 0
lambda
#>      [,1] [,2] [,3]
#> [1,]    1    0    0
#> [2,]    0    1    0
#> [3,]    0    0    1
theta
#>      [,1] [,2] [,3]
#> [1,]  0.2  0.0  0.0
#> [2,]  0.0  0.2  0.0
#> [3,]  0.0  0.0  0.2
theta_l # theta_l <- t(chol(theta))
#>           [,1]      [,2]      [,3]
#> [1,] 0.4472136 0.0000000 0.0000000
#> [2,] 0.0000000 0.4472136 0.0000000
#> [3,] 0.0000000 0.0000000 0.4472136

Parametric Bootstrap

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

We use the PBSSMOUFixed 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 Ornstein–Uhlenbeck model multiple times is computationally intensive.

library(bootStateSpace)
pb <- PBSSMOUFixed(
  R = R,
  path = path,
  prefix = prefix,
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  mu = mu,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  ncores = parallel::detectCores(),
  seed = 42
)
summary(pb)
#> Call:
#> PBSSMOUFixed(R = R, path = path, prefix = prefix, n = n, time = time, 
#>     delta_t = delta_t, mu0 = mu0, sigma0_l = sigma0_l, mu = mu, 
#>     phi = phi, sigma_l = sigma_l, nu = nu, lambda = lambda, theta_l = theta_l, 
#>     ncores = parallel::detectCores(), seed = 42)
#>                est     se R    2.5%   97.5%
#> phi_1_1    -0.3570 0.0915 5 -0.2905 -0.0677
#> phi_2_1     0.7710 0.2973 5  0.4056  1.1398
#> phi_3_1    -0.4500 0.1507 5 -0.6873 -0.3524
#> phi_1_2     0.0000 0.1110 5 -0.4812 -0.2093
#> phi_2_2    -0.5110 0.2813 5 -0.9066 -0.2762
#> phi_3_2     0.7290 0.1325 5  0.6434  0.9556
#> phi_1_3     0.0000 0.1592 5 -0.0033  0.3680
#> phi_2_3     0.0000 0.1793 5 -0.1667  0.2364
#> phi_3_3    -0.6930 0.0971 5 -0.8155 -0.5816
#> sigma_1_1   0.2446 0.0501 5  0.1767  0.3029
#> sigma_2_1   0.0220 0.0313 5 -0.0319  0.0398
#> sigma_3_1  -0.0500 0.0420 5 -0.1246 -0.0279
#> sigma_2_2   0.0707 0.0480 5  0.0221  0.1345
#> sigma_3_2   0.0154 0.0174 5  0.0133  0.0523
#> sigma_3_3   0.0755 0.0326 5  0.0253  0.1067
#> theta_1_1   0.2000 0.0195 5  0.1681  0.2093
#> theta_2_2   0.2000 0.0181 5  0.1939  0.2347
#> theta_3_3   0.2000 0.0118 5  0.1908  0.2162
#> mu0_1_1     0.0000 0.1985 5 -0.2841  0.1974
#> mu0_2_1     0.0000 0.2249 5 -0.4795  0.0903
#> mu0_3_1     0.0000 0.2002 5  0.2694  0.7245
#> sigma0_1_1  1.0000 0.6896 5  0.1709  1.6286
#> sigma0_2_1  0.2000 0.7082 5 -0.2206  1.3790
#> sigma0_3_1  0.2000 0.4829 5 -0.0541  1.1038
#> sigma0_2_2  1.0000 0.5403 5  0.1653  1.5218
#> sigma0_3_2  0.2000 0.4366 5 -0.1888  0.9029
#> sigma0_3_3  1.0000 0.4151 5  0.2797  1.3094
summary(pb, type = "bc")
#> Call:
#> PBSSMOUFixed(R = R, path = path, prefix = prefix, n = n, time = time, 
#>     delta_t = delta_t, mu0 = mu0, sigma0_l = sigma0_l, mu = mu, 
#>     phi = phi, sigma_l = sigma_l, nu = nu, lambda = lambda, theta_l = theta_l, 
#>     ncores = parallel::detectCores(), seed = 42)
#>                est     se R    2.5%   97.5%
#> phi_1_1    -0.3570 0.0915 5 -0.3056 -0.3056
#> phi_2_1     0.7710 0.2973 5  0.4387  1.1650
#> phi_3_1    -0.4500 0.1507 5 -0.6977 -0.3539
#> phi_1_2     0.0000 0.1110 5 -0.2025 -0.2025
#> phi_2_2    -0.5110 0.2813 5 -0.9622 -0.3598
#> phi_3_2     0.7290 0.1325 5  0.6402  0.9317
#> phi_1_3     0.0000 0.1592 5 -0.0252  0.3118
#> phi_2_3     0.0000 0.1793 5 -0.1261  0.2696
#> phi_3_3    -0.6930 0.0971 5 -0.8057 -0.5734
#> sigma_1_1   0.2446 0.0501 5  0.1843  0.3074
#> sigma_2_1   0.0220 0.0313 5 -0.0287  0.0406
#> sigma_3_1  -0.0500 0.0420 5 -0.1148 -0.0277
#> sigma_2_2   0.0707 0.0480 5  0.0431  0.1429
#> sigma_3_2   0.0154 0.0174 5  0.0132  0.0496
#> sigma_3_3   0.0755 0.0326 5  0.0326  0.1085
#> theta_1_1   0.2000 0.0195 5  0.1684  0.2098
#> theta_2_2   0.2000 0.0181 5  0.1937  0.2280
#> theta_3_3   0.2000 0.0118 5  0.1901  0.2162
#> mu0_1_1     0.0000 0.1985 5 -0.3199  0.0697
#> mu0_2_1     0.0000 0.2249 5 -0.1604  0.1127
#> mu0_3_1     0.0000 0.2002 5  0.2657  0.2657
#> sigma0_1_1  1.0000 0.6896 5  0.1630  1.5791
#> sigma0_2_1  0.2000 0.7082 5 -0.1815  1.4226
#> sigma0_3_1  0.2000 0.4829 5 -0.0835  0.6530
#> sigma0_2_2  1.0000 0.5403 5  0.6522  1.6013
#> sigma0_3_2  0.2000 0.4366 5 -0.2111  0.7772
#> sigma0_3_3  1.0000 0.4151 5  0.6142  1.3516

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