8.6 Exercises

1. Simulate the dynamic linear model assuming \(X_t \sim N(1, 0.1\sigma^2)\), \(w_t \sim N(0, 0.5\sigma^2)\), \(\mu_t \sim N(0, \sigma^2)\), \(\beta_0 = 1\), \(B_0 = 0.5\sigma^2\), \(\sigma^2 = 0.25\), and \(G_t = 1\), for \(t = 1, \dots, 100\). Then, perform the filtering recursion fixing \(\Sigma = 25 \times 0.25\), \(\Omega_1 = 0.5\Sigma\) (high signal-to-noise ratio) and \(\Omega_2 = 0.1\Sigma\) (low signal-to-noise ratio). Plot and compare the results.

2. Simulate the dynamic linear model \(y_t = \beta_t x_t + \mu_t\), \(\beta_t = \beta_{t-1} + w_t\), where \(x_t \sim N(1, 0.1\sigma^2)\), \(w_t \sim N(0, 0.5\sigma^2)\), \(\mu_t \sim N(0, \sigma^2)\), \(\beta_0 = 0\), \(B_0 = 0.5\sigma^2\), and \(\sigma^2 = 1\), for \(t = 1, \dots, 100\). Perform the filtering and smoothing recursions from scratch.

3. Simulate the process \(y_t = \alpha z_t + \beta_t x_t + \boldsymbol{h}^{\top}\boldsymbol{\epsilon}_t\), \(\beta_t = \beta_{t-1} + \boldsymbol{H}^{\top}\boldsymbol{\epsilon}_t\), where \(\boldsymbol{h}^{\top} = [1 \ 0]\), \(\boldsymbol{H}^{\top} = [0 \ 1/\tau]\), \(\boldsymbol{v}_t \sim N(\boldsymbol{0}_2, \sigma^2 \boldsymbol{I}_2)\), \(x_t \sim N(1, 2\sigma^2)\), \(z_t \sim N(0, 2\sigma^2)\), \(\alpha = 2\), \(\tau^2 = 5\), and \(\sigma^2 = 0.1\), for \(t = 1, \dots, 200\). Assume \(\pi({\beta}_0, {\alpha}, \sigma^2, {\tau}) = \pi({\beta}_0)\pi({\alpha})\pi(\sigma^2)\pi(\tau^2)\) where \(\sigma^2 \sim IG(\alpha_0/2, \delta_0/2)\), \(\tau^2 \sim G(v_{0}/2, v_{0}/2)\), \({\alpha} \sim N({a}_0, {A}_0)\), and \({\beta}_0 \sim N({b}_0, {B}_0)\) such that \(\alpha_0 = \delta_0 = 1\), \(v_0 = 5\), \(a_0 = 0\), \(A_0 = 1\), \(\beta_0 = 0\), \(B_0 = \sigma^2/\tau^2\). Program the MCMC algorithm including the simulation smoother.

4. Show that the posterior distribution of \(\boldsymbol{\phi} \mid \boldsymbol{\beta}, \sigma^2, \boldsymbol{y}, \boldsymbol{X}\) in the model \(y_t = \boldsymbol{x}_t^{\top} \boldsymbol{\beta} + \mu_t\) where \(\phi(L) \mu_t = \epsilon_t\) and \(\epsilon_t \stackrel{iid}{\sim} N(0, \sigma^2)\) is \(N(\boldsymbol{\phi}_n, \boldsymbol{\Phi}_n)\mathbb{1}(\boldsymbol{\phi} \in S_{\boldsymbol{\phi}})\), where \(\boldsymbol{\Phi}_n = (\boldsymbol{\Phi}_0^{-1} + \sigma^{-2} \boldsymbol{U}^{\top} \boldsymbol{U})\), \(\boldsymbol{\phi}_n = \boldsymbol{\Phi}_n (\boldsymbol{\Phi}_0^{-1} \boldsymbol{\phi}_0 + \sigma^{-2} \boldsymbol{U}^{\top} \boldsymbol{\mu})\), and \(S_{\boldsymbol{\phi}}\) is the stationary region of \(\boldsymbol{\phi}\).

5. Show that in the \(AR(2)\) stationary process, \(y_t = \mu + \phi_1 y_{t-1} + \phi_2 y_{t-2} + \epsilon_t\), where \(\epsilon_t \sim N(0, \sigma^2)\), \(\mathbb{E}[y_t] = \frac{\mu}{1 - \phi_1 - \phi_2}\), and \(\text{Var}[y_t] = \frac{\sigma^2(1 - \phi_2)}{1 - \phi_2 - \phi_1^2 - \phi_1^2 \phi_2 - \phi_2^2 + \phi_2^3}\).

6. Program a Hamiltonian Monte Carlo taking into account the stationary restrictions on \(\phi_1\) and \(\phi_2\), and \(\epsilon_0\) such that the acceptance rate is near 65%.

7. Stochastic volatility model

   • Program a sequential importance sampling (SIS) from scratch in the vanilla stochastic volatility model setting \(\mu = -10\), \(\phi = 0.95\), \(\sigma = 0.3\), and \(T = 250\). Check what happens with its performance.
   • Modify the sequential Monte Carlo (SMC) to perform multinomial resampling when the effective sample size is lower than 50% the initial number of particles.

8. Estimate the vanilla stochastic volatility model using the dataset 17ExcRate.csv, provided by Ramı́rez-Hassan and Frazier (2024), which contains the exchange rate log daily returns for USD/EUR, USD/GBP, and GBP/EUR from one year before and after the WHO declared the COVID-19 pandemic on 11 March 2020.

9. Simulate the VAR(1) process: \[ \begin{bmatrix} y_{1t}\\ y_{2t}\\ y_{3t}\\ \end{bmatrix} = \begin{bmatrix} 2.8\\ 2.2\\ 1.3\\ \end{bmatrix} + \begin{bmatrix} 0.5 & 0 & 0\\ 0.1 & 0.1 & 0.3\\ 0 & 0.2 & 0.3\\ \end{bmatrix} \begin{bmatrix} y_{1t-1}\\ y_{2t-1}\\ y_{3t-1}\\ \end{bmatrix} + \begin{bmatrix} \mu_{1t}\\ \mu_{2t}\\ \mu_{3t}\\ \end{bmatrix}, \] where \(\boldsymbol{\Sigma} = \begin{bmatrix} 2.25 & 0 & 0\\ 0 & 1 & 0.5\\ 0 & 0.5 & 0.74\\ \end{bmatrix}\).

   • Use vague independent priors setting \(\boldsymbol{\beta}_0 = \boldsymbol{0}\), \(\boldsymbol{B}_0 = 100\boldsymbol{I}\), \(\boldsymbol{V}_0 = 5\boldsymbol{I}\), \(\alpha_0 = 3\), and estimate a VAR(1) model using the rsurGibbs function from the package bayesm. Then, program from scratch

References

Ramı́rez-Hassan, Andrés, and David T. Frazier. 2024. “Testing Model Specification in Approximate Bayesian Computation Using Asymptotic Properties.” Journal of Computational and Graphical Statistics 33 (3): 1–14.