Reference no: EM133696218

Computational statistical inference

Question 1: The GARCH (1,1) model is a simple and popular tool used to model time series in economet- rics. The structure of the GARCH(1,1) model is as follows

y_t ∼ N (0, σ_t²)

σ_t² = ω + αy²_t-1 + βαy²_t-1

for t = 1, . . . , T. Thus observations (y₁, . . . , y_T) are modelled as being normally distributed around 0 with a variance σ₂ that is dependent on both the observation and the variance at the previous time point. The GARCH(1,1) model has four parameters θ = (ω, α, β, σ₁²).

1: Draw a conditional dependency diagram for this model with T = 4, including (y₁, y2, y3, y4), (σ₁², σ₂², σ₃², σ₄²)

and (ω, α, β).

2: Provide an expression for the log-likelihood l(θ|y), choose a suitable prior density for θ and determine the posterior density p(θ|y).

3: Write code in R or Matlab to simulate artificial data for arbitrary T and θ = (ω, α, β, σ₁²).

4: Use this code to simulate a time series (y₁, . . . , y_T ) of length T = 1000 for θ = (9, 0.1, 0.85, 9).

5: Implement an adaptive Metropolis sampler to sample N = 10000 values of θ from p(θ|y).

6: Using the slice sampler for individual coordinates, implement a Gibbs sampler to generate N = 10000 values of θ from p(θ|y).

7: Assess the convergence of the two samplers.

8: Plot auto-covariance functions for the two samplers and compare the performance of the samplers.

9: Plot marginal densities for each of ω, α, β and σ2 and comment on any differences between densities obtained using the two samplers.

Question 2: In this question, you will develop and assess ABC methods for analysing the data you simulated in Question 1

1: Propose four summary statistics for applying ABC methods to observations (y1, . . . , yT ), assuming a GARCH(1,1) model. Defend your proposal.

2: Propose a suitable discrepancy metric for comparing summary statistics. Defend your proposal.

3: Implement a basic ABC sampler for estimating θ using samples drawn from the prior distribution.

4: Implement an MCMC ABC sampler for the same problem.

5: Implement a sequential Monte Carlo ABC sampler for the same problem.

6: Compare your three samplers in terms of run times.

7: Assess the accuracy of the ABC samplers by comparing them to the MCMC samplers from Question

8: Use plots as part of your answer.

9: Apply regression adjustment and again assess the accuracy relative to ABC without regression ad- justment, and relative to MCMC samplers from Question 1.

10: Of all the samplers developed for the assignment, which do you prefer and why?