Getting started
Following example will estimate a simple Markov switching model with regime dependent intercept, exogenous variable and variance. The model is defined as follows:
\[\begin{align*} y_t &= \mu_s + \beta_s x_t + \epsilon_t, & \epsilon &\sim \mathbb{N}(0,\mathcal{\Sigma}_s) \\ \end{align*}\]
\[\begin{equation*} P(S_t = i | S_{t-1} = j) = \begin{bmatrix} p_1 & 1 - p_2\\ 1 - p_1 & p_2 \end{bmatrix} \end{equation*}\]
using MarSwitching
using Random
using Statistics
k = 2 # number of regimes
T = 400 # number of generated observations
μ = [1.0, -0.5] # regime-switching intercepts
β = [-1.5, 0.0] # regime-switching coefficient for β
σ = [1.1, 0.8] # regime-switching standard deviation
P = [0.9 0.05 # transition matrix (left-stochastic)
0.1 0.95]
Random.seed!(123)
# generate artificial data with given parameters
y, s_t, X = generate_msm(μ, σ, P, T, β = β)
# estimate the model
model = MSModel(y, k, intercept = "switching", exog_switching_vars = X[:,2])
# we may simulated data also from estimated model
# e.g. for calculating VaR:
quantile(generate_msm(model, 1000)[1], 0.05)
# or more interestingly, output summary table
summary_msm(model)Markov Switching Model with 2 regimes
=================================================================
# of observations: 400 AIC: 1115.629
# of estimated parameters: 8 BIC: 1147.561
Error distribution: Gaussian Instant. adj. R^2: 0.5955
Loglikelihood: -549.8 Step-ahead adj. R^2: 0.4432
-----------------------------------------------------------------
------------------------------
Summary of regime 1:
------------------------------
Coefficient | Estimate | Std. Error | z value | Pr(>|z|)
-------------------------------------------------------------------
β_0 | -0.463 | 0.047 | -9.828 | < 1e-3
β_1 | 0.048 | 0.047 | 1.011 | 0.312
σ | 0.755 | 0.019 | 38.897 | < 1e-3
-------------------------------------------------------------------
Expected regime duration: 27.73 periods
-------------------------------------------------------------------
------------------------------
Summary of regime 2:
------------------------------
Coefficient | Estimate | Std. Error | z value | Pr(>|z|)
-------------------------------------------------------------------
β_0 | 0.975 | 0.097 | 10.049 | < 1e-3
β_1 | -1.552 | 0.09 | -17.326 | < 1e-3
σ | 1.044 | 0.034 | 30.835 | < 1e-3
-------------------------------------------------------------------
Expected regime duration: 13.94 periods
-------------------------------------------------------------------
left-stochastic transition matrix:
| regime 1 | regime 2
---------------------------------------
regime 1 | 96.394% | 7.173% |
regime 2 | 3.606% | 92.827% |The estimated model has a following form:
\[y_t = \begin{cases} 0.82 - 1.48 \times x_t + \epsilon_{1,t} ,& \epsilon_1 &\sim \mathbb{N}(0,1.12), & \text{for } S_t = 1\\ -0.51 - 0.003 \times x_t + \epsilon_{2,t} ,& \epsilon_2 &\sim \mathbb{N}(0,0.84), & \text{for } S_t = 2 \end{cases}\]
\[\begin{equation*} P(S_t = i | S_{t-1} = j) = \begin{bmatrix} 91.18\% & 3.5\% \\ 8.82\% & 96.5\% \end{bmatrix} \end{equation*}\]
The package also provides functions for filtered transition probabilities $P(S_t = i | \Psi_t)$, as well as smoothed ones $P(S_t = i | \Psi_T)$. Essentially, the difference is that in order to calculate the smoothed probabilities the whole sample is used.
using Plots
plot(filtered_probs(model),
label = ["Regime 1" "Regime 2"],
title = "Regime probabilities",
linewidth = 2)
using Plots
plot(smoothed_probs(model),
label = ["Regime 1" "Regime 2"],
title = "Smoothed regime probabilities",
linewidth = 2)