Modeling the COVID-19 Infection Rates by Regime Switching Unobserved Components Models

This repository contains all necessary code to replicate the results, figures and tables of Haimerl, Paul, and Tobias Hartl. 2023. "Modeling COVID-19 Infection Rates by Regime-Switching Unobserved Components Models" Econometrics 11, no. 2: 10. https://doi.org/10.3390/econometrics11020010

If you use any parts of this code, please cite this paper.

Contents

1. Replicate_paper.Rmd: R-Notebook that guides you through all the necessary steps to replicate the paper
2. R: Folder containing R scripts
3. src: Folder holding Rcpp functions
4. Output: Model outputs (to be created)

Model specifications

To set up the regime-switching unobserved components (UC) model, let $i_t$ denote the daily reported COVID-19 cases and transform $y_t = log(i_t)$. The measurement equation of the state space form is then given by

$y_t = \mu_t + \gamma_t + c_t + \epsilon_t, \quad \epsilon_t \sim N(0, \sigma^2_{\epsilon})$.

The trend $\mu_t$ is defined as

$\mu_t = \mu_{t-1} + \nu_t + \xi_t, \quad \xi_t \sim i.i.d.N(0, \sigma^2_{\xi})$ with

$\nu_t = S_t \nu_1 + (1 - S_t) \nu_0$.

The seasonal component $\gamma_t$ admits to

$\gamma_t = -\sum^6_{j=1} \gamma_{t-j}$.

The cyclical component $c_t$ is given by

$(1 - L\phi_1 - L^2\phi_2)c_t = \eta_t, \quad \eta_t \sim i.i.d.N(0, \sigma^2_{\eta})$

where $L$ denotes the lag operator and all characteristic roots of the lag polynomial lie outside the unit circle.

The regime indicator $S_t \in { 0, 1 }$ governs $\nu_t$. By imposing $\nu_1 < 0$ we declare $S_t = 1$ as the infection down-turning regime. The regimes follow a first order stationary Markov chain with the time homogeneous transition probabilities

$\Pr(S_t = 0 | S_t = 0) = q, \ \Pr(S_t = 1 | S_t = 1) = p$.

To robustify our findings, we further introduce several extensions to this approach (see section 4).

Among other, we include a specification that substitutes the deterministic seasonal component $\gamma_t$ with a stochastic seasonal unit root process

$\gamma^{UR}t = -\sum^6{j=1} \gamma^{UR}{t-j} + x_t, \quad (1 - L^7) x_t = \omega_t, \quad \omega_t \sim i.i.d.N(0, \sigma^2{\omega})$.

Furthermore, we consider a regime process that incorporates a third state $S_t = 2$, which is characterized by $\nu_t = 0$ during time periods where this state is active.

Lastly, even though not shown in the paper, we investigate specifications that (i) allow for endogeneity between the trend and regime processes, (ii) regime-dependent heteroscedasticity in either $\xi_t$ or $\eta_t$, as well as (iii) a model where a second independent Markov chain $S_t^*$ governs this regime-induced innovation heteroscedasticity.

Turning to the endogenous specification, innovations to the trend and regime processes may exhibit a non-zero correlation $\rho$ as proposed in Kim et al. (2008); Kang (2014). For a previous contribution that employs regime dependent heteroscedasticity, i.e. $\sigma^2_{\xi, S_t^} = S_t^ \sigma^2_{\xi, S_t^=1} + (1 - S_t^) \sigma^2_{\xi, S_t^=0}$ where $S_t^$ may or may not equal $S_t$ (same applying to $\sigma^2_{\eta, S_t^*}$), we refer to Engel and Kim (1999).

For further details, see sections 2 and 4.

The following table provides an overview of all implemented models as well as their respective specifications. A ✅ indicates a non-zero value or the respective extension being implemented. To use any of these specifications, define modelSpec in Chunk 2 of the notebook as one of the abbreviations. Note that the trailing dot must be included. Model specific functions can be found in R/Models/

	$\gamma_t$	$\gamma^{UR}_t$	$\epsilon_t$	$c_t$	$S_t = 2$	$\rho$	2 Regime processes ($S_t \neq S_t^*$)	$\sigma^2_{\xi, S_t^* = 0} \neq \sigma^2_{\xi, S_t^* = 1}$	$\sigma^2_{\eta, S_t^* = 0} \neq \sigma^2_{\eta, S_t^* = 1}$
D.Seas.	✅		✅
D.Seas.C.	✅			✅
D.Seas.C.3St.	✅			✅	✅
D.Seas.2P.TVarSw.	✅		✅				✅	✅
UR.Seas.		✅	✅
UR.Seas.C.		✅		✅
UR.Seas.En.		✅	✅			✅
UR.Seas.TVarSw.		✅	✅					✅
UR.Seas.CVarSw.		✅		✅					✅
UR.Seas.2P.TVarSw.		✅	✅				✅	✅
UR.Seas.2P.CVarSw.		✅		✅			✅		✅

References

Engel, C., & Kim, C.-J. (1999). The Long-Run U.S./U.K. Real Exchange Rate. Journal of Money, Credit and Banking, 31(3), 335–356. https://doi.org/10.2307/2601115

Kang, K. H. (2014). Estimation of state-space models with endogenous Markov regime-switching parameters. The Econometrics Journal, 17(1), 56–82. http://www.jstor.org/stable/43697655

Kim, C.-J., Piger, J., & Startz, R. (2008). Estimation of Markov regime-switching regression models with endogenous switching. Journal of Econometrics, 143(2), 263–273. https://doi:10.1016/j.jeconom.2007.10.002