Time-dependent shrinkage of time-varying parameter regression models

Abstract

This article studies the time-varying parameter (TVP) regression model in which the regression coefficients are random walk latent states with time-dependent conditional variances. This TVP model is flexible to accommodate a wide variety of time variation patterns but requires effective shrinkage on the state variances to avoid over-fitting. A Bayesian shrinkage prior is proposed based on reparameterization that translates the variance shrinkage problem into a variable shrinkage one in a conditionally linear regression with fixed coefficients. The proposed prior allows strong shrinkage for the state variances while maintaining the flexibility to accommodate local signals. A Bayesian estimation method is developed that employs the ancilarity-sufficiency interweaving strategy to boost sampling efficiency. Simulation study and an empirical application to forecast inflation rate illustrate the benefits of the proposed approach.

KEYWORDS:

• ASIS
• Bayesian shrinkage
• horseshoe
• MCMC
• TVP

JEL Classification:

• C01
• C11
• C22
• E37

Acknowledgments

I would like to thank Professor Esfandiar Maasoumi (the editor), an AE and two referees for many invaluable comments that have greatly improved the article. All remaining errors are my own. The views in this article are solely the author’s responsibility and are not related to the company the author works in. The author reports that there are no competing interests to declare.

Notes

1 Other recent examples of shrinkage TVP models with homoskedastic latent states include Cadonna et al. (2020), Chan et al. (2020) etc.

2 Another strand of the literature allows heteroskedastic latent states by applying time-dependent spike-and-slab mixture priors for state variances (e.g. Giordani and Kohn (2008), Chan et al. (2012), Hauzenberger (2021), Rockova and McAlinn (2021)) but faces the computational hurdle due to the combinatorial complexity of sampling the mixture indicators of the spike-and-slab priors.

3 See Hauzenberger et al. (2020) for similar strategies for versions of TVP models where latent states follow independent Gaussian distributions rather than random walks.

4 To see this, let ${\beta }_t^{*}$ = β_t - β₀. The TVP model can be rewritten as y_t = $x_t^{\prime }{\beta }_0$ + $x_t^{\prime }{\beta }_t^{*}$ + ϵ_t, ${\beta }_t^{*}$ = ${\beta }_{t-1}^{*}$ + η_t and ${\beta }_0^{*}$ = 0.

5 Alternative shrinkage priors for linear regressions include the spike-and-slab one (George and McCulloch (1993), Ishwaran and Rao (2005)) and the normal-gamma one (Griffin and Brown (2010)) etc. A comprehensive comparison of the various shrinkage priors in the current TVP context is left for future research.

6 The density of an inverted beta distribution IB(a, b) is $p(x)=\frac{x^{a-1}{(1+x)}^{-a-b}}{B(a,b)}I\{x>0\}$ where $B(·,·)$ is the beta function and a and b are positive real numbers. If $x\sim IB(0.5,0.5)$, then $\sqrt{x}\sim C^{+}(0,1)$ and vice versa, where $C^{+}(0,1)$ is a standard half-Cauchy distribution with the density $p(z)=\frac{2}{\pi (1+z^2)}I\{z>0\}$.

7 If $\pm \sqrt{x}\sim N(0,a)$, then $x\sim G(0.5,2a)$ and vice versa, where the gamma distribution $G(\alpha ,\beta )$ has the density $\frac{1}{\Gamma (\alpha ){\beta }^{\alpha }}{x}^{\alpha -1}\hspace{0.17em}\exp \hspace{0.17em}(-\frac{x}{\beta })$.

8 See Johndrow et al. (2020) and Hauzenberger et al. (2020) for methods that use approximations to the exact algorithm of Bhattacharya et al. (2016).

9 Alternative approaches to simulate the latent states from a linear Gaussian state space system include Fruhwirth-Schnatter (1994), Rue (2001) and McCausland et al. (2011) etc.

10 In experiments, another two data generating processes are studied where the ratio of ${\sigma }^2$ to the variance of the dependent variable is 0.5 and 0.8 respectively. The estimation results are qualitatively similar.

11 Generating 1,000 posterior draws from the GHS and DGHS prior specifications takes about 20 and 26 seconds respectively on a standard desktop computer with a 3.0 GHz Intel Core i5 CPU, running in MATLAB R2020b.

12 The ESS is computed by the initial monotone sequence method of Geyer (1992) and is normalized by dividing by the number of posterior draws.

13 The data source is the FRED database of the U.S. federal reserve bank of St. Louis. The series names are CPILFESL, TB3MS and UNRATE for consumer price index, 3-month treasury bill rate and unemployment rate respectively. Quarterly average is computed as the average monthly values within each quarter.

14 Sampling from the GIG distribution is by adapting the Matlab function gigrnd written by Enes Makalic and Daniel Schimidt that implements an algorithm from Devroye (2014).

15 A Gamma distribution $G(0.5,2)$ is equivalent to a ${\chi }^2(1)$ distribution.

16 Sampling from the polya-gamma distribution is by the Matlab function pgdraw written by Enes Makalic and Daniel Schimidt that implements an algorithm from Windle (2013).