Theory Coherent Shrinkage of Time Varying Parameters in VARs: Abstract and Introduction

Table of Links

Abstract and Introduction

Theory coherent TVP-VAR

Forecasting with the TC-TVP-VAR

Response analysis at the ZLB with the TC-TVP-VAR

Conclusion and References

A Appendix

Abstract

Time-varying parameters Vector Autoregressive (TVP-VAR) models are frequently used in economics to capture evolving relationships among the macroeconomic variables. However, TVP-VARs have the tendency of overfitting the data, resulting in inaccurate forecasts and imprecise estimates of typical objects of interests such as the impulse response functions. This paper introduces a Theory Coherent Time-Varying Parameters Vector Autoregressive Model (TC-TVP-VAR), which leverages on an arbitrary theoretical framework derived by an underlying economic theory to form a prior for the time varying parameters. This “theory coherent” shrinkage prior significantly improves inference precision and forecast accuracy over the standard TVP-VAR. Furthermore, the TC-TVP-VAR can be used to perform indirect posterior inference on the deep parameters of the underlying economic theory. The paper reveals that using the classical 3-equation New Keynesian block to form a prior for the TVPVAR substantially enhances forecast accuracy of output growth and of the inflation rate in a standard model of monetary policy. Additionally, the paper shows that the TC-TVP-VAR can be used to address the inferential challenges during the Zero Lower Bound (ZLB) period.

J.E.L Classification Code: C32, C53

Keywords: Time Varying Parameters VARs, Bayesian Econometrics, DSGE-VARs

1 Introduction

Over the past four decades vector autoregressive models have become the leading tool for description, forecasting, structural inference and policy analysis of macroeconomic data (Sims 1980; Stock et al. 2001). A natural progression in the literature was to allow for time-varying parameters to capture changes in the complex dynamic interrelationship among the variables in the system (Cogley et al. 2002; Primiceri 2005; Cogley et al. 2005). On one side, this class of models known as Time Varying Parameters VARs (TVP-VARs) can be flexible enough to fit many different forms of structural instabilities and evolving nonlinear relationships among the macroeconomic variables. On the other side, due to the growing number of parameters, they can easily become too flexible with adverse consequences on the precision of inference and on the reliability of the forecasts.

In this paper I propose to use economic theory to sharpen inference in TVP-VARs. The approach consists in exploiting prior information coming from a more tightly parameterized model derived by an underlying economic theory about the macroeconomic variables in the system. This prior information provides a set of economically grounded (fuzzy) restrictions which are incorporated into a shrinkage prior for the TVP-VAR. The resulting model, that I label Theory Coherent TVP-VAR (TC-TVP-VAR), is a flexible statistical model for the data that leverages on economic theory to enhance inference about the time varying parameters. In the TC-TVP-VAR, two crucial hyper-parameters govern the behavior of the time varying coefficients: the first one determines their intrinsic time variation, while the other one determines their degree of theory coherence stemming from the amount of shrinkage towards the crossequation restrictions implied by the economic theory. Both hyper-parameters can be optimally tuned by maximizing the marginal data density of the TC-TVP-VAR which is available in closed form. Moreover, the TC-TVP-VAR can also be used as a tool for learning about the deep parameters from the underlying economic theory. As a matter of fact, the deep parameters from the economic theory are another set of hyper-parameters of the model that are indirectly estimated by projecting the TVP-VAR estimates onto the restrictions implied by the model from the economic theory. Thus, in this approach, learning about the deep parameters from the economic theory happens indirectly through learning about the TVP-VAR parameters.

In the paper I show that incorporating the restrictions implied by the economic theory into a prior for the coefficients of TVP-VARs can be beneficial to improve forecast accuracy and to obtain more precise estimates of typical objects of interest such as the impulse response functions. In particular, I find that encoding the restrictions from a conventional three equation New Keynesian model into a prior for the parameters of a trivariate TVP-VAR for output growth, inflation rate and the interest rate improves both point and density forecast accuracy of both output growth and the inflation rate at all the horizons considered (one quarter ahead, two quarters ahead and one year ahead). Then, I exploit the TC-TVP-VAR to investigate whether the US economy’s performance was affected by a binding zero lower bound (ZLB) constraint as predicted by a standard New Keynesian model. Indeed, according to a standard New-Keynesian model, the economy is expected to exhibit a different response to demand and supply shocks when the ZLB constraint is in effect. However, and more importantly, the short length of the ZLB period in the US makes the standard TVP-VAR unfit to detect the change in the responses predicted by the NK model (Benati et al. 2023). In other words, whether or not there was a change in the response of the economy during the ZLB as predicted by a standard NK model, cannot be directly inferred by using a standard TVP-VAR. Based on a simulation study, I show that the TC-TVP-VAR can in principle be used to solve this inferential problem. In particular, I exploit the time varying restriction functions implied by a medium scale NK model that accounts for forward guidance and the ZLB period to parametrize the shrinkage prior in the TC-TVP-VAR. I show that this approach allows to estimate more precisely the response of the economy to macroeconomic shocks inside and outside the ZLB period, solving the inferential problems of the standard TVP-VAR. Finally, estimating the model on US data, I find that there are convincing evidences supporting a change in the response of the economy during the ZLB period similar to the one predicted by the NK model.

Related Literature An increasing number of studies has recently focused on the issue of mitigating complexity and over-parametrization in TVP-VARs. One strand of literature has focused on identifying fixed versus time varying coefficients, by concentrating on the variance selection problem in the generic state equations of each of the TVP-VARs’ coefficients. (Fr¨uhwirth-Schnatter et al. 2010; Belmonte et al. 2014; Kalli et al. 2014; Bitto et al. 2019). This literature has produced shrinkage priors for variances aimed at “automatically” reducing time-varying coefficients to static ones if the model is overfitting.[1] While treating the coefficients of the model as independent stochastic processes and just focusing on the problem of tuning the optimal amount of time variation of the single coefficients, this strand of literature typically entirely neglects co-movement and correlation among the coefficients. However, in macroeconomic applications, the high degree of co-movement in the parameters is an empirical regularity. This fact was already found and stressed by Cogley et al. (2005) in one of the papers that introduced TVP-VARs in the field. In the same paper, the authors envisaged that the reduced-form parameters should move in a highly structured way because of the cross-equation restrictions suggesting that “a formal treatment of cross-equation restrictions with parameter drift is a priority for future work” (p. 274). More in line with these considerations, a smaller number of studies (Wind et al. 2014; Stevanovic 2016; Chan et al. 2020) proposed to use a factor structure to model the time variation of the parameters. Despite being compatible with the idea of the coefficients varying in a highly structured way because of the cross-equation restrictions associated with macroeconomic equations, this approach is purely statistical and abstracts from any macroeconomic theory disciplining the behavior of the coefficients.

This paper fills this gap in the literature by showing how to exploit prior information grounded on the basis of an economic theory to state a priori a plausible correlation structure among the time varying parameters of the model. In this aspect, the contribution conceptually borrows from ideas from the seminal work of Ingram et al. (1994) and operationally from the insights in Del Negro et al. (2004) which show how to exploit the non-linear cross equation restrictions implied by a DSGE to form a prior for the parameters of a constant parameters VAR model. Extending the framework of Del Negro et al. (2004) to TVP-VARs is important at least for two reasons. First, because in macroeconomic applications the assumption of constant coefficients is often restrictive. Indeed, instabilities in the autoregressive coefficients of VARs used to model the dynamics of key macroeconomic indicators such as output and inflation have been widely documented in the literature (Cogley et al. 2002; Primiceri 2005; D’Agostino et al. 2013). Especially in this setting, as the model becomes more flexible, additional shrinkage can be particularly beneficial to reduce overfitting. Second, because economic theories themselves might imply time varying moments for the data that result in time-varying restriction functions for the coefficients. For example, macroeconomic theories assuming rational expectations extended so as to allow some parameters to vary according to Markov process with given transition probabilities, lead to state space representation with time varying coefficients (Farmer et al. 2009). At the same time, solutions for linear stochastic rational expectations models in the face of a finite sequence of anticipated structural changes lead to state space representation with time varying coefficients (Cagliarini et al. 2013).[2] Likewise, outside the rational expectation framework, macroeconomic theories that assume learning can lead to state space representation with time varying coefficients (Milani 2007). In all those cases, the proposed TC-TVP-VAR can be used both to incorporate the implied time varying restrictions into a prior for the time varying coefficients of the VAR and to estimate the deep parameters of the underlying economic model. In this sense, the paper is also related to the strand of literature that exploits an auxiliary flexible statistical model for the data to make indirect inference on the deep parameters of a structural model from the economic theory (see for example Gallant et al. (2009) Fessler et al. (2019)).[3] Finally, the paper is more broadly related to the econometric literature showing that moment conditions from economic theory can successfully be exploited for forecasting macroeconomic and financial variables (Giacomini et al. (2014) and Carriero et al. (2021) most notably).

Outline The paper is organized as follows. In section 2 I introduce the TC-TVP-VAR, presenting an analytical derivation of the proposed theory coherent prior and a simulation from the prior to showcase its main properties. In addition, I discuss the fit-complexity trade-off linked to the calibration of the hyper-parameters determining the intrinsic persistence of the time varying-coefficients and the degree of shrinkage towards the restrictions from the theory. I conclude section 2 by presenting an MCMC sampler used for estimation of the parameters of the TC-TVP-VAR. Afterwards, in section 3 I exploit the well known 3-equations New Keynesian (NK) block to form a prior for the parameters of a trivariate TVP-VAR for GDP growth, inflation and the interest rate for the US economy. I compare the forecasts from a TC-TVP-VAR that encodes the restrictions from the NK model as a prior for the time varying coefficients to the forecasts from a standard TVP-VAR, showing that the former outperforms the latter both in terms of point and density forecast accuracy. Then, in section 4 I conduct a simulation study and show that a standard TVP-VAR struggles to detect the change in the response of the economy to macroeconomic shocks during the ZLB period as predicted by a standard NK model. I show that instead the TC-TVP-VAR can in principle be used to solve this inferential problem. Finally, I estimate a TC-TVP-VAR on US data to investigate whether the US economy was indeed affected by a binding ZLB. Section 5 concludes.

[1.] Similarly, from a frequentist perspective, Coulombe (2021) showed that time varying parameters can be framed as ridge regressions problems and used cross validation to tune the optimal amount of time variation in each of the state equations of the coefficients of the TVP-VAR.

[2.] Another notable case within the rational expectations framework are models log-linearized around time varying trend for inflation (Cogley et al. 2008; Ascari et al. 2014; Ascari et al. 2023).

[3.] In the TC-TVP-VAR the structural parameters from the underlying economic theory are estimated by implicitly minimizing the weighted discrepancy between the unrestricted TVP-VAR estimates and the restriction functions. This approach can be thought as a Bayesian version of Smith Jr. (1993) as in Del Negro et al. (2004).