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Lubik, Thomas A.; Christian Matthes,. "Time-Varying Parameter Vector Autoregressions: Specification, Estimation, and an Application." Economic Quarterly. Federal Reserve Bank of Richmond. 2015. HighBeam Research. 26 Apr. 2018 <https://www.highbeam.com>.
Lubik, Thomas A.; Christian Matthes,. "Time-Varying Parameter Vector Autoregressions: Specification, Estimation, and an Application." Economic Quarterly. 2015. HighBeam Research. (April 26, 2018). https://www.highbeam.com/doc/1G1-497731325.html
Lubik, Thomas A.; Christian Matthes,. "Time-Varying Parameter Vector Autoregressions: Specification, Estimation, and an Application." Economic Quarterly. Federal Reserve Bank of Richmond. 2015. Retrieved April 26, 2018 from HighBeam Research: https://www.highbeam.com/doc/1G1-497731325.html
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Time-varying parameter vector autoregressions (TVP-VARs) have become an increasingly popular tool for analyzing the behavior of macroeconomic time series. TVP-VARs differ from more standard fixed-coefficient VARs in that they allow for coefficients in an otherwise linear VAR model to vary over time following a specified law of motion. In addition, TVP-VARs often include stochastic volatility (SV), which allows for time variation in the variances of the error processes that affect the VAR.
The attractiveness of TVP-VARs is based on the recognition that many, if not most, macroeconomic time series exhibit some form of nonlinearity. For instance, the unemployment rate tends to rise much faster at the start of a recession than it declines at the onset of a recovery. Stock market indices exhibit occasional episodes where volatility, as measured by the variance of stock price movements, rises considerably. As a third example, many aggregate series show a distinct change in behavior in terms of their persistence and their volatility around the early 1980s when the Great Inflation of the 1970s turned into the Great Moderation, behavior that is akin to a structural shift in certain moments of interest. All these examples of nonlinearity in macroeconomic time series have potentially distinct underlying structural causes. But they can all potentially be captured by means of the flexible framework that is a TVP-VAR with SV.
A VAR is a simple time series model that explains the joint evolution of economic variables through their own lags. A TVP-VAR preserves this structure but in addition models the coefficients as stochastic processes. In the most common application, the maintained assumption is that the coefficients follow random walks, specifically the intercepts, the lag coefficients as well as the variance and covariances of the error terms in the regression. Conditional on the parameters, a TVP-VAR is still a linear VAR, but the overall model is highly nonlinear. While the assumption of random walk behavior may seem restrictive, it provides for a flexible functional form to capture various forms of nonlinearity.
The main challenge in applying TVP-VAR models is how to conduct inference. In this article, we therefore discuss the Bayesian approach to estimating a TVP-VAR with SV. (1) Bayesian inference in this class of models relies on the Gibbs sampler, which is designed to easily compute multivariate densities. The key insight is to break up a computationally intractable problem into sequences of feasible steps. We will discuss these steps in detail and show how they can be applied to TVP-VARs.
The article is structured as follows. We begin with a discussion of the specification of TVP-VARs and how they are developed from fixed-coefficient VARs. We show how to introduce stochastic volatility in the covariance matrix of the errors and present an argument for why time variation in the lag coefficients needs to be modeled jointly with stochastic volatility. The main body of the article presents the Gibbs sampling approach to conducting inference in Bayesian TVP-VARs, which we preamble with a short discussion of the thinking behind Bayesian methods. Finally, we illustrate the method by means of a simple application to data on inflation, unemployment, and the nominal interest rate for the United States.
1. SPECIFICATION
VARs are arguably the most important empirical tool for applied macroeconomists. They were introduced to the economics literature by Sims (1980) as a response to the then-prevailing large-scale macroeconometric modeling approach. What Sims memorably criticized were the incredible identification assumptions imposed in these models that stemmed largely from a lack of sound theoretical economic underpinnings and that hampered structural interpretation of their findings. In contrast, VARs are deceptively simple in that they are designed to simply capture the joint dynamics of economic time series without imposing ad-hoc identification restrictions.
More specifically, a VAR describes the evolution of a vector of n economic variables [y.sub.t] at time t as a linear function of its own lags up to order L and a vector e of unforecastable disturbances:
[y.sub.t] = [c.sub.t] + [L.summation over (j=1)] [A.sub.j][y.sub.t-j] + [e.sub.t]. (1)
It is convenient to assume that the error term et is Gaussian with mean 0 and covariance matrix [[OMEGA].sub.e]. [c.sub.t] is a vector of deterministic components, possibly including time trends, while the [A.sub.j] are conformable matrices that capture lag dynamics.
VAR models along the lines of (1) have proven to be remarkably popular for studying, for instance, the effects and implementation of monetary policy (see Christiano, Eichenbaum, and Evans 1999, for a comprehensive survey). However, VARs of this kind can only describe economic behavior that is approximately linear and does not exhibit substantial variation over time. The linear VAR in (1) contains a built-in notion of time invariance: conditional forecasts as of time t, such as [E.sub.t][y.sub.t+1], only depend on the last L values of the vector of observables but are otherwise independent of time. More strongly, the conditional one-step-ahead variance is fully independent of time: [E.sub.t][([y.sub.t+i] - [E.sub.t][y.sub.t+1])([y.sub.t+1] - [E.sub.t][y.sub.t+1])] = [[OMEGA].sub.e]
Yet, in contrast, a long line of research documents that conditional higher moments can vary over time, starting with the seminal ARCH model of Engle (1982). Moreover, research in macroeconomics, such as Lubik and Schorfheide (2004), has shown that monetary policy rules can change over time and can therefore introduce nonlinearities, such as breaks or shifts, into aggregate economic time series. (2) The first observation has motivated Uhlig (1997) to introduce time variation in [[OMEGA].sub.e]. The second observation stimulated the work by Cogley and Sargent (2002) to introduce time variation in [A.sub.j] and c in addition to stochastic volatility.
We will now describe how to model time variation in each of these sets of parameters separately. In the next step, we will discuss why researchers should model changes in both sets of parameters jointly. We then present the Gibbs sampling algorithm that is used for Bayesian inference in this class of models and which allows for easy combination of the approaches because of its modular nature.
A VAR with Random-Walk Time Variation in the Coefficients
Suppose a researcher wants to capture time variation in the data by using a parsimonious yet flexible model as in the VAR (1). The key question is how to model this time variation in the coefficients [A.sub.j] and c. One possibility is to impose a priori break points at specific dates. Alternatively, break points can be chosen endogenously as part of the estimation algorithm. Threshold VARs or VARs with Markov switching in the parameters (e.g., Sims and Zha 2006) are examples of this type of model, which is often useful in environments where the economic modeler may have some a priori information or beliefs about the underlying source of time variation, such as discrete changes in the behavior of the monetary authority. In general, however, a flexible framework with random time variation seems preferable for a wide range of nonlinear behavior in the data. Following Cogley and Sargent (2002), a substantial part of the literature has consequently opted for a flexible specification that can accommodate a large number of patterns of time variation.
The standard model of time variation in the coefficients starts with the VAR (1). In contrast to the fixed-coefficient version, the parameters of the intercept and of the lag coefficient matrix are allowed to vary over time in a prescribed manner. We thus specify the TVP-VAR:
[y.sub.t] = [c.sub.t] + [L.summation over (j=1)][A.sub.j,t][y.sub.t-j] + [e.sub.t]. (2)
It is convenient to collect the values of the lagged variables in a matrix and define [X'.sub.t] [equivalent to] = I [cross product] (1, [y'.sub.t-L] ..., [y'.sub.t-l]), where '[cross product]' denotes the Kronecker product. We also define [[theta].sub.t] to collect the VAR's time-varying coefficients in vectorized form, that is, [[theta].sub.t] = vec([[c.sub.t] [A.sub.1,t] [A.sub.2,t] ... [A.sub.L,t]]'). This allows us to rewrite (2) in the following form:
[y.sub.t] = [X'.sub.t][[theta].sub.t] + [e.sub.t]. (3)
The commonly assumed law of motion for [[theta].sub.t] is a random walk:
[[theta].sub.t] = [[theta].sub.t-1] + [u.sub.t]; (4)
where [u.sub.t]|N(0, Q) and is assumed to be independent of [e.sub.t]. A random-walk specification is parsimonious in that it can capture a large number of patterns without introducing additional parameters that need to be estimated. (3) This assumption is mainly one of convenience for reasons of parsimony and flexibility as (4) is rarely interpreted as the underlying data-generating process for the question at hand, but it can approximate it arbitrarily well (see Canova, Ferroni, and Matthes 2015).
Introducing Stochastic Volatility
A second source of time variation in time series can stem from variation in second or higher moments of the error terms. Stochastic volatility, or, specifically, time variation in variances and covariances, can be introduced into a model in a number of ways. Much of the recent literature on stochastic volatility in macroeconomics has chosen to follow the work of Kim, Shephard, and Chib (1998). It is built on a flexible model for volatility that uses an unobserved components approach. (4)
We start from the observation that we can always decompose a covariance matrix [[OMEGA].sub.e] as follows:
[[OMEGA].sub.e] = [[LAMBDA].sup.-1] [SIGMA][SIGMA]' ([[LAMBDA].sup.-1])'. (5)
A is a lower triangular matrix with ones on the main diagonal, while [SIGMA] is a diagonal matrix. Intuitively, the diagonal matrix [SIGMA][SIGMA]' collects the independent innovation variances, while the triangular matrix [[LAMBDA].sup.-1] collects the loadings of the innovations onto the VAR error term e, and thereby the covariation among the shocks. It has proven to be convenient to parameterize time variation in [[OMEGA].sub.e] directly by making the free elements of [LAMBDA] and [SIGMA] vary over time. While this decomposition is general, once priors on the elements of [SIGMA] and [LAMBDA] are imposed, the ordering of variables in the VAR matters for the estimation of the reduced-form parameters, which stands in contrast to the standard time-invariant VAR model (see Primiceri 2005).
We now define the element of [[LAMBDA].sub.t] in row i and column j as [[lambda].sup.ij.sub.t] and a representative free element j of the time-varying coefficient matrix [[SIGMA].sub.t] as [[sigma].sup.j.sub.t]. It has become the convention in the literature to model the coefficients [[sigma].sup.j.sub.t] as geometric random walks:
log [[sigma]. …
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