Silvia Miranda-Agrippino and Giovanni Ricco
Bayesian vector autoregressions (BVARs) are standard multivariate autoregressive models routinely used in empirical macroeconomics and finance for structural analysis, forecasting, and scenario analysis in an ever-growing number of applications.
A preeminent field of application of BVARs is forecasting. BVARs with informative priors have often proved to be superior tools compared to standard frequentist/flat-prior VARs. In fact, VARs are highly parametrized autoregressive models, whose number of parameters grows with the square of the number of variables times the number of lags included. Prior information, in the form of prior distributions on the model parameters, helps in forming sharper posterior distributions of parameters, conditional on an observed sample. Hence, BVARs can be effective in reducing parameters uncertainty and improving forecast accuracy compared to standard frequentist/flat-prior VARs.
This feature in particular has favored the use of Bayesian techniques to address “big data” problems, in what is arguably one of the most active frontiers in the BVAR literature. Large-information BVARs have in fact proven to be valuable tools to handle empirical analysis in data-rich environments.
BVARs are also routinely employed to produce conditional forecasts and scenario analysis. Of particular interest for policy institutions, these applications permit evaluating “counterfactual” time evolution of the variables of interests conditional on a pre-determined path for some other variables, such as the path of interest rates over a certain horizon.
The “structural interpretation” of estimated VARs as the data generating process of the observed data requires the adoption of strict “identifying restrictions.” From a Bayesian perspective, such restrictions can be seen as dogmatic prior beliefs about some regions of the parameter space that determine the contemporaneous interactions among variables and for which the data are uninformative. More generally, Bayesian techniques offer a framework for structural analysis through priors that incorporate uncertainty about the identifying assumptions themselves.
Silvia Miranda-Agrippino and Giovanni Ricco
Vector autoregressions (VARs) are linear multivariate time-series models able to capture the joint dynamics of multiple time series. Bayesian inference treats the VAR parameters as random variables, and it provides a framework to estimate “posterior” probability distribution of the location of the model parameters by combining information provided by a sample of observed data and prior information derived from a variety of sources, such as other macro or micro datasets, theoretical models, other macroeconomic phenomena, or introspection.
In empirical work in economics and finance, informative prior probability distributions are often adopted. These are intended to summarize stylized representations of the data generating process. For example, “Minnesota” priors, one of the most commonly adopted macroeconomic priors for the VAR coefficients, express the belief that an independent random-walk model for each variable in the system is a reasonable “center” for the beliefs about their time-series behavior. Other commonly adopted priors, the “single-unit-root” and the “sum-of-coefficients” priors are used to enforce beliefs about relations among the VAR coefficients, such as for example the existence of co-integrating relationships among variables, or of independent unit-roots.
Priors for macroeconomic variables are often adopted as “conjugate prior distributions”—that is, distributions that yields a posterior distribution in the same family as the prior p.d.f.—in the form of Normal-Inverse-Wishart distributions that are conjugate prior for the likelihood of a VAR with normally distributed disturbances. Conjugate priors allow direct sampling from the posterior distribution and fast estimation. When this is not possible, numerical techniques such as Gibbs and Metropolis-Hastings sampling algorithms are adopted.
Bayesian techniques allow for the estimation of an ever-expanding class of sophisticated autoregressive models that includes conventional fixed-parameters VAR models; Large VARs incorporating hundreds of variables; Panel VARs, that permit analyzing the joint dynamics of multiple time series of heterogeneous and interacting units. And VAR models that relax the assumption of fixed coefficients, such as time-varying parameters, threshold, and Markov-switching VARs.
The cointegrated VAR approach combines differences of variables with cointegration among them and by doing so allows the user to study both long-run and short-run effects in the same model. The CVAR describes an economic system where variables have been pushed away from long-run equilibria by exogenous shocks (the pushing forces) and where short-run adjustments forces pull them back toward long-run equilibria (the pulling forces). In this model framework, basic assumptions underlying a theory model can be translated into testable hypotheses on the order of integration and cointegration of key variables and their relationships. The set of hypotheses describes the empirical regularities we would expect to see in the data if the long-run properties of a theory model are empirically relevant.
Chao Gu, Han Han, and Randall Wright
The effects of news (i.e., information innovations) are studied in dynamic general equilibrium models where liquidity matters. As a leading example, news can be announcements about monetary policy directions. In three standard theoretical environments—an overlapping generations model of fiat currency, a new monetarist model accommodating multiple payment methods, and a model of unsecured credit—transition paths are constructed between an announcement and the date at which events are realized. Although the economics is different, in each case, news about monetary policy can induce volatility in financial and other markets, with transitions displaying booms, crashes, and cycles in prices, quantities, and welfare. This is not the same as volatility based on self-fulfilling prophecies (e.g., cyclic or sunspot equilibria) studied elsewhere. Instead, the focus is on the unique equilibrium that is stationary when parameters are constant but still delivers complicated dynamics in simple environments due to information and liquidity effects. This is true even for classically-neutral policy changes. The induced volatility can be bad or good for welfare, but using policy to exploit this in practice seems difficult because outcomes are very sensitive to timing and parameters. The approach can be extended to include news of real factors, as seen in examples.
Knut Are Aastveit, James Mitchell, Francesco Ravazzolo, and Herman K. van Dijk
Increasingly, professional forecasters and academic researchers in economics present model-based and subjective or judgment-based forecasts that are accompanied by some measure of uncertainty. In its most complete form this measure is a probability density function for future values of the variable or variables of interest. At the same time, combinations of forecast densities are being used in order to integrate information coming from multiple sources such as experts, models, and large micro-data sets. Given the increased relevance of forecast density combinations, this article explores their genesis and evolution both inside and outside economics. A fundamental density combination equation is specified, which shows that various frequentist as well as Bayesian approaches give different specific contents to this density. In its simplest case, it is a restricted finite mixture, giving fixed equal weights to the various individual densities. The specification of the fundamental density combination equation has been made more flexible in recent literature. It has evolved from using simple average weights to optimized weights to “richer” procedures that allow for time variation, learning features, and model incompleteness. The recent history and evolution of forecast density combination methods, together with their potential and benefits, are illustrated in the policymaking environment of central banks.
Alfred Duncan and Charles Nolan
In recent decades, macroeconomic researchers have looked to incorporate financial intermediaries explicitly into business-cycle models. These modeling developments have helped us to understand the role of the financial sector in the transmission of policy and external shocks into macroeconomic dynamics. They also have helped us to understand better the consequences of financial instability for the macroeconomy. Large gaps remain in our knowledge of the interactions between the financial sector and macroeconomic outcomes. Specifically, the effects of financial stability and macroprudential policies are not well understood.
Sushant Acharya and Paolo Pesenti
Global policy spillovers can be defined as the effect of policy changes in one country on economic outcomes in other countries. The literature has mainly focused on monetary policy interdependencies and has identified three channels through which policy spillovers can materialize. The first is the expenditure-shifting channel—a monetary expansion in one country depreciates its currency, making its goods cheaper relative to those in other countries and shifting global demand toward domestic tradable goods. The second is the expenditure-changing channel—expansionary monetary policy in one country raises both domestic and foreign expenditure. The third is the financial spillovers channel—expansionary monetary policy in one country eases financial conditions in other economies. The literature generally finds that the net transmission effect is positive but small. However, estimated spillovers vary widely across countries and over time. In the aftermath of the Great Recession, the policy debate has devoted special attention to the possibility that the magnitude and sign of international spillovers might have changed in an environment of low interest rates worldwide, as the expenditure-shifting channel becomes more relevant when the effective lower bound reduces the effectiveness of conventional monetary policies.
Home bias in international macroeconomics refers to the fact that investors around the world tend to allocate majority of their portfolios into domestic assets, despite the potential benefits to be had from international diversification. This phenomenon has been occurring across countries, over time, and across equity or bond portfolios. The bias towards domestic assets tends to be larger in developing countries relative to developed economies, with Europe characterized by the lowest equity home bias, while Central and South America—by the highest equity home bias. In addition, despite the secular decline in the level of equity home bias over time in all countries and regions, home bias still remains a robust feature of the data.
Whether home bias is a puzzle depends on the portfolio allocation that one uses as a theoretical benchmark. For instance, home bias in equity portfolio is a puzzle when assessed through the lens of a simple international capital asset pricing model (CAPM) with homogeneous investors. This model predicts that investors should hold world market portfolios, namely a portfolio with the share of domestic asset equal to the share of those assets in the world market portfolio. For instance, since the share of US equity in the world capitalization in 2016 was 56%, then US investors should allocate 56% of their equity portfolio into local assets, while investing the remaining 44% into foreign equities. Instead, foreign equity comprised just 23% of US equity portfolio in 2016, hence the equity home bias.
Alternative portfolio benchmark comes from the theories that emphasize costs for trading assets in international financial markets. These include transaction and information costs, differential tax treatments, and more broadly, differences in institutional environments. This research, however, has so far been unable to reach a consensus on the explanatory power of such costs.
Yet another theory argues that equity home bias can arise due to the hedging properties of local equity. In particular, local equity can provide insurance from real exchange rate risk and non-tradable income risk (such as labor income risk), and thus a preference towards home equities is not a puzzle, but rather an optimal response to such risks.
These theories, main advances and results in the macroeconomic literature on home bias are discussed in this article. It starts by presenting some empirical facts on the extent and dynamics of equity home bias in developed and developing countries. It is then shown how home bias can arise as an equilibrium outcome of the hedging demand in the model with real exchange rate and non-tradable labor income risk. Since solving models with portfolio choice is challenging, the recent advances in solving such models are also outlined in this article.
Integrating the portfolio dynamics into models that can generate realistic asset price and exchange rate dynamics remains a fruitful avenue for future research. A discussion of additional open questions in this research agenda and suggestions for further readings are also provided.
Chao Gu, Han Han, and Randall Wright
This article provides an introduction to New Monetarist Economics. This branch of macro and monetary theory emphasizes imperfect commitment, information problems, and sometimes spatial (endogenously) separation as key frictions in the economy to derive endogenously institutions like monetary exchange or financial intermediation. We present three generations of models in development of New Monetarism. The first model studies an environment in which agents meet bilaterally and lack commitment, which allows money to be valued endogenously as means of payment. In this setup both goods and money are indivisible to keep things tractable. Second-generation models relax the assumption of indivisible goods and use bargaining theory (or related mechanisms) to endogenize prices. Variations of these models are applied to financial asset markets and intermediation. Assets and goods are both divisible in third-generation models, which makes them better suited to policy analysis and empirical work. This framework can also be used to help understand financial markets and liquidity.
Many nonlinear time series models have been around for a long time and have originated outside of time series econometrics. The stochastic models popular univariate, dynamic single-equation, and vector autoregressive are presented and their properties considered. Deterministic nonlinear models are not reviewed. The use of nonlinear vector autoregressive models in macroeconometrics seems to be increasing, and because this may be viewed as a rather recent development, they receive somewhat more attention than their univariate counterparts. Vector threshold autoregressive, smooth transition autoregressive, Markov-switching, and random coefficient autoregressive models are covered along with nonlinear generalizations of vector autoregressive models with cointegrated variables. Two nonlinear panel models, although they cannot be argued to be typically macroeconometric models, have, however, been frequently applied to macroeconomic data as well. The use of all these models in macroeconomics is highlighted with applications in which model selection, an often difficult issue in nonlinear models, has received due attention. Given the large amount of nonlinear time series models, no unique best method of choosing between them seems to be available.