Cointegration and Error Correction Mechanisms for Singular Stochastic Vectors

Large-dimensional dynamic factor models and dynamic stochastic general equilibrium models, both widely used in empirical macroeconomics, deal with singular stochastic vectors, i.e., vectors of dimension r which are driven by a q-dimensional white noise, with q < r . The present paper studies cointegration and error correction representations for an I ( 1 ) singular stochastic vector y t . It is easily seen that y t is necessarily cointegrated with cointegrating rank c ≥ r − q . Our contributions are: (i) we generalize Johansen’s proof of the Granger representation theorem to I ( 1 ) singular vectors under the assumption that y t has rational spectral density; (ii) using recent results on singular vectors by Anderson and Deistler, we prove that for generic values of the parameters the autoregressive representation of y t has a finite-degree polynomial. The relationship between the cointegration of the factors and the cointegration of the observable variables in a large-dimensional factor model is also discussed.

Bayesian Vector Autoregressions: Applications

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.

EULER EQUATION ESTIMATION ON MICRO DATA

Consumption Euler equations are important tools in empirical macroeconomics. When estimated on micro data, they are typically linearized, so standard IV or GMM methods can be employed to deal with the measurement error that is endemic to survey data. However, linearization, in turn, may induce serious approximation bias. We numerically solve and simulate six different life-cycle models, and then use the simulated data as the basis for a series of Monte Carlo experiments in which we evaluate the performance of linearized Euler equation estimation. We sample from the simulated data in ways that mimic realistic data structures. The linearized Euler equation leads to biased estimates of the EIS, but that bias is modest when there is a sufficient time dimension to the data, and sufficient variation in interest rates. However, a sufficient time dimension can only realistically be achieved with a synthetic cohort. Estimates from synthetic cohorts of sufficient length, while often exhibiting small mean bias, are quite imprecise. We also show that in all data structures, estimates are less precise in impatient models.

Bayesian Methods for Empirical Macroeconomics with Big Data

Bayesian econometric methods are increasingly popular in empirical macroeconomics. They have been particularly popular among macroeconomists working with Big Data (where the number of variables under study is large relative to the number of observations). This paper, which is based on a keynote address at the Rimini Centre for Economic Analysis' 2016 Money-Macro-Finance Workshop, explains why this is so. It discusses the problems that arise with conventional econometric methods and how Bayesian methods can successfully overcome them either through use of prior shrinkage or through model averaging. The discussion is kept at a relatively non-technical level, providing the main ideas underlying and motivation for the models and methods used. It begins with single-equation models (such as regression) with many explanatory variables, then moves on to multiple equation models (such as Vector Autoregressive, VAR, models) before tacking the challenge caused by parameter change (e.g. changes in VAR coefficients or volatility). It concludes with an example of how the Bayesian can address all these challenges in a large multi-country VAR involving 133 variables: 7 variables for each of 19 countries.

Book Review: Real-World Decision Making: an Encyclopedia of Behavioral Economics

Altman is the Dean and Head of School of the Newcastle Business School, University of Newcastle in Callaghan, Australia. His areas of research include behavioral economics, x-inefficiency theory, institutional change, economics of cooperatives, economic history, methodology, and empirical macroeconomics. He has previously edited the Handbook of Contemporary Behavioral Economics (Routledge, 2006) and authored Behavioral Economics for Dummies (Wiley, 2012) and Economic Growth and the High Wage Economy (Routledge, 2012).