Recent Developments in Modelling Nonstationary Vector Autoregressions

In this paper we review some recent developments in the modelling of nonstationary vector autoregressions (VARs) which we feel have great potential for furthering applied researchers understanding of the relationships linking the variables making up a VAR. The developments surveyed are the use of model determination criteria in selecting lag length, trend order and cointegrating rank, causality testing in vector error correction models, FM-VAR estimation of levels VARS, common trends and cycles analysis, permanent and transitory decompositions, impulse response asymptotics, and the links between cointegrated VARs and structural models. The techniques are illustrated by applications to the modelling of U.K. equities, dividends and interest rates.     

Efficient Estimation Of Cointegrating Vectors and Testing for Causality in Vector Autoregressions

This paper surveys various methods of estimating cointegrating vectors and testing for causality in cointegrated VARs, and draws some implications for the applied researcher. In a single equation framework a number of estimators can be used, whose asymptotic efficiency depends on the extent to which they correct for possible endogeneity and serial correlation of the regressors. Such estimates are asymptotically equivalent to those obtained using full system methods, even if the cointegration space is multidimensional, provided there are no cross-equation restrictions. Using the triangular representation proposed by Phillips (1988), we show that one can employ in the context of an ECM a least squares estimator if weak exogeneity holds. If not, the alternatives are augmenting it by the leads of the regressors as in Stock and Watson (1993), or using the fully modified (FM) estimator due to Phillips and Hansen (1990). Other possibilities are the nonparametric approach developed by Bierens (1997), or the ARDL formulation due to Pesaran and Shin (1995). As for causality testing, we argue that it should be conducted within an ECM rather than a VAR formulation, as the limit distributions are much more likely to be standard in the former case. Alternatively, one can carry out statistical tests in the context of a VAR in levels estimated either by using the FM-VAR method as in Phillips (1995), or by augmenting the VAR as in Toda and Yamamoto (1995). Other, computationally easier tests have been introduced by Dolado and Lutkepohl (1996) and Saikkonen and Lütkepohl (1996).     

Forecasting with Medium and Large Bayesian VARS

This paper is motivated by the recent interest in the use of Bayesian VARs for forecasting, even in cases where the number of dependent variables is large. In such cases factor methods have been traditionally used, but recent work using a particular prior suggests that Bayesian VAR methods can forecast better. In this paper, we consider a range of alternative priors which have been used with small VARs, discuss the issues which arise when they are used with medium and large VARs and examine their forecast performance using a US macroeconomic dataset containing 168 variables. We find that Bayesian VARs do tend to forecast better than factor methods and provide an extensive comparison of the strengths and weaknesses of various approaches. Typically, we find that the simple Minnesota prior forecasts well in medium and large VARs, which makes this prior attractive relative to computationally more demanding alternatives. Our empirical results show the importance of using forecast metrics based on the entire predictive density, instead of relying solely on those based on point forecasts. Copyright © 2011 John Wiley & Sons, Ltd.     

Efficient estimation of Bayesian VARMAs with time‐varying coefficients

Empirical work in macroeconometrics has been mostly restricted to using vector autoregressions (VARs), even though there are strong theoretical reasons to consider general vector autoregressive moving averages (VARMAs). A number of articles in the last two decades have conjectured that this is because estimation of VARMAs is perceived to be challenging and proposed various ways to simplify it. Nevertheless, VARMAs continue to be largely dominated by VARs, particularly in terms of developing useful extensions. We address these computational challenges with a Bayesian approach. Specifically, we develop a Gibbs sampler for the basic VARMA, and demonstrate how it can be extended to models with time‐varying vector moving average (VMA) coefficients and stochastic volatility. We illustrate the methodology through a macroeconomic forecasting exercise. We show that in a class of models with stochastic volatility, VARMAs produce better density forecasts than VARs, particularly for short forecast horizons.     

Steady‐state modeling and macroeconomic forecasting quality

Vector autoregressions (VARs) with informative steady‐state priors are standard forecasting tools in empirical macroeconomics. This study proposes (i) an adaptive hierarchical normal‐gamma prior on steady states, (ii) a time‐varying steady‐state specification which accounts for structural breaks in the unconditional mean, and (iii) a generalization of steady‐state VARs with fat‐tailed and heteroskedastic error terms. Empirical analysis, based on a real‐time dataset of 14 macroeconomic variables, shows that, overall, the hierarchical steady‐state specifications materially improve out‐of‐sample forecasting for forecasting horizons longer than 1 year, while the time‐varying specifications generate superior forecasts for variables with significant changes in their unconditional mean.     

Institutional hypothesis of the long-run income velocity of money and parameter stability of the equilibrium relationship

It has recently been argued that when the conventional specification of M2 income velocity is extended to include proxies for two types of institutional change, as emphasized by Bordo and Jonung (1987, 1990), corresponding to the processes of monetization and increasing financial sophistication of financial developments, this extended model is stable in the sense that one can reject the null hypothesis of no cointegration against the alternative of a single cointegrating vector. There may be implications that such an equilibrium relation is a structural income velocity of money function. The evidence based on century-long data from 1880 to 1986 presented in this paper about parameter instability of the cointegrating vector of velocity with its determinants for Canada, Norway, Sweden, and the United Kingdom casts doubt on this interpretation. The evidence is based on using formal stability tests. Moreover, it has an ‘eyeball’ support from the sequential estimates of various parameters of the cointegrating relationship including income and interest semi-elasticities.     

Bayesian inference in a time varying cointegration model

There are both theoretical and empirical reasons for believing that the parameters of macroeconomic models may vary over time. However, work with time-varying parameter models has largely involved vector autoregressions (VARs), ignoring cointegration. This is despite the fact that cointegration plays an important role in informing macroeconomists on a range of issues. In this paper, we develop a new time varying parameter model which permits cointegration. We use a specification which allows for the cointegrating space to evolve over time in a manner comparable to the random walk variation used with TVP–VARs. The properties of our approach are investigated before developing a method of posterior simulation. We use our methods in an empirical investigation involving the Fisher effect.     

Nonparametric cointegration analysis of fractional systems with unknown integration orders

In this paper a nonparametric variance ratio testing approach is proposed for determining the cointegration rank in fractionally integrated systems. The test statistic is easily calculated without prior knowledge of the integration order of the data, the strength of the cointegrating relations, or the cointegration vector(s). The latter property makes it easier to implement than regression-based approaches, especially when examining relationships between several variables with possibly multiple cointegrating vectors. Since the test is nonparametric, it does not require the specification of a particular model and is invariant to short-run dynamics. Nor does it require the choice of any smoothing parameters that change the test statistic without being reflected in the asymptotic distribution. Furthermore, a consistent estimator of the cointegration space can be obtained from the procedure. The asymptotic distribution theory for the proposed test is non-standard but easily tabulated or simulated. Monte Carlo simulations demonstrate excellent finite sample properties, even rivaling those of well-specified parametric tests. The proposed methodology is applied to the term structure of interest rates, where, contrary to both fractional- and integer-based parametric approaches, evidence in favor of the expectations hypothesis is found using the nonparametric approach.     

Making measurement errors and interpreting path coefficients: a practical perspective

The purpose of this analysis is to provide a practical approach to the assessment of reliability. In particular, we examine the impact of random measurement error upon the magnitude and interpretation of standardized regression coefficients (or path coefficients) and the specification of regression models. With the proper research the relationship between measured and true values can be inferred by using path coefficients. Such inferences allow assessments of the specification of statistical models. Several examples illustrate how researchers can be misled without knowledge of the impact of measurement error.     

A sequential Monte Carlo approach to inference in multiple‐equation Markov‐switching models

Vector autoregressions with Markov‐switching parameters (MS‐VARs) offer substantial gains in data fit over VARs with constant parameters. However, Bayesian inference for MS‐VARs has remained challenging, impeding their uptake for empirical applications. We show that sequential Monte Carlo (SMC) estimators can accurately estimate MS‐VAR posteriors. Relative to multi‐step, model‐specific MCMC routines, SMC has the advantages of generality, parallelizability, and freedom from reliance on particular analytical relationships between prior and likelihood. We use SMC's flexibility to demonstrate that model selection among MS‐VARs can be highly sensitive to the choice of prior.     

Trend stationarity in the I(2) cointegration model

A vector autoregressive model for I(2) processes which allows for trend-stationary components and restricts the deterministic part of the process to be at most linear is defined. A two-step statistical analysis of the model is derived. The joint test of I(1) and I(2) cointegrating ranks is shown to be asymptotically similar with respect to the drift terms and the asymptotic distribution is tabulated. The cointegrating parameters are shown to be mixed Gaussian and an application for UK monetary data illustrates the proposed analysis.     

A semiparametric cointegrating regression: Investigating the effects of age distributions on consumption and saving

We consider a semiparametric cointegrating regression model, for which the disequilibrium error is further explained nonparametrically by a functional of distributions changing over time. The paper develops the statistical theories of the model. We propose an efficient econometric estimator and obtain its asymptotic distribution. A specification test for the model is also investigated. The model and methodology are applied to analyze how an aging population in the US influences the consumption level and the savings rate. We find that the impact of age distribution on the consumption level and the savings rate is consistent with the life-cycle hypothesis.     

Composite likelihood methods for large Bayesian VARs with stochastic volatility

Adding multivariate stochastic volatility of a flexible form to large vector autoregressions (VARs) involving over 100 variables has proved challenging owing to computational considerations and overparametrization concerns. The existing literature works with either homoskedastic models or smaller models with restrictive forms for the stochastic volatility. In this paper, we develop composite likelihood methods for large VARs with multivariate stochastic volatility. These involve estimating large numbers of parsimonious models and then taking a weighted average across these models. We discuss various schemes for choosing the weights. In our empirical work involving VARs of up to 196 variables, we show that composite likelihood methods forecast much better than the most popular large VAR approach, which is computationally practical in very high dimensions: the homoskedastic VAR with Minnesota prior. We also compare our methods to various popular approaches that allow for stochastic volatility using medium and small VARs involving up to 20 variables. We find our methods to forecast appreciably better than these as well.     

Bayesian Model Selection with an Uninformative Prior*

Bayesian model selection with posterior probabilities and no subjective prior information is generally not possible because of the Bayes factors being ill‐defined. Using careful consideration of the parameter of interest in cointegration analysis and a re‐specification of the triangular model of Phillips (Econometrica, Vol. 59, pp. 283–306, 1991), this paper presents an approach that allows for Bayesian comparison of models of cointegration with ‘ignorance’ priors. Using the concept of Stiefel and Grassman manifolds, diffuse priors are specified on the dimension and direction of the cointegrating space. The approach is illustrated using a simple term structure of the interest rates model.     

Behavioural response and form of the representative component of the indirect utility function in travel choice models

In the last ten years, discrete-choice modelling using techniques of multinomial logit and multinomial probit, have been increasingly used in a wide range of applications. One area of recent interest is the functional form of the representative component of the preference function. In this paper we draw together a number of approaches used to investigate alternative functional form, and then develop in some detail one procedure — known as the Box-Tukey statistical search approached. While this is not a substitute for direct behavioural specification, it does provide a useful complementary basis for identifying the sensitivity of key policy parameters (e.g., elasticities) to varying functional form. The empirical illustration is drawn from analysis of a mode choice data set.     

Forecasting economic time series using flexible versus fixed specification and linear versus nonlinear econometric models

Nine macroeconomic variables are forecast in a real-time scenario using a variety of flexible specification, fixed specification, linear, and nonlinear econometric models. All models are allowed to evolve through time, and our analysis focuses on model selection and performance. In the context of real-time forecasts, flexible specification models (including linear autoregressive models with exogenous variables and nonlinear artificial neural networks) appear to offer a useful and viable alternative to less flexible fixed specification linear models for a subset of the economic variables which we examine, particularly at forecast horizons greater than 1-step ahead. We speculate that one reason for this result is that the economy is evolving (rather slowly) over time. This feature cannot easily be captured by fixed specification linear models, however, and manifests itself in the form of evolving coefficient estimates. We also provide additional evidence supporting the claim that models which ‘win’ based on one model selection criterion (say a squared error measure) do not necessarily win when an alternative selection criterion is used (say a confusion rate measure), thus highlighting the importance of the particular cost function which is used by forecasters and ‘end-users’ to evaluate their models. A wide variety of different model selection criteria and statistical tests are used to illustrate our findings.     

Measuring underlying economic activity

Recently, interest in the methodology of constructing coincident economic indicators has been revived by the work of Stock and Watson (1989b). They adopt the framework of the state space form and Kalman filter in which to construct an optimal estimate of an unobserved component. This is interpreted as corresponding to underlying economic activity derived from a set of observed indicator variables. In this paper we apply the Stock and Watson approach to the UK where the observed indicator variables are those that make up the Central Statistical Office (CSO) coincident indicator. The time series properties of the indicator variables are examined and three of the five variables are first difference stationary and are cointegrated, the remaining two are stationary in levels. We then construct two alternative measures of economic activity, each of which deals with the different orders of stationarity of the variables. The first uses the levels of the observed component variables that allows for the cointegrating relationship. The second imposes stationarity on the I(1) variables before the estimation by taking first differences. The levels index is viewed as the preferred specification as it allows for the long-run relationships between the variables and has a superior statistical fit. ©1996 John Wiley & Sons, Ltd.     

Measuring output gap nowcast uncertainty

We propose a methodology for gauging the uncertainty in output gap nowcasts across a large number of commonly-deployed vector autoregressive (VAR) specifications for inflation and the output gap. Our approach utilises many output gap measures to construct ensemble nowcasts for inflation using a linear opinion pool. The predictive densities for the latent output gap utilise weights based on the ability of each specification to provide accurate probabilistic forecasts of inflation. In an application based on US real-time data, nowcasting over the out-of-sample evaluation period from 1991q2 to 2010q1, we demonstrate that a system of bivariate VARs produces well-calibrated ensemble densities for inflation, in contrast to univariate autoregressive benchmarks. The implied nowcast densities for the output gap are multimodal and indicate a considerable degree of uncertainty. For example, we assess the probability of a negative output gap at around 45% between 2004 and 2007. Despite the Greenspan policy regime, there still remained a substantial risk that the nowcast for output was below potential in real time. We extend our methodology to include distinct output gap measures, based on alternative filters, and show that, in our application, the nowcast density for the output gap is sensitive to the detrending method.     

A fractional cointegration var analysis of exchange rate dynamics

In their seminal work, Baillie and Bollerslev (1994) carried out an analysis of deviations from the cointegrating relationship of seven important exchange rates. They suggested that the exchange rate series possess long memory and therefore such processes could be well described as fractionally integrated processes. Hence, the influence of shocks to the equilibrium exchange rates may only vanish at very long horizons. In this work we analyze the cointegrating structure of five exchange rates to the US dollar, namely the British pound, the Euro, the Swedish Krona, the Canadian Dollar and the Swiss Franc. The series possess long memory and we show that they can be modeled through fractional integration. In fact, standard cointegration is rejected with the more traditional Johansen CVAR methodology. By using the recently introduced Fractionally Cointegrated VAR by Johansen and Nielsen (2012) we provide a cointegrating relationship taking into account fractional integration.     

Modelling non-stationary ‘Big Data’

‘Fat big data’ characterise data sets that contain many more variables than observations. We discuss the use of both principal components analysis and equilibrium correction models to identify cointegrating relations that handle stochastic trends in non-stationary fat data. However, most time series are wide-sense non-stationary—induced by the joint occurrence of stochastic trends and distributional shifts—so we also handle the latter by saturation estimation. Seeking substantive relationships when there are vast numbers of potentially spurious connections cannot be achieved by merely choosing the best-fitting equation or trying hundreds of empirical fits and selecting a preferred one, perhaps contradicted by others that go unreported. Conversely, fat big data are useful if they help ensure that the data generation process is nested in the postulated model, and increase the power of specification and mis-specification tests without raising the chances of adventitious significance. We model the monthly UK unemployment rate, using both macroeconomic and Google Trends data, searching across 3000 explanatory variables, yet identify a parsimonious, statistically valid, and theoretically interpretable specification.