Sparse Bayesian vector autoregressions in huge dimensions

We develop a Bayesian vector autoregressive (VAR) model with multivariate stochastic volatility that is capable of handling vast dimensional information sets. Three features are introduced to permit reliable estimation of the model. First, we assume that the reduced-form errors in the VAR feature a factor stochastic volatility structure, allowing for conditional equation-by-equation estimation. Second, we apply recently developed global–local shrinkage priors to the VAR coefficients to cure the curse of dimensionality. Third, we utilize recent innovations to sample efficiently from high-dimensional multivariate Gaussian distributions. This makes simulation-based fully Bayesian inference feasible when the dimensionality is large but the time series length is moderate. We demonstrate the merits of our approach in an extensive simulation study and apply the model to US macroeconomic data to evaluate its forecasting capabilities.     

Forecasting an aggregate of cointegrated disaggregates

This study examines the problem of forecasting an aggregate of cointegrated disaggregates. It first establishes conditions under which forecasts of an aggregate variable obtained from a disaggregate VECM will be equal to those from an aggregate, univariate time series model, and develops a simple procedure for testing those conditions. The paper then uses Monte Carlo simulations to show, for a finite sample, that the proposed test has good size and power properties and that whether a model satisfies the aggregation conditions is closely related to out‐of‐sample forecast performance. The paper then shows that ignoring cointegration and specifying the disaggregate model as a VAR in differences can significantly affect analyses of aggregation, with the VAR‐based test for aggregation possibly leading to faulty inference and the differenced VAR forecasts potentially understating the benefits of disaggregate information. Finally, analysis of an empirical problem confirms the basic results. Copyright © 2000 John Wiley & Sons, Ltd.     

Causality and forecasting in incomplete systems

In this paper we examine how causality inference and forecasting within a bivariate VAR, consisting of y(t) and x(t), are affected by the omission of a third variable, w(t), which causes (a) none, (b) one, and (c) both variables in the bivariate system. We also derive conditions under which causality inference and forecasting are invariant to the selection of a bivariate or a trivariate model. The most general condition for the invariance of both causality and forecasting to model selection is shown to require the omitted variable not to cause any of the variables in the bivariate system, although it allows the omitted variable to be caused by the other two. We also show that the conditions for one-way causality inference to be invariant to model selection are not sufficient to ensure that forecasting will also be invariant to the model selected. Finally, we present a numerical illustration of the potential losses, in terms of the variance of the forecast, as a function of the forecast horizon and for alternative parameter values—they can be rather large, as the omission of a variable can make the incomplete model unstable. © 1997 John Wiley & Sons, Ltd.     

Forecasting key US macroeconomic variables with a factor‐augmented Qual VAR

In this paper, we first extract factors from a monthly dataset of 130 macroeconomic and financial variables. These extracted factors are then used to construct a factor‐augmented qualitative vector autoregressive (FA‐Qual VAR) model to forecast industrial production growth, inflation, the Federal funds rate, and the term spread based on a pseudo out‐of‐sample recursive forecasting exercise over an out‐of‐sample period of 1980:1 to 2014:12, using an in‐sample period of 1960:1 to 1979:12. Short‐, medium‐, and long‐run horizons of 1, 6, 12, and 24 months ahead are considered. The forecast from the FA‐Qual VAR is compared with that of a standard VAR model, a Qual VAR model, and a factor‐augmented VAR (FAVAR). In general, we observe that the FA‐Qual VAR tends to perform significantly better than the VAR, Qual VAR and FAVAR (barring some exceptions relative to the latter). In addition, we find that the Qual VARs are also well equipped in forecasting probability of recessions when compared to probit models.     

Forecasts of inflation from var models

Why are forecasts of inflation from VAR models so much worse than their forecasts of real variables? This paper documents that relatively poor performance, and finds that the price equation of a VAR model fitted to US post-war data is poorly specified. Statistical work by other authors has found that coefficients in such price equations may not be constant. Based on specific monetary actions, two changes in monetary policy regimes are proposed. Accounting for those two shifts yields significantly more accurate forecasts and lessens the evidence of misspecification.     

Testing for Granger (non‐)causality in a time‐varying coefficient VAR model

In this paper we propose Granger (non‐)causality tests based on a VAR model allowing for time‐varying coefficients. The functional form of the time‐varying coefficients is a logistic smooth transition autoregressive (LSTAR) model using time as the transition variable. The model allows for testing Granger non‐causality when the VAR is subject to a smooth break in the coefficients of the Granger causal variables. The proposed test then is applied to the money–output relationship using quarterly US data for the period 1952:2–2002:4. We find that causality from money to output becomes stronger after 1978:4 and the model is shown to have a good out‐of‐sample forecasting performance for output relative to a linear VAR model. Copyright © 2008 John Wiley & Sons, Ltd.     

Comparing ex-ante forecasts from a sem and var model: An application to the italian economy

The predictive performance of a large-scale structural econometric model (SEM) of the Italian economy the Prometeia model is compared in this paper with a vector autoregressive (VAR) model estimated for a selection of six main variables of interest. The paper concentrates on the quarterly ex-ante forecasts of GDP growth rate and the annual forecasts of GDP growth and inflation rate, over the period 1980-85. It concludes that no forecaster is systematically better than the other. In particular, the VAR model outperforms the SEM in short-run forecasts, suggesting that, for the latter, more careful attention should be addressed to questions of dynamic specification. On the other hand, for longer intervals, the SEM forecasts are more accurate than the VAR forecasts, in that they can benefit from the judgemental interventions of the model users and the model can pick up the non-linearities of the economy which cannot be captured by the VAR. Given the different kinds of information that can be extracted from the two approaches, it seems more reasonable to consider them as complementary rather than alternative tools for modelling and forecasting. Therefore, rather than attempting to establish the superiority of one type of model over the other, this kind of comparisons should be seen as a useful diagnostic tool for detecting types of model misspecification.     

Bias‐corrected bootstrap prediction regions for vector autoregression

This paper examines small sample properties of alternative bias‐corrected bootstrap prediction regions for the vector autoregressive (VAR) model. Bias‐corrected bootstrap prediction regions are constructed by combining bias‐correction of VAR parameter estimators with the bootstrap procedure. The backward VAR model is used to bootstrap VAR forecasts conditionally on past observations. Bootstrap prediction regions based on asymptotic bias‐correction are compared with those based on bootstrap bias‐correction. Monte Carlo simulation results indicate that bootstrap prediction regions based on asymptotic bias‐correction show better small sample properties than those based on bootstrap bias‐correction for nearly all cases considered. The former provide accurate coverage properties in most cases, while the latter over‐estimate the future uncertainty. Overall, the percentile‐t bootstrap prediction region based on asymptotic bias‐correction is found to provide highly desirable small sample properties, outperforming its alternatives in nearly all cases. Copyright © 2004 John Wiley & Sons, Ltd.     

Beating the VAR: Improving Swedish GDP Forecasts Using Error and Intercept Corrections

This paper examines the forecast accuracy of an unrestricted vector autoregressive (VAR) model for GDP, relative to a comparable vector error correction model (VECM) that recognizes that the data are characterized by co‐integration. In addition, an alternative forecast method, intercept correction, is considered for further comparison. Recursive out‐of‐sample forecasts are generated for both models and forecast techniques. The generated forecasts for each model are objectively evaluated by a selection of evaluation measures and equal accuracy tests. The result shows that the VECM consistently outperforms the VAR models. Further, intercept correction enhances the forecast accuracy when applied to the VECM, whereas there is no such indication when applied to the VAR model. For certain forecast horizons there is a significant difference in forecast ability between the intercept corrected VECM compared to the VAR model. Copyright © 2015 John Wiley & Sons, Ltd.     

An integrated bayesian vector auto regression and error correction model for forecasting electricity consumption and prices

The analysis and forecasting of electricity consumption and prices has received considerable attention over the past forty years. In the 1950s and 1960s most of these forecasts and analyses were generated by simultaneous equation econometric models. Beginning in the 1970s, there was a shift in the modeling of economic variables from the structural equations approach with strong identifying restrictions towards a joint time-series model with very few restrictions. One such model is the vector auto regression (VAR) model. It was soon discovered that the unrestricted VAR models do not forecast well. The Bayesian vector auto regression (BVAR) approach as well the error correction model (ECM) and models based on the theory of co integration have been offered as alternatives to the simple VAR model. This paper argues that the BVAF., ECM, and co integration models are simply VAR models with various restrictions placed on the coefficients. Based on this notion of a restricted VAR model, a four-step procedure for specifying VAR forecasting models is presented and then applied to monthly data on US electricity consumption and prices.     

Forecasting US interest rates and business cycle with a nonlinear regime switching VAR model

This paper introduces a regime switching vector autoregressive model with time‐varying regime probabilities, where the regime switching dynamics is described by an observable binary response variable predicted simultaneously with the variables subject to regime changes. Dependence on the observed binary variable distinguishes the model from various previously proposed multivariate regime switching models, facilitating a handy simulation‐based multistep forecasting method. An empirical application shows a strong bidirectional predictive linkage between US interest rates and NBER business cycle recession and expansion periods. Due to the predictability of the business cycle regimes, the proposed model yields superior out‐of‐sample forecasts of the US short‐term interest rate and the term spread compared with the linear and nonlinear vector autoregressive (VAR) models, including the Markov switching VAR model.     

Structural,VAR and BVAR models of exchange rate determination: A comparison of their forecasting performance

This paper compares the out-of-sample forecasting accuracy of a wide class of structural, BVAR and VAR models for major sterling exchange rates over different forecast horizons. As representative structural models we employ a portfolio balance model and a modified uncovered interest parity model, with the latter producing the more accurate forecasts. Proper attention to the long-run properties and the short-run dynamics of structural models can improve on the forecasting performance of the random walk model. The structural model shows substantial improvement in medium-term forecasting accuracy, whereas the BVAR model is the more accurate in the short term. BVAR and VAR models in levels strongly out predict these models formulated in difference form at all forecast horizons.     

A large Bayesian VAR with a block-specific shrinkage: A forecasting application for Italian industrial production

This paper proposes a Bayesian vector autoregression (BVAR) model with the Kalman filter to forecast the Italian industrial production index in a pseudo real-time experiment. Minnesota priors are adopted as a general framework, but a different shrinkage pattern is imposed for both the VAR coefficients and the Kalman gain, depending on the informative contribution of each variable investigated at frequency level. Both a time-varying and a constant selection for the shrinkage are proposed. Overall, the new BVAR models significantly improve the forecasting performance in comparison with the more traditional versions based on standard Minnesota priors with a single shrinkage, equal for all the variables, and selected on the basis of some optimal criteria. Very promising results come out in terms of density forecasting as well.     

含氧有机物结构与毒性(log1/C)关系研究

为探寻含氧有机物结构与毒性(logl/C)的关系,应用分子顶点及顶点相互作用值对部分含氧有机物进行了结构表征.采用多元线性回归及逐步回归建立了2个定量结构-毒性关系模型.经过比较,发现模型(M2)具有最佳模拟结果,此时模型的复相关系数(R)为0.952,标准偏差(SD)为0.325.采用Jackknife法对模型进行了稳健性检验,结果表明,回归模型具有可接受的总体稳健性及良好的预测能力;另外,采用留一法(leave-one-out)对模型进行交互检验,复相关系数(RCV)为0.927,标准偏差(SDCV)为0.396,这也说明所建模型的稳定性与预测能力均较为理想.     

Asymptotic normal and bootstrap inference in structural VAR analysis

The aim of the paper is to examine the performance of bootstrap and asymptotic parametric inference methods in structural VAR analysis. The results obtained through a Monte Carlo experiment suggest that the two approaches are largely equivalent in most, but not all, cases. While the asymptotic method turns out to be surprisingly robust with respect to the distribution of the errors, the bootstrap does deliver results superior in terms of both length of the confidence interval and coverage when highly non-linear statistics (such as the components of the variance of the forecast error) are considered.     

Policy‐Oriented Macroeconomic Forecasting with Hybrid DGSE and Time‐Varying Parameter VAR Models

Micro‐founded dynamic stochastic general equilibrium (DSGE) models appear to be particularly suited to evaluating the consequences of alternative macroeconomic policies. Recently, increasing efforts have been undertaken by policymakers to use these models for forecasting, although this proved to be problematic due to estimation and identification issues. Hybrid DSGE models have become popular for dealing with some of the model misspecifications and the trade‐off between theoretical coherence and empirical fit, thus allowing them to compete in terms of predictability with VAR models. However, DSGE and VAR models are still linear and they do not consider time variation in parameters that could account for inherent nonlinearities and capture the adaptive underlying structure of the economy in a robust manner. This study conducts a comparative evaluation of the out‐of‐sample predictive performance of many different specifications of DSGE models and various classes of VAR models, using datasets for the real GDP, the harmonized CPI and the nominal short‐term interest rate series in the euro area. Simple and hybrid DSGE models were implemented, including DSGE‐VAR and factor‐augmented DGSE, and tested against standard, Bayesian and factor‐augmented VARs. Moreover, a new state‐space time‐varying VAR model is presented. The total period spanned from 1970:Q1 to 2010:Q4 with an out‐of‐sample testing period of 2006:Q1–2010:Q4, which covers the global financial crisis and the EU debt crisis. The results of this study can be useful in conducting monetary policy analysis and macro‐forecasting in the euro area. Copyright © 2016 John Wiley & Sons, Ltd.     

Forecasting under model uncertainty: Non-homogeneous hidden Markov models with Pòlya-Gamma data augmentation

We consider finite state-space non-homogeneous hidden Markov models for forecasting univariate time series. Given a set of predictors, the time series are modeled via predictive regressions with state-dependent coefficients and time-varying transition probabilities that depend on the predictors via a logistic/multinomial function. In a hidden Markov setting, inference for logistic regression coefficients becomes complicated and in some cases impossible due to convergence issues. In this paper, we aim to address this problem utilizing the recently proposed Pólya-Gamma latent variable scheme. Also, we allow for model uncertainty regarding the predictors that affect the series both linearly — in the mean — and non-linearly — in the transition matrix. Predictor selection and inference on the model parameters are based on an automatic Markov chain Monte Carlo scheme with reversible jump steps. Hence the proposed methodology can be used as a black box for predicting time series. Using simulation experiments, we illustrate the performance of our algorithm in various setups, in terms of mixing properties, model selection and predictive ability. An empirical study on realized volatility data shows that our methodology gives improved forecasts compared to benchmark models.     

Monthly data and short‐term forecasting: an assessment of monthly data in a VAR model

Use of monthly data for economic forecasting purposes is typically constrained by the absence of monthly estimates of GDP. Such data can be interpolated but are then prone to measurement error. However, the variance matrix of the measurement errors is typically known. We present a technique for estimating a VAR on monthly data, making use of interpolated estimates of GDP and correcting for the impact of measurement error. We then address the question how to establish whether the model estimated from the interpolated monthly data contains information absent from the analogous quarterly VAR. The techniques are illustrated using a bivariate VAR modelling GDP growth and inflation. It is found that, using inflation data adjusted to remove seasonal effects and the impacts of changes to indirect taxes, the monthly model has little to add to a quarterly model when projecting one quarter ahead. However, the monthly model has an important role to play in building up a picture of the current quarter once one or two months' hard data becomes available. Copyright © 1999 John Wiley & Sons, Ltd.     

Volatility forecasting with double Markov switching GARCH models

This paper investigates inference and volatility forecasting using a Markov switching heteroscedastic model with a fat‐tailed error distribution to analyze asymmetric effects on both the conditional mean and conditional volatility of financial time series. The motivation for extending the Markov switching GARCH model, previously developed to capture mean asymmetry, is that the switching variable, assumed to be a first‐order Markov process, is unobserved. The proposed model extends this work to incorporate Markov switching in the mean and variance simultaneously. Parameter estimation and inference are performed in a Bayesian framework via a Markov chain Monte Carlo scheme. We compare competing models using Bayesian forecasting in a comparative value‐at‐risk study. The proposed methods are illustrated using both simulations and eight international stock market return series. The results generally favor the proposed double Markov switching GARCH model with an exogenous variable. Copyright © 2008 John Wiley & Sons, Ltd.     

A state-dependent linear recurrent formula with application to time series with structural breaks

An underlying assumption in Multivariate Singular Spectrum Analysis (MSSA) is that the time series are governed by a linear recurrent continuation. However, in the presence of a structural break, multiple series can be transferred from one homogeneous state to another over a comparatively short time breaking this assumption. As a consequence, forecasting performance can degrade significantly. In this paper, we propose a state-dependent model to incorporate the movement of states in the linear recurrent formula called a State-Dependent Multivariate SSA (SD-MSSA) model. The proposed model is examined for its reliability in the presence of a structural break by conducting an empirical analysis covering both synthetic and real data. Comparison with standard MSSA, BVAR, VAR and VECM models shows the proposed model outperforms all three models significantly.