Bayesian model comparison for time‐varying parameter VARs with stochastic volatility

We develop importance sampling methods for computing two popular Bayesian model comparison criteria, namely, the marginal likelihood and the deviance information criterion (DIC) for time‐varying parameter vector autoregressions (TVP‐VARs), where both the regression coefficients and volatilities are drifting over time. The proposed estimators are based on the integrated likelihood, which are substantially more reliable than alternatives. Using US data, we find overwhelming support for the TVP‐VAR with stochastic volatility compared to a conventional constant coefficients VAR with homoskedastic innovations. Most of the gains, however, appear to have come from allowing for stochastic volatility rather than time variation in the VAR coefficients or contemporaneous relationships. Indeed, according to both criteria, a constant coefficients VAR with stochastic volatility outperforms the more general model with time‐varying parameters.     

Parametric pricing of higher order moments in S&P500 options

A general parametric framework based on the generalized Student t‐distribution is developed for pricing S&P500 options. Higher order moments in stock returns as well as time‐varying volatility are priced. An important computational advantage of the proposed framework over Monte Carlo‐based pricing methods is that options can be priced using one‐dimensional quadrature integration. The empirical application is based on S&P500 options traded on select days in April 1995, a total sample of over 100,000 observations. A range of performance criteria are used to evaluate the proposed model, as well as a number of alternative models. The empirical results show that pricing higher order moments and time‐varying volatility yields improvements in the pricing of options, as well as correcting the volatility skew associated with the Black–Scholes model. Copyright © 2004 John Wiley & Sons, Ltd.     

Testing for a unit root in the volatility of asset returns

It is now well established that the volatility of asset returns is time varying and highly persistent. One leading model that is used to represent these features of the data is the stochastic volatility model. The researcher may test for non-stationarity of the volatility process by testing for a unit root in the log-squared time series. This strategy for inference has many advantages, but is not followed in practice because these unit root tests are known to have very poor size properties. In this paper I show that new tests that are robust to negative MA roots allow a reliable test for a unit root in the volatility process to be conducted. In applying these tests to exchange rate and stock returns, strong rejections of non-stationarity in volatility are obtained. Copyright © 1999 John Wiley & Sons, Ltd.     

LONG‐RUN RISKS IN THE TERM STRUCTURE OF INTEREST RATES: ESTIMATION

This paper estimates a model in which persistent fluctuations in expected consumption growth, expected inflation, and their time‐varying volatility determine asset price variation. The model features Epstein–Zin recursive preferences, which determine the market price of macro risk factors. Analysis of the US nominal term structure data from 1953 to 2006 shows that agents dislike high uncertainty and demand compensation for volatility risks. Also, the time variation of the term premium is driven by the compensation for inflation volatility risk, which is distinct from consumption volatility risk. The central role of inflation volatility risk in explaining the time‐varying term premium is consistent with other empirical evidence including survey data. In contrast, the existing long‐run risks literature emphasizes consumption volatility risk and ignores inflation‐specific time‐varying volatility. The estimation results of this paper suggest that inflation‐specific volatility risk is essential for fitting the time series of the US nominal term structure data. Copyright © 2012 John Wiley & Sons, Ltd.     

What Drives Output Volatility? The Role of Demographics and Government Size Revisited

This paper studies the determinants of output volatility in a panel of 22 OECD countries. In contrast to the existing literature, we avoid ad hoc estimates of volatility based on rolling windows, and we account for possible non‐stationarity. Specifically, output volatility is modelled within an unobserved components model where the volatility series is the outcome of both macroeconomic determinants and a latent integrated process. A Bayesian model selection approach tests for the presence of the non‐stationary component. The results point to demographics and government size as important determinants of macroeconomic (in)stability. A larger share of prime‐age workers is associated with lower output volatility, while higher public expenditure increases volatility.     

Commodity prices and fiscal policy design: Procyclical despite a rule

Recent studies offer evidence of reduced fiscal procyclicality to commodity price changes in resource‐rich countries—a feature commonly attributed to the adoption of fiscal policy rules. We revisit this issue and find that, by controlling for global activity shocks while allowing for time‐varying changes in both fiscal policy and the volatility of shocks, this finding does not hold. To show this we develop a time‐varying dynamic factor model, allowing for a multiple of shocks, stochastic volatility and time‐varying parameters, and estimate it on data for Norway, whose handling of resource wealth is often cited as exemplary.     

Detecting nonlinearity in time series by model selection criteria

This article analyzes the use of model selection criteria for detecting nonlinearity in the residuals of a linear model. Model selection criteria are applied for finding the order of the best autoregressive model fitted to the squared residuals of the linear model. If the order selected is not zero, this is considered as an indication of nonlinear behavior. The BIC and AIC criteria are compared to some popular nonlinearity tests in three Monte Carlo experiments. We conclude that the BIC model selection criterion seems to offer a promising tool for detecting nonlinearity in time series. An example is shown to illustrate the performance of the tests considered and the relationship between nonlinearity and structural changes in time series.     

Macroeconomic forecasting and structural change

The aim of this paper is to assess whether modeling structural change can help improving the accuracy of macroeconomic forecasts. We conduct a simulated real‐time out‐of‐sample exercise using a time‐varying coefficients vector autoregression (VAR) with stochastic volatility to predict the inflation rate, unemployment rate and interest rate in the USA. The model generates accurate predictions for the three variables. In particular, the forecasts of inflation are much more accurate than those obtained with any other competing model, including fixed coefficients VARs, time‐varying autoregressions and the naïve random walk model. The results hold true also after the mid 1980s, a period in which forecasting inflation was particularly hard. Copyright © 2011 John Wiley & Sons, Ltd.     

时间偏好与居民住房需求的内生波动

从时间偏好内生化角度研究居民住房需求随年龄波动的微观机制,理论模型表明：即使没有其他外生冲击,年龄的变化也会引起居民住房需求的内生波动。具体来看,居民住房需求随年龄存在先上升后下降的倒“U”型波动趋势;进一步地,利用中国家庭住户收入项目调查数据（CHIPS）对理论模型结论进行了实证检验,该结论在修正样本选择性偏误和稳健性检验后仍然成立。因此在长期中必须注意居民住房需求波动对我国房地产市场发展的影响。     

Modeling Financial Sector Joint Tail Risk in the Euro Area

We develop a novel high‐dimensional non‐Gaussian modeling framework to infer measures of conditional and joint default risk for numerous financial sector firms. The model is based on a dynamic generalized hyperbolic skewed‐t block equicorrelation copula with time‐varying volatility and dependence parameters that naturally accommodates asymmetries and heavy tails, as well as nonlinear and time‐varying default dependence. We apply a conditional law of large numbers in this setting to define joint and conditional risk measures that can be evaluated quickly and reliably. We apply the modeling framework to assess the joint risk from multiple defaults in the euro area during the 2008–2012 financial and sovereign debt crisis. We document unprecedented tail risks between 2011 and 2012, as well as their steep decline following subsequent policy actions. Copyright © 2016 John Wiley & Sons, Ltd.     

An empirical analysis of alternative parametric ARCH models

This paper presents empirical evidence on the effectiveness of eight different parametric ARCH models in describing daily stock returns. Twenty‐seven years of UK daily data on a broad‐based value weighted stock index are investigated for the period 1971–97. Several interesting results are documented. Overall, the results strongly demonstrate the utility of parametric ARCH models in describing time‐varying volatility in this market. The parameters proxying for asymmetry in models that recognize the asymmetric behaviour of volatility are highly significant in each and every case. However, the ‘performance’ of the various parameterizations is often fairly similar with the exception of the multiplicative GARCH model that performs qualitatively differently on several dimensions of performance. The outperformance of any model(s) is not consistent across different sub‐periods of the sample, suggesting that the optimal choice of a model is period‐specific. The outperformance is also not consistent as we change from in‐sample inferences to out‐of‐sample inferences within the same period. Copyright © 2000 John Wiley & Sons, Ltd.     

Assessing the Transmission of Monetary Policy Using Time‐varying Parameter Dynamic Factor Models*

This article extends the current literature which questions the stability of the monetary transmission mechanism, by proposing a factor‐augmented vector autoregressive (VAR) model with time‐varying coefficients and stochastic volatility. The VAR coefficients and error covariances may change gradually in every period or be subject to abrupt breaks. The model is applied to 143 post‐World War II quarterly variables fully describing the US economy. I show that both endogenous and exogenous shocks to the US economy resulted in the high inflation volatility during the 1970s and early 1980s. The time‐varying factor augmented VAR produces impulse responses of inflation which significantly reduce the price puzzle. Impulse responses of other indicators of the economy show that the most notable changes in the transmission of unanticipated monetary policy shocks occurred for gross domestic product, investment, exchange rates and money.     

Modelling Inflation Volatility

This paper discusses estimation of US inflation volatility using time‐varying parameter models, in particular whether it should be modelled as a stationary or random walk stochastic process. Specifying inflation volatility as an unbounded process, as implied by the random walk, conflicts with priors beliefs, yet a stationary process cannot capture the low‐frequency behaviour commonly observed in estimates of volatility. We therefore propose an alternative model with a change‐point process in the volatility that allows for switches between stationary models to capture changes in the level and dynamics over the past 40 years. To accommodate the stationarity restriction, we develop a new representation that is equivalent to our model but is computationally more efficient. All models produce effectively identical estimates of volatility, but the change‐point model provides more information on the level and persistence of volatility and the probabilities of changes. For example, we find a few well‐defined switches in the volatility process and, interestingly, these switches line up well with economic slowdowns or changes of the Federal Reserve Chair. Moreover, a decomposition of inflation shocks into permanent and transitory components shows that a spike in volatility in the late 2000s was entirely on the transitory side and characterized by a rise above its long‐run mean level during a period of higher persistence. Copyright © 2015 John Wiley & Sons, Ltd.     

Macroeconomic Forecasting Performance under Alternative Specifications of Time‐Varying Volatility

This paper compares alternative models of time‐varying volatility on the basis of the accuracy of real‐time point and density forecasts of key macroeconomic time series for the USA. We consider Bayesian autoregressive and vector autoregressive models that incorporate some form of time‐varying volatility, precisely random walk stochastic volatility, stochastic volatility following a stationary AR process, stochastic volatility coupled with fat tails, GARCH and mixture of innovation models. The results show that the AR and VAR specifications with conventional stochastic volatility dominate other volatility specifications, in terms of point forecasting to some degree and density forecasting to a greater degree. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymmetric Quantile Persistence and Predictability: the Case of US Inflation

This article investigates the evidence of time‐variation and asymmetry in the persistence of US inflation. We compare the out‐of‐sample performance of different forecasting models and find that quantile forecasts from an Auto‐Regressive (AR) model with level‐dependent volatility are at least as accurate as the forecasts of the Quantile Auto‐Regressive model, in particular for the core inflation measures. Our results indicate that the persistence of core inflation has been relatively constant and high, but it declined for the headline inflation measures. We also find that the asymmetric persistence of inflation shocks can be mostly attributed to the positive relation between inflation level and its volatility.     

Volatility of Price Indices for Heterogeneous Goods with Applications to the Fine Art Market

Price indices for heterogeneous goods such as real estate or fine art constitute crucial information for institutional or private investors considering alternative investment decisions in times of financial markets turmoil. Classical mean‐variance analysis of alternative investments has been hampered by the lack of a systematic treatment of volatility in these markets. In this paper we propose a hedonic regression framework which explicitly defines an underlying stochastic process for the price index, allowing to treat the volatility parameter as the object of interest. The model can be estimated using maximum likelihood in combination with the Kalman filter. We derive theoretical properties of the volatility estimator and show that it outperforms the standard estimator. We show that extensions to allow for time‐varying volatility are straightforward using a local‐likelihood approach. In an application to a large data set of international blue chip artists, we show that volatility of the art market, although generally lower than that of financial markets, has risen after the financial crisis of 2008–09, but sharply decreased during the recent debt crisis. Copyright © 2013 John Wiley & Sons, Ltd.     

Combining Time Variation and Mixed Frequencies: an Analysis of Government Spending Multipliers in Italy

In this paper, we propose a time‐varying parameter vector autoregression (VAR) model with stochastic volatility which allows for estimation on data sampled at different frequencies. Our contribution is twofold. First, we extend the methodology developed by Cogley and Sargent (Drifts and volatilities: monetary policies and outcomes in the post WWII U.S. Review of Economic Studies 2005; 8 : 262–302) and Primiceri (Time varying structural vector autoregressions and monetary policy. Review of Economic Studies 2005; 72 : 821–852) to a mixed‐frequency setting. In particular, our approach allows for the inclusion of two different categories of variables (high‐frequency and low‐frequency) into the same time‐varying model. Second, we use this model to study the macroeconomic effects of government spending shocks in Italy over the 1988:Q4–2013:Q3 period. Italy—as well as most other euro area economies—is characterized by short quarterly time series for fiscal variables, whereas annual data are generally available for a longer sample before 1999. Our results show that the proposed time‐varying mixed‐frequency model improves on the performance of a simple linear interpolation model in generating the true path of the missing observations. Second, our empirical analysis suggests that government spending shocks tend to have positive effects on output in Italy. The fiscal multiplier, which is maximized at the 1‐year horizon, follows a U‐shape over the sample considered: it peaks at around 1.5 at the beginning of the sample; it then stabilizes between 0.8 and 0.9 from the mid 1990s to the late 2000s, before rising again to above unity during the recent crisis. Copyright © 2015 John Wiley & Sons, Ltd.     

Spurious Fixed Effects Regression*

This article shows that spurious regression results can occur for a fixed effects model with weak time series variation in the regressor and/or strong time series variation in the regression errors when the first‐differenced and Within‐OLS estimators are used. Asymptotic properties of these estimators and the related t‐tests and model selection criteria are studied by sending the number of cross‐sectional observations to infinity. This article shows that the first‐differenced and Within‐OLS estimators diverge in probability, that the related t‐tests are inconsistent, that R²s converge to zero in probability and that AIC and BIC diverge to ?∞ in probability. The results of the article warn that one should not jump to the use of fixed effects regressions without considering the degree of time series variations in the data.     

Modelling Regime‐Specific Stock Price Volatility*

Single‐state generalized autoregressive conditional heteroscedasticity (GARCH) models identify only one mechanism governing the response of volatility to market shocks, and the conditional higher moments are constant, unless modelled explicitly. So they neither capture state‐dependent behaviour of volatility nor explain why the equity index skew persists into long‐dated options. Markov switching (MS) GARCH models specify several volatility states with endogenous conditional skewness and kurtosis; of these the simplest to estimate is normal mixture (NM) GARCH, which has constant state probabilities. We introduce a state‐dependent leverage effect to NM‐GARCH and thereby explain the observed characteristics of equity index returns and implied volatility skews, without resorting to time‐varying volatility risk premia. An empirical study on European equity indices identifies two‐state asymmetric NM‐GARCH as the best fit of the 15 models considered. During stable markets volatility behaviour is broadly similar across all indices, but the crash probability and the behaviour of returns and volatility during a crash depends on the index. The volatility mean‐reversion and leverage effects during crash markets are quite different from those in the stable regime.     

Continuous‐time models,realized volatilities,and testable distributional implications for daily stock returns

We provide an empirical framework for assessing the distributional properties of daily speculative returns within the context of the continuous‐time jump diffusion models traditionally used in asset pricing finance. Our approach builds directly on recently developed realized variation measures and non‐parametric jump detection statistics constructed from high‐frequency intra‐day data. A sequence of simple‐to‐implement moment‐based tests involving various transformations of the daily returns speak directly to the importance of different distributional features, and may serve as useful diagnostic tools in the specification of empirically more realistic continuous‐time asset pricing models. On applying the tests to the 30 individual stocks in the Dow Jones Industrial Average index, we find that it is important to allow for both time‐varying diffusive volatility, jumps, and leverage effects to satisfactorily describe the daily stock price dynamics. Copyright © 2009 John Wiley & Sons, Ltd.