Bayesian hypothesis testing in latent variable models

Hypothesis testing using Bayes factors (BFs) is known not to be well defined under the improper prior. In the context of latent variable models, an additional problem with BFs is that they are difficult to compute. In this paper, a new Bayesian method, based on the decision theory and the EM algorithm, is introduced to test a point hypothesis in latent variable models. The new statistic is a by-product of the Bayesian MCMC output and, hence, easy to compute. It is shown that the new statistic is appropriately defined under improper priors because the method employs a continuous loss function. In addition, it is easy to interpret. The method is illustrated using a one-factor asset pricing model and a stochastic volatility model with jumps.     

Maximum likelihood estimation of partially observed diffusion models

This paper develops a maximum likelihood (ML) method to estimate partially observed diffusion models based on data sampled at discrete times. The method combines two techniques recently proposed in the literature in two separate steps. In the first step, the closed form approach of Aït-Sahalia (2008) is used to obtain a highly accurate approximation to the joint transition probability density of the latent and the observed states. In the second step, the efficient importance sampling technique of Richard and Zhang (2007) is used to integrate out the latent states, thereby yielding the likelihood function. Using both simulated and real data, we show that the proposed ML method works better than alternative methods. The new method does not require the underlying diffusion to have an affine structure and does not involve infill simulations. Therefore, the method has a wide range of applicability and its computational cost is moderate.     

Identification and estimation of Gaussian affine term structure models

This paper develops new results for identification and estimation of Gaussian affine term structure models. We establish that three popular canonical representations are unidentified, and demonstrate how unidentified regions can complicate numerical optimization. A separate contribution of the paper is the proposal of minimum-chi-square estimation as an alternative to MLE. We show that, although it is asymptotically equivalent to MLE, it can be much easier to compute. In some cases, MCSE allows researchers to recognize with certainty whether a given estimate represents a global maximum of the likelihood function and makes feasible the computation of small-sample standard errors.     

Examining macroeconomic models through the lens of asset pricing

We develop new methods for representing the asset-pricing implications of stochastic general equilibrium models. We provide asset-pricing counterparts to impulse response functions and the resulting dynamic value decompositions (DVDs). These methods quantify the exposures of macroeconomic cash flows to shocks over alternative investment horizons and the corresponding prices or investors’ compensations. We extend the continuous-time methods developed in Hansen and Scheinkman (2012) and Borovi?ka et al. (2011) by constructing discrete-time, state-dependent, shock-exposure and shock-price elasticities as functions of the investment horizon. Our methods are applicable to economic models that are nonlinear, including models with stochastic volatility.     

On the statistical identification of DSGE models

Dynamic Stochastic General Equilibrium (DSGE) models are now considered attractive by the profession not only from the theoretical perspective but also from an empirical standpoint. As a consequence of this development, methods for diagnosing the fit of these models are being proposed and implemented. In this article we illustrate how the concept of statistical identification, that was introduced and used by Spanos [Spanos, Aris, 1990. The simultaneous-equations model revisited: Statistical adequacy and identification. Journal of Econometrics 44, 87–105] to criticize traditional evaluation methods of Cowles Commission models, could be relevant for DSGE models. We conclude that the recently proposed model evaluation method, based on the DSGE–VAR(λ)$\left(\lambda \right)$, might not satisfy the condition for statistical identification. However, our application also shows that the adoption of a FAVAR as a statistically identified benchmark leaves unaltered the support of the data for the DSGE model and that a DSGE–FAVAR can be an optimal forecasting model.     

The affine arbitrage-free class of Nelson-Siegel term structure models

We derive the class of affine arbitrage-free dynamic term structure models that approximate the widely used Nelson-Siegel yield curve specification. These arbitrage-free Nelson-Siegel (AFNS) models can be expressed as slightly restricted versions of the canonical representation of the three-factor affine arbitrage-free model. Imposing the Nelson-Siegel structure on the canonical model greatly facilitates estimation and can improve predictive performance. In the future, AFNS models appear likely to be a useful workhorse representation for term structure research.     

Term structure models and the zero bound: An empirical investigation of Japanese yields

When Japanese short-term bond yields were near their zero bound, yields on long-term bonds showed substantial fluctuation, and there was a strong positive relationship between the level of interest rates and yield volatilities/risk premiums. We explore whether several families of dynamic term structure models that enforce a zero lower bound on short rates imply conditional distributions of Japanese bond yields consistent with these patterns. Multi-factor “shadow-rate” and quadratic-Gaussian models, evaluated at their maximum likelihood estimates, capture many features of the data. Furthermore, model-implied risk premiums track realized excess returns during extended periods of near-zero short rates. In contrast, the conditional distributions implied by non-negative affine models do not match their sample counterparts, and standard Gaussian affine models generate implausibly large negative risk premiums.     

Network VAR models to measure financial contagion

Financial contagion among countries can arise from different channels, the most important of which are financial markets and bank lending. The paper aims to build an econometric network approach to understand the extent to which contagion spillovers (from one country to another) aris from financial markets, from bank lending, or from both. To achieve this aim we consider a model specification strategy which combines Vector Autoregressive models with network models. The paper contributes to the contagion literature with a model that can consider bank exposures and financial market prices, jointly and not only separately. From an empirical viewpoint, our results show that both bilateral exposures and market prices act as contagion channels in the transmission of shocks arising from a country to other countries.     

A Gaussian approximation scheme for computation of option prices in stochastic volatility models

We consider European options on a price process that follows the log-linear stochastic volatility model. Two stochastic integrals in the option pricing formula are costly to compute. We derive a central limit theorem to approximate them. At parameter settings appropriate to foreign exchange data our formulas improve computation speed by a factor of 1000 over brute force Monte Carlo making MCMC statistical methods practicable. We provide estimates of model parameters from daily data on the Swiss Franc to Euro and Japanese Yen to Euro over the period 1999–2002.     

Decisionmetrics: A decision-based approach to econometric modelling

An approach to developing a possibly misspecified econometric model that will be used as the beliefs of an expected utility maximiser is proposed. This approach builds on a novel objective function that measures the value of predictive distributions in decision-making and is used in model estimation, selection and evaluation. The methods proposed also provide an econometric approach for developing arbitrary parametric action rules such as technical trading rules. The approach is compared in detail with existing methods and is applied in the context of a CARA investor's decision problem where analytical and empirical results suggest it is very effective.     

Spanning tests in return and stochastic discount factor mean–variance frontiers: A unifying approach

We propose new spanning tests that assess if the initial and additional assets share the economically meaningful cost and mean representing portfolios. We prove their asymptotic equivalence to existing tests under local alternatives. We also show that unlike two-step or iterated procedures, single-step methods such as continuously updated GMM yield numerically identical overidentifying restrictions test, so there is arguably a single spanning test. To prove these results, we extend optimal GMM inference to deal with singularities in the long run second moment matrix of the influence functions. Finally, we test for spanning using size and book-to-market sorted US stock portfolios.     

Empirical implementation of nonparametric first-price auction models

Nonparametric estimators provide a flexible means of uncovering salient features of auction data. Although these estimators are popular in the literature, many key features necessary for proper implementation have yet to be uncovered. Here we provide several suggestions for nonparametric estimation of first-price auction models. Specifically, we show how to impose monotonicity of the equilibrium bidding strategy; a key property of structural auction models not guaranteed in standard nonparametric estimation. We further develop methods for automatic bandwidth selection. Finally, we discuss how to impose monotonicity in auctions with differing numbers of bidders, reserve prices, and auction-specific characteristics. Finite sample performance is examined using simulated data as well as experimental auction data.     

Dynamic quantile models

This paper introduces the Dynamic Additive Quantile (DAQ) model that ensures the monotonicity of conditional quantile estimates. The DAQ model is easily estimable and can be used for computation and updating of the Value-at-Risk. An asymptotically efficient estimator of the DAQ is obtained by maximizing an objective function based on the inverse KLIC measure. An alternative estimator proposed in the paper is the Method of L-Moments estimator (MLM). The MLM estimator is consistent, but generally not fully efficient. Goodness-of-fit tests and diagnostic tools for the assessment of the model are also provided. For illustration, the DAQ model is estimated from a series of returns on the Toronto Stock Exchange (TSX) market index.     

Chi-squared tests for evaluation and comparison of asset pricing models

This paper presents a general statistical framework for estimation, testing and comparison of asset pricing models using the unconstrained distance measure of Hansen and Jagannathan (1997). The limiting results cover both linear and nonlinear models that could be correctly specified or misspecified. We propose modified versions of the existing model selection tests and new pivotal specification and model comparison tests with improved finite-sample properties. In addition, we provide formal tests of multiple model comparison. The excellent size and power properties of the proposed tests are demonstrated using simulated data from linear and nonlinear asset pricing models.     

How Do Alphas and Betas Move? Uncertainty,Learning and Time Variation in Risk Loadings*

I employ a parsimonious model with learning, but without conditioning information, to extract time‐varying measures of market‐risk sensitivities, pricing errors and pricing uncertainty. The evolution of these quantities has interesting implications for macroeconomic dynamics. Parameters estimated for US equity portfolios display significant low‐frequency fluctuations, along patterns that change across size and book‐to‐market stocks. Time‐varying betas display superior predictive accuracy for returns against constant and rolling‐window OLS estimates. As to the relationship of betas with business‐cycle variables, value stocks’ betas move pro‐cyclically, unlike those of growth stocks. Investment growth, rather than consumption, predicts the betas of value and small‐firm portfolios.     

A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation

This paper presents a Bayesian approach to bandwidth selection for multivariate kernel regression. A Monte Carlo study shows that under the average squared error criterion, the Bayesian bandwidth selector is comparable to the cross-validation method and clearly outperforms the bootstrapping and rule-of-thumb bandwidth selectors. The Bayesian bandwidth selector is applied to a multivariate kernel regression model that is often used to estimate the state-price density of Arrow–Debreu securities with the S&P 500 index options data and the DAX index options data. The proposed Bayesian bandwidth selector represents a data-driven solution to the problem of choosing bandwidths for the multivariate kernel regression involved in the nonparametric estimation of the state-price density pioneered by Aït-Sahalia and Lo [Aït-Sahalia, Y., Lo, A.W., 1998. Nonparametric estimation of state-price densities implicit in financial asset prices. The Journal of Finance, 53, 499, 547.]     

Nonlinear models for strongly dependent processes with financial applications

This paper is motivated by recent evidence that many univariate economic and financial time series have both nonlinear and long memory characteristics. Hence, this paper considers a general nonlinear, smooth transition regime autoregression which is embedded within a strongly dependent, long memory process. A time domain MLE$MLE$ with simultaneous estimation of the long memory, linear AR$AR$ and nonlinear parameters is shown to have desirable asymptotic properties. The Bayesian and Hannan–Quinn information criteria are shown to provide consistent model selection procedures. The paper also considers an alternative two step estimator where the original time series is fractionally filtered from an initial semi-parametric estimate of the long memory parameter. Simulation evidence indicates that the time domain MLE$MLE$ is generally superior to the two step estimator. The paper also includes some applications of the methodology and estimation of a fractionally integrated, nonlinear autoregressive-ESTAR$ESTAR$ model to forward premium and real exchange rates.     

A semiparametric stochastic volatility model

In this paper the correlation structure in the classical leverage stochastic volatility (SV) model is generalized based on a linear spline. In the new model the correlation between the return and volatility innovations is time varying and depends nonparametrically on the type of news arrived to the market. Theoretical properties of the proposed model are examined. The model estimation and comparison are conducted by Bayesian methods. The performance of the estimates are examined in simulations. The new model is fitted to daily and weekly US data and compared with the classical SV and GARCH models in terms of their in-sample and out-of-sample performances. Empirical results suggest evidence in favor of the proposed model. In particular, the new model finds strong evidence of time varying leverage effect in individual stocks when the classical model fails to identify the leverage effect.     

Inference with non-Gaussian Ornstein–Uhlenbeck processes for stochastic volatility

Continuous-time stochastic volatility models are becoming an increasingly popular way to describe moderate and high-frequency financial data. Barndorff-Nielsen and Shephard (2001a) proposed a class of models where the volatility behaves according to an Ornstein–Uhlenbeck (OU) process, driven by a positive Lévy process without Gaussian component. These models introduce discontinuities, or jumps, into the volatility process. They also consider superpositions of such processes and we extend that to the inclusion of a jump component in the returns. In addition, we allow for leverage effects and we introduce separate risk pricing for the volatility components. We design and implement practically relevant inference methods for such models, within the Bayesian paradigm. The algorithm is based on Markov chain Monte Carlo (MCMC) methods and we use a series representation of Lévy processes. MCMC methods for such models are complicated by the fact that parameter changes will often induce a change in the distribution of the representation of the process and the associated problem of overconditioning. We avoid this problem by dependent thinning methods. An application to stock price data shows the models perform very well, even in the face of data with rapid changes, especially if a superposition of processes with different risk premiums and a leverage effect is used.     

Extreme-quantile tracking for financial time series

Time series of financial asset values exhibit well-known statistical features such as heavy tails and volatility clustering. We propose a nonparametric extension of the classical Peaks-Over-Threshold method from extreme value theory to fit the time varying volatility in situations where the stationarity assumption may be violated by erratic changes of regime, say. As a result, we provide a method for estimating conditional risk measures applicable to both stationary and nonstationary series. A backtesting study for the UBS share price over the subprime crisis exemplifies our approach.