Maximum likelihood estimation of stochastic frontier models by the Fourier transform

The paper is concerned with several kinds of stochastic frontier models whose likelihood function is not available in closed form. First, with output-oriented stochastic frontier models whose one-sided errors have a distribution other than the standard ones (exponential or half-normal). The gamma and beta distributions are leading examples. Second, with input-oriented stochastic frontier models which are common in theoretical discussions but not in econometric applications. Third, with two-tiered stochastic frontier models when the one-sided error components follow gamma distributions. Fourth, with latent class models with gamma distributed one-sided error terms. Fifth, with models whose two-sided error component is distributed as stable Paretian and the one-sided error is gamma. The principal aim is to propose approximations to the density of the composed error based on the inversion of the characteristic function (which turns out to be manageable) using the Fourier transform. Procedures that are based on the asymptotic normal form of the log-likelihood function and have arbitrary degrees of asymptotic efficiency are also proposed, implemented and evaluated in connection with output-oriented stochastic frontiers. The new methods are illustrated using data for US commercial banks, electric utilities, and a sample from the National Youth Longitudinal Survey.     

A forecast evaluation of capital investment in agriculture

A stochastic coefficients model developed by Swamy and Tinsley is used to forecast agricultural investment. In two sets of out-of-sample forecasts, one for 5 years, the other for 10 years, the Swamy-Tinsley stochastic coefficients model outperforms competing fixed and stochastic coefficients empirical models of agricultural investment for a wide array of risk functions. The Swamy-Tinsley stochastic coefficients investment model forecasts continued declines in net investment for farm machinery, with greater declines toward the end of the forecast period. The Swamy-Tinsley method produced better predictions than both stochastic and fixed-coefficients competitors.     

Estimation and testing of stochastic frontier models using variational Bayes

We show how a wide range of stochastic frontier models can be estimated relatively easily using variational Bayes. We derive approximate posterior distributions and point estimates for parameters and inefficiency effects for (a) time invariant models with several alternative inefficiency distributions, (b) models with time varying effects, (c) models incorporating environmental effects, and (d) models with more flexible forms for the regression function and error terms. Despite the abundance of stochastic frontier models, there have been few attempts to test the various models against each other, probably due to the difficulty of performing such tests. One advantage of the variational Bayes approximation is that it facilitates the computation of marginal likelihoods that can be used to compare models. We apply this idea to test stochastic frontier models with different inefficiency distributions. Estimation and testing is illustrated using three examples.     

Calibration of stochastic volatility models: A Tikhonov regularization approach

We aim to calibrate stochastic volatility models from option prices. We develop a Tikhonov regularization approach with an efficient numerical algorithm to recover the risk neutral drift term of the volatility (or variance) process. In contrast to most existing literature, we do not assume that the drift term has any special structure. As such, our algorithm applies to calibration of general stochastic volatility models. An extensive numerical analysis is presented to demonstrate the efficiency of our approach. Interestingly, our empirical study reveals that the risk neutral variance processes recovered from market prices of options on S&P 500 index and EUR/USD exchange rate are indeed linearly mean-reverting.     

Conditions for the optimality of exponential smoothing forecast procedures

Exponential smoothing procedures, in particular those recommended byBrown [1962] are used extensively in many areas of economics, business and engineering. It is shown in this paper that:

1. Brown's forecasting procedures are optimal in terms of achieving minimum mean square error forecasts only if the underlying stochastic process is included in a limited subclass of ARIMA (p, d, q) processes. Hence, it is shown what assumptions are made when using these procedures.
2. The implication of point (i) is that the users ofBrown's procedures tacitly assume that the stochastic processes which occur in the real world are from the particular restricted subclass of ARIMA (p, d, q) processes. No reason can be found why these particular models should occur more frequently than others.
3. It is further shown that even if a stochastic process which would lead toBrown's model occurred, the actual methods used for making the forecasts are clumsy and much simpler procedures can be employed.

     

Universal Pareto dominance and welfare for plausible utility functions

We study Pareto efficiency in a setting that involves two kinds of uncertainty: Uncertainty over the possible outcomes is modeled using lotteries whereas uncertainty over the agents’ preferences over lotteries is modeled using sets of plausible utility functions. A lottery is universally Pareto undominated if there is no other lottery that Pareto dominates it for all plausible utility functions. We show that, under fairly general conditions, a lottery is universally Pareto undominated iff it is Pareto efficient for some vector of plausible utility functions, which in turn is equivalent to affine welfare maximization for this vector. In contrast to previous work on linear utility functions, we use the significantly more general framework of skew-symmetric bilinear (SSB) utility functions as introduced by Fishburn (1982). Our main theorem generalizes a theorem by Carroll (2010) and implies the ordinal efficiency welfare theorem. We discuss three natural classes of plausible utility functions, which lead to three notions of ordinal efficiency, including stochastic dominance efficiency, and conclude with a detailed investigation of the geometric and computational properties of these notions.     

Econometric specification of stochastic discount factor models

We consider the problem of derivative pricing when the stochastic discount factors are exponential-affine functions of underlying state variable. In particular we discuss the conditionally Gaussian framework and introduce semi-parametric pricing methods for models with path dependent drift and volatility. This approach is also applied to more complicated frameworks, such as pricing of a derivative written on an index, when the interest rate is stochastic.     

Can baby-boomers’ retirement increase stock prices?

In a dynamic asset-pricing model with Hyperbolic Absolute Risk Aversion preferences, investors who have Decreasing Relative Risk Aversion have an age dependent component in their optimal asset allocation rule. Unlike conventional models, this component affects equilibrium equity returns due to demographic trends in the same direction suggested by (Poterba, J. M. (2001). Demographic Structure and Asset Returns.The Review of Economics and Statistics, 83 (4) 565–584; Poterba, J. M. (2004). The Impact of Population Aging on Financial Markets, NBER Working Paper No. 10851). Calibration to US data between 1950 and 2050 reveals that between 1950 and 1970 this effect potentially added 0.15% to the 7.79% post-war long-term return (actual was 8.39%). As boomers joined the labor market (1970–2004) this positive effect turned negative (0.17–0.34%). Ignoring consumption and wealth effects, the model implies that this effect will draw 0.06% annually between 2005 and 2015 but add 0.22% between 2016 and 2050.     

New unit root asymptotics in the presence of deterministic trends

Recent work by Phillips (Econometrica 66 (1998) 1299) has shown that stochastic trends can be validly represented in empirical regressions in terms of deterministic functions of time. These representations offer an alternative mechanism for modelling stochastic trends. It is shown here that the alternate representations affect the asymptotics of all commonly used unit root tests in the presence of trends. In particular, the critical values of unit root tests diverge when the number of deterministic regressors K→∞ as the sample size n→∞. When they are appropriately recentered and standardized, unit root limit distributions are shown to be normal as K→∞.     

Endogenous participation risk in speculative markets

This paper analyses the dynamic implications of an asset-pricing model with incomplete participation due to entry costs. It is shown that heterogeneity in entry costs can lead to the existence of multiple stochastic sunspot equilibria, whereby the number of agents in the market and asset prices fluctuate endogeneously over time in the absence of fundamental uncertainty. Such asset-price fluctuations occur despite the uniqueness of the deterministic equilibrium, and thus bear no link to the usual notion of steady-state indeterminacy. In addition to excess volatility, the equilibria exhibit predictable and conditionally heteroskedastic returns.     

The method of endogenous gridpoints in theory and practice

The method of endogenous gridpoints (ENDG) significantly speeds up the solution to dynamic stochastic optimization problems with continuous state and control variables by avoiding repeated computations of expected outcomes while searching for optimal policy functions. I provide an interpolation technique for non-rectilinear grids that allow ENDG to be used in n-dimensional problems in an intuitive and computationally efficient way: the acceleration of ENDG with non-linear grid interpolation is nearly constant in the density of the grid. Further, ENDG has only been shown by example and has never been formally characterized. Using a theoretical framework for dynamic stochastic optimization problems, I formalize the method of endogenous gridpoints and present conditions for the class of models for which it can be used.     

Press freedom and jumps in stock prices

Press freedom varies substantially across countries. In a free environment, any news immediately becomes public knowledge through mediums including various electronic media and published materials. However, in an unfree environment, (economic) agents would have more discretionary powers to disclose good news immediately, while hiding bad news or releasing bad news slowly. We argue that this discretion affects stock prices and that stock markets in countries with a free press should be better processors of economic information. Using an equilibrium asset-pricing model in an economy under jump diffusion, we decompose the moments of the returns of international stock markets into a diffusive risk and a jump risk part. Using stock market data for a balanced panel of 50 countries, our results suggest that in countries with a free press, the better processing of bad news leads to more frequent negative jumps in stock prices. As a result, stock markets in those countries are characterized by higher volatility, driven by higher jump risk and more negative return asymmetry. The results are robust to the inclusion of various controls for governance and other country- or market-specific characteristics. We interpret these as good stock market characteristics because a free press improves welfare and increases economic growth.     

International borrowing without commitment and informational lags: Choice under uncertainty

A series of recent studies in economic growth theory have considered a class of models of international borrowing where, in the absence of a perfect investment commitment, the borrowing constraint depends on the historical performances of the country. Thus, a better level of past economic activity gives a higher reputation, thereby increasing the possibility of accessing the international credit market. This note considers this problem in a stochastic setting based on the volatility of the internal net capital. We study how the optimal consumption level and the maximal expected welfare depend on the combined influence of the trajectory of past economic variables and the volatile environment. In particular, we show how the strength of the history effect and the relative weight of the historical performance depend on the degree of risk.     

Moments structure of ? 1-stochastic volatility models

We consider Taylor??s stochastic volatility model (SVM) when the innovations of the hidden log-volatility process have a Laplace distribution (? ₁ exponential density), rather than the standard Gaussian distribution (? ₂) usually employed. Recently many investigations have employed ? ₁ metric to allow better modeling of the abrupt changes of regime observed in financial time series. However, the estimation of SVM is known to be difficult because it is a non-linear with an hidden markov process. Moreover, an additional difficulty yielded by the use of ? ₁ metric is the not differentiability of the likelihood function. An alternative consists in using a generalized or efficient method-of-moments (GMM/EMM) estimation. For this purpose, we derive here the moments and autocovariance function of such ? ₁-based stochastic volatility models.     

Optimal structure of joint inventory-pricing management with dual suppliers and different lead times

We consider a joint inventory-pricing control problem in a single-product, periodic-review, dual-supplier inventory system. The two suppliers have different lead times. One expedited supplier offers instantaneous replenishment, and one regular supplier requires an L-period lead time for delivery. The supply quantity is stochastic and the demand is price-dependent. For the expedited inventory replenishment, we characterize the optimal policy as a state-dependent almost-threshold policy by extending the stochastically linear in mid-point to a multidimensional setting. To investigate the optimal regular inventory replenishment and pricing policy, we propose the notions of partially stochastic translation (PST) and increasing partially stochastic translation (IPST), which help in obtaining the antimultimodularity preservation in dynamic programming problems. We provide properties, sufficient conditions, and examples for PST and IPST functions. By applying PST and IPST, we obtain the antimultimodularity of the profit functions. The antimultimodular profit functions ensure that the optimal regular ordering quantity and the optimal price are monotone in the current inventory level and outstanding order quantities. Moreover, we reveal that as the time interval increases, the effects of previous outstanding orders on the optimal regular ordering and pricing decisions are decreasing and increasing, respectively. PST and IPST also enable us to further characterize the optimal expedited ordering quantity as decreasing in the inventory level. However, the optimal expedited ordering quantity can be non-monotone with respect to the outstanding order quantities, as shown in the example.     

Envelope condition method with an application to default risk models

We develop an envelope condition method (ECM) for dynamic programming problems – a tractable alternative to expensive conventional value function iteration (VFI). ECM has two novel features: first, to reduce the cost of iteration on Bellman equation, ECM constructs policy functions using envelope conditions which are simpler to analyze numerically than first-order conditions. Second, to increase the accuracy of solutions, ECM solves for derivatives of value function jointly with value function itself. We complement ECM with other computational techniques that are suitable for high-dimensional problems, such as simulation-based grids, monomial integration rules and derivative-free solvers. The resulting value-iterative ECM method can accurately solve models with at least up to 20 state variables and can successfully compete in accuracy and speed with state-of-the-art Euler equation methods. We also use ECM to solve a challenging default risk model with a kink in value and policy functions.     

Stationary Markovian equilibrium in altruistic stochastic OLG models with limited commitment

We introduce a new class of infinite horizon altruistic stochastic OLG models with capital and labor, but without commitment between the generations. Under mild regularity conditions, for economies with either bounded or unbounded state spaces, continuous monotone Markov perfect Nash equilibrium (henceforth MPNE) are shown to exist, and form an antichain. Further, for each such MPNE, we can also construct a corresponding stationary Markovian equilibrium invariant distribution. We then show for many versions of our economies found in applied work in macroeconomics, unique MPNE exist relative to the space of bounded measurable functions. We also relate all of our results to those obtained by promised utility/continuation methods based upon the work of Abreu et al. (1990). As our results are constructive, we can provide characterizations of numerical methods for approximating MPNE, and we construct error bounds. Finally, we provide a series of examples to show the potential applications and limitations of our results.     

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.     

Asymmetric outcome in a symmetric dynamic duopoly

In a dynamic duopoly, will an initial asymmetry between firms increase or decrease over time? We examine this issue within a stochastic dynamic alternate-move duopoly model that explicitly accounts for action and reaction between firms. We consider two firms that are symmetric with regard to all primitives such as demand, cost and production functions, and which are subject to the same stochastic environment. The only asymmetry is with regard to their initial capital stocks which in turn asymmetrically influences their current and future profit and investment possibilities. We offer a characterization of the stochastic steady state and its supporting ergodic set for each firm. We are then able to identify the precise restrictions on the initial conditions under which the two firms either converge or diverge in the long run.     

Bayesian stochastic frontier analysis using WinBUGS

Markov chain Monte Carlo (MCMC) methods have become a ubiquitous tool in Bayesian analysis. This paper implements MCMC methods for Bayesian analysis of stochastic frontier models using the WinBUGS package, a freely available software. General code for cross-sectional and panel data are presented and various ways of summarizing posterior inference are discussed. Several examples illustrate that analyses with models of genuine practical interest can be performed straightforwardly and model changes are easily implemented. Although WinBUGS may not be that efficient for more complicated models, it does make Bayesian inference with stochastic frontier models easily accessible for applied researchers and its generic structure allows for a lot of flexibility in model specification.