CONFIDENCE INTERVALS FOR ABSORBING MARKOV CHAIN PROBABILITIES APPLIED TO LOAN PORTFOLIOS

The problem of estimating steady state absorption probabilities for first order stationary Markov chains having a finite state space is examined. As model parameters, these probabilities are analytic functions of transition probabilities Q and R, and they can be represented as P= (I-Q)^-1R. Estimators P̌ may be obtained by replacing the transition probabilities by their maximum likelihood estimators Q and Ř under multinomial theory. Using large sample multivariate normal theory, one can derive the asymptotic distribution of these estimators and can obtain large sample confidence intervals. Finally, an application related to estimating loss reserves for an installment loan portfolio assumed to satisfy a Markov chain is discussed.     

Analysis and Management of Periodic Review,Order‐Up‐To Level Inventory Systems with Order Crossover

In this article, we investigate the (R, S) periodic review, order‐up‐to level inventory control system with stochastic demand and variable leadtimes. Variable leadtimes can lead to order crossover, in which some orders arrive out of sequence. Most theoretical studies of order‐up‐to inventory systems under variable leadtimes assume that crossovers do not occur and, in so doing, overestimate the standard deviation of the realized leadtime distribution and prescribe policies that can inflate inventory costs. We develop a new analytic model of the expected costs associated with this system, making use of a novel approximation of the realized (reduced) leadtime standard deviation resulting from order crossovers. Extensive experimentation through simulation shows that our model closely approximates the true expected cost and can be used to find values of R and S that provide an expected cost close to the minimum cost. Taking account of, as opposed to ignoring, crossovers leads, on average, to substantial improvements in accuracy and significant cost reductions. Our results are particularly useful for managers seeking to reduce inventory costs in supply chains with variable leadtimes.     

Vertical Integration with Price‐Setting Competitive Newsvendors*

This article addresses the problem of distribution channel design under demand uncertainty. We consider two manufacturers, each producing a substitutable product and selling it through either a decentralized or an integrated retail store, which is modeled as a price‐setting newsvendor. A multiplicative demand function incorporating a random shock term is assumed. Of primary interest is how demand uncertainty and production cost affect the equilibrium distribution channel structure. Results indicate the following: On the one hand, if the random shock term is uniformly distributed on [0, x], then the equilibrium design does not depend on the variance of the shock. On the other hand, if the random shock term is uniformly distributed on [1 −r, 1 +r], then the equilibrium design does depend on the variance of the shock. In particular, an increase in r favors the integrated structure where both channels are integrated and hurts the decentralized configuration where both channels are decentralized. Additionally, we explain the qualitative similarities and differences between the structural properties of the equilibrium distribution channel structure when demand is, and is not, uncertain. We also establish that production cost always favors the decentralized structure, while it hurts the integrated one. One important managerial implication of our study is that, by reducing the number of decisions made in supply chains, the impact of demand uncertainty can be controlled to a certain extent.     

Aggregation and Market Demand: An Exterior Differential Calculus Viewpoint

We analyze under which conditions a given vector field can be disaggregated as a linear combination of gradients. This problem is typical of aggregation theory, as illustrated by the literature on the characterization of aggregate market demand and excess demand. We argue that exterior differential calculus provides very useful tools to address these problems. In particular, we show, using these techniques, that any analytic mapping in Rⁿ satisfying Walras Law can be locally decomposed as the sum of n individual, utility-maximizing market demand functions. In addition, we show that the result holds for arbitrary (price-dependent) income distributions, and that the decomposition can be chosen such that it varies continuously with the mapping. Finally, when income distribution can be freely chosen, then decomposition requires only n/2 agents.     

A Bias–Reduced Log–Periodogram Regression Estimator for the Long–Memory Parameter

In this paper, we propose a simple bias–reduced log–periodogram regression estimator, ^d_r, of the long–memory parameter, d, that eliminates the first– and higher–order biases of the Geweke and Porter–Hudak (1983) (GPH) estimator. The bias–reduced estimator is the same as the GPH estimator except that one includes frequencies to the power 2k for k=1,…,r, for some positive integer r, as additional regressors in the pseudo–regression model that yields the GPH estimator. The reduction in bias is obtained using assumptions on the spectrum only in a neighborhood of the zero frequency. Following the work of Robinson (1995b) and Hurvich, Deo, and Brodsky (1998), we establish the asymptotic bias, variance, and mean–squared error (MSE) of ^d_r, determine the asymptotic MSE optimal choice of the number of frequencies, m, to include in the regression, and establish the asymptotic normality of ^d_r. These results show that the bias of ^d_r goes to zero at a faster rate than that of the GPH estimator when the normalized spectrum at zero is sufficiently smooth, but that its variance only is increased by a multiplicative constant. We show that the bias–reduced estimator ^d_r attains the optimal rate of convergence for a class of spectral densities that includes those that are smooth of order s≥1 at zero when r≥(s−2)/2 and m is chosen appropriately. For s>2, the GPH estimator does not attain this rate. The proof uses results of Giraitis, Robinson, and Samarov (1997). We specify a data–dependent plug–in method for selecting the number of frequencies m to minimize asymptotic MSE for a given value of r. Some Monte Carlo simulation results for stationary Gaussian ARFIMA (1, d, 1) and (2, d, 0) models show that the bias–reduced estimators perform well relative to the standard log–periodogram regression estimator.     

THE STATIONARY BEER GAME

This paper presents a variant of the popular beer game. We call the new game the stationary beer game, which models the material and information flows in a production‐distribution channel serving a stationary market where the customer demands in different periods are independent and identically distributed. Different players, who all know the demand distribution, manage the different stages of the channel. Summarizing the initial experience with the stationary beer game, the paper provides compelling reasons why this game is an effective teaching tool.     

Revenue Management for Intermodal Transportation: The Role of Dynamic Forecasting

We study a joint capacity leasing and demand acceptance problem in intermodal transportation. The model features multiple sources of evolving supply and demand, and endogenizes the interplay of three levers—forecasting, leasing, and demand acceptance. We characterize the optimal policy, and show how dynamic forecasting coordinates leasing and acceptance. We find (i) the value of dynamic forecasting depends critically on scarcity, stochasticity, and volatility; (ii) traditional mean‐value equivalence approach performs poorly in volatile intermodal context; (iii) mean‐value‐based forecast may outperform stationary distribution‐based forecast. Our work enriches revenue management models and applications. It advances our understanding on when and how to use dynamic forecasting in intermodal revenue management.     

Dynamic Supply Risk Management with Signal‐Based Forecast,Multi‐Sourcing,and Discretionary Selling

We examine the critical role of advance supply signals—such as suppliers’ financial health and production viability—in dynamic supply risk management. The firm operates an inventory system with multiple demand classes and multiple suppliers. The sales are discretionary and the suppliers are susceptible to both systematic and operational risks. We develop a hierarchical Markov model that captures the essential features of advance supply signals, and integrate it with procurement and selling decisions. We characterize the optimal procurement and selling policy, and the strategic relationship between signal‐based forecast, multi‐sourcing, and discretionary selling. We show that higher demand heterogeneity may reduce the value of discretionary selling, and that the mean value‐based forecast may outperform the stationary distribution‐based forecast. This work advances our understanding on when and how to use advance supply signals in dynamic risk management. Future supply risk erodes profitability but enhances the marginal value of current inventory. A signal of future supply shortage raises both base stock and demand rationing levels, thereby boosting the current production and tightening the current sales. Signal‐based dynamic forecast effectively guides the firm's procurement and selling decisions. Its value critically depends on supply volatility and scarcity. Ignoring advance supply signals can result in misleading recommendations and severe losses. Signal‐based dynamic supply forecast should be used when: (a) supply uncertainty is substantial, (b) supply‐demand ratio is moderate, (c) forecast precision is high, and (d) supplier heterogeneity is high.     

A hybrid beam search looking-ahead algorithm for the circular packing problem

In this paper, we study the circular packing problem. Its objective is to pack a set of n circular pieces into a rectangular plate R of fixed dimensions L×W. Each piece’s type i, i=1,…,m, is characterized by its radius r _i and its demand b _i . The objective is to determine the packing pattern corresponding to the minimum unused area of R for the circular pieces placed. This problem is solved by using a hybrid algorithm that adopts beam search and a looking-ahead strategy. A node at a level ℓ of the beam-search tree contains a partial solution corresponding to the circles already placed inside R. Each node is then evaluated using a looking-ahead strategy, based on the minimum local-distance heuristic, by computing the corresponding complete solution. The nodes leading to the best solutions at level ℓ are then chosen for branching. A multi-start strategy is also considered in order to diversify the search space. The computational results show, on a set of benchmark instances, the effectiveness of the proposed algorithm.     

Newsvendor Problems with Demand Shocks and Unknown Demand Distributions

In today's competitive market, demand volume and even the underlying demand distribution can change quickly for a newsvendor seller. We refer to sudden changes in demand distribution as demand shocks. When a newsvendor seller has limited demand distribution information and also experiences underlying demand shocks, the majority of existing methods for newsvendor problems may not work well since they either require demand distribution information or assume stationary demand distribution. We present a new, robust, and effective machine learning algorithm for newsvendor problems with demand shocks but without any demand distribution information. The algorithm needs only an approximate estimate of the lower and upper bounds of demand range; no other knowledge such as demand mean, variance, or distribution type is necessary. We establish the theoretical bounds that determine this machine learning algorithm's performance in handling demand shocks. Computational experiments show that this algorithm outperforms the traditional approaches in a variety of situations including large and frequent shocks of the demand mean. The method can also be used as a meta‐algorithm by incorporating other traditional approaches as experts. Working together, the original algorithm and the extended meta‐algorithm can help manufacturers and retailers better adapt their production and inventory control decisions in dynamic environments where demand information is limited and demand shocks are frequent      

A Newsvendor Who Chooses Informational Effort

We study a newsvendor who can acquire the services of a forecaster, or, more generally, an information gatherer (IG) to improve his information about demand. When the IG's effort increases, does the average ex ante order quantity rise or fall? Do average ex post sales rise or fall? Improvements in information technology and in the services offered by forecasters provide motivation for the study of these questions. Much depends on our model of the IG and his efforts. We study an IG who sends a signal to a classic single‐period newsvendor. The signal defines the newsvendor's posterior probability distribution on the possible demands and the newsvendor uses that posterior to calculate the optimal order. Each of the possible posteriors is a scale/location transform of the same base distribution. When the IG works harder, the average scale parameter drops. Higher IG effort is always useful to the newsvendor. We show that there is a critical value of order cost. For costs on one side of this value more IG effort leads to a higher average ex ante order and for costs on the other side to a lower average order. But for all costs, more IG effort leads to higher average ex post sales. We obtain analogous results for a “regret‐averse” newsvendor who suffers a penalty that is a nonlinear function of the discrepancy between quantity ordered and true demand.     

Medial Axis and Optimal Locations for Min-Max Sphere Packing

We study the following min-max sphere packing problem originated from radiosurgical treatment planning using gamma knife (Bourland and Wu, 1996; Wu, 1996). Given an input (R, V), where R is a 3-dimensional (3D) bounded region and V a positive integer, find a packing of R using the minimum number of spheres (spheres may not be identical) such that the covered volume is at least V, and the number of points on the boundary of R touched by spheres is maximized. Bourland and Wu (1996) and Wu (1996), devised a greedy algorithm to solve the problem based on medial axis analysis. In particular, the algorithm places the center of each sphere on the medial axis of each subsequent region starting from R. While this approach has met with certain success, we show that medial axis does not always provide optimal locations for min-max sphere packing.     

Myopic Analysis for Multi‐Echelon Inventory Systems with Batch Ordering and Nonstationary/Time‐Correlated Demands

We provide an exact myopic analysis for an N‐stage serial inventory system with batch ordering, linear ordering costs, and nonstationary demands under a finite planning horizon. We characterize the optimality conditions of the myopic nested batching newsvendor (NBN) policy and the myopic independent batching newsvendor (IBN) policy, which is a single‐stage approximation. We show that echelon reorder levels under the NBN policy are upper bounds of the counterparts under both the optimal policy and the IBN policy. In particular, we find that the IBN policy has bounded deviations from the optimal policy. We further extend our results to systems with martingale model of forecast evolution (MMFE) and advance demand information. Moreover, we provide a recursive computing procedure and optimality conditions for both heuristics which dramatically reduces computational complexity. We also find that the NBN problem under the MMFE faced by one stage has one more dimension for the forecast demand than the one faced by its downstream stage and that the NBN policy is optimal for systems with advance demand information and stationary problem data. Numerical studies demonstrate that the IBN policy outperforms on average the NBN policy over all tested instances when their optimality conditions are violated.     

Multiperson Bargaining and Strategic Complexity

We investigate the effect of introducing costs of complexity in the n‐person unanimity bargaining game. As is well‐known, in this game every individually rational allocation is sustainable as a Nash equilibrium (also as a subgame perfect equilibrium if players are sufficiently patient and if n & 2). Moreover, delays in agreement are also possible in such equilibria. By limiting ourselves to a plausible notion of complexity that captures length of memory, we find that the introduction of complexity costs (lexicographically with the standard payoffs) does not reduce the range of possible allocations but does limit the amount of delay that can occur in any agreement. In particular, we show that in any n‐player game, for any allocation z, an agreement on z at any period t can be sustained as a Nash equilibrium of the game with complexity costs if and only if t≤n. We use the limit on delay result to establish that, in equilibrium, the strategies implement stationary behavior. Finally, we also show that ‘noisy Nash equilibrium’ with complexity costs sustains only the unique stationary subgame perfect equilibrium allocation.     

Admissibility in Games1

Suppose that each player in a game is rational, each player thinks the other players are rational, and so on. Also, suppose that rationality is taken to incorporate an admissibility requirement—that is, the avoidance of weakly dominated strategies. Which strategies can be played? We provide an epistemic framework in which to address this question. Specifically, we formulate conditions of rationality and mth‐order assumption of rationality (RmAR) and rationality and common assumption of rationality (RCAR). We show that (i) RCAR is characterized by a solution concept we call a “self‐admissible set”; (ii) in a “complete” type structure, RmAR is characterized by the set of strategies that survive m+1 rounds of elimination of inadmissible strategies; (iii) under certain conditions, RCAR is impossible in a complete structure.     

Robust Priors in Nonlinear Panel Data Models

Many approaches to estimation of panel models are based on an average or integrated likelihood that assigns weights to different values of the individual effects. Fixed effects, random effects, and Bayesian approaches all fall into this category. We provide a characterization of the class of weights (or priors) that produce estimators that are first‐order unbiased. We show that such bias‐reducing weights will depend on the data in general unless an orthogonal reparameterization or an essentially equivalent condition is available. Two intuitively appealing weighting schemes are discussed. We argue that asymptotically valid confidence intervals can be read from the posterior distribution of the common parameters when N and T grow at the same rate. Next, we show that random effects estimators are not bias reducing in general and we discuss important exceptions. Moreover, the bias depends on the Kullback–Leibler distance between the population distribution of the effects and its best approximation in the random effects family. Finally, we show that, in general, standard random effects estimation of marginal effects is inconsistent for large T, whereas the posterior mean of the marginal effect is large‐T consistent, and we provide conditions for bias reduction. Some examples and Monte Carlo experiments illustrate the results.     

Replenishment Policies for Multi‐Product Stochastic Inventory Systems with Correlated Demand and Joint‐Replenishment Costs

This study analyzes optimal replenishment policies that minimize expected discounted cost of multi‐product stochastic inventory systems. The distinguishing feature of the multi‐product inventory system that we analyze is the existence of correlated demand and joint‐replenishment costs across multiple products. Our objective is to understand the structure of the optimal policy and use this structure to construct a heuristic method that can solve problems set in real‐world sizes/dimensions. Using an MDP formulation we first compute the optimal policy. The optimal policy can only be computed for problems with a small number of product types due to the curse of dimensionality. Hence, using the insight gained from the optimal policy, we propose a class of policies that captures the impact of demand correlation on the structure of the optimal policy. We call this class (s, c, d, S)‐policies, and also develop an algorithm to compute good policies in this class, for large multi‐product problems. Finally using an exhaustive set of computational examples we show that policies in this class very closely approximate the optimal policy and can outperform policies analyzed in prior literature which assume independent demand. We have also included examples that illustrate performance under the average cost objective.     

Optimal Bandwidth Selection in Heteroskedasticity–Autocorrelation Robust Testing

This paper considers studentized tests in time series regressions with nonparametrically autocorrelated errors. The studentization is based on robust standard errors with truncation lag M=bT for some constant b∈(0, 1] and sample size T. It is shown that the nonstandard fixed‐b limit distributions of such nonparametrically studentized tests provide more accurate approximations to the finite sample distributions than the standard small‐b limit distribution. We further show that, for typical economic time series, the optimal bandwidth that minimizes a weighted average of type I and type II errors is larger by an order of magnitude than the bandwidth that minimizes the asymptotic mean squared error of the corresponding long‐run variance estimator. A plug‐in procedure for implementing this optimal bandwidth is suggested and simulations (not reported here) confirm that the new plug‐in procedure works well in finite samples.     

Hamiltonicity of hypercubes with a constraint of required and faulty edges

Let R and F be two disjoint edge sets in an n-dimensional hypercube Q _n . We give two constructing methods to build a Hamiltonian cycle or path that includes all the edges of R but excludes all of F. Besides, considering every vertex of Q _n incident to at most n−2 edges of F, we show that a Hamiltonian cycle exists if (A) |R|+2|F|≤2n−3 when |R|≥2, or (B) |R|+2|F|≤4n−9 when |R|≤1. Both bounds are tight. The analogous property for Hamiltonian paths is also given. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. Lih-Hsing Hsu’s research project is partially supported by NSC 95-2221-E-233-002. Shu-Chung Liu’s research project is partially supported by NSC 90-2115-M-163-003 and 95-2115-M-163-002. Yeong-Nan Yeh’s research project is partially supported by NSC 95-2115-M-001-009.     

Sign Restrictions,Structural Vector Autoregressions,and Useful Prior Information

This paper makes the following original contributions to the literature. (i) We develop a simpler analytical characterization and numerical algorithm for Bayesian inference in structural vector autoregressions (VARs) that can be used for models that are overidentified, just‐identified, or underidentified. (ii) We analyze the asymptotic properties of Bayesian inference and show that in the underidentified case, the asymptotic posterior distribution of contemporaneous coefficients in an n‐variable VAR is confined to the set of values that orthogonalize the population variance–covariance matrix of ordinary least squares residuals, with the height of the posterior proportional to the height of the prior at any point within that set. For example, in a bivariate VAR for supply and demand identified solely by sign restrictions, if the population correlation between the VAR residuals is positive, then even if one has available an infinite sample of data, any inference about the demand elasticity is coming exclusively from the prior distribution. (iii) We provide analytical characterizations of the informative prior distributions for impulse‐response functions that are implicit in the traditional sign‐restriction approach to VARs, and we note, as a special case of result (ii), that the influence of these priors does not vanish asymptotically. (iv) We illustrate how Bayesian inference with informative priors can be both a strict generalization and an unambiguous improvement over frequentist inference in just‐identified models. (v) We propose that researchers need to explicitly acknowledge and defend the role of prior beliefs in influencing structural conclusions and we illustrate how this could be done using a simple model of the U.S. labor market.