General Distributional Assumptions Research Articles

Risk assessment and optimal scheduling of serial projects

The valuation and planning of complex projects are becoming increasingly challenging with rising market uncertainty and the deregulation of many industries, which have also raised the need for efficient risk management. We take the perspective of a private firm interested in sequential capacity expansion of a project and develop a framework for measuring the downside risk of the serial project and optimising the sequence of the stages. Under general distributional assumptions for the duration of each stage, we present an accurate representation of the project’s net present value (NPV) distribution based on a Pearson curve fit, leading to closed-form expressions for the associated risk measures. We then assess the impact of duration variability on the value at risk and demonstrate its role in stochastic project scheduling. We also account for the trade-off between maximising the expected NPV and minimising the risk exposure, and obtain the optimal schedule for risk-averse decision-makers. It becomes obvious that both the duration variability of each stage and the decision-makers’ risk preferences can significantly affect the optimal sequence of the stages and that high duration variability is not always undesirable, even for risk-averse decision-makers.

Exact and asymptotic identification-robust inference for dynamic structural equations with an application to New Keynesian Phillips Curves

Many models in econometrics involve endogeneity and lagged dependent variables. We start by observing that usual identification-robust (IR) tests are unreliable when model variables are nonstationary or nearly nonstationary. We propose IR methods which are also robust to nonstationarity: one Anderson-Rubin type procedure and two split-sample procedures. Our procedures are also robust to missing instruments. For distributional theory, three different sets of assumptions are considered. First, on assuming Gaussian structural errors, we show that two of the proposed statistics follow the standard F distribution. Second, for more general cases, we assume that the distribution of errors is completely specified up to an unknown scale factor, allowing the Monte Carlo test method to be applied. This assumption enables one to deal with non-Gaussian error distributions. For example, even when errors follow heavy-tailed distribution, such as the Cauchy distribution or more generally the family of stable distributions—which may not have moments and thus make inference difficult—our procedures provide simple and exact solutions. Third, we establish the asymptotic validity of our procedures under quite general distributional assumptions. We present simulation results showing that our procedures control their level correctly and have good power properties. The methods are applied to an empirical example, the New Keynesian Phillips curve, in which both weak identification and nonstationarity present challenges. The results of this empirical study suggest forward-looking behavior of U.S. inflation.

Stressing of Migration Matrices for IFRS 9 and ICAAP Calculations

Rating transition matrices have become a workhorse of the IFRS 9 expected credit loss and ICAAP stress test modelling. The standard method to stress a through-the-cycle transition matrix is based on a single factor Gaussian model with a correlation parameter that is usually estimated on the level of a product pool. The goal of the paper is to generalize the model allowing for more general distributional assumptions and to test empirically the sensitivity of the results with respect to these assumptions and different possible approaches to the correlation parameter estimation. We are not aware of any such empirical study in the literature. The results show that this dependence is very strong with the standard approach underestimating the results, as we argue, of a more precise calculation many times. Therefore, there is a significant model risk that needs to be taken into account in ICAAP/IFRS 9 implementation and dealt with in further research.

Asymptotic behavior of a critical fluid model for a multiclass processor sharing queue via relative entropy

This work concerns the asymptotic behavior of critical fluid model solutions for a multiclass processor sharing queue under general distributional assumptions. Such critical fluid model solutions are measure-valued functions of time. We prove that critical fluid model solutions converge to the set of invariant states as time goes to infinity, uniformly for all initial conditions lying in certain relatively compact sets. This generalizes an earlier single-class result of Puha and Williams to the more complex multiclass setting. In particular, several new challenges are overcome, including formulation of a suitable relative entropy functional and identifying a convenient form of the time derivative of the relative entropy applied to trajectories of critical fluid model solutions.

The multiple filter test for change point detection in time series

A framework for the detection of change points in the expectation in sequences of random variables is presented. Specifically, we investigate time series with general distributional assumptions that may show an unknown number of change points in the expectation occurring on multiple time scales and that may also contain change points in other parameters. To that end we propose a multiple filter test (MFT) that tests the null hypothesis of constant expectation and, in case of rejection of the null hypothesis, an algorithm that estimates the change points. The MFT has three important benefits. First, it allows for general distributional assumptions in the underlying model, assuming piecewise sequences of i.i.d. random variables, where also relaxations with regard to identical distribution or independence are possible. Second, it uses a MOSUM type statistic and an asymptotic setting in which the MOSUM process converges weakly to a functional of a Brownian motion which is then used to simulate the rejection threshold of the statistical test. This approach enables a simultaneous application of multiple MOSUM processes which improves the detection of change points that occur on different time scales. Third, we also show that the method is practically robust against changes in other distributional parameters such as the variance or higher order moments which might occur with or even without a change in expectation. A function implementing the described test and change point estimation is available in the R package MFT.

A stochastic program with time series and affine decision rules for the reservoir management problem

This paper proposes a multi-stage stochastic programming formulation for the reservoir management problem. Our problem specifically consists in minimizing the risk of floods over a fixed time horizon for a multi-reservoir hydro-electrical complex. We consider well-studied linear time series models and enhance the approach to consider heteroscedasticity. Using these stochastic processes under very general distributional assumptions, we efficiently model the support of the joint conditional distribution of the random inflows and update these sets as new data are assimilated. Using robust optimization techniques and affine decision rules, we embed these time series in a tractable convex program. This allows us to obtain good quality solutions rapidly and test our model in a realistic simulation framework using a rolling horizon approach. Finally, we study a river system in western Québec and perform various numerical experiments based on different inflow generators.

On the Ornstein–Zernike equation for stationary cluster processes and the random connection model

Abstract In the first part of this paper we consider a general stationary subcritical cluster model in ℝd. The associated pair-connectedness function can be defined in terms of two-point Palm probabilities of the underlying point process. Using Palm calculus and Fourier theory we solve the Ornstein–Zernike equation (OZE) under quite general distributional assumptions. In the second part of the paper we discuss the analytic and combinatorial properties of the OZE solution in the special case of a Poisson-driven random connection model.

Efficient Computation of Bayesian Optimal Discriminating Designs

ABSTRACTAn efficient algorithm for the determination of Bayesian optimal discriminating designs for competing regression models is developed, where the main focus is on models with general distributional assumptions beyond the “classical” case of normally distributed homoscedastic errors. For this purpose, we consider a Bayesian version of the Kullback–Leibler (KL). Discretizing the prior distribution leads to local KL-optimal discriminating design problems for a large number of competing models. All currently available methods either require a large amount of computation time or fail to calculate the optimal discriminating design, because they can only deal efficiently with a few model comparisons. In this article, we develop a new algorithm for the determination of Bayesian optimal discriminating designs with respect to the Kullback–Leibler criterion. It is demonstrated that the new algorithm is able to calculate the optimal discriminating designs with reasonable accuracy and computational time in situations where all currently available procedures are either slow or fail.

The size of the cliff at the border

The scale of international migration flows depends on moving costs that are, in turn, influenced by host-country policies. This paper shows how to estimate the influence of policies upon bilateral migration flows to multiple destinations. We rely on a Poisson pseudo-maximum likelihood estimator to derive estimates that are consistent under more general distributional assumptions on the underlying RUM model than the ones commonly adopted in the literature. We derive bounds for the estimated direct and indirect effects of migration policies that reflect the uncertainty connected to the use of aggregate data, and we show that bilateral migration flows can be highly sensitive to the immigration policies set by other destination countries, an externality that we are able to quantify.

Bootstrap corrections of treatment effect estimates following selection

Bias of treatment effect estimators can occur when the maximum effect of several treatments is to be determined or the effect of the selected treatment or subgroup has to be estimated. Since those estimates may contribute to the decision as to whether to continue a drug development program, to select a specific dose or a specific subgroup of patients, methods should be applied that ensure a realistic rather than an overoptimistic estimator of a treatment effect following selection. Selection bias is well studied for normally distributed variables and to a lesser extent for other types of distributions. However, many methods developed for bias correction apply primarily to specific distributions. Since there is always uncertainty about the underlying distribution of data, a more generally applicable method is of interest. The bootstrap has been developed among others to estimate the bias under fairly general distributional assumptions. The potential of the bootstrap in reducing estimator bias after selection is investigated.

On the estimation of asset pricing models using univariate betas

We derive asymptotic standard errors of risk premia estimates based on the popular two-pass cross-sectional regression methodology developed by Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973) when univariate betas are used as regressors. Our standard errors are robust to model misspecification and allow for general distributional assumptions. In testing whether the beta risk of a given factor is priced, our misspecification robust standard error can lead to economically different conclusions from those based on the Jagannathan and Wang (1998) standard error which is derived under the correctly specified model.

On the Estimation of Asset Pricing Models Using Univariate Betas

We derive asymptotic standard errors of risk premia estimates based on the popular two-pass cross-sectional regression methodology developed by Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973) when univariate betas are used as regressors. Our standard errors are robust to model misspecification and allow for general distributional assumptions. In testing whether the beta risk of a given factor is priced, our misspecification robust standard error can lead to economically different conclusions from those based on the Jagannathan and Wang (1998) standard error which is derived under the correctly specified model.

A Note on the Estimation of Asset Pricing Models Using Simple Regression Betas

Working Paper 2009-12 March 2009 Abstract: Since Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973), the two-pass cross-sectional regression (CSR) methodology has become the most popular tool for estimating and testing beta asset pricing models. In this paper, we focus on the case in which simple regression betas are used as regressors in the second-pass CSR. Under general distributional assumptions, we derive asymptotic standard errors of the risk premia estimates that are robust to model misspecification. When testing whether the beta risk of a given factor is priced, our misspecification robust standard error and the Jagannathan and Wang (1998) standard error (which is derived under the correctly specified model) can lead to different conclusions. JEL classification: G12 Key words: two-pass cross-sectional regressions, risk premia, model misspecification, simple regression betas, multivariate betas Introduction In the empirical asset pricing literature, the popular two-pass cross-sectional regression (CSR) methodology developed by Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973) is often used for estimating risk premia and testing pricing models that relate expected security returns to security betas on economic factors (beta pricing models). Although there are many variations of this two-pass methodology, its basic approach always involves two steps. In the first pass, the betas of the test assets are estimated using the usual ordinary least squares (OLS) time series regression of returns on some common factors. In the second pass, the returns on test assets are regressed on the estimated betas obtained from the first pass. By running this second-pass CSR on a period-by-period basis, we obtain time series of the intercept and the slope coefficients. The average values of the intercept and the slope coefficients are then used as estimates of the zero-beta rate and the risk premia. Usually, asset betas are defined as the OLS slope coefficients in the multiple regression of asset returns on factors and are referred to as multiple regression or multivariate betas. However, there is a potential issue with the use of multiple regression betas: unless the factors are uncorrelated, the beta of an asset with respect to a particular factor in general depends on what other factors are included in the first-pass time series OLS regression. As a result, a factor can possess additional explanatory power for the cross-sectional differences in expected returns but yet have a zero risk premium in a model with multiple factors. This makes it problematic to use the risk premium of a factor for the purpose of model selection. To overcome this problem, Chen, Roll, and Ross (1986) and Jagannathan and Wang (1996, 1998) define the beta of an asset with respect to a given factor as the OLS slope coefficient in a simple regression of its return on the factor. These betas are normally referred to as simple regression or univariate betas. In models with simple regression betas, adding or deleting a factor in a model will not change the values of the betas corresponding to the other factors and selecting models based on risk premia becomes more meaningful. (1) Jagannathan and Wang (1998) present an asymptotic theory for models with simple regression betas. (2) However, their asymptotic results rest on the assumption that expected returns are exactly linear in the betas, i.e., the beta pricing model is correctly specified. It is difficult to justify this assumption when estimating the zero-beta rate and risk premia parameters for many different models because some (if not all) of the models are bound to be misspecified. Since asset pricing models are, at best, approximations of reality, it is inevitable that we will often, knowingly or unknowingly (since asset pricing tests have limited power), estimate an expected return relation that departs from exact linearity in the betas. …

Seat biases of apportionment methods under general distributional assumptions

Apportionment methods round vote proportions to integer numbers of seats in parliament. Seat biases of an apportionment method are expected differences between the seats actually apportioned to the parties and their ideal fractional shares of seats. In this letter it is shown that, for many popular apportionment methods, asymptotic seat biases for an increasing number of seats in parliament do not depend on the distributional assumption for the vote counts.

Median‐unbiased Estimation and Exact Inference Methods for First‐order Autoregressive Models with Conditional Heteroscedasticity of Unknown Form

Abstract. Consider the first‐order autoregressive model yt = φyt−1 + ɛt, t = 1,…, T, with arbitrary initial non‐zero value y0. Assuming that the error terms ɛt are independently distributed according to median‐zero distributions [Zieliński (1999)Journal of Time Series Analysis, Vol. 20, p. 477] shows that the estimator conjectured by Hurwicz (1950)Statistical Inference in Dynamic Economic Models. New York, NY: Wiley – the median of the consecutive ratios yt/yt−1– is an exactly median‐unbiased estimator of the autoregressive parameter φ. This paper shows that the Hurwicz estimator remains median‐unbiased under more general distributional assumptions, without assuming statistical independence. In particular, no restrictions are placed on the degree of heterogeneity and dependence of the conditional variance process. A computationally efficient method is also proposed to build exact confidence intervals for the autoregressive parameter which are valid in finite samples for any value of φ on the real line.

Discrete-Time Financial Planning Models Under Loss-Averse Preferences

We consider a dynamic asset allocation problem formulated as a mean-shortfall model in discrete time. A characterization of the solution is derived analytically under general distributional assumptions for serially independent risky returns. The solution displays risk taking under shortfall, as well as a specific form of time diversification. Also, for a representative stock-return distribution, risk taking increases monotonically with the number of decision moments given a fixed horizon. This is related to the well-known casino effect arising in a downside-risk and expected return framework. As a robustness check, we provide results for a modified objective with a quadratic penalty on shortfall. An analytical solution for a single-stage setup is derived, and numerical results for the two-period model and time diversification are provided.

Ancillaries and Conditional Inference

Sufficiency has long been regarded as the primary reduction pro- cedure to simplify a statistical model, and the assessment of the procedure involves an implicit global repeated sampling principle. By contrast, condi- tional procedures are almost as old and yet appear only occasionally in the central statistical literature. Recent likelihood theory examines the form of a general large sample statistical model and finds that certain natural condi- tional procedures provide, in wide generality, the definitive reduction from the initial variable to a variable of the same dimension as the parameter, a variable that can be viewed as directly measuring the parameter. We begin with a discussion of two intriguing examples from the literature that compare conditional and global inference methods, and come quite extraordinarily to opposite assessments concerning the appropriateness and validity of the two approaches. We then take two simple normal examples, with and with- out known scaling, and progressively replace the restrictive normal location assumption by more general distributional assumptions. We find that suffi- ciency typically becomes inapplicable and that conditional procedures from large sample likelihood theory produce the definitive reduction for the analy- sis. We then examine the vector parameter case and find that the elimination of nuisance parameters requires a marginalization step, not the commonly profferred conditional calculation that is based on exponential model struc- ture. Some general conditioning and modelling criteria are then introduced. This is followed by a survey of common ancillary examples, which are then assessed for conformity to the criteria. In turn, this leads to a discussion of the place for the global repeated sampling principle in statistical inference. It is argued that the principle in conjunction with various optimality criteria has been a primary factor in the longstanding attachment to the sufficiency approach and in the related neglect of the conditioning procedures based di- rectly on available evidence.

Tests of Mean-Variance Spanning

In this paper, we conduct a comprehensive study of tests for mean-variance spanning. Under the popular regression framework of Huberman and Kandel (1987), we provide geometric interpretations of three asymptotic tests (likelihood ratio, Wald, and Lagrange multiplier) of mean-variance spanning. Under normality assumption, we present their exact distributions and analyze their power comprehensively. Under general distributional assumptions, we review spanning tests based on the generalized method of moments (GMM), provide new GMM spanning test, and evaluate their performance. In addition, we compare the performance of various spanning tests in the regression framework with those cast in the stochastic discount factor framework. Our results suggest that the two set-ups have similar properties when returns are normally distributed, but the regression framework performs better when returns follow a multivariate Student-t distribution.

On stochastic machine scheduling with general distributional assumptions

A set of stochastic jobs is to be processed on a single machine which is subject to breakdowns and repairs. Under quite general conditions on the job's processing times, the uptimes and the repairtimes of the machine, we find a simple optimal permutation policy for the weighted sum of an increasing function of the completion times. Some optimal policies for two special cases, which have not been investigated in the literature, are given. Moreover, an upper bound is established on the loss incurred when a processing policy is adopted under a simplifying assumption of exponential processing requirements

Improved invariant set estimation for general-scale families

Confidence intervals for any power of a scale parameter of a distribution are constructed. We first state the needed general distributional assumptions and show how to construct the minimum length location-scale invariant interval, having a predetermined coverage coefficient 1− α. If we then relax the invariance restriction, we obtain an improved interval. Using information relating the size of the location parameter to that of the scale parameter, we shift the minimum length interval closer to zero, simultaneously bringing the endpoints closer to each other. These intervals have guaranteed coverage probability uniformly greater than a predetermined value 1− α, and have uniformly shorter length than the shortest location scale invariant interval. We illustrate our method with some examples.