https://hdl.handle.net/2123/8124 Sun, 07 Jun 2026 08:43:49 GMT 2026-06-07T08:43:49Z https://hdl.handle.net/2123/34743 Corrected Forecast Combinations Vasnev, Andrey; Liu, Chu-An This paper proposes corrected forecast combinations when the original combined forecast errors are serially dependent. Motivated by the classic Bates and Granger (1969) example, we show that combined forecast errors can be strongly autocorrelated and that a simple correction – adding a fraction of the previous combined error to the next-period combined forecast – can deliver sizable improvements in forecast accuracy, often exceeding the original gains from combining. We formalize the approach within the conditional-risk framework of Gibbs and Vasnev (2024), in which the combined error decomposes into a predictable component (measurable at the forecast origin) and an innovation. We then link this correction to efficient estimation of combination weights under time-series dependence via GLS, allowing joint estimation of weights and an error-covariance structure. Using the U.S. Survey of Professional Forecasters for major macroeconomic indices across various subsamples (including pre/post-2000, GFC, and COVID), we find that a parsimonious correction of the mean forecast with a coefficient around 0.5 is a robust starting point and often yields material improvements in forecast accuracy. For optimal-weight forecasts, the correction substantially mitigates the forecast combination puzzle by turning poorly performing out-of-sample optimal-weight combinations into competitive forecasts. Thu, 01 Jan 2026 00:00:00 GMT https://hdl.handle.net/2123/34743 2026-01-01T00:00:00Z https://hdl.handle.net/2123/31836 Combining simple multivariate HAR-like models for portfolio construction Clements, Adam; Vasnev, Andrey L. Forecasts of the covariance matrix of returns is a crucial input into portfolio construction. In recent years multivariate version of the Heterogenous AutoRegressive (HAR) models have been designed to utilise realised measures of the covariance matrix to generate forecasts. This paper shows that combining forecasts from simple HAR-like models provide more coefficients estimates, stable forecasts and lower portfolio turnover. The economic benefits of the combination approach become crucial when transactions costs are taken into account. This combination approach also provides benefits in the context of direct forecasts of the portfolio weights. Economic benefits are observed at both 1-day and 1-week ahead forecast horizons. Sun, 01 Jan 2023 00:00:00 GMT https://hdl.handle.net/2123/31836 2023-01-01T00:00:00Z https://hdl.handle.net/2123/31835 The role of data and priors in estimating climate sensitivity Ikefuji, Masako; Magnus, Jan R.; Vasnev, Andrey L. In Bayesian theory, the data together with the prior produce a posterior. We show that it is also possible to follow the opposite route, that is, to use data and posterior information (both of which are observable) to reveal the prior (which is not observable). We then apply the theory to equilibrium climate sensitivity as reported by the Intergovernmental Panel on Climate Change in an attempt to get some insight into the prior beliefs of the IPCC scientists. It appears that the data contain much less information than one might think, due to the presence of correlation. We conclude that the prior in the fifth IPCC report was too low, and in the sixth report too high. Sun, 01 Jan 2023 00:00:00 GMT https://hdl.handle.net/2123/31835 2023-01-01T00:00:00Z https://hdl.handle.net/2123/30211 Base-Stock Policies with Constant Lead Time: Closed-Form Solutions and Applications LI, ZHAOLIN (ERICK); LIANG, GUITIAN; FU, QI (GRACE); TEO, CHUNG-PIAW We study stationary base-stock policies for multiperiod dynamic inventory systems with a constant lead time and independently and identically distributed (iid) demands. When ambiguities in the underlying demand distribution arise, we derive the robust optimal base-stock level in closed forms using only the mean and variance of the iid demands. This simple solution performs exceptionally well in numerical experiments, and has important applications for several classes of problems in Operations Management. More important, we propose a new distribution-free method to derive robust solutions for multiperiod dynamic inventory systems. We formulate a zero-sum game in which the firm chooses a base-stock level to minimize its cost while Nature (which is the firm’s opponent) chooses an iid two-point distribution to maximize the firm’s time-average cost in the steady state. By characterizing the steady-state equilibrium, we demonstrate how lead time can affect the firm’s equilibrium strategy (i.e., the firm’s robust base-stock level), Nature’s equilibrium strategy (i.e., the firm’s most unfavorable distribution), and the value of the zero-sum game (i.e., the firm’s optimized worst-case time-average cost). With either backorders or lost sales, our numerical study shows that superior performance can be obtained using our robust base-stock policies, which mitigate the consequence of distribution mis-specification. Wed, 15 Mar 2023 00:00:00 GMT https://hdl.handle.net/2123/30211 2023-03-15T00:00:00Z https://hdl.handle.net/2123/29354 Global combinations of expert forecasts Qian, Yilin; Thompson, Ryan; Vasnev, Andrey L Expert forecast combination—the aggregation of individual forecasts from multiple subjectmatter experts— is a proven approach to economic forecasting. To date, research in this area has exclusively concentrated on local combination methods, which handle separate but related forecasting tasks in isolation. Yet, it has been known for over two decades in the machine learning community that global methods, which exploit taskrelatedness, can improve on local methods that ignore it. Motivated by the possibility for improvement, this paper introduces a framework for globally combining expert forecasts. Through our framework, we develop global versions of several existing forecast combinations. To evaluate the efficacy of these new global forecast combinations, we conduct extensive comparisons using synthetic and real data. Our real data comparisons, which involve expert forecasts of core economic indicators in the Eurozone, are the first empirical evidence that the accuracy of global combinations of expert forecasts can surpass local combinations. Sat, 01 Jan 2022 00:00:00 GMT https://hdl.handle.net/2123/29354 2022-01-01T00:00:00Z https://hdl.handle.net/2123/27307 On the uncertainty of a combined forecast: The critical role of correlation Magnus, Jan; Vasnev, Andrey The purpose of this paper is to show that the effect of the zero-correlation assumption in combining forecasts can be huge, and that ignoring (positive) correlation can lead to confidence bands around the forecast combination that are much too narrow. In the typical case where three or more forecasts are combined, the estimated variance increases without bound when correlation increases. Intuitively, this is because similar forecasts provide little information if we know that they are highly correlated. Although we concentrate on forecast combinations and confidence bands, our theory applies to any statistic where the observations are linearly combined. We apply our theoretical results to explain why forecasts by Central Banks (in our case, the Bank of Japan) are so frequently misleadingly precise. In most cases, a correlation above 0.7 is required to produce reasonable confidence bands. Fri, 01 Jan 2021 00:00:00 GMT https://hdl.handle.net/2123/27307 2021-01-01T00:00:00Z https://hdl.handle.net/2123/25045 Forecast combination puzzle in the HAR model Clements, Adam; Vasnev, Andrey The Heterogeneous Autoregressive (HAR) model of Corsi (2009) has become the benchmark model for predicting realized volatility given its simplicity and consistent empirical performance. Many modifications and extensions to the original model have been proposed that often only provide incremental forecast improvements. In this paper, we take a step back and view the HAR model as a forecast combination that combines three predictors: previous day realization (or random walk forecast), previous week average, and previous month average. When applying the Ordinary Least Squares (OLS) to combine the predictors, the HAR model uses optimal weights that are known to be problematic in the forecast combination literature. In fact, the simple average forecast often outperforms the optimal combination in many empirical applications. We investigate the performance of the simple average forecast for the realized volatility of the Dow Jones Industrial Average equity index. We find dramatic improvements in forecast accuracy across all horizons and different time periods. This is the first time the forecast combination puzzle is identified in this context. Fri, 01 Jan 2021 00:00:00 GMT https://hdl.handle.net/2123/25045 2021-01-01T00:00:00Z https://hdl.handle.net/2123/23695 Two-Stage Stochastic and Robust Optimization for Non-Adaptive Group Testing Ho-Nguyen, Nam We consider the problem of detecting defective items amongst a large collection, by conducting tests of individual or groups of items. Group testing offers improvements over the naive individual testing scheme by potentially certifying multiple individual items as non-defective with a single test. The group testing problem aims to design a group testing plan to detect the defective items using as few tests as possible. We propose novel two-stage stochastic and robust optimization formulations for the design of group testing plans in the noiseless non-adaptive setting. Our formulations enable us to certify optimality for existing group testing schemes, as well as model complex grouping constraints, a feature that is not discussed in the existing literature. Wed, 28 Oct 2020 00:00:00 GMT https://hdl.handle.net/2123/23695 2020-10-28T00:00:00Z https://hdl.handle.net/2123/22956 Too similar to combine? On negative weights in forecast combination Radchenko, Peter; Vasnev, Andrey; Wang, Wendun This paper provides the first thorough investigation of the negative weights that can emerge when combining forecasts. The usual practice in the literature is to ignore or trim negative weights, i.e., set them to zero. This default strategy has its merits, but it is not optimal. We study the problem from a variety of different angles, and the main conclusion is that negative weights emerge when highly correlated forecasts with similar variances are combined. In this situation, the estimated weights have large variances, and trimming reduces the variance of the weights and improves the combined forecast. The threshold of zero is arbitrary and can be improved. We propose an optimal trimming threshold, i.e., an additional tuning parameter to improve forecasting performance. The effects of optimal trimming are demonstrated in simulations. In the empirical example using the European Central Bank Survey of Professional Forecasters, we find that the new strategy performs exceptionally well and can deliver improvements of more than 10% for inflation, up to 20% for GDP growth, and more than 20% for unemployment forecasts relative to the equal-weight benchmark. Wed, 01 Jan 2020 00:00:00 GMT https://hdl.handle.net/2123/22956 2020-01-01T00:00:00Z https://hdl.handle.net/2123/20406 Predicting China’s Monetary Policy with Forecast Combinations Pauwels, Laurent China’s monetary policy is unconventional and constantly evolving as a result of its rapid economic development. This paper proposes to use forecast combinations to predict the People’s Bank of China’s monetary policy stance with a large set of 73 macroeconomic and financial predictors covering various aspects of China’s economy. The multiple instruments utilised by the People’s Bank of China are aggregated into a Monetary Policy Index (MPI). The intention is to capture the overall monetary policy stance of the People’s Bank of China into a single variable that can be forecasted. Forecast combination assign weights to predictors according to their forecasting performance to produce a consensus forecast. The out-of-sample forecast results demonstrate that optimal forecast combinations are superior in predicting the MPI over other models such as the Taylor rule and simple autoregressive models. The corporate goods price index and the US nominal effective exchange rate are the most important predictors. Tue, 14 May 2019 00:00:00 GMT https://hdl.handle.net/2123/20406 2019-05-14T00:00:00Z https://hdl.handle.net/2123/20386 Fundamental Moments Imbs, Jean; Pauwels, Laurent Global trade can give rise to global hubs, centers of activity whose influence on the global economy is large enough that local disturbances have consequences in the aggregate. This paper investigates the nature, existence, and rise of such hubs using the World Input-Output Tables (WIOT) to evaluate the importance of vertical trade in creating global hubs that significantly affect countries volatility and their co-movement. Our results suggest that the world has become more granular since 1995, with significant consequences on GDP volatility and co-movements especially in developed countries. These consequences are well explained by international trade. Wed, 08 May 2019 00:00:00 GMT https://hdl.handle.net/2123/20386 2019-05-08T00:00:00Z https://hdl.handle.net/2123/20203 A New Family of Copulas, with Application to Estimation of a Production Frontier System Amsler, Christine; Prokhorov, Artem; Schmidt, Peter In this paper we propose a new family of copulas for which the copula arguments are uncorrelated but dependent. Specifically, if w1 and w2 are the uniform random variables in the copula, they are uncorrelated, but w1 is correlated with |w2 - ½|. We show how this family of copulas can be applied to the error structure in an econometric production frontier model. We also generalize the family of copulas to three or more dimensions, and we give an empirical application. Mon, 25 Mar 2019 00:00:00 GMT https://hdl.handle.net/2123/20203 2019-03-25T00:00:00Z https://hdl.handle.net/2123/20204 Moment Redundancy Test with Application to Efficiency-Improving Copulas Hao, Bowen; Prokhorov, Artem; Qian, Hailong Moment redundancy as defined by Breusch et al. (1999) is a testable hypothesis. We propose a simple test of the hypothesis in the context of copula-based pseudo-maximum likelihood estimation considered by Prokhorov and Schmidt (2009b). A robust and efficiency-improving parametric copula permits sizable improvement in precision at no cost in terms of bias and the proposed test can be used to select such copulas. Mon, 25 Mar 2019 00:00:00 GMT https://hdl.handle.net/2123/20204 2019-03-25T00:00:00Z https://hdl.handle.net/2123/20178 Asymptotic Theory for Rotated Multivariate GARCH Models Asai, Manabu; Chang, Chia-Lin; McAleer, Michael; Pauwels, Laurent In this paper, we derive the statistical properties of a two step approach to estimating multivariate GARCH rotated BEKK (RBEKK) models. By the definition of rotated BEKK, we estimate the unconditional covariance matrix in the first step in order to rotate observed variables to have the identity matrix for its sample covariance matrix. In the second step, we estimate the remaining parameters via maximizing the quasi-likelihood function. For this two step quasi-maximum likelihood (2sQML) estimator, we show consistency and asymptotic normality under weak conditions. While second-order moments are needed for consistency of the estimated unconditional covariance matrix, the existence of finite sixth-order moments are required for convergence of the second-order derivatives of the quasi-log-likelihood function. We also show the relationship of the asymptotic distributions of the 2sQML estimator for the RBEKK model and the variance targeting (VT) QML estimator for the VT-BEKK model. Monte Carlo experiments show that the bias of the 2sQML estimator is negligible, and that the appropriateness of the diagonal specification depends on the closeness to either of the Diagonal BEKK and the Diagonal RBEKK models. Wed, 20 Mar 2019 00:00:00 GMT https://hdl.handle.net/2123/20178 2019-03-20T00:00:00Z https://hdl.handle.net/2123/20176 Equivalence of optimal forecast combinations under affine constraints Chan, Felix; Pauwels, Laurent Forecasts are usually produced from models and expert judgements. The reconciliation of different forecasts presents an interesting challenge for managerial decisions. Mean absolute deviations and mean squared errors scoring rules are commonly employed as the criteria of optimality to aggregate or combine multiple forecasts into a consensus forecast. While much is known about mean squared errors in the context of forecast combination, little attention has been given to the mean absolute deviation. This paper establishes the first-order condition and the optimal solutions from minimizing mean absolute deviation. With this result, the paper derives the conditions in which the optimal solutions for minimizing mean absolute deviation and mean squared error loss functions are equivalent. More generally, this paper derives a sufficient condition which ensures the equivalence of optimal solutions of minimizing different loss functions under the same affine constraint that each feasible solution must sum to one. A simulation study and an illustration using expert forecasts data corroborate the theoretical findings. Interestingly, the numerical analysis shows that even with skewness in the data, the equivalence is unaffected. However, when outliers are presented in the data, mean absolute deviation is more robust than the mean squared error in small samples, which is consistent with the conventional belief relating the two loss functions. Tue, 19 Mar 2019 00:00:00 GMT https://hdl.handle.net/2123/20176 2019-03-19T00:00:00Z https://hdl.handle.net/2123/20175 Higher Moment Constraints for Predictive Density Combinations Pauwels, Laurent; Radchenko, Peter; Vasnev, Andrey The majority of financial data exhibit asymmetry and heavy tails, which makes forecasting the entire density critically important. Recently, a forecast combina- tion methodology has been developed to combine predictive densities. We show that combining individual predictive densities that are skewed and/or heavy-tailed results in significantly reduced skewness and kurtosis. We propose a solution to over- come this problem by deriving optimal log score weights under Higher-order Moment Constraints (HMC). The statistical properties of these weights are investigated the- oretically and through a simulation study. Consistency and asymptotic distribution results for the optimal log score weights with and without high moment constraints are derived. An empirical application that uses the S&P 500 daily index returns illustrates that the proposed HMC weight density combinations perform very well relative to other combination methods. Tue, 19 Mar 2019 00:00:00 GMT https://hdl.handle.net/2123/20175 2019-03-19T00:00:00Z https://hdl.handle.net/2123/16205 Speeding up MCMC by Efficient Data Subsampling Quiroz, Matias; Villani, Mattias; Kohn, Robert; Tran, Minh-Ngoc We propose Subsampling MCMC, a Markov Chain Monte Carlo (MCMC) framework where the likelihood function for n observations is estimated from a random subset of m observations. We introduce a general and highly efficient unbiased estimator of the log-likelihood based on control variates obtained from clustering the data. The cost of computing the log-likelihood estimator is much smaller than that of the full log-likelihood used by standard MCMC. The likelihood estimate is bias-corrected and used in two correlated pseudo-marginal algorithms to sample from a perturbed posterior, for which we derive the asymptotic error with respect to n and m, respectively. A practical estimator of the error is proposed and we show that the error is negligible even for a very small m in our applications. We demonstrate that Subsampling MCMC is substantially more efficient than standard MCMC in terms of sampling efficiency for a given computational budget, and that it outperforms other subsampling methods for MCMC proposed in the literature. Fri, 01 Jan 2016 00:00:00 GMT https://hdl.handle.net/2123/16205 2016-01-01T00:00:00Z https://hdl.handle.net/2123/15839 Matrix Neural Networks Gao, Junbin; Guo, Yi; Wang, Zhiyong Traditional neural networks assume vectorial inputs as the network is arranged as layers of single line of computing units called neurons. This special structure requires the non-vectorial inputs such as matrices to be converted into vectors. This process can be problematic. Firstly, the spatial information among elements of the data may be lost during vectorisation. Secondly, the solution space becomes very large which demands very special treatments to the network parameters and high computational cost. To address these issues, we propose matrix neural networks (MatNet), which takes matrices directly as inputs. Each neuron senses summarised information through bilinear mapping from lower layer units in exactly the same way as the classic feed forward neural networks. Under this structure, back prorogation and gradient descent combination can be utilised to obtain network parameters e ciently. Furthermore, it can be conveniently extended for multimodal inputs. We apply MatNet to MNIST handwritten digits classi cation and image super resolution tasks to show its e ectiveness. Without too much tweaking MatNet achieves comparable performance as the state-of-the-art methods in both tasks with considerably reduced complexity. Wed, 02 Nov 2016 00:00:00 GMT https://hdl.handle.net/2123/15839 2016-11-02T00:00:00Z https://hdl.handle.net/2123/14745 Estimation of Hierarchical Archimedean Copulas as a Shortest Path Problem Matsypura, Dmytro; Neo, Emily; Prokhorov, Artem We formulate the problem of finding and estimating the optimal hierarchical Archimedean copula as an amended shortest path problem. The standard network flow problem is amended by certain constraints specific to copulas, which limit scalability of the problem. However, we show in dimensions as high as twenty that the new approach dominates the alternatives which usually require recursive estimation or full enumeration. Sat, 16 Apr 2016 00:00:00 GMT https://hdl.handle.net/2123/14745 2016-04-16T00:00:00Z https://hdl.handle.net/2123/14641 Efficient estimation of parameters in marginal in semiparametric multivariate models Panchenko, Valentyn; Prokhorov, Artem We consider a general multivariate model where univariate marginal distributions are known up to a common parameter vector and we are interested in estimating that vector without assuming anything about the joint distribution, except for the marginals. If we assume independence between the marginals and maximize the resulting quasi-likelihood, we obtain a consistent but inefficient estimate. If we assume a parametric copula (other than independence) we obtain a full MLE, which is efficient but only under correct copula specification and badly biased if the copula is misspecified. Instead we propose a sieve MLE estimator which improves over OMLE but does not suffer the drawbacks of the full MLE. We model the unknown part of the joint distribution using the Bernstein-Kantorovich polynomial copula and assess the resulting improvement over QMLE and over misspecified FMLE in terms of relative efficiency and robustness. We derive the asymptotic distribution of the new estimator and show that it reaches the semiparametric efficiency bound. Simulations suggest that the sieve MLE can be almost as efficient as FMLE relative to QMLE provided there is enough dependence between the marginals. An application using insurance company loss and expense data demonstrates empirical relevance of the estimator. Tue, 01 Mar 2016 00:00:00 GMT https://hdl.handle.net/2123/14641 2016-03-01T00:00:00Z https://hdl.handle.net/2123/14595 Block-Wise Pseudo-Marginal Metropolis-Hastings Tran, M.-N.; Kohn, R.; Quiroz, M.; Villani, M. The pseudo-marginal Metropolis-Hastings approach is increasingly used for Bayesian inference in statistical models where the likelihood is analytically intractable but can be estimated unbiasedly, such as random effects models and state-space models, or for data subsampling in big data settings. In a seminal paper, Deligiannidis et al. (2015) show how the pseudo-marginal Metropolis-Hastings (PMMH) approach can be made much more e cient by correlating the underlying random numbers used to form the estimate of the likelihood at the current and proposed values of the unknown parameters. Their proposed approach greatly speeds up the standard PMMH algorithm, as it requires a much smaller number of particles to form the optimal likelihood estimate. We present a closely related alternative PMMH approach that divides the underlying random numbers mentioned above into blocks so that the likelihood estimates for the proposed and current values of the likelihood only di er by the random numbers in one block. Our approach is less general than that of Deligiannidis et al. (2015), but has the following advantages. First, it provides a more direct way to control the correlation between the logarithms of the estimates of the likelihood at the current and proposed values of the parameters. Second, the mathematical properties of the method are simplified and made more transparent compared to the treatment in Deligiannidis et al. (2015). Third, blocking is shown to be a natural way to carry out PMMH in, for example, panel data models and subsampling problems. We obtain theory and guidelines for selecting the optimal number of particles, and document large speed-ups in a panel data example and a subsampling problem. Wed, 30 Mar 2016 00:00:00 GMT https://hdl.handle.net/2123/14595 2016-03-30T00:00:00Z https://hdl.handle.net/2123/14594 Fast Inference for Intractable Likelihood Problems using Variational Bayes Gunawan, David; Tran, Minh-Ngoc; Kohn, Robert Variational Bayes (VB) is a popular statistical method for Bayesian inference. The existing VB algorithms are restricted to cases where the likelihood is tractable, which precludes their use in many interesting models. Tran et al. (2015) extend the scope of application of VB to cases where the likelihood is intractable but can be estimated unbiasedly, and name the method “Variational Bayes with Intractable Likelihood (VBIL)”. This paper presents a version of VBIL, named Variational Bayes with Intractable Log-Likelihood (VBILL), that is useful for cases, such as big data and big panel data models, where only unbiased estimators of the log-likelihood are available. In particular, we develop an estimation approach, based on subsampling and the MapReduce programming technique, for analysing massive datasets which cannot fit into a single desktop’s memory. The proposed method is theoretically justified in the sense that, apart from an extra Monte Carlo error which can be controlled, it is able to produce estimators as if the true log-likelihood or full data were used. The proposed methodology is robust in the sense that it works well when only highly variable estimates of the log-likelihood are available. The method is illustrated empirically using several simulated datasets and a big real dataset based on the arrival time status of U. S. airlines. Keywords. Pseudo Marginal Metropolis-Hastings, Debiasing Approach, Big Data, Panel Data, Difference Estimator. Wed, 30 Mar 2016 00:00:00 GMT https://hdl.handle.net/2123/14594 2016-03-30T00:00:00Z https://hdl.handle.net/2123/14490 A New Measure of Vector Dependence, with an Application to Financial Contagion Medovikov, Ivan; Prokhorov, Artem We propose a new nonparametric measure of association between an arbitrary number of random vectors. The measure is based on the empirical copula process for the multivariate marginals, corresponding to the vectors, and is insensitive to the within-vector dependence. It is bounded by the [0, 1] interval, covering the entire range of dependence from vector independence to a vector version of a monotone relationship. We study the properties of the new measure under several well-known copulas and provide a nonparametric estimator of the measure, along with its asymptotic theory, under fairly general assumptions. To illustrate the applicability of the new measure, we use it to assess the degree of interdependence between equity markets in North and South America, Europe and Asia, surrounding the financial crisis of 2008. We find strong evidence of previously unknown contagion patterns, with selected regions exhibiting little dependence before and after the crisis and a lot of dependence during the crisis period. Fri, 11 Mar 2016 00:00:00 GMT https://hdl.handle.net/2123/14490 2016-03-11T00:00:00Z https://hdl.handle.net/2123/13839 Exact ABC using Importance Sampling Tran, Minh-Ngoc; Kohn, Robert Approximate Bayesian Computation (ABC) is a powerful method for carrying out Bayesian inference when the likelihood is computationally intractable. However, a draw- back of ABC is that it is an approximate method that induces a systematic error because it is necessary to set a tolerance level to make the computation tractable. The issue of how to optimally set this tolerance level has been the subject of extensive research. This paper proposes an ABC algorithm based on importance sampling that estimates expec- tations with respect to the exact posterior distribution given the observed summary statistics. This overcomes the need to select the tolerance level. By exact we mean that there is no systematic error and the Monte Carlo error can be made arbitrarily small by increasing the number of importance samples. We provide a formal justifica- tion for the method and study its convergence properties. The method is illustrated in two applications and the empirical results suggest that the proposed ABC based esti- mators consistently converge to the true values as the number of importance samples increases. Our proposed approach can be applied more generally to any importance sampling problem where an unbiased estimate of the likelihood is required. Wed, 23 Sep 2015 00:00:00 GMT https://hdl.handle.net/2123/13839 2015-09-23T00:00:00Z https://hdl.handle.net/2123/13795 GEL Estimation for Heavy-Tailed GARCH Models with Robust Empirical Likelihood Inference Hill, Jonathan B.; Prokhorov, Artem We construct a Generalized Empirical Likelihood estimator for a GARCH(1,1) model with a possibly heavy tailed error. The estimator imbeds tail-trimmed estimating equations allowing for over-identifying conditions, asymptotic normality, efficiency and empirical likelihood based confidence regions for very heavy-tailed random volatility data. We show the implied probabilities from the tail-trimmed Continuously Updated Estimator elevate weight for usable large values, assign large but not maximum weight to extreme observations, and give the lowest weight to non-leverage points. We derive a higher order expansion for GEL with imbedded tail-trimming (GELITT), which reveals higher order bias and efficiency properties, available when the GARCH error has a finite second moment. Higher order asymptotics for GEL without tail-trimming requires the error to have moments of substantially higher order. We use first order asymptotics and higher order bias to justify the choice of the number of trimmed observations in any given sample. We also present robust versions of Generalized Empirical Likelihood Ratio, Wald, and Lagrange Multiplier tests, and an efficient and heavy tail robust moment estimator with an application to expected shortfall estimation. Finally, we present a broad simulation study for GEL and GELITT, and demonstrate profile weighted expected shortfall for the Russian Ruble - US Dollar exchange rate. We show that tail-trimmed CUE-GMM dominates other estimators in terms of bias, mse and approximate normality. AMS classifications : 62M10 , 62F35. JEL classifications : C13 , C49. Fri, 11 Sep 2015 00:00:00 GMT https://hdl.handle.net/2123/13795 2015-09-11T00:00:00Z https://hdl.handle.net/2123/13800 Bayesian Semi-parametric Realized-CARE Models for Tail Risk Forecasting Incorporating Range and Realized Measures Gerlach, Richard; Wang, Chao A new framework named Realized Conditional Autoregressive Expectile (Realized- CARE) is proposed, through incorporating a measurement equation into the conventional CARE model, in a framework analogous to Realized-GARCH. The Range and realized measures (Realized Variance and Realized Range) are employed as the dependent variables of the measurement equation, since they have proven more efficient than return for volatility estimation. The dependence between Range & realized measures and expectile can be modelled with this measurement equation. The grid search accuracy of the expectile level will be potentially improved with introducing this measurement equation. In addition, through employing a quadratic fitting target search, the speed of grid search is significantly improved. Bayesian adaptive Markov Chain Monte Carlo is used for estimation, and demonstrates its superiority compared to maximum likelihood in a simulation study. Furthermore, we propose an innovative sub-sampled Realized Range and also adopt an existing scaling scheme, in order to deal with the micro-structure noise of the high frequency volatility measures. Compared to the CARE, the parametric GARCH and the Realized-GARCH models, Value-at-Risk and Expected Shortfall forecasting results of 6 indices and 3 assets series favor the proposed Realized-CARE model, especially the Realized-CARE model with Realized Range and sub-sampled Realized Range. Fri, 11 Sep 2015 00:00:00 GMT https://hdl.handle.net/2123/13800 2015-09-11T00:00:00Z https://hdl.handle.net/2123/13799 Fat tails and copulas: limits of diversification revisited Ibragimov, Rustam; Prokhorov, Artem; Mo, Jingyuan We consider the problem of portfolio risk diversification in a Value-at-Risk framework with heavy-tailed risks and arbitrary dependence captured by a copula function. We use the power law for modelling the tails and investigate whether the benefits of diversification persist when the risks in consideration are allowed to have extremely heavy tails with tail indices less than one and when their copula describes wide classes of dependence structures. We show that for asymptotically large losses with the Eyraud-Farlie-Gumbel-Morgenstern copula, the threshold value of tail indices at which diversification stops being beneficial is the same as for independent losses. We further extend this result to a wider range of dependence structures which can be approximated using power-type copulas and their approximations. This range of dependence structures includes many well known copula families, among which there are comprehensive, Archimedian, asymmetric and tail dependent copulas. In other words, diversification increases Value-at-Risk for tail indices less than one regardless of the nature of dependence between portfolio components within these classes. A wide set of simulations supports these theoretical results. Fri, 11 Sep 2015 00:00:00 GMT https://hdl.handle.net/2123/13799 2015-09-11T00:00:00Z https://hdl.handle.net/2123/13797 Supplemental Material for GEL Estimation for Heavy-Tailed GARCH Models with Robust Empirical Likelihood Inference Hill, Jonathan B.; Prokhorov, Artem The following supplemental material contains an omitted simulation experiment, and omitted proofs of theorems and preliminary lemmata. Section S contains simulation results, and Section A contains an appendix with omitted proofs. Fri, 11 Sep 2015 00:00:00 GMT https://hdl.handle.net/2123/13797 2015-09-11T00:00:00Z https://hdl.handle.net/2123/13798 Generalized Information Matrix Tests for Copulas Prokhorov, Artem; Schepsmeier, Ulf; Zhu, Yajing We propose a family of goodness-of-fit tests for copulas. The tests use generalizations of the information matrix (IM) equality of White (1982) and so relate to the copula test proposed by Huang and Prokhorov (2014). The idea is that eigenspectrum-based statements of the IM equality reduce the degrees of freedom of the test's asymptotic distribution and lead to better size-power properties, even in high dimensions. The gains are especially pronounced for vine copulas, where additional benefits come from simplifications of score functions and the Hessian. We derive the asymptotic distribution of the generalized tests, accounting for the non-parametric estimation of the marginals and apply a parametric bootstrap procedure, valid when asymptotic critical values are inaccurate. In Monte Carlo simulations, we study the behavior of the new tests, compare them with several Cramer-von Mises type tests and confirm the desired properties of the new tests in high dimensions. Fri, 11 Sep 2015 00:00:00 GMT https://hdl.handle.net/2123/13798 2015-09-11T00:00:00Z https://hdl.handle.net/2123/13263 Generalized Variance: A Robust Estimator of Stock Price Volatility Sutton, M; Vasnev, A; Gerlach, R This paper proposes an ex-post volatility estimator, called generalized variance, that uses high frequency data to provide measurements robust to the idiosyncratic noise of stock markets caused by market microstructures. The new volatility estimator is analyzed theoretically, examined in a simulation study and evaluated empirically against the two currently dominant measures of daily volatility: realized volatility and realized range. The main finding is that generalized variance is robust to the presence of microstructures while delivering accuracy superior to realized volatility and realized range in several circumstances. The empirical study features Australian stocks from the ASX 20. Thu, 30 Apr 2015 00:00:00 GMT https://hdl.handle.net/2123/13263 2015-04-30T00:00:00Z https://hdl.handle.net/2123/12755 Endogeneity in Stochastic Frontier Models Amsler, Christine; Artem, Prokhorov; Peter, Schmidt Stochastic frontier models are typically estimated by maximum likelihood (MLE) orcorrected ordinary least squares. The consistency of either estimator depends on exogeneity of the explanatory variables (inputs, in the production frontier setting). We will investigate the case that one or more of the inputs is endogenous, in the simultaneous equation sense of endogeneity. That is, we worry that there is correlation between the inputs and statistical noise or inefficiency. In a standard regression setting, simultaneity is handled by a number of procedures that are numerically or asymptotically equivalent. These include 2SLS; using the residual from the reduced form equations for the endogenous variables as a control function; and MLE of the system that contains the equation of interest plus the unrestricted reduced form equations for the endogenous variables (LIML). We will consider modifications of these standard procedures for the stochastic frontier setting. The paper is mostly a survey and combination of existing results from the stochastic frontier literature and the classic simultaneous equations literature, but it also contains some new results. Tue, 17 Feb 2015 00:00:00 GMT https://hdl.handle.net/2123/12755 2015-02-17T00:00:00Z https://hdl.handle.net/2123/12235 Forecasting risk via realized GARCH, incorporating the realized range Richard, Gerlach; Chao, Wang The realized GARCH framework is extended to incorporate the realized range, and the intra-day range, as potentially more efficient series of information than re- alized variance or daily returns, for the purpose of volatility and tail risk forecasting in a financial time series. A Bayesian adaptive Markov chain Monte Carlo method is employed for estimation and forecasting. Compared to a range of well known parametric GARCH models, predictive log-likelihood results across six market in- dex return series favor the realized GARCH models incorporating the realized range. Further, these same models also compare favourably for tail risk forecasting, both during and after the global financial crisis. Fri, 07 Nov 2014 00:00:00 GMT https://hdl.handle.net/2123/12235 2014-11-07T00:00:00Z https://hdl.handle.net/2123/12060 Bayesian Tail Risk Forecasting using Realised GARCH Contino, Christian; Gerlach, Richard A Realised Volatility GARCH model is developed within a Bayesian framework for the purpose of forecasting Value at Risk and Conditional Value at Risk. Student-t and Skewed Student-t return distributions are combined with Gaussian and Student-t distributions in the measurement equation in a GARCH framework to forecast tail risk in eight international equity index markets over a four year period. Three Realised Volatility proxies are considered within this framework. Realised Volatility GARCH models show a marked improvement compared to ordinary GARCH for both Value at Risk and Conditional Value at Risk forecasting. This improvement is consistent across a variety of data, volatility model speci_cations and distributions, and demonstrates that Realised Volatility is superior when producing volatility forecasts. Realised Volatility models implementing a Skewed Student-t distribution for returns in the GARCH equation are favoured. Fri, 10 Oct 2014 00:00:00 GMT https://hdl.handle.net/2123/12060 2014-10-10T00:00:00Z https://hdl.handle.net/2123/11816 Bayesian Assessment of Dynamic Quantile Forecasts Gerlach, Richard; Chen, Cathy W.S.; Lin, Edward M.H. Methods for Bayesian testing and assessment of dynamic quantile forecasts are proposed. Specifically, Bayes factor analogues of popular frequentist tests for independence of violations from, and for correct coverage of a time series of, quantile forecasts are developed. To evaluate the relevant marginal likelihoods involved, analytic integration methods are utilised when possible, otherwise multivariate adaptive quadrature methods are employed to estimate the required quantities. The usual Bayesian interval estimate for a proportion is also examined in this context. The size and power properties of the proposed methods are examined via a simulation study, illustrating favourable comparisons both overall and with their frequentist counterparts. An empirical study employs the proposed methods, in comparison with standard tests, to assess the adequacy of a range of forecasting models for Value at Risk (VaR) in several financial market data series. Wed, 10 Sep 2014 00:00:00 GMT https://hdl.handle.net/2123/11816 2014-09-10T00:00:00Z https://hdl.handle.net/2123/11773 Consistent Estimation of Linear Regression Models Using Matched Data Prokhorov, Artem; Hirukawa, Masayuki Economists often use matched samples, especially when dealing with earnings data where a number of missing observations need to be imputed. In this paper, we demonstrate that the ordinary least squares estimator of the linear regression model using matched samples is inconsistent and has a non-standard convergence rate to its probability limit. If only a few variables are used to impute the missing data then it is possible to correct for the bias. We propose two semi-parametric bias-corrected estimators and explore their asymptotic properties. The estimators have an indirectinference interpretation and their convergence rates depend on the number of variables used in matching. We can attain the parametric convergence rate if that number is no greater than three. Monte Carlo simulations confirm that the bias correction works very well in such cases. Fri, 05 Sep 2014 00:00:00 GMT https://hdl.handle.net/2123/11773 2014-09-05T00:00:00Z https://hdl.handle.net/2123/10457 Semi-parametric Expected Shortfall Forecasting Gerlach, Richard; Chen, Cathy W.S. Intra-day sources of data have proven effective for dynamic volatility and tail risk estimation. Expected shortfall is a tail risk measure, that is now recommended by the Basel Committee, involving a conditional expectation that can be semi-parametrically estimated via an asymmetric sum of squares function. The conditional autoregressive expectile class of model, used to indirectly model expected shortfall, is generalised to incorporate information on the intra-day range. An asymmetric Gaussian density model error formulation allows a likelihood to be developed that leads to semiparametric estimation and forecasts of expectiles, and subsequently of expected shortfall. Adaptive Markov chain Monte Carlo sampling schemes are employed for estimation, while their performance is assessed via a simulation study. The proposed models compare favourably with a large range of competitors in an empirical study forecasting seven financial return series over a ten year period. Tue, 01 Apr 2014 00:00:00 GMT https://hdl.handle.net/2123/10457 2014-04-01T00:00:00Z https://hdl.handle.net/2123/9943 Confidence Levels for CVaR Risk Measures and Minimax Limits* Anderson, Edward; Xu, Huifu; Zhang, Dali Conditional value at risk (CVaR) has been widely used as a risk measure in finance. When the confidence level of CVaR is set close to 1, the CVaR risk measure approximates the extreme (worst scenario) risk measure. In this paper, we present a quantitative analysis of the relationship between the two risk measures and it’s impact on optimal decision making when we wish to minimize the respective risk measures. We also investigate the difference between the optimal solutions to the two optimization problems with identical objective function but under constraints on the two risk measures. We discuss the benefits of a sample average approximation scheme for the CVaR constraints and investigate the convergence of the optimal solution obtained from this scheme as the sample size increases. We use some portfolio optimization problems to investigate teh performance of the CVaR approximation approach. Our numerical results demonstrate how reducing the confidence level can lead to a better overall performance. Wed, 01 Jan 2014 00:00:00 GMT https://hdl.handle.net/2123/9943 2014-01-01T00:00:00Z https://hdl.handle.net/2123/9293 Two-Sample Nonparametric Estimation of Intergenerational Income Mobility Murtazashvili, Irina; Liu, Di; Prokhorov, Artem We estimate intergenerational income mobility in the USA and Sweden. To measure the degree to which income status is transmitted from one generation to another we propose a nonparametric estimator, which is particularly relevant for cross-country comparisons. Our approach allows intergenerational mobility to vary across observable family characteristics. Furthermore, it ts situations when data on fathers and sons come from di fferent samples. Finally, our estimator is consistent in the presence of measurement error in fathers' long-run economic status. We fi nd that family background captured by fathers' education matters for intergenerational income persistence in the USA more than in Sweden suggesting that the character of inequality in the two countries is rather di fferent. Wed, 07 Aug 2013 00:00:00 GMT https://hdl.handle.net/2123/9293 2013-08-07T00:00:00Z https://hdl.handle.net/2123/9071 Competing for contracts with buyer uncertainty: Choosing price and quality variables Anderson, Edward; Qian, Cheng We model a situation in which a single firm evaluates competing suppliers and selects just one. Suppliers submit bids involving both price and quality variables. The buyer makes a choice which from the supplier's perspective appears to contain a stochastic element - for example the buyer may have information, which is not shared with the suppliers, and that gives one supplier an advantage in the final choice. We use a discrete choice model of buyer choice (e.g. multinomial logit). Our main result is that the supplier's choice of the quality variables is not affected by the competitive environment. Thus the suppliers compete only on price. We compare this with a second model in which the buyer's weighting on different quality variables is uncertain at the time bids are made. Thu, 09 May 2013 00:00:00 GMT https://hdl.handle.net/2123/9071 2013-05-09T00:00:00Z https://hdl.handle.net/2123/8963 Multiple Event Incidence and Duration Analysis for Credit Data Incorporating Non-Stochastic Loan Maturity Vasnev, Andrey; Gerlach, Richard; Watkins, John Applications of duration analysis in Economics and Finance exclusively employ methods for events of stochastic duration. In application to credit data, previous research incorrectly treats the time to pre-determined maturity events as censored stochastic event times. The medical literature has binary parametric ‘cure rate’ models that deal with populations that never experienced the modelled event. We propose and develop a Multinomial parametric incidence and duration model, incorporating such populations. In the class of cure rate models, this is the first fully parametric multinomial model and is the first framework to accommodate an event with pre-determined duration. The methodology is applied to unsecured personal loan credit data provided by one of Australia’s largest financial services organizations. This framework is shown to be more flexible and predictive through a simulation and empirical study that reveals: simulation results of estimated parameters with a large reduction in bias; superior forecasting of duration; explanatory variables can act in different directions upon incidence and duration; and, variables exist that are statistically significant in explaining only incidence or duration. Sat, 01 Dec 2012 00:00:00 GMT https://hdl.handle.net/2123/8963 2012-12-01T00:00:00Z https://hdl.handle.net/2123/8964 Practical use of sensitivity in econometrics with an illustration to forecast combinations Vasnev, Andrey; Magnus, Jan R Sensitivity analysis is important for its own sake and also in combination with diagnostic testing. We consider the question how to use sensitivity statistics in practice, in particular how to judge whether sensitivity is large or small. For this purpose we distinguish between absolute and relative sensitivity and highlight the context-dependent nature of any sensitivity analysis. Relative sensitivity is then applied in the context of forecast combination and sensitivity-based weights are introduced. All concepts are illustrated through the European yield curve. In this context it is natural to look at sensitivity to autocorrelation and normality assumptions. Different forecasting models are combined with equal, fit-based and sensitivity-based weights, and compared with the multivariate and random walk benchmarks. We show that the fit-based weights and the sensitivity-based weights are complementary. For long-term maturities the sensitivity-based weights perform better than other weights. Fri, 01 Mar 2013 00:00:00 GMT https://hdl.handle.net/2123/8964 2013-03-01T00:00:00Z https://hdl.handle.net/2123/8965 Forecast combination for U.S. recessions with real-time data Vasnev, Andrey; Pauwels, Laurent This paper proposes the use of forecast combination to improve predictive accuracy in forecasting the U.S. business cycle index, as published by the Business Cycle Dating Committee of the NBER. It focuses on one-step ahead out-of-sample monthly forecast utilising the well-established coincident indicators and yield curve models, allowing for dynamics and real-time data revisions. Forecast combinations use logscore and quadratic-score based weights, which change over time. This paper finds that forecast accuracy improves when combining the probability forecasts of both the coincident indicators model and the yield curve model, compared to each model's own forecasting performance. Fri, 01 Mar 2013 00:00:00 GMT https://hdl.handle.net/2123/8965 2013-03-01T00:00:00Z https://hdl.handle.net/2123/8933 Forecast combination for U.S. recessions with real-time data Vasnev, Andrey; Pauwels, Laurent This paper proposes the use of forecast combination to improve predictive accuracy in forecasting the U.S. business cycle index as published by the Business Cycle Dating Committee of the NBER. It focuses on one-step ahead out-of-sample monthly forecast utilising the well-established coincident indicators and yield curve models, allowing for dynamics and real-time data revisions. Forecast combinations use logscore and quadratic-score based weights, which change over time. This paper finds that forecast accuracy improves when combining the probability forecasts of both the coincident indicators model and the yield curve model, compared to each model's own forecasting performance. Tue, 01 Jan 2013 00:00:00 GMT https://hdl.handle.net/2123/8933 2013-01-01T00:00:00Z https://hdl.handle.net/2123/8932 Practical considerations for optimal weights in density forecast combination Vasnev, Andrey; Pauwels, Laurent The problem of finding appropriate weights to combine several density forecasts is an important issue currently debated in the forecast combination literature. Recently, a paper by Hall and Mitchell (IJF, 2007) proposes to combine density forecasts with optimal weights obtained from solving an optimization problem. This paper studies the properties of this optimization problem when the number of forecasting periods is relatively small and finds that it often produces corner solutions by allocating all the weight to one density forecast only. This paper’s practical recommendation is to have an additional training sample period for the optimal weights. While reserving a portion of the data for parameter estimation and making pseudo-out-of-sample forecasts are common practices in the empirical literature, employing a separate training sample for the optimal weights is novel, and it is suggested because it decreases the chances of corner solutions. Alternative log-score or quadratic-score weighting schemes do not have this training sample requirement. January Tue, 01 Jan 2013 00:00:00 GMT https://hdl.handle.net/2123/8932 2013-01-01T00:00:00Z https://hdl.handle.net/2123/8337 Maximum likelihood estimation of time series models: the Kalman filter and beyond Proietti, Tommaso; Luati, Alessandra The purpose of this chapter is to provide a comprehensive treatment of likelihood inference for state space models. These are a class of time series models relating an observable time series to quantities called states, which are characterized by a simple temporal dependence structure, typically a first order Markov process. The states have sometimes substantial interpretation. Key estimation problems in economics concern latent variables, such as the output gap, potential output, the non-accelerating-inflation rate of unemployment, or NAIRU, core inflation, and so forth. Time-varying volatility, which is quintessential to finance, is an important feature also in macroeconomics. In the multivariate framework relevant features can be common to different series, meaning that the driving forces of a particular feature and/or the transmission mechanism are the same. The objective of this chapter is reviewing this algorithm and discussing maximum likelihood inference, starting from the linear Gaussian case and discussing the extensions to a nonlinear and non Gaussian framework. Tue, 01 May 2012 00:00:00 GMT https://hdl.handle.net/2123/8337 2012-05-01T00:00:00Z https://hdl.handle.net/2123/8161 Survival Analysis for Credit Scoring: Incidence and Latency Watkins, John; Vasnev, Andrey; Gerlach, Richard Duration analysis is an analytical tool for time-to-event data that has been borrowed from medicine and engineering to be applied by econometricians to investigate typical economic and finance problems. In applications to credit data, time to the pre-determined maturity events have been treated as censored observations for the events with stochastic latency. A methodology, motivated by the cure rate model framework, is developed in this paper to appropriately analyse a set of mutually exclusive terminal events where at least one event may have a predetermined latency. The methodology is applied to a set of personal loan data provided by one of Australia's largest financial services institutions. This is the first framework to simultaneously model prepayment, write off and maturity events for loans. Furthermore, in the class of cure rate models it is the first fully parametric multinomial model and the first to accommodate for an event with pre-determined latency. The simulation study found this model performed better than the two most common applications of survival analysis to credit data. In addition, the result of the application to personal loans data reveals particular explanatory variables can act in different directions upon incidence and latency of an event and variables exist that may be statistically significant in explaining only incidence or latency. Sun, 01 Nov 2009 00:00:00 GMT https://hdl.handle.net/2123/8161 2009-11-01T00:00:00Z https://hdl.handle.net/2123/8159 Bayesian time-varying quantile forecasting for Value-at-Risk in financial markets Gerlach, Richard; Chen, Cathy W.S.; Chan, Nancy Y. C. Recently, Bayesian solutions to the quantile regression problem, via the likelihood of a Skewed-Laplace distribution, have been proposed. These approaches are extended and applied to a family of dynamic conditional autoregressive quantile models. Popular Value at Risk models, used for risk management in finance, are extended to this fully nonlinear family. An adaptive Markov chain Monte Carlo sampling scheme is adapted for estimation and inference. Simulation studies illustrate favourable performance, compared to the standard numerical optimization of the usual nonparametric quantile criterion function, in finite samples. An empirical study generating Value at Risk forecasts for ten major financial stock indices finds significant nonlinearity in dynamic quantiles and evidence favoring the proposed model family, for lower level quantiles, compared to a range of standard parametric volatility models, a semi-parametric smoothly mixing regression and some nonparametric risk measures, in the literature. Sat, 01 Aug 2009 00:00:00 GMT https://hdl.handle.net/2123/8159 2009-08-01T00:00:00Z https://hdl.handle.net/2123/8169 Bayesian Semi-parametric Expected Shortfall Forecasting in Financial Markets Gerlach, Richard; Chen, Cathy W.S.; Lin, Liou-Yan Bayesian semi-parametric estimation has proven effective for quantile estimation in general and specifically in financial Value at Risk forecasting. Expected short-fall is a competing tail risk measure, involving a conditional expectation beyond a quantile, that has recently been semi-parametrically estimated via asymmetric least squares and so-called expectiles. An asymmetric Gaussian density is proposed allowing a likelihood to be developed that leads to Bayesian semi-parametric estimation and forecasts of expectiles and expected shortfall. Further, the conditional autoregressive expectile class of model is generalised to two fully nonlinear families. Adaptive Markov chain Monte Carlo sampling schemes are employed for estimation in these families. The proposed models are clearly favoured in an empirical study forecasting eleven financial return series: clear evidence of more accurate expected shortfall forecasting, compared to a range of competing methods is found. Further, the most favoured models are those estimated by Bayesian methods. Sun, 01 Jan 2012 00:00:00 GMT https://hdl.handle.net/2123/8169 2012-01-01T00:00:00Z https://hdl.handle.net/2123/8173 Margining Option Portfolios by Network Flows Matsypura, D.; Timkovsky, V.G. As shown in [Rudd and Schroeder, 1982], the problem of margining option portfolios where option spreads with two legs are used for offsetting can be solved in polynomial time by network flow algorithms. However, spreads with only two legs do not provide sufficient accuracy in measuring risk. Therefore, margining practice also employs spreads with three and four legs. A polynomial time solution to the extension of the problem where option spreads with three and four legs are also used for offsetting is not known. In this paper we propose a heuristic network flow algorithm for this extension and present a computational study that proves high efficiency of this algorithm in margining practice. Wed, 01 Sep 2010 00:00:00 GMT https://hdl.handle.net/2123/8173 2010-09-01T00:00:00Z https://hdl.handle.net/2123/8166 Stochastic trends and seasonality in economic time series: new evidence from Bayesian stochastic model specification search Proietti, Tommaso; Grassi, Stefano An important issue in modelling economic time series is whether key unobserved components representing trends, seasonality and calendar components, are deterministic or evolutive. We address it by applying a recently proposed Bayesian variable selection methodology to an encompassing linear mixed model that features, along with deterministic effects, additional random explanatory variables that account for the evolution of the underlying level, slope, seasonality and trading days. Variable selection is performed by estimating the posterior model probabilities using a suitable Gibbs sampling scheme. The paper conducts an extensive empirical application on a large and representative set of monthly time series concerning industrial production and retail turnover. We find strong support for the presence of stochastic trends in the series, either in the form of a time-varying level, or, less frequently, of a stochastic slope, or both. Seasonality is a more stable component: only in 70% of the cases we were able to select at least one stochastic trigonometric cycle out of the six possible cycles. Most frequently the time variation is found in correspondence with the fundamental and the first harmonic cycles. An interesting and intuitively plausible finding is that the probability of estimating time-varying components increases with the sample size available. However, even for very large sample sizes we were unable to find stochastically varying calendar effects. Thu, 01 Sep 2011 00:00:00 GMT https://hdl.handle.net/2123/8166 2011-09-01T00:00:00Z