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Several novel statistical methods have been developed to estimate large integrated volatility matrices based on high-frequency financial data. To investigate their asymptotic behaviors, they require a sub-Gaussian or finite high-order moment assumption for observed log-returns, which cannot account for the heavy-tail phenomenon of stock returns. Recently, a robust estimator was developed to handle heavy-tailed distributions with some bounded fourth-moment assumption. However, we often observe that log-returns have heavier tail distribution than the finite fourth-moment and that the degrees of heaviness of tails are heterogeneous across asset and over time. In this paper, to deal with the heterogeneous heavy-tailed distributions, we develop an adaptive robust integrated volatility estimator that employs pre-averaging and truncation schemes based on jump-diffusion processes. We call this an adaptive robust pre-averaging realized volatility (ARP) estimator. We show that the ARP estimator has a sub-Weibull tail concentration with only finite 2$\alpha$-th moments for any $\alpha>1$. In addition, we establish matching upper and lower bounds to show that the ARP estimation procedure is optimal. To estimate large integrated volatility matrices using the approximate factor model, the ARP estimator is further regularized using the principal orthogonal complement thresholding (POET) method. The numerical study is conducted to check the finite sample performance of the ARP estimator., Comment: 64 pages, 7 figures

Ranked choice voting (RCV) is growing in popularity as an alternative to traditional U.S. election systems, as it is widely believed to result in more proportional outcomes for minority groups. However, little is known about the negative impacts of ballot truncation (i.e. voters not ranking every candidate either by choice or under election rules) on the signature benefits of RCV. In this study, we introduce new statistical models of ballot truncation based on novel insights from historical election data, and use these models to measure the effect of ballot truncation on proportional representation in simulated elections.

The logit-weighted reduced mixture of experts model (LRMoE) is a flexible yet analytically tractable non-linear regression model. While it has shown usefulness in modeling insurance loss frequencies and severities, model calibration becomes challenging when censored and truncated data are involved, which is common in actuarial practice. In this paper, we present an extended Expectation-Conditional-Maximization (ECM) algorithm that efficiently fits the LRMoE to random censored and random truncated regression data. The effectiveness of the proposed algorithm is empirically examined through a simulation study. Using real automobile insurance data sets, the usefulness and importance of the proposed algorithm are demonstrated through two actuarial applications: individual claim reserving and deductible ratemaking.

We discuss estimation of so-called long vector autoregressions for multivariate series exhibiting possibly time-varying mean and (co)variances. In applied work, such changes often escape undetected, and we ask how standard tools (least squares estimation, point forecasts, and estimated impulse responses) are affected when ignoring the changes altogether. Keeping the order of the autoregression fixed is known to lead to asymptotic bias in autoregressive parameter estimators in the presence of ignored changes in the mean. Yet we show that allowing the complexity of the model to increase with the sample size leads to consistent estimators of the AR coefficient matrices individually. The fitted long VAR models appear to have unit root behavior, in spite of the absence of any stochastic trend in the model, and may even mimic cointegration; but, in spite of the structural change in the data generating process, out-of-sample forecasts based on long VARs are consistent. These findings hold under constant as well as under time-varying covariances. In what concerns estimated impulse responses, their sampling behavior depends primarily on whether the residual covariance matrix is employed for identification of the structural shocks or not. While MA coefficient matrices obtained by inversion of the fitted long VAR model are consistent for the true coefficients under mild additional restrictions even under time-varying error (co)variances, the residual covariance matrix estimator converges to an "average" covariance matrix, such that localized estimators may be more suitable for a precise identification. Monte Carlo simulations and empirical illustration support our theoretical findings. Empirical relevance of the theory is illustrated through two illustrations: (i) international dynamics of inflation and (ii) uncertainty and economics activity.

Having observed a cluster of jumps in the discrete prices of a financial asset, we study and quantify the probability that the cluster is going to produce further jumps. Modeling the jump arrival times with an exponential Hawkes process, we provide some bounds for the future stochastic jump intensity under different hypotheses about the concentration of the past jumps arrivals. Moreover, we formalize the stochastic increasing property of the durations between two consecutive jumps and study the relation between the intensity decay time and the length of the possible clusters. Consequently, we provide formulas for the probability that, conditionally on the observation of a cluster of jumps, the generating phenomenon will produce a further jump, and bounds for the probability of observing a given number of consecutive jumps. As an empirical exercise, we apply our results to a record of the JPM asset prices. Firstly we show that the identified jumps display dependence and clustering behavior. Secondly, we find that, under the exponential Hawkes model delivering the best QQplot, the probability that a cluster of more than 1 jump produces a further jump is mostly close to 1.

We derive the formal expansion of the price of a VIX future (and more generally a VIX power payoff) in various Bergomi models at any order in powers of the volatility-of-volatility. We introduce the notion of volatility of the VIX squared implied by the VIX future, which we call "VIX2 implied volatility", and also expand this quantity at any order. We cover the one-factor, two-factor, and skewed two-factor Bergomi models and allow for maturity-dependent and/or time-dependent parameters. When the initial term-structure of variance swaps is flat, the expansions are in closed form; otherwise, they involve one-dimensional integrals which are extremely fast to compute. Our extensive numerical experiments show that both the price and implied volatility expansions provide very accurate approximations for a wide range of model parameters which covers the values typically used in the equity and FX derivatives markets and even well beyond---in particular, the expansions are accurate even for very large values of the volatility-of-volatility. They also show that the implied volatility expansion converges much faster than the price expansion. Its truncation at order 1 is already so accurate that it leads to a simple formula for the price of the VIX future that can virtually be considered exact. The derivation of the expansions naturally involves the (classical or dual bivariate) Hermite polynomials and exploits their orthogonality properties; it is interesting in itself. The expansions allow us to precisely pinpoint the roles of all the model parameters (volatility-of-volatility, mean reversions, correlations, mixing fraction) in the formation of the prices of VIX futures and VIX power payoffs in Bergomi models. We also use them together with the Bergomi-Guyon expansion of the S&P 500 smile to (instantaneously) calibrate the two-factor Bergomi model jointly to the term-structures of S&P 500 at-the-money skew and VIX2 implied volatility. Our tests and the new expansions shed more light on the inability of traditional stochastic volatility models to jointly fit S&P 500 and VIX market data. The (imperfect but decent) joint fit requires much larger values of volatility-of-volatility and fast mean reversion than the ones previously reported in Bergomi (2005, 2016).

This paper introduces a new R package, LRMoE, a statistical software tailor-made for actuarial applications which allows actuarial researchers and practitioners to model and analyze insurance loss frequencies and severities using the Logit-weighted Reduced Mixture-of-Experts (LRMoE) model. LRMoE offers several new distinctive features which are motivated by various actuarial applications and mostly cannot be achieved using existing packages for mixture models. Key features include a wider coverage on frequency and severity distributions and their zero inflation, the flexibility to vary classes of distributions across components, parameter estimation under data censoring and truncation, and a collection of insurance rate making and reserving functions. The package also provides several model evaluation and visualization functions to help users easily analyze the performance of the fitted model and interpret the model in insurance contexts.

Heteroskedasticity and autocorrelation-robust (HAR) inference in time series regression typically involves kernel estimation of the long-run variance. Conventional wisdom holds that, for a given kernel, the choice of truncation parameter trades off a test’s null rejection rate and power, and that this tradeoff differs across kernels. We formalize this intuition: using higher-order expansions, we provide a unified size-power frontier for both kernel and weighted orthonormal series tests using nonstandard “fixed-b” critical values. We also provide a frontier for the subset of these tests for which the fixed-b distribution is t or F. These frontiers are respectively achieved by the QS kernel and equal-weighted periodogram. The frontiers have simple closed-form expressions, which upon evaluation show that the price paid for restricting attention to tests with t and F critical values is small. The frontiers are derived for the Gaussian multivariate location model, but simulations suggest the qualitative findings extend to stochastic regressors.

We study an online knapsack problem where the items arrive sequentially and must be either immediately packed into the knapsack or irrevocably discarded. Each item has a different size and the objective is to maximize the total size of items packed. We focus on the class of randomized algorithms which initially draw a threshold from some distribution, and then pack every fitting item whose size is at least that threshold. Threshold policies satisfy many desiderata including simplicity, fairness, and incentive-alignment. We derive two optimal threshold distributions, the first of which implies a competitive ratio of 0.432 relative to the optimal offline packing, and the second of which implies a competitive ratio of 0.428 relative to the optimal fractional packing. We also consider the generalization to multiple knapsacks, where an arriving item has a different size in each knapsack and must be placed in at most one. This is equivalent to the AdWords problem where item truncation is not allowed. We derive a randomized threshold algorithm for this problem which is 0.214-competitive. We also show that any randomized algorithm for this problem cannot be more than 0.461-competitive, providing the first upper bound which is strictly less than 0.5. This online knapsack problem finds applications in many areas, like supply chain ordering, online advertising, and healthcare scheduling, refugee integration, and crowdsourcing. We show how our optimal threshold distributions can be naturally implemented in the warehouses for a Latin American chain department store. We run simulations on their large-scale order data, which demonstrate the robustness of our proposed algorithms.

The validity of the IV estimator relies on the orthogonality with respect to the random disturbance. However, in cases of endogenously truncated data as well as in other instances (e.g., censored data) which is very frequently the nature of data used in empirical research, there exists severe contamination in the disturbance due to the endogenous selection process. Such a contamination implies that even if the instrumental variable and the random disturbance are unconditionally independent, they are yet conditionally jointly dependent given the selection variable. The rationale is that the endogenous selection process generates a comovement between the IV and the disturbance which is related to the variation in the selection equation’s covariates. This contamination propagates additional bias introduced into the parameter estimates of the various covariates. Consequently, not only does the conventional IV not solve the problem it is intended to, but rather it introduces additional bias into the parameter estimates of the various covariates of the substantive equation. Our empirical implementation shows that even under mild correlation between the random disturbances, the resulting bias in the estimated parameter of the endogenous covariate in the substantive equation can amount to almost tenfold the true parameter value. We offer a semiparametric Fourier-dependent Sieve IV (SPIV) estimator correcting for both truncation as well as endogeneity biases. The Fourier estimator is a functional of the orthonormal polynomial sequence family. The most attractive feature of this estimator for our purposes is that it intrinsically prevents potential multicollinearity problems, a feature the kernel estimator does not possess. The proposed estimator removes the hurdle which prevents orthogonality under truncation or other misspecifications. Using Monte Carlo simulations attest to very high accuracy of our offered semiparametric Sieve IV estimator as well as high efficiency as reflected by √n consistency. These results have been verified by utilizing 2,000,000 different distribution functions, generating 100 million realizations to construct the various data sets.

Semiparametric correction for a sample selection bias in the presence of endogenous truncation is known to be much more difficult in the case of a binary selection variable than in the case of a continuous selection variable. This paper proposes a simple bandwidth-free semiparametric methodology to correct for a self-selection bias in a truncated sample, without any prior knowledge of the marginal density functions of the selection model’s random disturbances. Each of the unknown marginal density functions is estimated using Sieve estimator, utilizing Hermite polynomials as basis functions. The aforementioned procedure is appropriate for both binary and continuous selection variables cases under the covariate shift assumption. We consider a double hurdle model, which is a combination of two selection rules. The first is propagated by a truncation in the dependent variable of the substantive equation. The second is propagated by endogenous self-selection. The suggested correction procedure produces estimates that are of high accuracy and consistent based on Monte Carlo simulations. The random disturbances are not restricted to being symmetric and their marginal distribution functions are unknown. Thus, in the data generation process we verify the applicability of our procedure to cases in which the disturbances are neither jointly nor marginally normally distributed. These disturbances are constructed as realizations of non-symmetric distribution functions.

The rank-based measures of intergenerational mobility have become increasingly popular in economics literature. Recent evidence shows that rank-based measures are less affected by measurement error and life-cycle bias compared with other standard measures such as intergenerational regression coefficient and intergenerational correlation. However, most of the available household surveys suffer from sample truncation, because coresidency is used to define household membership. There is no evidence on how sample truncation affects the rank-based mobility estimates relative to intergenerational regression coefficient and intergenerational correlation. This paper provides evidence on this in the context of intergenerational schooling persistence, using two exceptional surveys from India and Bangladesh that include all children irrespective of residency status. The analysis shows that the measures of relative mobility (slopes) are biased downward in coresident samples, but the average bias in rank correlation is less than half of that in intergenerational regression coefficient, and comparable to that in intergenerational correlation in magnitude. The intercept estimates are biased upward, with the largest bias found in the intercept of the regression used for intergenerational correlation. Truncation bias in rank-based absolute mobility estimates is the lowest in most cases. The results strengthen the case for rank-based measures of intergenerational mobility when working with the standard household surveys.

Despite the growing interest in global inequality, assessing inequality trends is a major challenge becauseindividual data on income or consumption is not often available. Nevertheless, the periodic release of cer-tain summary statistics of the income distribution has become increasingly common. Hence, groupeddata in form of income shares have been conventionally used to construct inequality trends based onlower bound approximations of inequality measures. This approach introduces two potential sourcesof measurement error: first, these estimates are constructed under the assumption of equality of incomeswithin income shares; second, the highest income earners are not included in the household surveysfrom which grouped data is obtained. In this paper, we propose to deploy a flexible parametric model,which addresses these two issues in order to obtain a reliable representation of the income distributionand accurate estimates of inequality measures. This methodology is used to estimate the recent evolutionof global interpersonal inequality from 1990 to 2015 and to examine the effect of survey under-coverageof top incomes on the level and direction of global inequality. Overall, we find that item non-response atthe top of the distribution substantially biases global inequality estimates, but, more importantly, itmight also affect the direction of the trends. The authors hereby acknowledge UNU-WIDER and the project World Inequality where an earlier version of this study was published. The authors are grateful to Stephen Jenkins, Branko Milanovic, Nora Lustig, Juan Gabriel Rodriguez, Gustavo Marrero, Roy Van der Weide and participants at the UM Sustainability and Development Conference, Seventh ECINEQ Meeting, the 33th Annual Congress of the European Economic Association, and UNU-WIDER internal seminar series for helpful comments on earlier versions of this paper. Vanesa Jorda wishes to acknowledge financial support from the Ministerio de Economía y Competitividad (Project ECO2016-76203-C2-1-P).

In econometrics, long memory models for variance modeling like FIGARCH or FIAPARCH are characterized by a Fractional Differencing term. In order to estimate and apply these models, the infinite MacLaurin expansion of the differencing term has to be truncated at a certain level. We transfer the recently introduced fast fractional differencing that utilizes fast Fourier transforms (FFT) to long memory conditional variance models and show that this FFT approach offers immense speed-ups. This allows to further increase the truncation lag while ensuring a feasible computation time. We demonstrate how calculation times of parameter estimations of these models benefit from this new approach, relative to sample length and truncation lag. In this simulation study, allowing for higher truncation lags implies better and more precise results in parameter estimations. In order to emphasize the importance for practitioners and research in risk management, we carry out different time consuming rolling-window analyses for WTI and Brent crude oil returns and show that total computation times can be reduced by a factor 20 to 30 for FIGARCH. The speed-ups for FIAPARCH are found to be significantly higher. The FFT approach offers a computational advantage to all ARCH(infinity)-representations of widely-used long memory models like FIGARCH, FIAPARCH, HYGARCH, and FIEGARCH, especially for large data sets which are common in high frequency analyses.

This study proposes a new methodology with which the effectiveness of linear extrapolation (LE) of Jiang and Tian (2005) for the implied moment estimators of Bakshi et al. (2003) can be evaluated, even when the true moments are unknown. Using S&P 500 index options data and truncation sensitivity functions, this study suggests that although LE is effective for all three estimators, the implied skewness and kurtosis estimators can remain sensitive to truncation, i.e., the complete unavailability of option prices for a part of deep-out-of-the-money or deep-in-the-money region of strike price domain, even when LE is employed to mitigate truncation.

Statistical analyses on actual data depict operational risk as an extremely heavy-tailed phenomenon, able to generate losses so extreme as to suggest the use of infinite-mean models. But no loss can actually destroy more than the entire value of a bank or of a company, and this upper bound should be considered when dealing with tail-risk assessment. Introducing what we call the dual distribution, we show how to deal with heavy-tailed phenomena with a remote yet finite upper bound. We provide methods to compute relevant tail quantities such as the Expected Shortfall, which is not available under infinite-mean models, allowing adequate provisioning and capital allocation. This also permits a measurement of fragility. The main difference between our approach and a simple truncation is in the smoothness of the transformation between the original and the dual distribution. Our methodology is useful with apparently infinite-mean phenomena, as in the case of operational risk, but it can be applied in all those situations involving extreme fat tails and bounded support.

We generalize model calibration for a multivariate Tweedie distribution to allow for censored observations; estimation is based on the method of moments. The multivariate Tweedie distribution we consider incorporates dependence in a pool of lives via a common stochastic component. Pools may be interpreted in various ways, from nation-wide cohorts to employer-based pension annuity portfolios. In general, the common stochastic component is representative of systematic longevity risk, which is not accounted for in standard life tables and actuarial models used for annuity pricing and reserving.

This Appendix contains additional empirical results with respect to the published article. In Section 1, the posterior results for the HDP parameters of the IHMS- ARMA models are presented for the U.S. GDP growth rate and inflation series. In Section 2, we report additional in-sample and forecasting results for the same series. In Section 3, some results for a different truncation choice of the number of regimes in the approximate model are reported. Full paper available at: https://ssrn.com/abstract=2965441

Patent application data is subject to a truncation-bias since application information is released only after the patent is granted. To address this bias, patent application numbers are typically adjusted using historical patterns. The NBER patents database, which is widely used in extant research, was recently updated, extending the sample end-date of 2006 to 2010. The updated sample, with four more years of data, is largely free of truncation-bias over the period covered by the NBER-2006 sample, allowing us to evaluate the bias-adjustment method and its implications for existing studies. We find that adjustments perform poorly towards the end of NBER-2006 sample. Furthermore, the bias is correlated with firm attributes, such as stock-liquidity. We revisit the findings of Fang et al. (2014) and show that their result that stock-market-liquidity has a negative causal impact on innovation is not supported in the updated, relatively bias-free sample. Our findings suggest that studies using patent-application data should drop observations over 3-5 years from the end-date.

In this addendum to Verbelen et al. (2015), we present several additional examples of the calibration procedure for fitting multivariate mixtures of Erlangs to censored and truncated data.

We present a calibration procedure for fitting mixtures of Erlangs to censored and truncated data by iteratively using the EM algorithm. Mixtures of Erlangs form a very versatile, yet analytically tractable, class of distributions making them suitable for modeling purposes.The effectiveness of the proposed algorithm is demonstrated on simulated data as well as real data sets.

We develop a new method for pricing options on discretely sampled arithmetic average in exponential L'evy models. The main idea is the reduction to a backward induction procedure for the difference W_n between the Asian option with averaging over n sampling periods and the price of the European option with maturity one period. This allows for an efficient truncation of the state space. At each step of backward induction, W_n is calculated accurately and fast using a piece-wise interpolation or splines, fast convolution and either iFFT or the parabolic inverse Fourier transform. Numerical results demonstrate the advantages of the method.

This paper addresses challenges of estimating operational risk regulatory capital when a loss sample is truncated from below at a data collection threshold. Recent operational risk literature reports that the attempts to estimate loss distributions by the maximum likelihood method are not always successful under the truncation approach that accounts for the existence of censored losses - the likelihood surface is sometimes ascending with no global solution. The literature o ers an alternative called the shifting approach, which estimates the loss distribution without taking into account censored losses. We present a necessary and su cient condition for the existence of the global solution to the likelihood maximization problem under the truncation approach when the true loss distribution is lognormal, and derive a practically explicit expression for the global solution. We show by a simulation study that, as the sample size increases, the capital bias by the truncation approach declines while the bias by the shifting approach does not.

This paper studies the asymptotic validity of sieve bootstrap for nonstationary panel factor series. Two main results are shown. Firstly, a bootstrap Invariance Principle is derived pointwise in i, obtaining an upper bound for the order of truncation of the AR polynomial that depends on n and T. Consistent estimation of the long run variances is also studied for (n,T)→∞. Secondly, joint bootstrap asymptotics is also studied, investigating the conditions under which the bootstrap is valid. The extent of cross sectional dependence which can be allowed for is investigated, showing that, in the presence of strong cross dependence, consistent estimation of the long run variance (and therefore validity of the bootstrap) is no longer possible. The paper also considers extensions to the case of a mixture of stationary and nonstationary common factors.

The trading game invariance hypothesis of Kyle and Obizhaeva (2011a) is tested using the Trades and Quotes (“TAQ”) dataset. Over the period 1993-2001, the estimated monthly regression coefficients of the log of trade arrival rate on the log of trading activity has an almost constant value of 0.690, slightly higher than the value of 2/3 predicted by the invariance hypotheses. Over the period 2001-2008, the coefficient estimates rise almost linearly, with an average value of 0.787. Average trade size, normalized for trading activity, falls dramatically over the period 1993-2008. The distribution of trade size adjusted for trading activity resembles a log-normal more closely in 1993 than in 2001 or 2008, with truncation below the 100-share odd-lot boundary becoming a more prominent feature over time. These results suggests that the 2001 reduction in minimum tick size to one cent and the subsequent increase in algorithmic trading have resulted in more intense order shredding in actively traded stocks than inactively traded stocks. The invariance hypothesis explains 91% of the cross-sectional variation in print arrival rates and average print size.

In operational risk measurement, the estimation of severity distribution parameters is the main driver of capital estimates, yet this remains a non-trivial challenge for many reasons. Maximum likelihood estimation (MLE) does not adequately meet this challenge because of its well-documented non-robustness to modest violations of idealized textbook model assumptions, specifically that the data are independent and identically distributed (i.i.d.), which is clearly violated by operational loss data. Yet even under i.i.d. data, capital estimates based on MLE are biased upwards, sometimes dramatically, due to Jensen’s inequality. This overstatement of the true risk profile increases as the heaviness of the severity distribution tail increases, so dealing with data collection thresholds by using truncated distributions, which have thicker tails, increases MLE-related capital bias considerably. Truncation also augments correlation between a distribution’s parameters, and this exacerbates the non-robustness of MLE. This paper derives influence functions for MLE for a number of severity distributions, both truncated and not, to analytically demonstrate its non-robustness and its sometimes counterintuitive behavior under truncation. Empirical influence functions are then used to compare MLE against robust alternatives such as the Optimally Bias-Robust Estimator (OBRE) and the Cramer-von Mises (CvM) estimator. The ultimate focus, however, is on the economic and regulatory capital estimates generated by these three estimators. The mean adjusted single-loss approximation (SLA) is used to translate these parameter estimates into Value-at-Risk (VaR) based estimates of regulatory and economic capital. The results show that OBRE estimators are very promising alternatives to MLE for use with actual operational loss event data, whether truncated or not, when the ultimate goal is to obtain accurate (non-biased) and robust capital estimates.

This paper constructs estimators for panel data regression models with individual specific heterogeneity and two-sided censoring and truncation. Following Powell (1986) the estimation strategy is based on moment conditions constructed from re-censored or re-truncated residuals. While these moment conditions do not identify the parameter of interest, they can be used to motivate objective functions that do. We apply one of the estimators to study the effect of a Danish tax reform on household portfolio choice. The idea behind the estimators can also be used in a cross sectional setting.

We propose a semi-parametric approach for testing orthogonality and causality between two infinite-order co-integrated vector auto-regressive IVAR(1) series. The procedures considered can be viewed as extensions of classical methods proposed by Haugh (1976, JASA) and Hong (1996, Biometrika) for testing independence between stationary univariate time series. The tests are based on the residuals of long auto-regressions, hence allowing for computational simplicity, weak assumptions on the form of the underlying process, and a direct interpretation of the results in terms of innovations (or reduced-form shocks). The test statistics are standardized versions of the sum of weighted squares of residual cross-correlation matrices. The weights depend on a kernel function and a truncation parameter. The asymptotic distributions of the test statistics under the null hypothesis are derived, and consistency is established against fixed alternatives of serial cross-correlation of unknown form. Apart from standardization factors, the multivariate portmanteau statistic which takes into account a fixed number of lags, can be viewed as a special case of our procedure based on the truncated uniform kernel. A simulation study is presented which indicates that the proposed tests have good size and power properties in finite samples. The proposed procedures are applied to study interactions between Canadian and American monetary quarterly variables associated with monetary policy (money, interest rates, prices, aggregate output). The empirical results clearly allow to reject the absence of correlation between the shocks in both countries, and indicate a unidirectional Granger causality running from the U.S. variables to the Canadian ones.

When estimating and forecasting realized volatility in the presence of jumps, a form of bias-variance tradeoff is present in the selection of the truncation threshold. We propose an optimal method for threshold selection that minimizes the out-of-sample forecasting loss. The use of a forecasting framework is fundamentally different from the testing framework in the literature. We find that a priori large truncation thresholds may not be optimal from a forecasting perspective and smaller thresholds should be used. An extensive simulation study and an empirical application to S&P 500 futures demonstrate the effectiveness of the proposed method.

In the present article we deal with the problem of computing the first and second moments for the rectangularly double truncated multivariate normal density. Our primary aim is to extend the derivation of Tallis (1961) to general mean and covariance for double truncation. Indeed we also deduce a simple computer algorithm for computing the first two moments and the bivariate marginal density.

After performing a Least Trimmed Square procedure at 1%, Kraft, Leone, and Wasley (2006) find an inverted-U relation between Accruals or net operating assets (NOA) and subsequent one-year abnormal returns. They argue that this opposes behavioral explanations for the Accrual and NOA anomalies. We show that the inverted-U relation is a spurious consequence of the truncation bias noted in Kothari, Sabino, and Zach (2005). LTS trimming for these anomalies effectively trims observations by the value of the dependent variable (returns). We show that because returns skewness varies systematically with the accounting predictor, the LTS trimming induces a truncation bias that varies systematically across the accounting predictors. The variation in returns skewness is related to the variation in the incidence of loss firms across the accounting predictors. Among profit firms, which have less skewed returns and so are less subject to the truncation bias, the negative monotonic relation between accounting predictors and subsequent abnormal returns is robust to trimming. Thus, ex post non-random trimming of returns can spuriously induce evidence against either the efficient market hypothesis or behavioral theories. Additionally, this paper shows that the anomalies survive trimming, despite the truncation bias, when a larger set of asset pricing controls and test methods that control for cross-sectional dependence are used.

Dynamic Stochastic General Equilibrium models are often tested against empirical VARs or estimated by minimizing the distance between the model's and the VAR impulse response functions. These methodologies require that the data-generating process consistent with the DSGE theoretical model has a VAR representation. This paper discusses the assumptions needed for a finite-order VAR(p) representation of any subset of a DSGE model variables to exist. When a VAR(p) is only an approximation to the true VAR, the paper shows that the truncated VAR(p) may return largely incorrect estimates of the impulse response function. The results do not hinge on an incorrect identification strategy or on small sample bias. But the bias introduced by truncation can lead to bias in the identification of the structural shocks. Identification strategies that are equivalent in the true VAR representation perform differently in the approximating VAR.

Using Logistic Normal regressions, we model the price-setting behaviour for a large sample of Belgian consumer prices over the January 1989 - January 2001 period. Our results indicate that time-dependent features are very important, particularly an infinite mixture of Calvo pricing rules and truncation at specific horizons. Truncation is mainly a characteristic of pricing in the service sector where it mostly takes the form of annual Taylor contracts typically renewed at the end of December. Several other variables, including some that can be considered as state variables, are also found to be statistically significant. This is particularly so for accumulated sectoral inflation since the last price change. Once heterogeneity and the role of accumulated inflation are acknowledged, hazard functions become mildly upward-sloping, even in a low inflation regime. The contribution of the state-dependent variables to the pseudo-R² of our equations is, however, not particularly important.

We study bank discrimination against private firms in transition countries. Theoretically, we show that banks may discriminate for non-profit reasons, but this discrimination diminishes with a bank's incentives and human capital. Employing matching bank-firm data from China, we empirically examine the extent, sources and consequences of discrimination. Our unique survey design allows us to disentangle sample truncation, omitted variable bias, and endogeneity issues. Our empirical findings confirm the theoretical predictions. We also find that as a result of discrimination, private firms resort to more expensive trade credits.

We consider cross-validation strategies for the SNP nonparametric density estimator, which is a truncation (or sieve) estimator based upon a Hermite series expansion. Our main focus is on the use of SNP density estimators as an adjunct to EMM structural estimation. It is known that for this purpose a desirable truncation point occurs at the last point at which the MSE curve of the SNP density estimate declines abruptly. We study the determination of the MSE curve on a per sample basis for iid data by means of leave-one-out cross-validation and hold-out-sample cross-validation through an examination of their performance over the Marron-Wand test suite and models related to asset pricing and auction applications. We find that both methods are informative as to the location of abrupt drops. The hold-out-sample method is cheaper to compute because it requires fewer nonlinear optimizations. The minimum of the hold-out-sample cross-validation curve also seems to be a better indicator of the minimum of the true MSE curve. We consider the asymptotic justification of hold-out-sample cross-validation. For this purpose, we establish rates of convergence of the SNP estimator under the Hellinger norm that are of interest in their own right.

This paper analyzes autoregressive time series models where the errors are assumed to be martingale difference sequences that satisfy an additional symmetry condition on their fourth order moments. Under these conditions Quasi Maximum Likelihood estimators of the autoregressive parameters are no longer efficient in the GMM sense. The main result of the paper is the construction of efficient semiparametric instrumental variables estimators for the autoregressive parameters. The optimal instruments are linear functions of the innovation sequence. It is shown that a frequency domain approximation of the optimal instruments leads to an estimator which only depends on the data periodogram and an unknown linear filter. Semiparametric methods to estimate the optimal filter are proposed. The procedure is equivalent to GMM estimators where lagged observations are used as instruments. Due to the additional symmetry assumption on the fourth moments the number of instruments is allowed to grow at the same rate as the sample. No lag truncation parameters are needed to implement the estimator which makes it particularly appealing from an applied point of view.