The discrepancy distribution of stationary multiplier rules for rounding probabilities

The problem of rounding finitely many (continuous) probabilities to (discrete) proportions N _i/n is considered, for some fixed rounding accuracy n. It is well known that the rounded proportions need not sum to unity, and instead may leave a nonzero discrepancy D=(∑N _i) −n. We determine the distribution of D, assuming that the rounding function used is stationary and that the original probabilities follow a uniform distribution. Received: April 1999     

Rounding behaviour of professional macro-forecasters

The rounding of point forecasts of CPI inflation and the unemployment rate by U.S. Professional Forecasters is modest. There is little evidence that forecasts are rounded to a greater extent in response to higher perceived uncertainty surrounding future outcomes. There is clear evidence that the probability of decline forecasts are rounded: over half of the forecast probabilities of decline in the current quarter are multiples of ten. It is found here that the rounding of these probabilities correlates with worse accuracy, although it is also of note here that worse (less accurate) forecasters might round more as opposed to the degree of rounding per se worsening accuracy. By simulating the loss from rounding for a set of efficient forecasters, it is demonstrated that the explanation that respondents round otherwise efficient forecasts is implausible, and that the contribution of rounding is of minor importance.     

Regression discontinuity inference with specification error

A regression discontinuity (RD) research design is appropriate for program evaluation problems in which treatment status (or the probability of treatment) depends on whether an observed covariate exceeds a fixed threshold. In many applications the treatment-determining covariate is discrete. This makes it impossible to compare outcomes for observations “just above” and “just below” the treatment threshold, and requires the researcher to choose a functional form for the relationship between the treatment variable and the outcomes of interest. We propose a simple econometric procedure to account for uncertainty in the choice of functional form for RD designs with discrete support. In particular, we model deviations of the true regression function from a given approximating function—the specification errors—as random. Conventional standard errors ignore the group structure induced by specification errors and tend to overstate the precision of the estimated program impacts. The proposed inference procedure that allows for specification error also has a natural interpretation within a Bayesian framework.     

On the role of weights rounding in applications of resampling based on pseudopopulations

Resampling methods are widely studied and increasingly employed in applied research and practice. When dealing with complex sampling designs, common resampling techniques require adjusting noninteger sampling weights in order to construct the so called “pseudopopulation” in order to perform the actual resampling. The practice of rounding, however, has been empirically shown to be harmful under general designs. In this paper, we present asymptotic results concerning, in particular, the practice of rounding resampling weights to the nearest integer, an approach that is commonly adopted by virtue of its reduced computational burden, as opposed to randomization‐based alternatives. We prove that such approach leads to nonconsistent estimation of the distribution function of the survey variable; we provide empirical evidence of the practical consequences of the nonconsistency when the point estimation of the variance of complex estimators is of interest.     

Statistical Disclosure Control Methods for Census Frequency Tables

This paper provides a review of common statistical disclosure control (SDC) methods implemented at statistical agencies for standard tabular outputs containing whole population counts from a census (either enumerated or based on a register). These methods include record swapping on the microdata prior to its tabulation and rounding of entries in the tables after they are produced. The approach for assessing SDC methods is based on a disclosure risk–data utility framework and the need to find a balance between managing disclosure risk while maximizing the amount of information that can be released to users and ensuring high quality outputs. To carry out the analysis, quantitative measures of disclosure risk and data utility are defined and methods compared. Conclusions from the analysis show that record swapping as a sole SDC method leaves high probabilities of disclosure risk. Targeted record swapping lowers the disclosure risk, but there is more distortion of distributions. Small cell adjustments (rounding) give protection to census tables by eliminating small cells but only one set of variables and geographies can be disseminated in order to avoid disclosure by differencing nested tables. Full random rounding offers more protection against disclosure by differencing, but margins are typically rounded separately from the internal cells and tables are not additive. Rounding procedures protect against the perception of disclosure risk compared to record swapping since no small cells appear in the tables. Combining rounding with record swapping raises the level of protection but increases the loss of utility to census tabular outputs. For some statistical analysis, the combination of record swapping and rounding balances to some degree opposing effects that the methods have on the utility of the tables.     

Partial identification of probability distributions with misclassified data

This paper addresses the problem of data errors in discrete variables. When data errors occur, the observed variable is a misclassified version of the variable of interest, whose distribution is not identified. Inferential problems caused by data errors have been conceptualized through convolution and mixture models. This paper introduces the direct misclassification approach. The approach is based on the observation that in the presence of classification errors, the relation between the distribution of the ‘true’ but unobservable variable and its misclassified representation is given by a linear system of simultaneous equations, in which the coefficient matrix is the matrix of misclassification probabilities. Formalizing the problem in these terms allows one to incorporate any prior information into the analysis through sets of restrictions on the matrix of misclassification probabilities. Such information can have strong identifying power. The direct misclassification approach fully exploits it to derive identification regions for any real functional of the distribution of interest. A method for estimating the identification regions and construct their confidence sets is given, and illustrated with an empirical analysis of the distribution of pension plan types using data from the Health and Retirement Study.     

Bias reduction when data are rounded

Analytical bias reduction methods are developed for univariate rounded data for the first time. Extensions are given to rounding of multivariate data, and to smooth functionals of several distributions. As a by‐product, we give for the first time the relation between rounded and unrounded multivariate cumulants. Estimators obtained by analytical bias reduction are compared with bootstrap and jackknife estimators by simulation.     

A score test for non‐nested hypotheses with applications to discrete data models

In this paper it is shown that a convenient score test against non‐nested alternatives can be constructed from the linear combination of the likelihood functions of the competing models. This is essentially a test for the correct specification of the conditional distribution of the variable of interest. Given its characteristics, the proposed test is particularly attractive to check the distributional assumptions in models for discrete data. The usefulness of the test is illustrated with an application to models for recreational boating trips. Copyright © 2001 John Wiley & Sons, Ltd.     

A consistent model specification test with mixed discrete and continuous data

In this paper we propose a nonparametric kernel-based model specification test that can be used when the regression model contains both discrete and continuous regressors. We employ discrete variable kernel functions and we smooth both the discrete and continuous regressors using least squares cross-validation (CV) methods. The test statistic is shown to have an asymptotic normal null distribution. We also prove the validity of using the wild bootstrap method to approximate the null distribution of the test statistic, the bootstrap being our preferred method for obtaining the null distribution in practice. Simulations show that the proposed test has significant power advantages over conventional kernel tests which rely upon frequency-based nonparametric estimators that require sample splitting to handle the presence of discrete regressors.     

Anomalies and Unusual Patterns in Reported Earnings: Japanese Managers Round Earnings

Using Benford's law, this study documents pervasive evidence that managers of Japanese firms tend to engage in earnings manipulative activities of rounding earnings numbers to achieve key reference points. Similar to Carslaw (1988) and Thomas (1989) , we find that the first digit of earnings numbers is often emphasized by the management. We also find that key reference points are not limited to the first digit. The second, third, or even fourth digits are sometimes used as the reference points of the rounding earnings behavior. Finally, our results show that the incentives of rounding earnings numbers are negatively associated with the distance of pre‐rounded earnings to the next reference point.     

Labor market entry and earnings dynamics: Bayesian inference using mixtures‐of‐experts Markov chain clustering

This paper analyzes patterns in the earnings development of young labor market entrants over their life cycle. We identify four distinctly different types of transition patterns between discrete earnings states in a large administrative dataset. Further, we investigate the effects of labor market conditions at the time of entry on the probability of belonging to each transition type. To estimate our statistical model we use a model‐based clustering approach. The statistical challenge in our application comes from the difficulty in extending distance‐based clustering approaches to the problem of identifying groups of similar time series in a panel of discrete‐valued time series. We use Markov chain clustering, which is an approach for clustering discrete‐valued time series obtained by observing a categorical variable with several states. This method is based on finite mixtures of first‐order time‐homogeneous Markov chain models. In order to analyze group membership we present an extension to this approach by formulating a probabilistic model for the latent group indicators within the Bayesian classification rule using a multinomial logit model. Copyright © 2011 John Wiley & Sons, Ltd.     

Heterogeneous Treatment Under Regression Discontinuity Design: Application to Female High School Enrolment

This paper undertakes a regression discontinuity (RD) framework with multiple cutoffs unlike typical RD setting where researchers normalize the score variable and pool all the observations. This paper explores this heterogeneity in the effect of Islamic mayor on female secular high schooling in Turkey using the multiple cutoff RD framework developed in Cattaneo et al. (2016). The presence of many parties in the 1994 municipality election in Turkey means that vote share of the strongest opponent party can vary substantially leading to different cutoffs. Meyersson (2014) finds that Islamic mayors of 1994 promoted female high schooling using a normalized and pooled RD framework, which averages the effect across all the different cutoffs. We extend his work by segregating the effect of Islamic mayor across different opponent party's vote shares. Our results suggest that the positive effect on female secular high school attainment was more pronounced in municipalities where the strongest opponent party was secular than where the opponent was conservative. This heterogeneity can be attributed to a policy change in 1999, which restricted religious high school graduates from entering universities.     

Inverse-probability-weighted logrank test for stratified survival data with missing measurements

The stratified logrank test can be used to compare survival distributions of several groups of patients, while adjusting for the effect of some discrete variable that may be predictive of the survival outcome. In practice, it can happen that this discrete variable is missing for some patients. An inverse-probability-weighted version of the stratified logrank statistic is introduced to tackle this issue. Its asymptotic distribution is derived under the null hypothesis of equality of the survival distributions. A simulation study is conducted to assess behavior of the proposed test statistic in finite samples. An analysis of a medical dataset illustrates the methodology.     

ROUNDING,FOCAL POINT ANSWERS AND NONRESPONSE TO SUBJECTIVE PROBABILITY QUESTIONS

We develop a panel data model explaining answers to subjective probabilities about binary events and estimate it using data from the Health and Retirement Study on six such probabilities. The model explicitly accounts for several forms of ‘reporting behavior’: rounding, focal point ‘50%’ answers and item nonresponse. We find observed and unobserved heterogeneity in the tendencies to report rounded values or a focal answer, explaining persistency in 50% answers over time. Focal 50% answers matter for some of the probabilities. Incorporating reporting behavior does not have a large effect on the estimated distribution of the genuine subjective probabilities. Copyright © 2013 John Wiley & Sons, Ltd.     

Using simulation as a tool in selecting a retirement age under defined benefit pension plans

This paper examines how simulation modeling can be used to select a retirement age under defined benefit pension plans. This approach construes the variables affecting pension benefits as probabilistic variables. Simulations are then run to generate probabilistic values for the real value of pension benefits for alternative retirement ages. By construing variables affecting pension benefits as probability distributions, this approach reflects the uncertainty facing individuals contemplating retirement. By generating estimates of retirement benefits as probability distributions rather than as single deterministic values, the model provides individuals with a more realistic and complete frame of reference for making the retirement decision. The author is grateful to an anonymous referee and Joachim Zietz. JEF editor, for helpful comments.     

Mandatory Retirement Rules and the Retirement Decisions of University Professors in Canada

We examine the impact of mandatory retirement on the retirement decisions of professors in Canada using administrative data. Estimation of a discrete time hazard model indicates that faculty members at universities with mandatory retirement at age 65 have exit rates at age 65 that are around 30 to 38 percentage points higher than those of their counterparts at universities without mandatory retirement. This overall difference in exit rates is found when the sample is restricted by discipline, professional rank and type of university. Similar results are found for both men and women; however, the magnitude of this effect is somewhat smaller for women. Restricting the analysis to include faculty members who received their highest degree at age 34 or older does not affect the magnitude of the difference in exit rates between faculty at universities without mandatory retirement and those at universities with mandatory retirement. The estimated survival probabilities indicate that only 22.7% of faculty members employed at age 64 at universities without mandatory retirement will continue to be employed at the same university at age 72.     

Retirement policies for the 21st century

As we rapidly approach the 21st century, current human resource management policies emphasizing early retirement may prove shortsighted. Changing demographics and projected labor shortages in certain critical occupations may require a reversal of present early retirement policies. This article reports the views of a national sample of 897 executives on work and retirement. We report their reactions to pressures for early retirement, their responses to proposed incentives to delay retirmeent, and finally, their planned post retirement activities. Their responses have important implications for revising human resource policies in the years ahead.     

Functional-coefficient models for nonstationary time series data

This paper studies functional coefficient regression models with nonstationary time series data, allowing also for stationary covariates. A local linear fitting scheme is developed to estimate the coefficient functions. The asymptotic distributions of the estimators are obtained, showing different convergence rates for the stationary and nonstationary covariates. A two-stage approach is proposed to achieve estimation optimality in the sense of minimizing the asymptotic mean squared error. When the coefficient function is a function of a nonstationary variable, the new findings are that the asymptotic bias of its nonparametric estimator is the same as the stationary covariate case but convergence rate differs, and further, the asymptotic distribution is a mixed normal, associated with the local time of a standard Brownian motion. The asymptotic behavior at boundaries is also investigated.     

A nonparametric test for equality of distributions with mixed categorical and continuous data

In this paper we consider the problem of testing for equality of two density or two conditional density functions defined over mixed discrete and continuous variables. We smooth both the discrete and continuous variables, with the smoothing parameters chosen via least-squares cross-validation. The test statistics are shown to have (asymptotic) normal null distributions. However, we advocate the use of bootstrap methods in order to better approximate their null distribution in finite-sample settings and we provide asymptotic validity of the proposed bootstrap method. Simulations show that the proposed tests have better power than both conventional frequency-based tests and smoothing tests based on ad hoc smoothing parameter selection, while a demonstrative empirical application to the joint distribution of earnings and educational attainment underscores the utility of the proposed approach in mixed data settings.     

The importance of financial incentives on retirement choices: New evidence for Italy

This study exploits a new dataset to quantify the effect of financial incentives on retirement choices. This dataset contains—for the first time for Italy—information on seniority. The effects of marginal incentives and social security wealth (SSW) on retirement go in the expected direction; when employees become eligible for pension benefits, the change in financial incentives they experience is so great that their retirement probability increases by 30 percentage points.We also find that the procedure used in previous Italian studies to impute seniority leads to a considerable overestimation of that variable and of SSW. We show that, due to these measurement errors, the estimate of the SSW coefficient takes the wrong sign. A comparison of retirement studies across countries (see Gruber and Wise [Gruber, J., and Wise, D., (2004). Social Security Programs and Retirement Around the World: Micro-Estimation, NBER. The University of Chicago Press, Chicago and London.]) provides prima facie evidence that a lack of good quality data often leads to wrongly signed estimates of the SSW coefficient.