According to the conventional asymptotic theory, the two-step Generalized Method of Moments (GMM) estimator and test perform as least as well as the one-step estimator and test in large samples. The conventional asymptotic theory, as elegant and convenient as it is, completely ignores the estimation uncertainty in the weighting matrix, and as a result it may not reflect finite sample situations well. In this paper, we employ the fixed-smoothing asymptotic theory that accounts for the estimation uncertainty, and compare the performance of the one-step and two-step procedures in this more accurate asymptotic framework. We show the two-step procedure outperforms the one-step procedure only when the benefit of using the optimal weighting matrix outweighs the cost of estimating it. This qualitative message applies to both the asymptotic variance comparison and power comparison of the associated tests. A Monte Carlo study lends support to our asymptotic results.
{"title":"Should We Go One Step Further? An Accurate Comparison of One-Step and Two-Step Procedures in a Generalized Method of Moments Framework","authors":"Jungbin Hwang, Yixiao Sun","doi":"10.2139/ssrn.2646326","DOIUrl":"https://doi.org/10.2139/ssrn.2646326","url":null,"abstract":"According to the conventional asymptotic theory, the two-step Generalized Method of Moments (GMM) estimator and test perform as least as well as the one-step estimator and test in large samples. The conventional asymptotic theory, as elegant and convenient as it is, completely ignores the estimation uncertainty in the weighting matrix, and as a result it may not reflect finite sample situations well. In this paper, we employ the fixed-smoothing asymptotic theory that accounts for the estimation uncertainty, and compare the performance of the one-step and two-step procedures in this more accurate asymptotic framework. We show the two-step procedure outperforms the one-step procedure only when the benefit of using the optimal weighting matrix outweighs the cost of estimating it. This qualitative message applies to both the asymptotic variance comparison and power comparison of the associated tests. A Monte Carlo study lends support to our asymptotic results.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"2014 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129673705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a procedure for testing simple hypotheses on a subset of the structural parameters in linear instrumental variables models. Our test is valid uniformly over a large class of distributions allowing for identification failure and heteroskedasticity. The large-sample distribution of our test statistic is shown to depend on a key quantity that cannot be consistently estimated. Under our proposed procedure, we construct a confidence set for this key quantity and then maximize, over this confidence set, the appropriate quantile of the large-sample distribution of the test statistic. This maximum is used as the critical value and Bonferroni correction is used to control the overall size of the test. Monte Carlo simulations demonstrate the advantage of our test over the projection method in finite samples.
{"title":"A New Method for Uniform Subset Inference of Linear Instrumental Variables Models","authors":"Yinchu Zhu","doi":"10.2139/ssrn.2620552","DOIUrl":"https://doi.org/10.2139/ssrn.2620552","url":null,"abstract":"We propose a procedure for testing simple hypotheses on a subset of the structural parameters in linear instrumental variables models. Our test is valid uniformly over a large class of distributions allowing for identification failure and heteroskedasticity. The large-sample distribution of our test statistic is shown to depend on a key quantity that cannot be consistently estimated. Under our proposed procedure, we construct a confidence set for this key quantity and then maximize, over this confidence set, the appropriate quantile of the large-sample distribution of the test statistic. This maximum is used as the critical value and Bonferroni correction is used to control the overall size of the test. Monte Carlo simulations demonstrate the advantage of our test over the projection method in finite samples.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132019059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper shows how to compare the size of the middle class in income distributions using a polarization index that do not account for identification. We derive a class of polarization indices where the antagonism function is constant in identification. The comparison of distributions using an index from this class motivates the introduction of an alienation dominance surface, which is a function of an alienation threshold. We first prove that a distribution has a large alienation component in polarization compared to another if the former always has a larger dominance surface than the latter regardless of the value of the alienation threshold. Then, we show that the distribution with large dominance surface is more concentrated in the tails and has a smaller middle class than the other distribution. We implement statistical inference and test dominance between pairs of distributions using the asymptotic theory and Intersection Union tests. Our methodology is illustrated in comparing the declining of the middle class across pairwise distributions of twenty-two countries from the Luxembourg Income Study data base.
{"title":"Comparing the Size of the Middle Class Using the Alienation Component of Polarization","authors":"André-Marie Taptué","doi":"10.2139/ssrn.2610844","DOIUrl":"https://doi.org/10.2139/ssrn.2610844","url":null,"abstract":"This paper shows how to compare the size of the middle class in income distributions using a polarization index that do not account for identification. We derive a class of polarization indices where the antagonism function is constant in identification. The comparison of distributions using an index from this class motivates the introduction of an alienation dominance surface, which is a function of an alienation threshold. We first prove that a distribution has a large alienation component in polarization compared to another if the former always has a larger dominance surface than the latter regardless of the value of the alienation threshold. Then, we show that the distribution with large dominance surface is more concentrated in the tails and has a smaller middle class than the other distribution. We implement statistical inference and test dominance between pairs of distributions using the asymptotic theory and Intersection Union tests. Our methodology is illustrated in comparing the declining of the middle class across pairwise distributions of twenty-two countries from the Luxembourg Income Study data base.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123605686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the context of polarized societies, income homogeneity is linked to the frequency and the intensity of social unrest. Most homogenous countries exhibit a lower frequency of intense social conflicts and less homogeneous countries show a higher frequency of moderate social conflicts. This paper develops a methodology to compare the degree of homogeneity of two income distributions. We use for that purpose and index of polarization that does not account for alienation. This index is the identification component of polarization that measures the degree to which individuals feel alike in an income distribution. This development leads to identification dominance curves and derives first-order and higher-order stochastic dominance conditions. First-order stochastic dominance is performed through identification dominance curves drawn on a support of identification thresholds. These curves are used to determine whether identification, homogeneity, or similarity of individuals is greater in one distribution than in another for general classes of polarization indices and ranges of possible identification thresholds. We also derive the asymptotic sampling distribution of identification dominance curves and test dominance between two distributions using Intersection Union tests and bootstrapped p-values. Our methodology is illustrated by comparing pairs of distributions of eleven countries drawn from the Luxembourg Income Study database.
{"title":"Comparing the Homogeneity of Income Distributions Using Polarization Indices","authors":"André-Marie Taptué","doi":"10.2139/ssrn.2610860","DOIUrl":"https://doi.org/10.2139/ssrn.2610860","url":null,"abstract":"In the context of polarized societies, income homogeneity is linked to the frequency and the intensity of social unrest. Most homogenous countries exhibit a lower frequency of intense social conflicts and less homogeneous countries show a higher frequency of moderate social conflicts. This paper develops a methodology to compare the degree of homogeneity of two income distributions. We use for that purpose and index of polarization that does not account for alienation. This index is the identification component of polarization that measures the degree to which individuals feel alike in an income distribution. This development leads to identification dominance curves and derives first-order and higher-order stochastic dominance conditions. First-order stochastic dominance is performed through identification dominance curves drawn on a support of identification thresholds. These curves are used to determine whether identification, homogeneity, or similarity of individuals is greater in one distribution than in another for general classes of polarization indices and ranges of possible identification thresholds. We also derive the asymptotic sampling distribution of identification dominance curves and test dominance between two distributions using Intersection Union tests and bootstrapped p-values. Our methodology is illustrated by comparing pairs of distributions of eleven countries drawn from the Luxembourg Income Study database.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131847418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper propose a new panel stochastic dominance (SD) test-PDD test, the asymptotic properties are derived, which extends Davidson and Duclos (DD) SD test to a panel context. The PDD test also contributes to settle one of the demerits while working with financial derivatives time series: that the standard individual tests for Stochastic Dominance in time series are unsatisfactory in terms of power when the sample size is too small, and typically the financial derivatives have a limited life, in particular, stock options and covered warrants. This is because the pairwise SD tests are nonparametric, and nonparametric tests require large sample size, in this case, the individual tests for financial derivative time series may not distinguish between the null and the alternative hypotheses for each series, and lead to retain the null hypothesis, even if the alternative is true. Hence the PDD test would improve the power of individual SD tests: a panel test gathers all the information of all the series, and then increases the power compared to its corresponding individual test. This paper also extends the classical likelihood ratio (LR) information efficiency test to a panel framework to get more powerful new tests. A bootstrap methodology is developed to correct the size distortion of the LR test.
{"title":"Panel Stochastic Dominance Test and Panel Informational Efficiency LR Test","authors":"C. de Peretti, Chia-Ying Chan, W. Wong, C. Siani","doi":"10.2139/ssrn.2604662","DOIUrl":"https://doi.org/10.2139/ssrn.2604662","url":null,"abstract":"This paper propose a new panel stochastic dominance (SD) test-PDD test, the asymptotic properties are derived, which extends Davidson and Duclos (DD) SD test to a panel context. The PDD test also contributes to settle one of the demerits while working with financial derivatives time series: that the standard individual tests for Stochastic Dominance in time series are unsatisfactory in terms of power when the sample size is too small, and typically the financial derivatives have a limited life, in particular, stock options and covered warrants. This is because the pairwise SD tests are nonparametric, and nonparametric tests require large sample size, in this case, the individual tests for financial derivative time series may not distinguish between the null and the alternative hypotheses for each series, and lead to retain the null hypothesis, even if the alternative is true. Hence the PDD test would improve the power of individual SD tests: a panel test gathers all the information of all the series, and then increases the power compared to its corresponding individual test. This paper also extends the classical likelihood ratio (LR) information efficiency test to a panel framework to get more powerful new tests. A bootstrap methodology is developed to correct the size distortion of the LR test.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131841282","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The modified Sharpe ratio is commonly used to evaluate the risk-adjusted performance of an investment with non-normal returns, such as hedge funds. In this note, a test for equality of modified Sharpe ratios of two investments is developed. A simulation study demonstrates the good size and power properties of the test. An application illustrates the complementarity between the Sharpe ratio and modified Sharpe ratio test for performance testing on hedge fund return data.
{"title":"Testing Equality of Modified Sharpe Ratios","authors":"David Ardia, Kris Boudt","doi":"10.2139/ssrn.2516591","DOIUrl":"https://doi.org/10.2139/ssrn.2516591","url":null,"abstract":"The modified Sharpe ratio is commonly used to evaluate the risk-adjusted performance of an investment with non-normal returns, such as hedge funds. In this note, a test for equality of modified Sharpe ratios of two investments is developed. A simulation study demonstrates the good size and power properties of the test. An application illustrates the complementarity between the Sharpe ratio and modified Sharpe ratio test for performance testing on hedge fund return data.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132853030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider here the problem of testing the effect of a subset of predictors for a regression model with predictor dimension fixed but ultra high dimensional responses. Because the response dimension is ultra high, the classical method of likelihood ratio test is no longer applicable. To solve the problem, we propose a novel solution, which decomposes the original problem into many testing problems with univariate responses. Subsequently, the usual residual sum of squares (RSS) type test statistics can be obtained. Those statistics are then integrated together across different responses to form an overall and powerful test statistic. Under the null hypothesis, the resulting test statistic is asymptotically standard normal after some appropriate standardization. Numerical studies are presented to demonstrate the finite sample performance of the test statistic and a real example about paid search advertising is analyzed for illustration purpose.
{"title":"Testing Predictor Significance with Ultra High Dimensional Multivariate Responses","authors":"Yingying Ma, Wei Lan, Hansheng Wang","doi":"10.2139/ssrn.2533353","DOIUrl":"https://doi.org/10.2139/ssrn.2533353","url":null,"abstract":"We consider here the problem of testing the effect of a subset of predictors for a regression model with predictor dimension fixed but ultra high dimensional responses. Because the response dimension is ultra high, the classical method of likelihood ratio test is no longer applicable. To solve the problem, we propose a novel solution, which decomposes the original problem into many testing problems with univariate responses. Subsequently, the usual residual sum of squares (RSS) type test statistics can be obtained. Those statistics are then integrated together across different responses to form an overall and powerful test statistic. Under the null hypothesis, the resulting test statistic is asymptotically standard normal after some appropriate standardization. Numerical studies are presented to demonstrate the finite sample performance of the test statistic and a real example about paid search advertising is analyzed for illustration purpose.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132049658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Testing data for conformity to Benford's law is used not only by auditors exploiting a numerical phenomenon to detect fraudulently reported data. Operationally goodness-of-fit tests are used to conclude if data that should, does indeed comply with Benford's law. Naturally, not all statistical tests share the same sensitivity for detecting departures from the null-hypothesis, and thus the test choice is of central importance. This study compares seven tests for Benford's law common in literature. These tests are presented together with the critical values required for statistical hypothesis testing. The procedures are compared in terms of their power, at a significance level of 5%, versus 16 alternative distributions covering a wide range of possible deviations. Even though no test consistently dominated all other tests, results show, amongst other findings, that the current method of choice, the Chi^2-test, is consistently outperformed by Watson's-U^2 statistic.
{"title":"Testing for Benford's Law: A Monte Carlo Comparison of Methods","authors":"D. Joenssen","doi":"10.2139/ssrn.2545243","DOIUrl":"https://doi.org/10.2139/ssrn.2545243","url":null,"abstract":"Testing data for conformity to Benford's law is used not only by auditors exploiting a numerical phenomenon to detect fraudulently reported data. Operationally goodness-of-fit tests are used to conclude if data that should, does indeed comply with Benford's law. Naturally, not all statistical tests share the same sensitivity for detecting departures from the null-hypothesis, and thus the test choice is of central importance. This study compares seven tests for Benford's law common in literature. These tests are presented together with the critical values required for statistical hypothesis testing. The procedures are compared in terms of their power, at a significance level of 5%, versus 16 alternative distributions covering a wide range of possible deviations. Even though no test consistently dominated all other tests, results show, amongst other findings, that the current method of choice, the Chi^2-test, is consistently outperformed by Watson's-U^2 statistic.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"607 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116392002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper shows that the problem of testing hypotheses in moment condition models without any assumptions about identification may be considered as a problem of testing with an infinite‐dimensional nuisance parameter. We introduce a sufficient statistic for this nuisance parameter in a Gaussian problem and propose conditional tests. These conditional tests have uniformly correct asymptotic size for a large class of models and test statistics. We apply our approach to construct tests based on quasi‐likelihood ratio statistics, which we show are efficient in strongly identified models and perform well relative to existing alternatives in two examples.
{"title":"Conditional Inference with a Functional Nuisance Parameter","authors":"Isaiah Andrews, Anna Mikusheva","doi":"10.2139/ssrn.2500534","DOIUrl":"https://doi.org/10.2139/ssrn.2500534","url":null,"abstract":"This paper shows that the problem of testing hypotheses in moment condition models without any assumptions about identification may be considered as a problem of testing with an infinite‐dimensional nuisance parameter. We introduce a sufficient statistic for this nuisance parameter in a Gaussian problem and propose conditional tests. These conditional tests have uniformly correct asymptotic size for a large class of models and test statistics. We apply our approach to construct tests based on quasi‐likelihood ratio statistics, which we show are efficient in strongly identified models and perform well relative to existing alternatives in two examples.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114075474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We examine the relationship between exports and output for a small developing country Nepal which has received little attention in such literature. Using multivariate cointegration and vector error correction model for the data of 1975 to 2011 we find long-run relationship between real exports, investment, and real GDP and bidirectional long run and short run causality between real exports and real GDP. This implies that in case of Nepal, exports in an 'engine of the growth' and growth is also 'the engine of exports'.
{"title":"Testing Export-Led Growth Hypothesis in Nepal Using Multivariate Cointegration Approach","authors":"Nayan Krishna Joshi, M. P. Dahal, B. Paudel","doi":"10.2139/ssrn.2497647","DOIUrl":"https://doi.org/10.2139/ssrn.2497647","url":null,"abstract":"We examine the relationship between exports and output for a small developing country Nepal which has received little attention in such literature. Using multivariate cointegration and vector error correction model for the data of 1975 to 2011 we find long-run relationship between real exports, investment, and real GDP and bidirectional long run and short run causality between real exports and real GDP. This implies that in case of Nepal, exports in an 'engine of the growth' and growth is also 'the engine of exports'.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128557181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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