This paper introduces a new test for error cross-sectional independence in large panel data models with exogenous regressors having heterogenous slope coefficients. The proposed statistic, LM_{RMT}, is based on the Lagrange Multiplier (LM) principle and the sample correlation matrix R_{N} of the model's residuals. Since in large panels R_{N} poorly estimates its population counterpart, results from Random Matrix Theory are used to establish the high-dimensional limiting distribution of LM_{RMT} under heteroskedastic normal errors and assuming that both the panel size N and the sample size T grow to infinity in comparable magnitude. Simulation results support our theoretical findings, with LM_{RMT} being correctly sized (except for some small values of N and T). Further, the small sample size and power outcomes show robustness of our statistic to deviations from the assumptions of normality for the error terms and regressors, of strict exogeneity for the regressors, as well as of heterogeneity for their slope coefficients. The test has comparable small sample properties to related tests in the literature which have been developed under different asymptotic theory.
{"title":"A Lagrange-Multiplier Test for Large Heterogeneous Panel Data Models","authors":"Natalia Bailey, Dandan Jiang, Jianfeng Yao","doi":"10.2139/ssrn.3804164","DOIUrl":"https://doi.org/10.2139/ssrn.3804164","url":null,"abstract":"This paper introduces a new test for error cross-sectional independence in large panel data models with exogenous regressors having heterogenous slope coefficients. The proposed statistic, LM_{RMT}, is based on the Lagrange Multiplier (LM) principle and the sample correlation matrix R_{N} of the model's residuals. Since in large panels R_{N} poorly estimates its population counterpart, results from Random Matrix Theory are used to establish the high-dimensional limiting distribution of LM_{RMT} under heteroskedastic normal errors and assuming that both the panel size N and the sample size T grow to infinity in comparable magnitude. Simulation results support our theoretical findings, with LM_{RMT} being correctly sized (except for some small values of N and T). Further, the small sample size and power outcomes show robustness of our statistic to deviations from the assumptions of normality for the error terms and regressors, of strict exogeneity for the regressors, as well as of heterogeneity for their slope coefficients. The test has comparable small sample properties to related tests in the literature which have been developed under different asymptotic theory.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133832865","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 this paper we present a new asymptotically normal test for out-of-sample evaluation in nested models. Our approach is a simple modification of a traditional encompassing test that is commonly known as Clark and West test (CW). The key point of our strategy is to introduce an independent random variable that prevents the traditional CW test from becoming degenerate under the null hypothesis of equal predictive ability. Using the approach developed by West (1996), we show that in our test the impact of parameter estimation uncertainty vanishes asymptotically. Using a variety of Monte Carlo simulations in iterated multi-step-ahead forecasts we evaluate our test and CW in terms of size and power. These simulations reveal that our approach is reasonably well-sized even at long horizons when CW may present severe size distortions. In terms of power, results are mixed but CW has an edge over our approach. Finally, we illustrate the use of our test with an empirical application in the context of the commodity currencies literature.
{"title":"\"Go Wild for a While!\": A New Asymptotically Normal Test for Forecast Evaluation in Nested Models","authors":"Pablo M. Pincheira, Nicolás Hardy, Felipe Muñoz","doi":"10.2139/ssrn.3770402","DOIUrl":"https://doi.org/10.2139/ssrn.3770402","url":null,"abstract":"In this paper we present a new asymptotically normal test for out-of-sample evaluation in nested models. Our approach is a simple modification of a traditional encompassing test that is commonly known as Clark and West test (CW). The key point of our strategy is to introduce an independent random variable that prevents the traditional CW test from becoming degenerate under the null hypothesis of equal predictive ability. Using the approach developed by West (1996), we show that in our test the impact of parameter estimation uncertainty vanishes asymptotically. Using a variety of Monte Carlo simulations in iterated multi-step-ahead forecasts we evaluate our test and CW in terms of size and power. These simulations reveal that our approach is reasonably well-sized even at long horizons when CW may present severe size distortions. In terms of power, results are mixed but CW has an edge over our approach. Finally, we illustrate the use of our test with an empirical application in the context of the commodity currencies literature.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127566964","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 investigate a test of conditional predictive ability described in Giacomini and White (2006; Econometrica). Our main goal is simply to demonstrate existence of the null hypothesis and, in doing so, clarify just how unlikely it is for this hypothesis to hold. We do so using a simple example of point forecasting under quadratic loss. We then provide simulation evidence on the size and power of the test. While the test can be accurately sized we find that power is typically low.
{"title":"Tests of Conditional Predictive Ability: Existence, Size, and Power","authors":"Michael W. McCracken","doi":"10.20955/wp.2020.050","DOIUrl":"https://doi.org/10.20955/wp.2020.050","url":null,"abstract":"We investigate a test of conditional predictive ability described in Giacomini and White (2006; Econometrica). Our main goal is simply to demonstrate existence of the null hypothesis and, in doing so, clarify just how unlikely it is for this hypothesis to hold. We do so using a simple example of point forecasting under quadratic loss. We then provide simulation evidence on the size and power of the test. While the test can be accurately sized we find that power is typically low.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120997272","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}
Z. Fang, Andrés Santos, A. Shaikh, Alexander Torgovitsky
This paper considers the problem of testing whether there exists a non‐negative solution to a possibly under‐determined system of linear equations with known coefficients. This hypothesis testing problem arises naturally in a number of settings, including random coefficient, treatment effect, and discrete choice models, as well as a class of linear programming problems. As a first contribution, we obtain a novel geometric characterization of the null hypothesis in terms of identified parameters satisfying an infinite set of inequality restrictions. Using this characterization, we devise a test that requires solving only linear programs for its implementation, and thus remains computationally feasible in the high‐dimensional applications that motivate our analysis. The asymptotic size of the proposed test is shown to equal at most the nominal level uniformly over a large class of distributions that permits the number of linear equations to grow with the sample size.
{"title":"Inference for Large-Scale Linear Systems with Known Coefficients","authors":"Z. Fang, Andrés Santos, A. Shaikh, Alexander Torgovitsky","doi":"10.2139/ssrn.3695284","DOIUrl":"https://doi.org/10.2139/ssrn.3695284","url":null,"abstract":"This paper considers the problem of testing whether there exists a non‐negative solution to a possibly under‐determined system of linear equations with known coefficients. This hypothesis testing problem arises naturally in a number of settings, including random coefficient, treatment effect, and discrete choice models, as well as a class of linear programming problems. As a first contribution, we obtain a novel geometric characterization of the null hypothesis in terms of identified parameters satisfying an infinite set of inequality restrictions. Using this characterization, we devise a test that requires solving only linear programs for its implementation, and thus remains computationally feasible in the high‐dimensional applications that motivate our analysis. The asymptotic size of the proposed test is shown to equal at most the nominal level uniformly over a large class of distributions that permits the number of linear equations to grow with the sample size.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124488242","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}
Abhay Kumar, R. Soni, Iqbal Thonse Hawaldar, Meghna J. Vyas, V. Yadav
The purpose of this study is to test whether the Indian pharmaceutical companies support efficient market hypotheses (EMH) and examine the efficiency of the Indian stock market in three forms, i.e., the weak, the semi-strong, and the strong form of market efficiency. For testing the weak form of efficiency, researchers collected stock price data of 10 listed pharmaceutical companies for the past six years, from 2012 to 2017 from the NSE website, and conducted a run test. To test the efficiency of semi-strong form, researchers collected data on the announcement of events like buyback, stock split, rights issue, dividend, bonus issue. They conducted an event study on the data. For testing the strong form of efficiency, researchers collected data consisting of NAV of some mutual funds (pharmaceutical funds) and the returns of a benchmarking index to compare. The study concludes that the pharmaceutical companies and Indian stock market is efficient in the weak form of EMH and not efficient in the semi-strong and strong form of EMH.
{"title":"The Testing of Efficient Market Hypotheses: A Study of Indian Pharmaceutical Industry","authors":"Abhay Kumar, R. Soni, Iqbal Thonse Hawaldar, Meghna J. Vyas, V. Yadav","doi":"10.32479/ijefi.9764","DOIUrl":"https://doi.org/10.32479/ijefi.9764","url":null,"abstract":"The purpose of this study is to test whether the Indian pharmaceutical companies support efficient market hypotheses (EMH) and examine the efficiency of the Indian stock market in three forms, i.e., the weak, the semi-strong, and the strong form of market efficiency. For testing the weak form of efficiency, researchers collected stock price data of 10 listed pharmaceutical companies for the past six years, from 2012 to 2017 from the NSE website, and conducted a run test. To test the efficiency of semi-strong form, researchers collected data on the announcement of events like buyback, stock split, rights issue, dividend, bonus issue. They conducted an event study on the data. For testing the strong form of efficiency, researchers collected data consisting of NAV of some mutual funds (pharmaceutical funds) and the returns of a benchmarking index to compare. The study concludes that the pharmaceutical companies and Indian stock market is efficient in the weak form of EMH and not efficient in the semi-strong and strong form of EMH.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122115330","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 considers a p-dimensional time series model of the form x(t)=Π x(t-1)+Σ^(1/2)y(t), 1≤t≤T, where y(t)=(y(t1),...,y(tp))^T and Σ is the square root of a symmetric positive definite matrix. Here Π is a symmetric matrix which satisfies that ∥Π ∥_2≤ 1 and T(1-∥Π ∥_min) is bounded. The linear processes Y(tj) is of the form ∑_{k=0}^∞b(k)Z(t-k,j) where ∑_{i=0}^∞|b(i)| < ∞ and {Z(ij) } are are independent and identically distributed (i.i.d.) random variables with E Z ij =0, E|Z(ij)|²=1 and E|Z(ij)|^4< ∞. We first investigate the asymptotic behavior of the first k largest eigenvalues of the sample covariance matrices of the time series model. Then we propose a new estimator for the high-dimensional near unit root setting through using the largest eigenvalues of the sample covariance matrices and use it to test for near unit roots. Such an approach is theoretically novel and addresses some important estimation and testing issues in the high-dimensional near unit root setting. Simulations are also conducted to demonstrate the finite-sample performance of the proposed test statistic.
{"title":"Estimation and Testing for High-dimensional Near Unit Root Time Series","authors":"Bo Zhang, Jiti Gao, G. Pan","doi":"10.2139/ssrn.3579168","DOIUrl":"https://doi.org/10.2139/ssrn.3579168","url":null,"abstract":"This paper considers a p-dimensional time series model of the form x(t)=Π x(t-1)+Σ^(1/2)y(t), 1≤t≤T, where y(t)=(y(t1),...,y(tp))^T and Σ is the square root of a symmetric positive definite matrix. Here Π is a symmetric matrix which satisfies that ∥Π ∥_2≤ 1 and T(1-∥Π ∥_min) is bounded. The linear processes Y(tj) is of the form ∑_{k=0}^∞b(k)Z(t-k,j) where ∑_{i=0}^∞|b(i)| < ∞ and {Z(ij) } are are independent and identically distributed (i.i.d.) random variables with E Z ij =0, E|Z(ij)|²=1 and E|Z(ij)|^4< ∞. We first investigate the asymptotic behavior of the first k largest eigenvalues of the sample covariance matrices of the time series model. Then we propose a new estimator for the high-dimensional near unit root setting through using the largest eigenvalues of the sample covariance matrices and use it to test for near unit roots. Such an approach is theoretically novel and addresses some important estimation and testing issues in the high-dimensional near unit root setting. Simulations are also conducted to demonstrate the finite-sample performance of the proposed test statistic.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"9 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114038516","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}
Applied research requires the usage of the proper statistics for hypothesis testing. Constrained optimization problems provide a framework that enables the researcher to build a statistic that fits his data and hypothesis at hand. In this paper I show some of the necessary conditions to obtain a Lagrange Multiplier test as well as some popular applications in order to highlight the usefulness of the test when the researcher must rely in asymptotic theory and to help the reader in the construction of a test in applied work.
{"title":"Lagrange Multiplier Tests in Applied Research","authors":"J. Astaiza-Gómez","doi":"10.2139/ssrn.3669884","DOIUrl":"https://doi.org/10.2139/ssrn.3669884","url":null,"abstract":"Applied research requires the usage of the proper statistics for hypothesis testing. Constrained optimization problems provide a framework that enables the researcher to build a statistic that fits his data and hypothesis at hand. In this paper I show some of the necessary conditions to obtain a Lagrange Multiplier test as well as some popular applications in order to highlight the usefulness of the test when the researcher must rely in asymptotic theory and to help the reader in the construction of a test in applied work.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123865530","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}
Neither existing theory nor prior empirical work can tell us the impact of non-normality on required sample sizes for Student-t tests of the mean in U.S. stock returns. Prior empirical work and bounds from a modified Berry-Esseen theorem do suggest, however, that the answer should vary with market capitalization, driven by third moments. For two-tailed nominally 5%-sized one-sample tests, we find that at least 100 observations are needed for large-capitalization stocks, and at least 200 observations are needed for small-capitalization stocks. Larger sample sizes are required for significance levels below 5%, or if one-tailed tests are used with skewed data.
{"title":"U.S. Stock Returns, the Berry-Esseen Theorem, and Statistical Testing","authors":"T. Crack, L. Mcalevey, Anindya Sen","doi":"10.2139/ssrn.3641266","DOIUrl":"https://doi.org/10.2139/ssrn.3641266","url":null,"abstract":"Neither existing theory nor prior empirical work can tell us the impact of non-normality on required sample sizes for Student-t tests of the mean in U.S. stock returns. Prior empirical work and bounds from a modified Berry-Esseen theorem do suggest, however, that the answer should vary with market capitalization, driven by third moments. For two-tailed nominally 5%-sized one-sample tests, we find that at least 100 observations are needed for large-capitalization stocks, and at least 200 observations are needed for small-capitalization stocks. Larger sample sizes are required for significance levels below 5%, or if one-tailed tests are used with skewed data.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116197138","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 statistical power of a hit rates test of taste-based discrimination varies markedly with the parameters of the application. For the presentation of multiple tests, power should be reported alongside p-values. Tests should also adjust for differences in search intensity whenever possible. If not, differential search intensity across groups will bias results. Theoretical bounds on the bias are wider than commonly observed differences in hit rates, but a simple empirical adjustment provides a valid test when data contain a discrete indicator of search intensity.
{"title":"Statistical Power and Search Intensity in Hit Rates Tests of Discrimination","authors":"Alex Lundberg","doi":"10.2139/ssrn.3454878","DOIUrl":"https://doi.org/10.2139/ssrn.3454878","url":null,"abstract":"The statistical power of a hit rates test of taste-based discrimination varies markedly with the parameters of the application. For the presentation of multiple tests, power should be reported alongside <i>p</i>-values. Tests should also adjust for differences in search intensity whenever possible. If not, differential search intensity across groups will bias results. Theoretical bounds on the bias are wider than commonly observed differences in hit rates, but a simple empirical adjustment provides a valid test when data contain a discrete indicator of search intensity.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123244206","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}
Traditionally, the development of investment strategies has required domain-specific knowledge and access to restricted datasets. These two barriers exist by design: (a) Financial knowledge is hoarded by firms, and protected as trade secrets, and (b) Financial data is expensive, making it inaccessible to the broad scientific community. This presentation explores how these two barriers impact the quality of quantitative research, and how investment tournaments can help deliver better investment outcomes by overcoming those two barriers.
{"title":"The Past and Future of Quantitative Research (Presentation Slides)","authors":"Marcos M. López de Prado","doi":"10.2139/ssrn.3447561","DOIUrl":"https://doi.org/10.2139/ssrn.3447561","url":null,"abstract":"Traditionally, the development of investment strategies has required domain-specific knowledge and access to restricted datasets. These two barriers exist by design: (a) Financial knowledge is hoarded by firms, and protected as trade secrets, and (b) Financial data is expensive, making it inaccessible to the broad scientific community. \u0000 \u0000This presentation explores how these two barriers impact the quality of quantitative research, and how investment tournaments can help deliver better investment outcomes by overcoming those two barriers.","PeriodicalId":425229,"journal":{"name":"ERN: Hypothesis Testing (Topic)","volume":"79 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127856399","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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