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Lecture 7: Common Factor Tests and Stability Tests
Lecture 7: Common Factor Tests and Stability Tests R.G. Pierse 1 Common factor Tests Consider again the regression model with first order autoregressive errors from lecture 4: Yt = β1 + β2X2t + ··· + βkXkt + ut ; t = 1; ··· ;T (1.1) and ut = ρut−1 + "t ; t = 1; ··· ;T (1.2) where −1 < ρ < 1 : The transformed version of this equation is Yt − ρYt−1 = (1 − ρ)β1 + β2(X2t − ρX2;t−1) + ··· + βk(Xkt − ρXk;t−1) + "t (1.3) where "t obeys all the assumptions of the classical model. Note that this model has k + 1 parameters. Compare this with the unrestricted dynamic model: Yt = γ1 + ρYt−1 + β2X2t + ··· + βkXkt + γ2X2;t−1 + ··· + γkXk;t−1 + "t : (1.4) which has 2k parameters. Equation (1.3) is a special case of equation (1.4) that satisfies the k − 1 nonlinear restrictions γj = −ρβj ; j = 2; ··· ; k : This suggests that the autoregressive model can be viewed as a restricted form of a dynamic model, satisfying a particular type of parameter restriction known as common factor restrictions. Evidence of serial correlation from the residuals from estimating equation (1.1) may indicate omission of any or all of the variables Yt−1, X2;t−1, ··· , Xk;t−1 rather than indicating the AR(1) model (1.3). 1 Sargan (1964) and Hendry and Mizon (1978) suggest a test of common factor restrictions in the model (1.4). This test is based on the statistic RSS T log r RSSu where RSSu is the residual sum of squares from the unrestricted model (1.4) and RSSr is the residual sum of squares from the restricted model (1.3). -
Three Statistical Testing Procedures in Logistic Regression: Their Performance in Differential Item Functioning (DIF) Investigation
Research Report Three Statistical Testing Procedures in Logistic Regression: Their Performance in Differential Item Functioning (DIF) Investigation Insu Paek December 2009 ETS RR-09-35 Listening. Learning. Leading.® Three Statistical Testing Procedures in Logistic Regression: Their Performance in Differential Item Functioning (DIF) Investigation Insu Paek ETS, Princeton, New Jersey December 2009 As part of its nonprofit mission, ETS conducts and disseminates the results of research to advance quality and equity in education and assessment for the benefit of ETS’s constituents and the field. To obtain a PDF or a print copy of a report, please visit: http://www.ets.org/research/contact.html Copyright © 2009 by Educational Testing Service. All rights reserved. ETS, the ETS logo, GRE, and LISTENING. LEARNING. LEADING. are registered trademarks of Educational Testing Service (ETS). SAT is a registered trademark of the College Board. Abstract Three statistical testing procedures well-known in the maximum likelihood approach are the Wald, likelihood ratio (LR), and score tests. Although well-known, the application of these three testing procedures in the logistic regression method to investigate differential item function (DIF) has not been rigorously made yet. Employing a variety of simulation conditions, this research (a) assessed the three tests’ performance for DIF detection and (b) compared DIF detection in different DIF testing modes (targeted vs. general DIF testing). Simulation results showed small differences between the three tests and different testing modes. However, targeted DIF testing consistently performed better than general DIF testing; the three tests differed more in performance in general DIF testing and nonuniform DIF conditions than in targeted DIF testing and uniform DIF conditions; and the LR and score tests consistently performed better than the Wald test. -
Testing for INAR Effects
Communications in Statistics - Simulation and Computation ISSN: 0361-0918 (Print) 1532-4141 (Online) Journal homepage: https://www.tandfonline.com/loi/lssp20 Testing for INAR effects Rolf Larsson To cite this article: Rolf Larsson (2020) Testing for INAR effects, Communications in Statistics - Simulation and Computation, 49:10, 2745-2764, DOI: 10.1080/03610918.2018.1530784 To link to this article: https://doi.org/10.1080/03610918.2018.1530784 © 2019 The Author(s). Published with license by Taylor & Francis Group, LLC Published online: 21 Jan 2019. Submit your article to this journal Article views: 294 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=lssp20 COMMUNICATIONS IN STATISTICS - SIMULATION AND COMPUTATIONVR 2020, VOL. 49, NO. 10, 2745–2764 https://doi.org/10.1080/03610918.2018.1530784 Testing for INAR effects Rolf Larsson Department of Mathematics, Uppsala University, Uppsala, Sweden ABSTRACT ARTICLE HISTORY In this article, we focus on the integer valued autoregressive model, Received 10 April 2018 INAR (1), with Poisson innovations. We test the null of serial independ- Accepted 25 September 2018 ence, where the INAR parameter is zero, versus the alternative of a KEYWORDS positive INAR parameter. To this end, we propose different explicit INAR model; Likelihood approximations of the likelihood ratio (LR) statistic. We derive the limit- ratio test ing distributions of our statistics under the null. In a simulation study, we compare size and power of our tests with the score test, proposed by Sun and McCabe [2013. Score statistics for testing serial depend- ence in count data. -
Wald (And Score) Tests
Wald (and Score) Tests 1 / 18 Vector of MLEs is Asymptotically Normal That is, Multivariate Normal This yields I Confidence intervals I Z-tests of H0 : θj = θ0 I Wald tests I Score Tests I Indirectly, the Likelihood Ratio tests 2 / 18 Under Regularity Conditions (Thank you, Mr. Wald) a.s. I θbn → θ √ d −1 I n(θbn − θ) → T ∼ Nk 0, I(θ) 1 −1 I So we say that θbn is asymptotically Nk θ, n I(θ) . I I(θ) is the Fisher Information in one observation. I A k × k matrix ∂2 I(θ) = E[− log f(Y ; θ)] ∂θi∂θj I The Fisher Information in the whole sample is nI(θ) 3 / 18 H0 : Cθ = h Suppose θ = (θ1, . θ7), and the null hypothesis is I θ1 = θ2 I θ6 = θ7 1 1 I 3 (θ1 + θ2 + θ3) = 3 (θ4 + θ5 + θ6) We can write null hypothesis in matrix form as θ1 θ2 1 −1 0 0 0 0 0 θ3 0 0 0 0 0 0 1 −1 θ4 = 0 1 1 1 −1 −1 −1 0 θ5 0 θ6 θ7 4 / 18 p Suppose H0 : Cθ = h is True, and Id(θ)n → I(θ) By Slutsky 6a (Continuous mapping), √ √ d −1 0 n(Cθbn − Cθ) = n(Cθbn − h) → CT ∼ Nk 0, CI(θ) C and −1 p −1 Id(θ)n → I(θ) . Then by Slutsky’s (6c) Stack Theorem, √ ! n(Cθbn − h) d CT → . −1 I(θ)−1 Id(θ)n Finally, by Slutsky 6a again, −1 0 0 −1 Wn = n(Cθb − h) (CId(θ)n C ) (Cθb − h) →d W = (CT − 0)0(CI(θ)−1C0)−1(CT − 0) ∼ χ2(r) 5 / 18 The Wald Test Statistic −1 0 0 −1 Wn = n(Cθbn − h) (CId(θ)n C ) (Cθbn − h) I Again, null hypothesis is H0 : Cθ = h I Matrix C is r × k, r ≤ k, rank r I All we need is a consistent estimator of I(θ) I I(θb) would do I But it’s inconvenient I Need to compute partial derivatives and expected values in ∂2 I(θ) = E[− log f(Y ; θ)] ∂θi∂θj 6 / 18 Observed Fisher Information I To find θbn, minimize the minus log likelihood. -
Comparison of Wald, Score, and Likelihood Ratio Tests for Response Adaptive Designs
Journal of Statistical Theory and Applications Volume 10, Number 4, 2011, pp. 553-569 ISSN 1538-7887 Comparison of Wald, Score, and Likelihood Ratio Tests for Response Adaptive Designs Yanqing Yi1∗and Xikui Wang2 1 Division of Community Health and Humanities, Faculty of Medicine, Memorial University of Newfoundland, St. Johns, Newfoundland, Canada A1B 3V6 2 Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 Abstract Data collected from response adaptive designs are dependent. Traditional statistical methods need to be justified for the use in response adaptive designs. This paper gener- alizes the Rao's score test to response adaptive designs and introduces a generalized score statistic. Simulation is conducted to compare the statistical powers of the Wald, the score, the generalized score and the likelihood ratio statistics. The overall statistical power of the Wald statistic is better than the score, the generalized score and the likelihood ratio statistics for small to medium sample sizes. The score statistic does not show good sample properties for adaptive designs and the generalized score statistic is better than the score statistic under the adaptive designs considered. When the sample size becomes large, the statistical power is similar for the Wald, the sore, the generalized score and the likelihood ratio test statistics. MSC: 62L05, 62F03 Keywords and Phrases: Response adaptive design, likelihood ratio test, maximum likelihood estimation, Rao's score test, statistical power, the Wald test ∗Corresponding author. Fax: 1-709-777-7382. E-mail addresses: [email protected] (Yanqing Yi), xikui [email protected] (Xikui Wang) Y. Yi and X. -
Robust Score and Portmanteau Tests of Volatility Spillover Mike Aguilar, Jonathan B
Journal of Econometrics 184 (2015) 37–61 Contents lists available at ScienceDirect Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom Robust score and portmanteau tests of volatility spillover Mike Aguilar, Jonathan B. Hill ∗ Department of Economics, University of North Carolina at Chapel Hill, United States article info a b s t r a c t Article history: This paper presents a variety of tests of volatility spillover that are robust to heavy tails generated by Received 27 March 2012 large errors or GARCH-type feedback. The tests are couched in a general conditional heteroskedasticity Received in revised form framework with idiosyncratic shocks that are only required to have a finite variance if they are 1 September 2014 independent. We negligibly trim test equations, or components of the equations, and construct heavy Accepted 1 September 2014 tail robust score and portmanteau statistics. Trimming is either simple based on an indicator function, or Available online 16 September 2014 smoothed. In particular, we develop the tail-trimmed sample correlation coefficient for robust inference, and prove that its Gaussian limit under the null hypothesis of no spillover has the same standardization JEL classification: C13 irrespective of tail thickness. Further, if spillover occurs within a specified horizon, our test statistics C20 obtain power of one asymptotically. We discuss the choice of trimming portion, including a smoothed C22 p-value over a window of extreme observations. A Monte Carlo study shows our tests provide significant improvements over extant GARCH-based tests of spillover, and we apply the tests to financial returns data. Keywords: Finally, based on ideas in Patton (2011) we construct a heavy tail robust forecast improvement statistic, Volatility spillover which allows us to demonstrate that our spillover test can be used as a model specification pre-test to Heavy tails Tail trimming improve volatility forecasting. -
Statistical Asymptotics Part II: First-Order Theory
First-Order Asymptotic Theory Statistical Asymptotics Part II: First-Order Theory Andrew Wood School of Mathematical Sciences University of Nottingham APTS, April 15-19 2013 Andrew Wood Statistical Asymptotics Part II: First-Order Theory First-Order Asymptotic Theory Structure of the Chapter This chapter covers asymptotic normality and related results. Topics: MLEs, log-likelihood ratio statistics and their asymptotic distributions; M-estimators and their first-order asymptotic theory. Initially we focus on the case of the MLE of a scalar parameter θ. Then we study the case of the MLE of a vector θ, first without and then with nuisance parameters. Finally, we consider the more general setting of M-estimators. Andrew Wood Statistical Asymptotics Part II: First-Order Theory First-Order Asymptotic Theory Motivation Statistical inference typically requires approximations because exact answers are usually not available. Asymptotic theory provides useful approximations to densities or distribution functions. These approximations are based on results from probability theory. The theory underlying these approximation techniques is valid as some quantity, typically the sample size n [or more generally some measure of information], goes to infinity, but the approximations obtained are often accurate even for small sample sizes. Andrew Wood Statistical Asymptotics Part II: First-Order Theory First-Order Asymptotic Theory Test statistics Consider testing the null hypothesis H0 : θ = θ0, where θ0 is an arbitrary specified point in Ωθ. If desired, we may -
Lectures on Structural Change
Lectures on Structural Change Eric Zivot Department of Economics, University of Washington April 4, 2001 1 Overview of Testing for and Estimating Struc- tural Change in Econometric Models 2 Some Preliminary Asymptotic Theory Reference: Stock, J.H. (1994) “Unit Roots, Structural Breaks and Trends,” in Handbook of Econometrics, Vol. IV. 3 Tests of Parameter Constancy in Linear Mod- els 3.1 Rolling Regression Reference: Zivot, E. and J. Wang (2002). Modeling Financial Time Series with S-PLUS. Springer-Verlag, New York. For a window of width k<n<T,the rolling linear regression model is yt(n) = Xt(n)βt(n) + εt(n),t= n,...,T (n 1) (n k) (k 1) (n 1) × × × × Observations in yt(n)andXt(n)aren most recent values from times • t n +1tot − OLS estimates are computed for sliding windows of width n and increment • m Poor man’s time varying regression model • 3.1.1 Application: Simulated Data Consider the linear regression model yt = α + βxt + εt,t=1,...,T =200 xt iid N(0, 1) ∼ 2 εt iid N(0,σ ) ∼ 1 No structural change parameterization: α =0,β =1,σ =0.5 Structural change cases Break in intercept: α =1fort>100 • Break in slope: β =3fort>100 • Break in variance: σ =0.25 for t>100 • Random walk in slope: β = βt = βt 1 +ηt,ηt iid N(0, 0.1) and β0 =1. • − ∼ (show simulated data) 3.1.2 Application: Exchange rate regressions References: 1. Bailey, R. and T. Bollerslev (???) 2. Sakoulis, G. and E. Zivot (2001). “Time Variation and Structural Change in the Forward Discount: Implications for the Forward Rate Unbiasedness Hypothesis,” unpublished manuscript, Department of Eco- nomics, University of Washington. -
Power Analysis for the Wald, LR, Score, and Gradient Tests in a Marginal Maximum Likelihood Framework: Applications in IRT
Power analysis for the Wald, LR, score, and gradient tests in a marginal maximum likelihood framework: Applications in IRT Felix Zimmer1, Clemens Draxler2, and Rudolf Debelak1 1University of Zurich 2The Health and Life Sciences University March 9, 2021 Abstract The Wald, likelihood ratio, score and the recently proposed gradient statistics can be used to assess a broad range of hypotheses in item re- sponse theory models, for instance, to check the overall model fit or to detect differential item functioning. We introduce new methods for power analysis and sample size planning that can be applied when marginal maximum likelihood estimation is used. This avails the application to a variety of IRT models, which are increasingly used in practice, e.g., in large-scale educational assessments. An analytical method utilizes the asymptotic distributions of the statistics under alternative hypotheses. For a larger number of items, we also provide a sampling-based method, which is necessary due to an exponentially increasing computational load of the analytical approach. We performed extensive simulation studies in two practically relevant settings, i.e., testing a Rasch model against a 2PL model and testing for differential item functioning. The observed distribu- tions of the test statistics and the power of the tests agreed well with the predictions by the proposed methods. We provide an openly accessible R package that implements the methods for user-supplied hypotheses. Keywords: marginal maximum likelihood, item response theory, power analysis, Wald test, score test, likelihood ratio, gradient test We wish to thank Carolin Strobl for helpful comments in the course of this research. -
Package 'Lmtest'
Package ‘lmtest’ September 9, 2020 Title Testing Linear Regression Models Version 0.9-38 Date 2020-09-09 Description A collection of tests, data sets, and examples for diagnostic checking in linear regression models. Furthermore, some generic tools for inference in parametric models are provided. LazyData yes Depends R (>= 3.0.0), stats, zoo Suggests car, strucchange, sandwich, dynlm, stats4, survival, AER Imports graphics License GPL-2 | GPL-3 NeedsCompilation yes Author Torsten Hothorn [aut] (<https://orcid.org/0000-0001-8301-0471>), Achim Zeileis [aut, cre] (<https://orcid.org/0000-0003-0918-3766>), Richard W. Farebrother [aut] (pan.f), Clint Cummins [aut] (pan.f), Giovanni Millo [ctb], David Mitchell [ctb] Maintainer Achim Zeileis <[email protected]> Repository CRAN Date/Publication 2020-09-09 05:30:11 UTC R topics documented: bgtest . .2 bondyield . .4 bptest . .6 ChickEgg . .8 coeftest . .9 coxtest . 11 currencysubstitution . 13 1 2 bgtest dwtest . 14 encomptest . 16 ftemp . 18 gqtest . 19 grangertest . 21 growthofmoney . 22 harvtest . 23 hmctest . 25 jocci . 26 jtest . 28 lrtest . 29 Mandible . 31 moneydemand . 31 petest . 33 raintest . 35 resettest . 36 unemployment . 38 USDistLag . 39 valueofstocks . 40 wages ............................................ 41 waldtest . 42 Index 46 bgtest Breusch-Godfrey Test Description bgtest performs the Breusch-Godfrey test for higher-order serial correlation. Usage bgtest(formula, order = 1, order.by = NULL, type = c("Chisq", "F"), data = list(), fill = 0) Arguments formula a symbolic description for the model to be tested (or a fitted "lm" object). order integer. maximal order of serial correlation to be tested. order.by Either a vector z or a formula with a single explanatory variable like ~ z. -
Wald (And Score) Tests
Wald (and score) tests • MLEs have an approximate multivariate normal sampling distribution for large samples (Thanks Mr. Wald.) • Approximate mean vector = vector of true parameter values for large samples • Asymptotic variance-covariance matrix is easy to estimate • H0: Cθ = h (Linear hypothesis) H0 : Cθ = h Cθ h is multivariate normal as n − → ∞ Leads! to a straightforward chisquare test • Called a Wald test • Based on the full model (unrestricted by any null hypothesis) • Asymptotically equivalent to the LR test • Not as good as LR for smaller samples • Very convenient sometimes Example of H0: Cθ=h Suppose θ = (θ1, . θ7), with 1 1 H : θ = θ , θ = θ , (θ + θ + θ ) = (θ + θ + θ ) 0 1 2 6 7 3 1 2 3 3 4 5 6 θ1 θ2 1 1 0 0 0 0 0 θ 0 − 3 0 0 0 0 0 1 1 θ = 0 − 4 1 1 1 1 1 1 0 θ 0 − − − 5 θ6 θ 7 Multivariate Normal Univariate Normal • 1 1 (x µ)2 – f(x) = exp − σ√2π − 2 σ2 (x µ)2 ! " – −σ2 is the squared Euclidian distance between x and µ, in a space that is stretched by σ2. 2 (X µ) 2 – − χ (1) σ2 ∼ Multivariate Normal • 1 1 1 – f(x) = 1 k exp 2 (x µ)#Σ− (x µ) Σ 2 (2π) 2 − − − | | 1 # $ – (x µ)#Σ− (x µ) is the squared Euclidian distance between x and µ, −in a space that− is warped and stretched by Σ. 1 2 – (X µ)#Σ− (X µ) χ (k) − − ∼ Distance: Suppose Σ = I2 2 1 d = (X µ)!Σ− (X µ) − − 1 0 x1 µ1 = x1 µ1, x2 µ2 − − − 0 1 x2 µ2 # $ # − $ ! " x1 µ1 = x1 µ1, x2 µ2 − − − x2 µ2 # − $ = !(x µ )2 + (x µ ")2 1 − 1 2 − 2 2 2 d = (x1 µ1) + (x2 µ2) − − (x1, x2) % x µ 2 − 2 (µ1, µ2) x µ 1 − 1 86 APPENDIX A. -
Permutation Methods for Chow Test Analysis an Alternative for Detecting Structural Break in Linear Models
Advances in Science, Technology and Engineering Systems Journal Vol. 4, No. 4, 12-20 (2019) ASTESJ www.astesj.com ISSN: 2415-6698 Permutation Methods for Chow Test Analysis an Alternative for Detecting Structural Break in Linear Models Aronu, Charles Okechukwu1,*, Nworuh, Godwin Emeka2 1Chukwuemeka Odumegwu Ojukwu University (COOU), Department of Statistics, Uli, Campus, 431124, Nigeria 2Federal University of Technology Owerri (FUTO), Department of Statistics,460114, Nigeria A R T I C L E I N F O A B S T R A C T Article history: This study examined the performance of two proposed permutation methods for Chow test Received:22 February, 2019 analysis and the Milek permutation method for testing structural break in linear models. Accepted:19 June, 2019 The proposed permutation methods are: (1) permute object of dependent variable and (2) Online: 09 July, 2019 permute object of the predicted dependent variable. Simulation from gamma distribution and standard normal distribution were used to evaluate the performance of the methods. Keywords: Also, secondary data were used to illustrate a real-life application of the methods. The Chow Test findings of the study showed that method 1(permute object of dependent variable) and Economic Growth Method2 (permute object of the predicted dependent variable) performed better than the Structural Break traditional Chow test analysis while the Chow test analysis was found to perform better Permutation Method than the Milek permutation for structural break. The methods were used to test whether the introduction of Nigeria Electricity Regulatory Commission (NERC) in the year 2005 has significant impact on economic growth in Nigeria.