Implementations of tests on the exogeneity of selected variables and their performance in practice - Thesis

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)

UvA-DARE (Digital Academic Repository)

Implementations of tests on the exogeneity of selected variables and their

performance in practice

Pleus, M.

Publication date

2015

Document Version

Final published version

Link to publication

Citation for published version (APA):

Pleus, M. (2015). Implementations of tests on the exogeneity of selected variables and their

performance in practice. Tinbergen Institute.

General rights

It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons).

Disclaimer/Complaints regulations

If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.

Implementations of tests on the

exogeneity of selected variables

and their performance in practice

Milan Pleus

In order to consistently estimate a causal economic relationship at least as many exogenous non-explanatory instrumental variables are required as there are endogenous explanatory variables. This thesis studies various techniques that can be used to classify selected variables as either exogenous or endogenous. These techniques are of great importance as they provide guidance regarding avoiding inconsistency and enhancing efficiency of estimators of the causal effects. Various implementations are first examined in the context of models for cross-section data and subsequently for dynamic panel data models. In addition to standard techniques some more refined alternatives are proposed. The results are used to re-examine various popular studies. Milan Pleus (1987) obtained both his bachelor’s and master’s degree in econometrics at the University of Amsterdam. It is at this same university that he started as a PhD student in 2011. His research interests include testing procedures, simultaneity and panel data models.

Implementations of tests on the exogeneity of selected variables and their performance in practice

Implementations of tests on the exogeneity of selected

variables and their performance in practice

ISBN 978 90 316 0437 1

Cover design: Crasborn Graphic Designers bno, Valkenburg a.d. Geul

This book is no. 617 of the Tinbergen Institute Research Series, established through cooperation between Thela Thesis and the Tinbergen Institute. A list of books which already appeared in the series can be found in the back.

Implementations of tests on the exogeneity of selected

variables and their performance in practice

ACADEMISCH PROEFSCHRIFT

ter verkrijging van de graad van doctor

aan de Universiteit van Amsterdam

op gezag van de Rector Magnificus

prof. dr. D.C. van den Boom

ten overstaan van een door het college voor promoties

ingestelde commissie,

in het openbaar te verdedigen in de Agnietenkapel

op vrijdag 29 mei 2015, te 14:00 uur

door

Milan Pleus

Promotiecommissie:

Promotor: Prof. dr. J.F. Kiviet

Overige leden: Prof. dr. H.P. Boswijk Dr. M.J.G. Bun Prof. dr. J.-M. Dufour Prof. dr. F. Kleibergen Prof. dr. F. Windmeijer

Faculteit Economie en Bedrijfskunde

Dit onderzoek werd mede mogelijk gemaakt door steun van de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) in het programma ”Statistical inference meth-ods regarding effectivity of endogenous policy measures”.

Acknowledgements

This thesis marks the end of four years of research done at the University of Amsterdam, a period I mostly reflect on with much pleasure. Although every PhD project has its struggles, I consider myself lucky to have encountered relatively few of them. One of the more important reasons for this is the encouragement and distraction offered by various people. It is here that I would like to take the opportunity to thank them.

First of all I would like to thank my supervisor Jan Kiviet. I cannot begin to imagine these years without his guidance. Apart from Jan being an excellent supervisor and researcher, I can only speak highly of him as a person. Shortly before starting this project it became clear that Jan would accept a position at Nanyang Technological University in Singapore. This entailed that he would only be in Amsterdam for three to six months a year. Consequently, I have made several trips to Singapore, for which I am very grateful. During these stays Jan and his wife Rose never failed to make me feel most welcome.

I want to thank the other members of the committee, Peter Boswijk, Maurice Bun, Jean-Marie Dufour, Frank Kleibergen and Frank Windmeijer, for reading the manuscript and providing me with helpful comments.

Of course I must thank my fellow PhD student and friend Rutger Poldermans, with whom I have shared many experiences over the years. Although we have distracted each other on many occasions, I have found our discussions to be of great help and inspiration. I want to thank my other colleagues at the University of Amsterdam and in particular: Jan Bogers, Simon Broda, Cees Diks, Kees Jan van Garderen, Noud van Giersbergen, Art¯uras Juodis, Dick Kerver, Herman ten Napel, Kees Nieuwland, Hans van Ophem, Andrew Pua and Roald Ramer. Daan in 't Veld was kind enough to help me with typesetting this thesis, which is greatly appreciated. It has always been my intention to thank Mars Cramer for his conversation and it saddens me that this is no longer possible.

Furthermore I would like to thank the Netherlands Organisation for Scientic Research (NWO) for granting financial support to the project ”Statistical inference methods re-garding effectivity of endogenous policy measures”. Without their support this thesis would not have materialized.

Outside of the academic community I am grateful to my friends, in particular Dick Lorier, Jaap Lorier and Dani¨ella Brals. I would like to thank the members of futsal team Oranje, whose company I always enjoy. A special thanks also to Monique van Buul and Paul Kamphuis for their support.

I am privileged to have experienced the loving care of my parents, Jan and Mone. They have always supported me and were available in times of need. My brother Timo is one of the most talented persons I know and I am grateful to have him as a friend. Finally, I would like to thank Marte for her love and support. She makes me happy.

Contents

1 Introduction 1

1.1 Outline of the thesis . . . 4

2 The performance of tests on endogeneity of subsets of explanatory vari-ables scanned by simulation 7 2.1 Introduction . . . 7

2.2 Testing the orthogonality of subsets of explanatory variables . . . 10

2.2.1 The model and setting . . . 10

2.2.2 The source of any estimator discrepancy . . . 12

2.2.3 Testing based on the source of any discrepancy . . . 15

2.2.4 Testing based on the discrepancy as such . . . 19

2.2.5 Testing based on covariance of structural and reduced form distur-bances . . . 21

2.2.6 Testing by an incremental Sargan test . . . 22

2.2.7 Concerns for practitioners . . . 23

2.3 Earlier Monte Carlo designs and results . . . 23

2.4 A more comprehensive Monte Carlo design . . . 26

2.4.1 The simulated data generating process . . . 26

2.4.2 Simulation design parameter space . . . 27

2.5 Simulation findings on rejection probabilities . . . 31

2.5.1 At least one exogenous regressor . . . 32

2.5.2 Both regressors endogenous . . . 36

2.6 Results for bootstrapped tests . . . 38

2.6.1 A bootstrap routine for subset DWH test statistics . . . 38

2.6.2 Simulation results for bootstrapped test statistics . . . 39

2.7 Empirical case study . . . 41

3 On overidentifying restrictions tests and their incremental versions 53

3.1 Introduction . . . 53

3.2 Testing overidentifying restrictions . . . 55

3.2.1 Test statistics and distributions . . . 56

3.2.2 Power properties . . . 61

3.3 Neglecting heteroskedasticity . . . 65

3.4 A higher order refinement to the Sargan test . . . 67

3.5 Simulation design . . . 69

3.6 Simulation findings on rejection probabilities . . . 71

3.7 Conclusions . . . 73

Appendix 3.A Proofs of theorems . . . 75

Appendix 3.B Details on the corrected statistic . . . 77

4 Accuracy and efficiency of various GMM inference techniques in dy-namic micro panel data models: theory 85 4.1 Introduction . . . 85

4.2 Basic GMM results for linear models . . . 88

4.2.1 Model and estimators . . . 88

4.2.2 Some algebraic peculiarities . . . 90

4.2.3 Particular test procedures . . . 91

4.3 Implementations for dynamic micro panel models . . . 93

4.3.1 Model and assumptions . . . 93

4.3.2 Removing individual effects by first differencing . . . 94

4.3.3 Respecting the equation in levels as well . . . 100

4.3.4 Coefficient restriction tests . . . 105

4.3.5 Tests of overidentification restrictions . . . 107

4.3.6 Modified GMM . . . 109

4.4 Intermediate conclusions . . . 111

Appendix 4.A Corrected variance estimation for 2-step GMM . . . 112

Appendix 4.B Partialling out and GMM . . . 114

Appendix 4.C Extracting redundant moment conditions . . . 115

5 Accuracy and efficiency of various GMM inference techniques in dy-namic micro panel data models: practice 117 5.1 Introduction . . . 117

5.2 Simulation design . . . 118

5.3.1 DGPs under effect stationarity . . . 128

5.3.2 Nonstationarity . . . 142

5.4 Empirical results . . . 146

5.5 Major findings . . . 151

Appendix 5.A Derivations for (5.17) . . . 153

6 Refined exogeneity tests in dynamic panel data models 155 6.1 Introduction . . . 155

6.2 Exogeneity tests . . . 157

6.2.1 Estimators and assumptions . . . 157

6.2.2 Incremental Sargan-Hansen test . . . 159

6.2.3 Hausman test . . . 161

6.3 Some possible refinements . . . 162

6.3.1 Diagonal Sargan-Hansen test . . . 162

6.3.2 Finite sample corrected variance for the Hausman test . . . 164

6.4 Testing exogeneity in dynamic panel data models . . . 166

6.4.1 Model and assumptions . . . 166

6.4.2 Full comprehensive internal instrument matrices . . . 168

6.4.3 Estimators . . . 169

6.4.4 Establishing endogeneity . . . 171

6.4.5 Establishing weak exogeneity . . . 173

6.5 Simulation design . . . 174

6.6 Simulation results . . . 176

6.6.1 Results under strict exogeneity . . . 178

6.6.2 Results under weak exogeneity . . . 180

6.6.3 Results under endogeneity . . . 180

6.7 Empirical case study . . . 181

6.8 Conclusions . . . 184

Appendix 6.A Non-negativeness of ˜JI(2) . . . 186

Appendix 6.B Proof of Theorem 6.1 . . . 188

Appendix 6.C Estimating the variance of the vector of contrasts . . . 189

Appendix 6.D Correcting ˜H(2) _{. . . 190}

Bibliography 197

Chapter 1

Introduction

In any econometrics class the estimation method of Ordinary Least Squares (OLS) is usually the first students get acquainted with. Benefits of this method are its nice statis-tical properties and the fact that it is relatively easy to derive and comprehend. One of the assumptions needed for consistency of the OLS estimator is that the right-hand side variables (explanatory variables) are not endogenous. Contrary to exogenous and prede-termined explanatory variables, endogenous explanatory variables are contemporaneously correlated with the disturbance term. Endogeneity may arise for one of various reasons: (i ) some of the explanatory variables are subject to measurement error; (ii ) some of the explanatory variables and the dependent variable are jointly determined by a system of simultaneous equations; (iii ) some parts of the model are overlooked or unobserved and correlated with some explanatory variables. These causes are frequently encountered in practice and call for alternative estimation techniques. An example in which this plays a role is when one investigates the effectiveness of policy measures, as they are often endogenous with respect to their targets.

A popular method that allows for endogeneity of explanatory variables is Instrumental Variables (IV) or Two-Stage Least Squares (2SLS). At least as many external instrumen-tal variables as there are endogenous regressors have to be found. These instrumeninstrumen-tal variables must satisfy two properties: (i ) they should be valid, meaning that they are ex-ogenous or predetermined with respect to the disturbance term in the equation of interest; (ii ) they should be sufficiently correlated with the endogenous regressors. As can be seen, the IV estimator trades one orthogonality condition for another to regain consistency. However, as for instance Bazzi and Clemens (2013) state, the most valid instruments could be the weakest, and the strongest could be the least valid. Both requirements have received much attention in the literature. The last three decades much progress has been made with respect to inference techniques that are robust to weak instruments. Influential

papers on this subject are for instance Anderson and Rubin (1949), Nelson and Startz (1990a,b), Bound et al. (1995), Staiger and Stock (1997), Kleibergen (2002, 2005) and Moreira (2003), whereas Stock et al. (2002) provide a survey.

Tests on the validity of the instruments are generally called overidentifying restrictions tests for the following reason. As already indicated, consistent estimation of the causal relationship requires at least as many valid external instruments as there are endogenous explanatory variables. An instrument is external if it has coefficient zero in the causal relationship and is therefore rightfully excluded from it. The model parameters are said to be just identified if the number of external instruments equals the number of endogenous explanatory variables. By estimating the parameters of a just identified model, IV imposes uncorrelatedness of all instruments with the disturbances. Hence, in the just identified case the uncorrelatedness cannot be tested. The model parameters are overidentified if the number of external instruments exceeds the number of endogenous explanatory variables. Only whether the instruments yielded by this excess of required exclusion restrictions are correlated with the disturbances can be tested. In fact, testing the exclusion from the model of the overidentifying external instruments is equivalent to testing the validity of the corresponding external instruments. The literature regarding these tests generally dates further back to papers by for instance Sargan (1958), Basmann (1960), Hwang (1980a) Hausman and Taylor (1981), Hansen (1982), Magdalinos (1985, 1994) and Newey (1985). More recently, Bowsher (2002) has investigated the performance of overidentifying restrictions tests in dynamic panel data models. Parente and Silva (2012) warn that an insignificant test outcome may provide little comfort. Hahn et al. (2011) have proposed a test that is robust to certain forms of weak instruments and Chao et al. (2014) have developed a test that is robust to many instruments and heteroskedasticity.

The asymptotic variance of the limiting distribution of the IV estimator depends on the strength of the external instruments. OLS has the smallest asymptotic variance as it instruments the explanatory variables by themselves, yielding the best possible fit. Super-fluously treating explanatory variables as endogenous therefore increases the asymptotic variance and makes the study more reliant on the availability of external instruments. Tests on the orthogonality of all possibly endogenous explanatory variables have been suggested by Durbin (1954), Wu (1973), Revankar and Hartley (1973), Revankar (1978) and Hausman (1978) and are often referred to as Durbin-Wu-Hausman (DWH) tests. Tests on the orthogonality of subsets of potentially endogenous regressors have been dis-cussed by Hwang (1980b), Spencer and Berk (1981), Wu (1983), Smith (1984, 1985), Hwang (1985) and Newey (1985). Some of the proposed tests are asymptotically or even algebraically equivalent.

Sharp-eyed readers may have formed a conjecture by now regarding the subject of this thesis. It are the tests regarding the validity of instruments and the exogeneity of explanatory variables that will receive our interest and in particular their subset versions. These so-called exogeneity tests have in common that they test whether particular vari-ables are endogenous or predetermined regarding their correlation with the disturbance term. More generally stated, they all infer about the validity of moment conditions. It is therefore no surprise that all tests on subsets of possibly endogenous explanatory vari-ables can be applied to subsets of external instruments too. These tests are of great importance to practitioners as they provide guidance regarding consistency and efficiency of estimators. As shown above, tests for the orthogonality of subsets of variables have received substantial attention. However, generally accepted rules for best practice on how to approach this problem do not seem available and are not yet supported by any simulation evidence. In the first part of this thesis properties of various tests are re-viewed and their small sample behaviour is investigated in the context of cross-sectional data. The second part of this thesis deals with inference techniques in dynamic panel data models and in particular with testing for exogeneity in that special context. Panel data offer some meaningful advantages over single-indexed data. The fact that for every individual several time-series observations are available offers the possibility to deal with unobserved individual effects that are correlated with some of the explanatory variables, possibly eliminating the above as third mentioned source of endogeneity. In standard applications these unobserved individual effects have to be assumed constant over time. Several transformations exist that get rid of the unobserved individual effects. In case of dynamic relationships, when the number of individuals is relatively large while covering just a few time periods, the analysis is often based on the generalized method of moments (GMM). This class of estimators, proposed by Hansen (1982), includes both OLS and IV as special cases. GMM is often praised for its flexibility, generality, ease of use and efficiency. As the transformations of the data that get rid of the individual effects render the lagged dependent variable endogenous with respect to the transformed disturbance term, instrumental variables are required. The popular first difference transformation also induces serial dependence in the disturbances in which case GMM offers efficiency gains over IV when taking this into account. Another benefit of panel data is the almost free availability of internal instruments. First noted by Anderson and Hsiao (1981) the lagged values of the explanatory variables establish instruments, potentially rendering the search for external instruments superfluous. Several studies have proposed additional instruments. These include Arellano and Bond (1991), Ahn and Schmidt (1995), Arellano and Bover (1995) and Blundell and Bond (1998). Although yielding an asymptotically

efficient estimator, its bias may be substantial if the number of instruments grows large relative to the width of the panel, see for instance Newey and Windmeijer (2009). Not only the bias of the estimator is affected. Bowsher (2002) reports simulation findings on overidentifying restrictions tests and finds that their rejection probabilities approach zero when the number of instruments grows large. Findings like these have led practitioners to consider instrument reduction methods, which include omitting long lags and collaps-ing. This last technique reduces the number of instruments by taking a particular linear transformation of reduced rank. The two reduction methods can also be combined.

Whereas it is common practice to test the overidentifying restrictions after IV or GMM estimation, their subset versions are less frequently used. There is a noticeable exception in dynamic panel data models. The popular Blundell-Bond GMM estimator employs an additional set of instruments with respect to the Arellano-Bond GMM estimator. These additional instruments are only valid under effect stationarity. Therefore Blundell-Bond estimates are often supplied with a subset test on these instruments to validate its use. Also in adequately specified dynamic panel data models the subset tests can be used to classify explanatory variables. In this context the methodological difference between testing the orthogonality of explanatory variables and instruments further fades. Classification of the explanatory variables implies which lags constitute valid instruments. These lags have a logical ordering, the most recent ones are naturally assumed to be the strongest and the ones furthest away are assumed to be the weakest. Wrongly classifying explanatory variables may render a subset of instruments either invalid or cause the practitioner to abstain from using some of the strongest instruments available. As only a subset of instruments is rendered invalid by misclassification, one can directly test this subset instead of relying on the standard overidentifying restrictions test often reported by popular software packages.

Many studies on econometric theory are accompanied by results on Monte Carlo sim-ulations. These Monte Carlo simulations allow researchers to infer about finite sample properties of techniques that are often too complicated to derive analytically. As simula-tion results are much less general than theoretical results, a well designed simulasimula-tion study is essential. In this thesis a lot of attention has been paid to designing these simulations. For more information on this topic see Kiviet (2012).

1.1

Outline of the thesis

Tests for classification as endogenous or predetermined of arbitrary subsets of explana-tory variables are investigated in Chapter 2, which is based on Kiviet and Pleus (2014).

These tests are formulated as significance tests in auxiliary IV regressions and their rela-tionships with various more classic test procedures are examined and critically compared with statements in the literature. Then simulation experiments are designed by solving the data generating process parameters from salient econometric features, namely: degree of simultaneity and multicollinearity of regressors, and individual and joint strength of external instrumental variables. Next, for various test implementations, a wide class of relevant cases is scanned for flaws in performance regarding type I and II errors. Substan-tial size distortions occur, but these can be cured remarkably well through bootstrapping, except when instruments are weak. The power of the subset tests is such that they es-tablish an essential addition to the well-known classic full-set DWH tests in a data based classification of individual explanatory variables.

The performance of overidentifying restrictions tests is examined in Chapter 3. Well-known tests such as the Sargan (1958) test, the test by Basmann (1960) and a Hausman-type test are examined. Power properties of these tests and their incremental versions are derived and recent papers that state that these tests provide little comfort in testing the actual moment conditions as these tests may have power equal to size for specific cases are clarified. On the other hand it is possible that a significant test outcome is not the result of invalid moment conditions. In the presence of conditional heteroskedasticity, the variance of the moment conditions may either be underestimated or overestimated. A rarely applied Cornish Fisher correction for the Sargan test proposed by Magdalinos (1985) is re-examined. Additionally, simulation experiments are used to compare the different test statistics and we conclude that in small samples the Cornish Fisher corrected test outperforms the others in terms of size control. However, with respect to the power properties of the various tests we find that the corrected Sargan test performs less well than the other statistics when instruments are weak.

The following two chapters are based on Kiviet et al. (2014). Although separated due to their joint length, they are inextricably linked. Both chapters are about the popular Arellano-Bond and Blundell-Bond GMM estimation techniques for single linear dynamic panel data models. Studies employing these techniques are growing exponentially in number. However, for researchers it is hard to make a reasoned choice between many different possible implementations of these estimators and associated tests. Chapter 4, which focuses on theory and techniques, explicates in a systematic way many options regarding: (i ) reducing, extending or modifying the set of instruments; (ii ) specifying the weighting matrix in relation to the type of heteroskedasticity; (iii ) using (robustified) one-step or (corrected) step variance estimators; (iv ) employing one-step or two-step residuals in Sargan-Hansen overall or incremental overidentification restrictions tests.

This is all done for models in which some regressors may be either strictly exogenous, predetermined or endogenous.

Chapter 5 examines the performance in finite samples of the inference techniques discussed in Chapter 4 by simulation. The root mean squared errors of the coefficient estimators are compared and also the size of tests on coefficient values and of different implementations of overidentification restriction tests. Also the size and power of tests on the validity of the additional orthogonality conditions exploited by the Blundell-Bond technique are assessed over a pretty wide grid of relevant cases. Surprisingly, particular asymptotically optimal and relatively robust weighting matrices are found to be superior in finite samples to ostensibly more appropriate versions. Most of the variants of tests for overidentification restrictions show serious deficiencies. A recently developed modi-fication of GMM is found to have potential when the cross-sectional heteroskedasticity is pronounced and the time-series dimension of the sample not too small. Finally all techniques are employed to actual data and lead to some profound insights.

In chapter 6 we investigate the performance of subset tests in adequately specified lin-ear dynamic panel data models, estimated by GMM. Because in that context usually just internal instruments are being exploited, misclassification of explanatory variables renders either a specific subset of instruments invalid or yields inefficient estimates. Rather than testing all overidentifying restrictions by the Sargan-Hansen test, the focus is on subsets using either the incremental Sargan-Hansen test or a Hausman test. Although it is known in the literature that the Sargan-Hansen test suffers when using many instruments, it is yet unclear in what way the incremental test is affected. Therefore, test statistics are considered in which the number of employed instruments is deliberately restricted. Two possible refinements are proposed. Recently Hayakawa (2014) has proposed a method which forces a block diagonal structure on the weighting matrix in order to reduce prob-lems stemming from taking its inverse. This method is generalized to the incremental test. A finite sample corrected variance estimate for the vector of contrasts is derived from which two new Hausman test statistics are constructed. Simulation is used to in-vestigate finite sample performance. The corrected Hausman statistics outperform the standard Hausman implementation uniformly. The test by Hayakawa (2014) and its in-cremental version are found to yield little benefits and often perform less well than the standard implementations. Collapsing is found to improve performance regarding size control as is estimating variances under the null hypothesis. The results are illustrated using a study on the effect of deterrence on crime.

Chapter 2

The performance of tests on

endogeneity of subsets of

explanatory variables scanned by

simulation

2.1

Introduction

In this chapter various test procedures are derived and examined for the classification of arbitrary subsets of explanatory variables as either endogenous or exogenous with re-spect to a single adequately specified structural equation. Correct classification is highly important because misclassification leads to either inefficient or inconsistent estimation. The derivations, which in essence are based on employing Hausman’s principle of exam-ining the discrepancy between two alternative estimators, formulate the various tests as joint significance tests of additional regressors in auxiliary IV regressions. Their relation-ships are demonstrated with particular forms of classic tests such as Durbin-Wu-Hausman orthogonality tests, Revankar-Hartley covariance tests and Sargan-Hansen overidentifica-tion restricoveridentifica-tion tests. Various different and some under the null hypothesis asymptotically equivalent implementations follow. The latter vary only regarding degrees of freedom ad-justments and the type of disturbance variance estimator employed. We run simulations over a wide class of relevant cases, to find out which versions have best control over type I error probabilities and to get an idea of the power of these tests. This should help to use these tests effectively in practice when trying to avoid both evils of inconsistency and inef-ficiency. To that end a simulation approach is developed by which relevant data generating processes (DGPs) are designed by deriving the values for their parameters from chosen

salient features of the system, namely: degree of simultaneity of individual explanatory variables, degree of multicollinearity between explanatory variables, and individual and joint strength of employed external instrumental variables. This allows scanning the rele-vant parameter space of wide model classes for flaws in performance regarding type I and II errors of all implementations of the tests and their bootstrapped versions. We find that testing orthogonality by standard methods is impeded for weakly identified regressors. Like bootstrapped tests require resampling under the null, we find here that testing for orthogonality by auxiliary regressions benefits from estimating variances under the null, as in Lagrange multiplier tests, rather than under the alternative, as in Wald-type tests. However, after proper size correction we find that the Wald-type tests exhibit the best power properties.

Procedures for testing the orthogonality of all possibly endogenous regressors regarding the error term have been developed by Durbin (1954), Wu (1973), Revankar and Hartley (1973), Revankar (1978) and Hausman (1978). Mutual relationships between these are discussed in Nakamura and Nakamura (1981) and Hausman and Taylor (1981). This test problem has been put into a likelihood framework under normality by Holly (1982) and Smith (1983). Most of the papers just mentioned, and in particular Davidson and MacKinnon (1989, 1990), provide a range of implementations for these tests that can easily be obtained from auxiliary regressions. Although this type of inference problem does address one of the basic fundaments of the econometric analysis of observational data, relatively little evidence on the performance of the available tests in finite samples is available. Monte Carlo studies on the performance of some of the implementations in static linear models can be found in Wu (1974), Meepagala (1992), Chmelarova and Hill (2010), Jeong and Yoon (2010), Hahn et al. (2011) and Doko Tchatoka (2014), whereas such results for linear dynamic models are presented in Kiviet (1985).

The more subtle problem of deriving a test for the orthogonality of subsets of the regressors not involving all of the possibly endogenous regressors has also received sub-stantial attention over the last three decades. Nevertheless, generally accepted rules for best practice on how to approach this problem do not seem available yet, or are confusing as we shall see, and not yet supported by any simulation evidence. Self-evidently, though, the situation where one is convinced of the endogeneity of a few of the regressors, but wants to test some other regressors for orthogonality, is of high practical relevance. If orthogo-nality is established, this permits to use these regressors as instrumental variables, which (if correct) improves the efficiency and the identification situation, because it makes the analysis less dependent on the availability of external instruments. This is important in particular when available external instruments are weak or of doubtful exogeneity status.

Testing the orthogonality of subsets of the possibly endogenous regressors was addressed first by Hwang (1980b) and next by Spencer and Berk (1981, 1982), Wu (1983), Smith (1984, 1985), Hwang (1985) and Newey (1985), who all suggest various test procedures, some of them asymptotically or even algebraically equivalent. So do Pesaran and Smith (1990), who also provide theoretical arguments regarding an ordering of the power of the various tests, although they are asymptotically equivalent under the null and under local alternatives. Various of the possible subset test implementations are paraphrased in Ruud (1984, 2000), Davidson and MacKinnon (1993) and in Baum et al. (2003), and occasion-ally their relationships with particular forms of Sargan-Hansen (partial-)overidentification test statistics are examined. As we shall show, a few particular situations still call for fur-ther analysis and formal proofs, and sometimes results from the studies mentioned above have to be corrected. As far as we know, there are no published simulation results yet on the actual qualities of tests for the exogeneity for arbitrary subsets of the regressors in finite samples.

In this chapter we shall try to elucidate the various forms of available test statistics for the endogeneity of subsets of the regressors, demonstrate their origins and their rela-tionships, and also produce solid Monte Carlo results on their performance in single static linear models with endogenous regressors and IID disturbances. That yet no simulation results are available on subset tests may be due to the fact that it is not straight-forward how one should design a range of appealing and representative experiments. We believe that in this respect the present study, which closely follows the rules set out in Kiviet (2012), may claim originality. Besides exploiting some invariance properties, we choose the remaining parameter values for the DGP indirectly from the inverse relationships be-tween the DGP parameter values and fundamental orthogonal econometric notions. The latter constitute an insightful base for the relevant nuisance parameter space. The present design can easily be extended to cover cases with a more realistic degree of overidentifi-cation and number of jointly dependent regressors. Other obvious extensions would be: to include recently developed tests which are specially built to cope with weak instru-ments, to consider non Gaussian and non IID disturbances, to examine dynamic models, to include tests for the validity (orthogonality) of instruments which are not included in the regression, etc. Regarding all these aspects the present study just offers an initial reference point.

The structure of the chapter is as follows. In Section 2.2, we first define the model’s maintained properties and the hypothesis to be tested. Next, in a series of subsections, various routes to develop test procedures are followed and their resulting test statistics are discussed and compared analytically. Section 2.3 reviews earlier Monte Carlo designs

and results regarding orthogonality tests. In Section 2.4 we set out our approach to obtain DGP parameter values from chosen basic econometric characteristics. A simulation design is obtained to parametrize a synthetic single linear static regression model including two possibly endogenous regressors with an intercept and involving two external instruments. For this design Section 2.5 presents simulation results for a selection of practically relevant parametrizations. Section 2.6 produces similar results for bootstrapped versions of the tests, Section 2.7 provides an empirical case study and Section 2.8 concludes.

2.2

Testing the orthogonality of subsets of

explana-tory variables

2.2.1

The model and setting

We consider the single linear equation with endogenous regressors

y = Xβ + u, (2.1)

with IID unobserved disturbancesu∼ (0, σ2_I

n), K-element unknown coefficient vector β,

ann× K regressor matrix X and n × 1 regressand y. We also have an n × L matrix Z

containing sample observations on identifying instrumental variables, so

E(Zu) = 0, rank(Z) = L, rank(X) = K and rank(ZX) = K. (2.2)

In addition, we make asymptotic regularity assumptions to guarantee asymptotic identi-fication of all elements ofβ too and consistency of its IV (or 2SLS) estimator

ˆ

β = (XPZX)−1XPZy, (2.3)

wherePZ=Z(ZZ)−1Z. Hence, we assume that

plimn−1ZZ = ΣZZ and plimn−1ZX = ΣZX (2.4)

are finite and have full column rank, whereas ˆβ has limiting normal distribution n1/2_{( ˆβ− β)→ N}d _{0, σ}2_[Σ

ZXΣ−1ZZΣZX]−1 

. (2.5)

therefore be partitioned as

X = (Y Z₁) andZ = (Z₁Z₂), (2.6)

whereZj has Lj columns forj = 1, 2. Because the number of columns in Y is K− L1>

0 we find from L = L₁+L₂ ≥ K that L₂ > 0, but we allow L₁ ≥ 0, so Z₁ may be void. Throughout this chapter the model just defined establishes the maintained unrestrained1_{hypothesis, which allowsY to contain endogenous variables. Below we will}

examine particular further curbed versions of the maintained hypothesis and develop tests to verify these further limitations. These are not parametric restraints regardingβ but

involve orthogonality conditions in addition to theL maintained orthogonality conditions

embedded inE(Zu) = 0. All these extra orthogonality conditions concern regressors and

not further external instrumental variables. Therefore, we consider a partitioning ofY in KeandKocolumns

Y = (YeYo), (2.7)

where the variablesYeare maintained as possibly e

¯ndogenous, whereas for theKovariables Yotheir possible o

¯rthogonality will be examined, i.e. whetherE(Y



ou) = 0 seems to hold. We define then× (L + Ko) matrix

Zr= (Z Yo), (2.8)

which relates to all the orthogonality conditions in the r

¯estrained model. Note that (2.2)

implies thatZrhas full column rank, providedn≥ L + Ko. Now the null and alternative hypotheses that we will examine can be expressed as

H0 : y = Xβ + u, u∼ (0, σ2I), E(Zru) = 0, and (2.9)

H1 : y = Xβ + u, u∼ (0, σ2I), E(Zu) = 0, E(You)= 0.

Hence,H0 _assumesE(Y

ou) = 0.

Under the extended set of orthogonality conditions E(Zru) = 0, i.e. under H0, the restrained IV estimator is

ˆ

βr= (XPZrX)−1XPZry. (2.10)

IfH0_{is valid this estimator is consistent and, provided plimn}−1_Z

rZr= ΣZ_rZr exists and

is invertible, its limiting normal distribution has varianceσ2_[Σ

Z_rXΣ −1

Z_rZrΣZrX]−1, which in-1_{The terms ”restrained” and ”unrestrained” are used rather than ”restricted” and ”unrestricted”.}

This is done in order to avoid confusion as the latter terms are often used in the context of coefficient restrictions.

volves an asymptotic efficiency gain over (2.5). However, under the alternative hypothesis

H1_{estimator ˆβ}

ris inconsistent.

A test for (2.9) should (as always) have good control over its type I error probability2

and preferably also have high power, in order to prevent the acceptance of an inconsistent estimator. In practice inference on (2.9) usually establishes just one link in a chain of tests to decide on the adequacy of model specification (2.1) and the maintained instrumentsZ,

see for instance Godfrey and Hutton (1994) and Guggenberger (2010). Many of the firm results obtained below require to make the very strong assumptions embedded in (2.1) and (2.2) and leave it to the practitioner to make a balanced use of them within an actual modelling context.

In the derivations to follow we make use of the following three properties of projection matrices, which for any full column rank matrix A are denoted as PA = A(AA)−1A: For a full column rank matrixC = (A B) one has (i) PA =PCPA =PAPC; (ii) PC =

PA+PM_AB=P(A MAB), where MA=I−PA; (iii) forC∗= (A∗B), where A∗=A−BD and

D an arbitrary matrix of appropriate dimensions, PC∗=PB+PMBA∗=PB+PMBA=PC.

2.2.2

The source of any estimator discrepancy

A test based on the Hausman principle focusses on the discrepancy vector ˆ

β− ˆβr = (XPZX)−1XPZy− (XPZrX)−1XPZry

= (XPZX)−1XPZ[I− X(XPZrX)−1XPZr]y

= (XPZX)−1(PZX)ˆur

= (XPZX)−1(PZYePZYoZ1)uˆr, (2.11)

where ˆur=y−X ˆβrdenotes the IV residuals obtained underH0. Although testing whether the discrepancy between these two coefficient estimators is significantly different from zero is not equivalent to testingH0_{, we will show that in fact all existing test procedures employ}

the outcome of this discrepancy to infer on the (in)validity ofH0_{. Because (X}_P

ZX)−1is non-singular ˆβ− ˆβris close to zero if and only if theK× 1 vector (PZYePZYo Z1)uˆris. So, we will examine now when its three sub-vectors

YePZuˆr, YoPZuˆrandZ₁uˆr (2.12) 2_{An actual type I error probability much larger than the chosen nominal value would more often than}

intended lead to using an inefficient estimator. A much lower actual type I error than the nominal level would deprive the test from its power hampering the detection of estimator inconsistency.

will jointly be close to zero. Note that due to the identification assumptions both PZYe andPZYowill have full column rank so cannot beO.

For the IV residuals ˆur we haveXPZruˆr = 0, and since PZrX = (PZrYeYo Z1), this

yields

YePZruˆr= 0, Youˆr= 0 andZ₁uˆr= 0. (2.13)

Note that the third vector of (2.12) is always zero according to the third equality from (2.13). Using projection matrix property (ii) and the first equality of (2.13), we find for the first vector of (2.12) that

YePZuˆr=Ye(PZr− PMZYo)ˆur=−YePMZYouˆr, so

YePZuˆr=−YeMZYo(YoMZYo)−1YoMZˆur. (2.14) ThisKeelement vector will be close to zero when theKoelement vectorYoMZuˆris. Due to the occurrence of theKe× KomatrixYeMZYoas a first factor in the right-hand side of (2.14), it seems possible thatYePZuˆrmay be close to zero too in cases whereYoMZuˆr= 0; we will return to that possibility below. For the second vector of (2.12) we find, upon using the second equality of (2.13), that

YoPZuˆr=−YoMZuˆr. (2.15) Hence, the second vector of (2.12) will be close to zero if and only if the vectorYoMZuˆr is close to zero.

From the above it follows that YoMZuˆr being close to zero is both necessary and sufficient for the full discrepancy vector (2.11) to be small. Checking whetherYoMZuˆris close to zero corresponds to examining to what degree the variablesMZYo do obey the orthogonality conditions, while using ˆuras a proxy foru, which is asymptotically valid under the extended set of orthogonality conditions. Note that by focussing onMZYothe tested variables Yohave been purged from their components spanned by the columns of

Z. Since these are maintained to be orthogonal with respect to u, they should better be

excluded from the test indeed.

Since the inverse matrix in the right-hand side of (2.11) is positive definite, the proba-bility limits of ˆβ and ˆβrwill be similar if and only if plimn−1YoMZuˆr= 0. Regarding the power of any discrepancy based test of (2.9) it is now of great interest to examine whether it could happen underH1_{to have plimn}−1_Y

reduced form equations

Yj=ZΠj+ (uγj+Vj), for j∈ {e, o}, (2.16)

where Πj is an L× Kj matrix of reduced form parameters, γj is a Kj× 1 vector that parametrizes the simultaneity andVjestablishes the components of the zero mean reduced form disturbances which are uncorrelated withu and of course with Z. After this further

parametrization the hypotheses (2.9) can now be expressed asH0_:γ

o= 0 andH1:γo= 0. Let (L + Ko)× (L + Ko) matrix Ψ be such that ΨΨ= (ZrZr)−1. From

YoMZuˆr = −YoPZ[In− X(XPZrX)−1XPZr]u

= −Y_oPZ[PZr− PZrX(XPZrX)−1XPZr]u

= −Y_oPZZrΨ[IL+Ko− PΨZ_rX]ΨZru (2.17) it follows that plimn−1YoMZˆur = 0 if (L + Ko)× 1 vector plim n−1Zru = σ2(0 γo) is in the column space spanned by plimn−1ZrX = ΣZ_rX. This is obviously the case when

γo = 0. However, it cannot occur for γo = 0, because (L + Ko)× 1 vector ΣZ_rXc, with

c a K× 1 vector, has its first L ≥ K elements equal to zero only for c = 0, due to

the identification assumptions. This excludes the existence of a vector c = 0 yielding

ΣZ_rXc = σ2(0γo) whenγo= 0, so under asymptotic identification the discrepancy will be nonzero asymptotically whenYocontains an endogenous variable.

Cases in which the asymptotic identification assumptions are violated are Πe=Ce/

√ n

and/or Πo=Co/√n, where Ce andCo are matrices of appropriate dimensions with full column rank and all elements fixed and finite.3 _{Examining Σ}

Z_rXc closer yields ΣZ_rXc =  ΣZZΠe Π_oΣZZΠe+ ΣV_oVe+σ2γoγe  c1+  ΣZZΠo ΣY_oYo  c2+  ΣZZ1 Π_oΣZZ1  c3, (2.18)

where c = (c₁ c₂ c₃) and ΣV_oVe = plimn−1VoVe. If only Πo = Co/

√

n, so when all the

instruments Z are weak and asymptotically irrelevant for the set of regressors Yo whose orthogonality is tested, we can set c₁ = 0 andc₃ = 0 and then for c₂ = σ2_Σ−1

Y_oYoγo =

σ2_(σ2_γ

oγo+ ΣV_oVo)−1γo= 0 we have ΣZ_rXc = σ2(0γo)= 0, demonstrating that the test will have no asymptotic power. If only Πe=Ce/

√

n, thus all the instruments Z are weak

forYe, a solution c= 0 can be found upon taking c2 = 0, c3 = 0 andc1= 0, provided

ΣV_oVe+σ2γoγe= O or YeandYoare asymptotically not uncorrelated. Onlyc3has to be

set at zero to find a solution whenZ is weak for both YoandYe. From (2.18) it can also 3_{Doko Tchatoka (2014) considers a similar situation for the special case Ke= 0 and Ko= 1.}

be established that when fromZ₂ at leastKe+Ko instruments are not weak forY the discrepancy will always be different from zero asymptotically whenγo= 0.

Using (2.16) we also find plimn−1YeMZYo= ΣV_oVe+σ2γeγo, which demonstrates that the first vector of (2.12) would forγo= 0 tend to zero also when γe= 0 while the reduced form disturbances ofYeandYoare uncorrelated. This indicates the plausible result that a discrepancy based test may loose power whenYeis unnecessarily treated as endogenous andYo is establishing a weak instrument forYeafter partialing outZ.

2.2.3

Testing based on the source of any discrepancy

Next we examine the implementation of testing closeness to zero ofYoMZuˆrin an auxiliary regression. Consider

y = Xβ + PZYoζ + u∗, (2.19)

whereu∗=u− PZYoζ. Its estimation by IV employing the instruments Zryields coeffi-cients that can be obtained by applying OLS to the second-stage regression ofy on PZrX

andPZrPZYo=PZYo. For ζ partitioned regression yields

ˆ

ζ = (YoPZMP_ZrXPZYo)−1YoPZMP_ZrXy, (2.20)

where, using rule (i),YoPZMP_ZrXy = YoPZ[I−X(XPZrX)−1XPZr]y = YoPZuˆr. Thus, by testingζ = 0 in (2.19) we in fact examine whether YoPZuˆr=−YoMZuˆrdiffers significantly from a zero vector, which is indeed what we aim for.4

Alternatively, consider the auxiliary regression

y = Xβ + MZYoξ + v∗, (2.21)

wherev∗=u−MZYoξ. Using the instruments Zrinvolves here applying OLS to the second-stage regression ofy on PZrX and PZrMZYo=PZrYo− PZrPZYo=Yo− PZYo=MZYo. This yields

ˆ

ξ = (YoMZMP_ZrXMZYo)−1YoMZMP_ZrXy, (2.22)

4_{This procedure provides the explicit solution to the exercise posed in (Davidson and MacKinnon,}

where

YoMZMP_ZrXy = YoMZ[I− PZrX(XPZrX)−1XPZr]y

= Yo[I− X(XPZrX)−1XPZr]y− YoPZ[I− X(XPZrX)−1XPZr]y

= YoMZuˆr. (2.23)

Thus, like testing ζ = 0 in (2.19), testing ξ = 0 in auxiliary regression (2.21) examines

the magnitude ofYoMZuˆr. The estimator for β resulting from (2.21) is

ˆ

βr∗= (XPZrMM_ZYoPZrX)−1XPZrMM_ZYoy.

BecausePZrMMZYo =PZr− PZrPMZYo=PZr− (PZ+PMZYo)PMZYo =PZr− PMZYo=PZ, we find ˆβr∗= ˆβ. Hence, the IV estimator of β exploiting the extended set of instruments in the auxiliary model (2.21) equals the unrestrained IV estimator ˆβ. Many text books

mention this result for the special caseKe= 0.

From the above we find that testing whether included possibly endogenous variables

Yocan actually be used effectively as valid extra instruments, can be done as follows: Add them toZ, so use Zras instruments, and add at the same time the regressorsMZYo(the reduced form residuals of the alleged endogenous variablesYo in the maintained model) to the model, and then test their joint significance. When testingξ = 0 in (2.21) by a

Wald-type statistic, and assuming for the moment thatσ2_{is known, the test statistic is}

σ−2yPM_{PZr X}M_ZYoy = σ−2y(MA− MC)y, (2.24)

where A = PZrX, B = MZYo and C = (A B). Hence, yPM_{PZr X}MZYoy is equal to the

difference between the OLS residual sums of squares of the restricted (byξ = 0) and the

unrestricted second stage regressions (2.21). One easily finds that testingζ = 0 in (2.19)

by a Wald-type test yields in the numerator

yPM_{PZr X}PZYoy = y(MA− MC∗)y,

with againA = PZrX = (PZrYeYoZ1), but C∗= (A B∗) withB∗=PZYo. Although C∗=

C, both span the same subspace, so MC =MC∗ and thus the two auxiliary regressions

lead to numerically equivalent Wald-type test statistics.

Of course, σ2 _{is in fact unknown and should be replaced by an estimator that is}

consistent under the null. There are various options for this. Two rather obvious choices would be ˆσ2_{= ˆu}_{u/n or ˆˆ σ}2

alternatives) asymptotically equivalent test statistics, both with χ2_(K

o) asymptotic null distribution. Further asymptotically equivalent variants can be obtained by employing a degrees of freedom correction in the estimation ofσ2_{and/or by dividing the test statistic}

byKo and then confronting it with critical values from an F distribution with Ko and

n− l degrees of freedom with l some finite number, possibly K + Ko.

Testing the orthogonality ofYoandu, while maintaining the endogeneity of Ye, by a simple χ2_{-form statistic and using as in a Wald-type test the estimate ˆσ}2 _{(without any}

degrees of freedom correction) from the unrestrained model, will be indicated by Wo. When using the uncorrected restrained estimator ˆσ2

r, the statistic will be denoted here as

Do. So we have the two archetype test statistics

Wo=yPM_{PZr X}M_ZYoy/ˆσ2 andDo=yPM_{PZr X}M_ZYoy/ˆσ2r. (2.25)

Using the restrainedσ2_{estimator, as in a Lagrange-multiplier-type test under normality,}

was already suggested in Durbin (1954, p.27), whereKe=L1= 0 andKo=L2= 1.

Before we discuss further options for estimating σ2 _{in general subset tests, we shall}

first focus on the special caseKe= 0, where the full set of endogenous regressors is tested. Then ˆσ2

r =yMXy/n = n−K_n s2 stems from OLS. Wu (1973) suggested for this case four

test statistics, indicated asT1, ..., T4, where

T₄=n− 2Ko− L1 n 1 Ko DoandT3= n− 2Ko− L1 n 1 Ko Wo. (2.26)

On the basis of his simulation results Wu recommended to use the monotonic transfor-mation ofT4(orDo) T2= T4 1− Ko n−2Ko−L1T4 = n− 2Ko− L1 n 1 Ko Do 1− Do/n . (2.27)

He showed that under normality of both structural and reduced form disturbances the null distribution ofT₂isF (Ko, n− 2Ko− L1) in finite samples.5 BecauseKe= 0 implies

MP_ZrX=MX we find from (2.24) that in this case

Do 1− Do/n =n y _P MXMZYoy y(MX− PMXMZYo)y =ny _P MXMZYoy yM(X MZYo)y =y _P MXMZYoy ¨ σ2 .

Hence, from the final expression we see that T2 estimates σ2 by ¨σ2 = yM(X MZYo)y/n,

which is the OLS residual variance of auxiliary regression (2.21). Like ˆσ2 _{and ˆσ}2

r, ¨σ2 is 5_{Wu’s T1test for case Ke= 0, which under normality has a F (Ko, L2− Ko) distribution, has a poor}

consistent under the null, because plimn−1YoMZuˆr= 0 implies, after substituting (2.23) in (2.22), that plim ˆξ = 0.

Pesaran and Smith (1990) show that under the alternative plim ˆσ2_{≥ plim ˆσ}2

r≥ plim ¨σ2

and then invoke arguments due to Bahadur to expect thatT2 (which uses ¨σ2) has better

power thanT₄ (which uses ˆσ2

r), whereas bothT2 andT4 are expected to outperformT3

(which uses ˆσ2_{). However, they did not verify this experimentally. Moreover, becauseT} 2

is a simple monotonic transformation ofT₄whenKe= 0, we also know that after a fully successful size correction both should have equivalent power.

Following the same lines of thought for cases whereKe> 0, we expect (after proper size correction)Do to do better thanWo, but Pesaran and Smith (1990) suggest that an even better result may be expected from formally testingξ = 0 in the auxiliary regression

(2.21) while exploiting instrumentsZr. This amounts to the χ2(Ko) test statisticTo, which (omitting its degrees of freedom correction) generalizes Wu’sT₂ for cases whereKe≥ 0, and is given by

To=yPM_{PZr X}MZYoy/¨σ2=y(MA− MC)y/¨σ2, (2.28)

with

¨

σ2= (y− X ˆβ − MZYoξ)ˆ(y− X ˆβ − MZYoξ)/n.ˆ (2.29)

Actually, it seems that Pesaran and Smith (1990, p.49) employ a slightly different esti-mator forσ2_{, namely}

(y− X ˆβ − MZYoξˆ∗)(y− X ˆβ − MZYoξˆ∗)/n (2.30) with

ˆ

ξ∗= (YoMZYo)−1YoMZ(y− X ˆβ). (2.31)

However, because OLS residuals are orthogonal to the regressors we haveYoMZ(y−X ˆβ−

MZYoξ) = 0, from which it follows that ˆˆ ξ = ˆξ∗, so their test is equivalent with To. WhenKe> 0 the three tests Wo, DoandToare not simple monotonic transformations of each other, so they may have genuinely different size and power properties in finite samples. In particular, we find that for

Do 1− Do/n = y _P Cy− yPAy (ˆuruˆr− yPCy + yPAy)/n ,

the denominator in the right-hand expression differs from ¨σ2_(unlessK

e= 0).6Using that ˆ

ξ is given by (2.31) we find from (2.29) that ¨σ2_{= ˆu}_M

MZYou/nˆ ≤ ˆσ2, so

Wo≤ To, (2.32)

whereasDocan be at either side ofWoandTo.

2.2.4

Testing based on the discrepancy as such

Direct application of the Hausman (1978) principle yields the test statistic

Ho= ( ˆβ− ˆβr)[ˆσ2(XPZX)−1− ˆσr2(XPZrX)−1]−( ˆβ− ˆβr), (2.33) which uses a generalized inverse for the matrix in square brackets. Whenσ2_{were known}

the matrix in square brackets would certainly be singular though semi-positive definite. Using two different σ2 _{estimates might lead to nonsingularity but could yield negative}

test statistics. As is obvious from the above, (2.33) will not converge to aχ2

Kdistribution under H0_{, but in our framework to one with K}

o degrees of freedom, cf. Hausman and Taylor (1981). Some further analysis leads to the following.

Letβ have separate components as follows from the decompositions

Xβ = Yeβe+Yoβo+Z1β1=Y βeo+Z1β1, (2.34)

whereas (XPZX)−1 has blocks Ajk, j, k = 1, 2, where A11 is a Keo× Keo matrix with

Keo=Ke+Ko. Then we find from (2.11) and (2.13) that

ˆ β− ˆβr= (XPZX)−1  YPZuˆr 0  =  A₁₁ A21  YPZuˆr, ˆ βeo− ˆβeo,r=A11YPZuˆr. (2.35)

Hence, the discrepancy vector of the two coefficient estimates of just the regressors inY,

but also those of the full regressor matrixX, are both linear transformations of rank Keo of the vectorYPZuˆr. Therefore it is obvious that the Hausman-type test statistic (2.33) 6_{Therefore, the test statistic (54) suggested in Baum et al. (2003, p.26), although asymptotically}

equivalent to the tests suggested here, is built on an inappropriate analogy with the K_e = 0 case. Moreover, in their formulas (53) and (54) Q∗should be the difference between the residual sums of squares of second-stage regressions, precisely as in (2.25). The line below (54) suggests that Q∗is a difference between squared IV residuals (which would mean that Q∗could be negative) of the (un)restricted auxiliary regressions, although their footnote 25 seems to suggest otherwise.

can also be obtained from

Ho= ( ˆβeo− ˆβeo,r)[ˆσ2(YPM_Z1Z2Y )−1− ˆσr2(YPM_Z1(Z2Yo)Y )−1]−( ˆβeo− ˆβeo,r). (2.36) Both test statistics are algebraically equivalent, because of the unique linear relationship

ˆ β− ˆβr=  IKeo A₂₁A−1₁₁  ( ˆβeo− ˆβeo,r). (2.37)

Calculating (2.36) instead of (2.33) just mitigates the numerical problems.

One now wonders whether an equivalent Hausman-type test can be calculated on the basis of the discrepancy between the estimated coefficients for just the regressorsYo. This is not the case, because a relationship of the form ( ˆβeo− ˆβeo,r) =G( ˆβo− ˆβo,r), where G is a Keo× Ko matrix, cannot be found7. However, a matrix G can be found such that ( ˆβeo− ˆβeo,r) =G ˆξ, indicating that test Ho can be made equivalent to the three distinct tests of the foregoing subsection, provided similar σ2 _{estimates are being used. Using}

(2.14) and (2.15) in (2.35) we obtain ˆ βeo− ˆβeo,r = A11YPZuˆr = −A₁₁  YeMZYo(YoMZYo)−1 IKo  (YoMZMP_ZrXMZYo) ˆξ, (2.38)

because (2.22) and (2.23) yieldYoMZuˆr = (YoMZMP_ZrXMZYo) ˆξ. So, under the null hy-pothesis particular implementations ofWo, Do, To andHo are equivalent.8 WhenHo is used with two differentσ2_{estimates it may come close to a hybrid implementation ofW}

o andDo where the two residual sums of squares in the numerator are scaled by different

σ2_{estimates as in} W Do= yMP_ZrXy ˆ σ2 r −yM(P_ZrX M_ZYo)y ˆ σ2 . (2.39)

7_{Note that Wu (1983) and Hwang (1985) start off by analyzing a test based on the discrepancy}

ˆ

β_{o− ˆβo,r}. Both Wu (1983) and Ruud (1984, p.236) wrongly suggest equivalence of such a test with (2.33)

and (2.36).

8_{This generalizes the equivalence result mentioned below (22.27) in Ruud (2000, p.581), which just}

treats the case Ke= 0. Note, however, that because Ruud starts off from the full discrepancy vector, the transformation he presents is in fact singular and therefore the inverse function mentioned in his footnote 24 is non-unique (the zero matrix may be replaced with any other matrix of the same dimensions). To obtain a unique inverse transformation, one should start off from the coefficient discrepancy for just the regressors Y, and this is found to be nonsingular for Ke= 0 only.

2.2.5

Testing based on covariance of structural and reduced

form disturbances

Auxiliary regression (2.21) is used to detect correlation of u and Vo (the reduced form disturbances ofYo) by examing the covariance of the residuals ˆurandMZYo. This might perhaps be done in a more direct way by augmenting regression (2.1) by the actual reduced form disturbances, giving

y = Xβ + (Yo− ZΠo)φ + w∗, (2.40)

wherew∗=u− (Yo− ZΠo)φ with φ a Ko× 1 vector. Let ZΠo =Z1Πo1+Z2Πo2, then (2.40) can be written as

y = Yeβe+Yo(βo+φ) + Z1(β1− Πo1φ)− Z2Πo2φ + w∗

= Xβ∗+Z2φ∗+w∗ (2.41)

in which we may assume that E(Zw∗) = 0, though E(Yew∗) = 0. However, testing

φ∗ = 0, which corresponds to φ = 0 in (2.40), through estimating (2.41) consistently is

not an option, unlessKe= 0. ForKe> 0, which is the case of our primary interest here, (2.41) contains all available instruments as regressors, so we cannot instrumentYe.

For the caseKe= 0 the test ofφ∗= 0 yields the test of Revankar and Hartley (1973), which is an exact test under normality. WhenKo=L2 (just identification) it specializes

to Wu’sT₂.9 _WhenL

2> Ko(overidentification) Revankar (1978) argues that testing the

Korestrictionsφ = 0 by testing the L2restrictionsφ∗= 0 is inefficient. He then suggests

to testφ = 0 by a quadratic form in the difference of the least-squares estimator of βo+φ in (2.41) and the IV estimator ofβo.10

From the above we see that the tests on the covariance of disturbances do not have a straight-forward generalization for the case Ke> 0. However, a test that comes close to it replaces theL− L₁ columns ofZ₂in (2.41) by a set ofL− K regressors Z₂∗ which span a subspace ofZ2, such that (PZYeZ1Z2∗) spans the same space asZ. Testing these

L− K exclusion restrictions yields the familiar Sargan-Hansen test for testing all the

so-called overidentification restrictions of model (2.1). It is obvious that this test will have power for alternatives in whichZ₂ andu are correlated, possibly because some of

9_{This is proved as follows: Both tests have regressors X under the null, and under the alternative the}

full column rank matrices (X PZYo) and (X Z2) respectively. These matrices span the same space when

X = (Y_oZ₁) and Z = (Z1Z2) have the same number of columns.

10_{Meepagala (1992) produces numerical results indicating that the discrepancy based tests have lower}

power than the Revankar and Hartley (1973) test when instruments are weak and than the Revankar (1978) test when the instruments are strong.

the variables in Z₂ are actually omitted regressors. In practical situations this type of test, and also Hausman type tests for the orthogonality of particular instruments not included as regressors in the specification11_{, are very useful. However, we do not consider}

such implementations here, because right from the beginning we have chosen a context in which all instrumentsZ are assumed to be uncorrelated with u. This allows focus on tests

serving only the second part of the two-part testing procedure as exposed by Godfrey and Hutton (1994), who also highlight the asymptotic independence of these two parts.

2.2.6

Testing by an incremental Sargan test

The original test of overidentifying restrictions initiated by Sargan (1958) does not enable to infer directly on the orthogonality of individual instrumental variables, but a so-called incremental Sargan test does. It builds on the maintained hypothesisE(Zu) = 0 and can

test the orthogonality of additional potential instrumental variables. Choosing for these the included regressorsYoyields a test statistic for the hypotheses (2.9) which is given by

So= ˆ urPZruˆr ˆ σ2 r −uˆPZuˆ ˆ σ2 . (2.42)

When using for both separate Sargan statistics the same σ2 _{estimate, and employing}

PZu = (Pˆ Z− PP_ZX)y, the numerator would be ˆ

urPZruˆr− ˆuPZu = yˆ (PZr− PP_ZrX− PZ+PPZX)y = y(PMZYo+PPZX− PP_ZrX)y = y(P_(PZX MZYo)− PP_ZrX)y,

whereas that of Wo andDo in (2.24) is given byy(PC− PA)y, where C = (A B) with

A = PZrX and B = MZYo. Equivalence12 is proved by using general result (iii) on

projection matrices, upon takingA∗ = PZX. Using PZr = PZ+PM_ZYo, we have A∗ =

A−PBX = A−B(BB)−1BX, so D = (BB)−1BX. Thus P(A B)=P(A∗B)=P(PZX MZYo)

giving

ˆ

urPZruˆr− ˆuPZu = yˆ (P(PZrX MZYo)− PP_ZrX)y. (2.43) Hence, in addition to theHo statistic,Soestablishes yet another hybrid form combining elements of bothWoandDo, but different from (2.39).

11_{See Hahn et al. (2011) for a study on its behaviour under weak instruments.}

12_{Ruud (2000, p.582) proves this just for the special case Ke= 0. Newey (1985, p.238), (Baum et al.,}

2003, p.23 and formula 55) and Hayashi (2000) mention equivalence for K_{e≥ 0, but do not provide a} proof.