We investigate the validity of the standard specification tests for assessing the exogeneity of subvectors in the linear IV regression. Our results show that ignoring the endogeneity of the regressors whose exogeneity is not being tested leads to invalid tests (level is not controlled). When the fitted values from the first stage regression of these regressors are used as instruments under the partial null hypothesis of interest, as suggested Hausman and Taylor (1980, 1981), some versions of these tests are invalid when identification is weak and the number of instruments is moderate. However, all tests are overly conservative and have no power when the number of instruments increases, even for moderate identification

strength

Doko Tchatoka, F

We investigate the validity of the standard specification tests for assessing the
exogeneity of subvectors in the linear IV regression. Our results show that
ignoring the endogeneity of the regressors whose
exogeneity is not being tested leads to invalid tests (level is not controlled).
When the fitted values from the first stage regression of these  regressors  are used as instruments
 under the partial null hypothesis of interest, as suggested
 Hausman and Taylor (1980, 1981), some versions of
these tests are invalid when identification is weak and the number of instruments is  moderate. However,
all tests are overly conservative and have no power  when the  number of instruments increases,
even for moderate identification strength

Doko Tchatoka, Firmin

Munich RePEc Personal Archive

MPRAMunich Personal RePEc ArchiveOn the Validity of Durbin-Wu-HausmanTests for Assessing Partial ExogeneityHypotheses with Possibly WeakInstrumentsFirmin Doko TchatokaSchool of Economics and Finance, University of Tasmania20. July 2012Online at http://mpra.ub.uni-muenchen.de/40184/MPRA Paper No. 40184, posted 20. July 2012 10:58 UTCOn the Validity of Durbin-Wu-Hausman Tests for AssessingPartial Exogeneity Hypotheses with Possibly WeakInstrumentsFirmin Doko TchatokaSchool of Economics and FinanceUniversity of Tasmania, Private Bag 85, Hobart TAS 7001Tel: +613 6226 7226; Fax:+61 3 6226 7587; e-mail: Firmin.dokotchatoka@utas.edu.auJuly 6, 2012ABSTRACTWe investigate the validity of the standard specification tests for assessing the exogeneityof subvectors in the linear IV regression. Our results show that ignoring the endogeneityof the regressors whose exogeneity is not being tested leads to invalid tests (level is notcontrolled). When the fitted values from the first stage regression of these regressors areused as instruments under the partial null hypothesis of interest, as suggested Hausmanand Taylor (1980, 1981), some versions of these tests are invalid when identification is weakand the number of instruments is moderate. However, all tests are overly conservative andhave no power when the number of instruments increases, even for moderate identificationstrength.Key words: Partial exogeneity; sized distortions; weak identification.JEL classification: C12; C15; C3.1. IntroductionExogeneity tests of the type proposed by proposed by Durbin (1954), Wu (1973), andHausman (1978), henceforth DWH tests, are widely used in applied work to decide whetherordinary least squares (OLS) or instrumental variable (IV) methods is appropriate. Itis now well known that if the null hypothesis of interest is specified on the whole set ofsupposedly endogenous regressors, DWH tests are valid even when instruments are weak[see Staiger and Stock (1997), Hahn et al. (2010), Guggenberger (2010), and Doko Tchatokaand Dufour (2011a, 2011b)]. However, their validity when testing for partial exogeneity isunclear, especially when identification is weak.Many economic questions often involve more than one supposedly endogenous variable,but researcher may want to challenge the exogeneity of only a subset of them. This may beparticularly motivated by some economic theory. Furthermore, efficient estimation of modelparameters requires to use available instruments only for regressors that are endogenous.Thus, assessing individual exogeneity of supposedly endogenous regressors is an importantissue.In this paper, we focus on the linear IV regression and question the validity of two testingpractices, based on DWH type statistics, for assessing partial exogeneity hypotheses. Thefirst is the use of Sargan-Hansen C-type tests ignoring the endogeneity of the regressorswhose exogeneity is not being tested [see Chaudhuri and Rose (2009)]. The second is theextension of DWH tests to partial exogeneity hypotheses, proposed by Hausman and Taylor(1980, 1981). Theses procedures use the fitted values from the first stage regression of theregressors whose exogeneity is not being investigated as instruments under the partial nullhypothesis of interest. Both practices are widely used in applied work.In both cases, we examine the size and power of the corresponding tests through aMonte Carlo experiment, allowing for the presence of weak instruments. We find that theuse of either C-tests or Hausman and Taylor (1980, 1981) extension may lead to misleadingconclusions even when identification strength is moderate.The remainder of the paper is organized as follows. Section 2 formulates the modeland presents the statistical problem of interest. Section 3 analyze the behavior of the teststhrough a Monte Carlo experiment. Conclusions are drawn in Section 4.12. Model, statistical problem and test statisticsWe consider the following linear IV regression modely = Y β +Xθ + u , (2.1)Y = ZΠ+ υ, X = ZΓ+ ξ (2.2)where y ∈ Rn is a vector of observations on a dependent variable, Y ∈ Rn×my andX ∈ Rn×mx (my + mx = m ≥ 1) are two matrices of (possibly) endogenous explana-tory variables, Z ∈ Rn×l is a matrix of exogenous instruments, u = (u1, . . . , un)′ ∈ Rn isthe vector of structural disturbances, υ ∈ Rn×my and ξ ∈ Rn×mx are matrices of reducedform disturbances, β ∈ Rmy and θ ∈ Rmx are unknown structural parameter vectors, whileΠ ∈ Rl×my and Γ ∈ Rl×mx are unknown reduced form coefficient matrices. Model (2.1)-(2.2) can be modified to include exogenous variables Z1. If so, our results do not alterqualitatively if we replace the variables that are currently in (2.1)-(2.2) by the residualsthat result from their projection onto Z1. We shall assume that the instrument matrix Zhas full-column rank l with probability one and l ≥ m.The usual necessary and sufficient condition for identification of model (2.1)-(2.2) isrank([Π, Γ]) = m. If [Π, Γ] is close not to have full rank, (β′, θ′)′ is not identifiable butsome of its linear combinations are ill-determined by the data, a situation often called “weakidentification” in this type of setup [see Andrews and Stock (2006), Dufour (2003)]. When[Π, Γ] = 0, the instruments Z are irrelevant and (θ′, β′)′ is completely unidentified.We will study in turn the problem of testing the partial exogeneity of Y , i.e. thehypothesisHp0 : cov(Y, u) = συu = 0 (2.3)where the regressors X whose exogeneity is not being tested may be endogenous. Haus-man and Taylor (1980) [see also Hausman and Taylor (1981, p.1389, footnote 10)] pro-pose to assess Hp0 via the standard DWH specification tests upon comparing the two stageleast squares estimator (βˆ′0,2SLS, θˆ′0,2SLS)′ obtained using [Y, Xˆ] as instruments, to those(βˆ′1,2SLS, θˆ′1,2SLS)′ obtained from [Yˆ , Xˆ], where [Yˆ , Xˆ ] = PZ [Y, X] and PZ = Z(Z′Z)−1Z ′.2This is mathematically equivalent1 to test the exogeneity of Y in modely = Y β + Xˆθ + e , (2.4)Y = ZΠ+ υ. (2.5)Which means that the corresponding DWH statistics can be written as a quadratic formin the subvector of differences of the 2SLS estimates of β alone. An important question iswhether the tests obtained in such a way are still valid, especially when identification is weak.The problem stems from the fact that when instruments are weak, Xˆ is not independent ofu even in large-sample. So, using Xˆ as instruments under Hp0 may be problematic.From (2.4)-(2.5), we can express βˆ0,2SLS and βˆ1,2SLS as: βˆ0,2SLS =(Y ′MXˆY )−1Y ′MXˆy, βˆ1,2SLS = (Yˆ′MXˆ Yˆ )−1Yˆ ′MXˆy, where MXˆ = In − PXˆ . Weconsider four alternative DWH type statistics written in the following unified formulation:Wj = κj(βˆ1,2SLS − βˆ0,2SLS)′Σˆ−1j (βˆ1,2SLS − βˆ0,2SLS), j = 1, 2, 3, 4 (2.6)Σˆ1 = σ˜22∆ˆ, ∆ˆ = Ωˆ−1IV − Ωˆ−1LS , Σˆ2 = σ˜2Ωˆ−1IV − σˆ2Ωˆ−1LS , Σˆ3 = σ˜2∆ˆ, Σˆ4 = σˆ2∆ˆ,ΩˆIV = Yˆ′MXˆ Yˆ /n, ΩˆLS = Y′MXˆY/n, σ˜2 = (y − Y βˆ1,2SLS)′MXˆ(y − Y βˆ1,2SLS)/n,σ˜22 = σˆ2 − (βˆ1,2SLS − βˆ0,2SLS)′∆ˆ−1(βˆ1,2SLS − βˆ0,2SLS), κ1 = (n− 2my)/myσˆ2 = (y − Y βˆ0,2SLS)′MXˆ(y − Y βˆ0,2SLS)/n, κi = n, j = 2, 3, 4.The statistics in (2.6) differ only through the variance estimators of the errors in (2.4)-(2.5) and the scaling factors κj . σˆ2 and σ˜2 are the usual IV-based estimators (withoutcorrection for degrees of freedom), while σ˜22 can be interpreted as an alternative IV-basedscaling factor. The statistic W1 is an extension of Wu (1973) T2-statistic. Wj (j ≥ 2)are analogues to alternative Hausman (1978) type-statistics studied by Staiger and Stock(1997), Guggenberger (2010), and Doko Tchatoka and Dufour (2011a, 2011b). The Sargan-Hansen C-tests ignoring the endogeneity of X consist of replacing Xˆ by X in (2.6), so thatβˆ0,2SLS collapses to the OLS estimator of β in (2.1).1See Baum et al. (2003) for further details.3Following Hausman and Taylor (1980, 1981), Hp0 is rejected if the corresponding statisticis greater than the desired critical value of a χ2(my)-distribution. This however requires Xˆbe independent of the structural error, which is not guaranteed when instruments are weak.We shall now analyze the size and power of the tests through a Monte Carlo experiment,in both strong and weak identification setups.3. Simulation resultsIn each of the following experiments, data are generated from model (2.1)-(2.2) with threeendogenous regressors (Y : n× 2, W : n× 1) by settingui = (1 + ρ2υ1 + ρ2υ2 + ρ2ξ)−1/2(ε1i + ρυ1ε2i + ρυ2ε3i + ρξε4i),υi = [(1 + ρ2υ1)−1/2(ρυ1ε1i + ε2i), (1 + ρ2υ2)−1/2(ρυ2ε1i + ε3i)],ξi = (1 + ρ2ξ)−1/2(ρξε1i + ε4i), (ε1i, ε2i, ε3i, ε4i)′ ∼ IIDN (0, I4) (3.7)for all i = 1, . . . , n, ρυ1 ∈ {−.9, −.2, 0, .3, .8} , ρυ2 = ρυ1/√3 and ρξ = 0.4. The re-sults are qualitatively the same for alternative choices of ρξ and ρυ2 . The exogeneity ofY is then expressed as Hp0 : ρυ1 = 0. Z contains l ∈ {3, 20} instruments each generatedIIDN(0, 1) and is kept fix within experiment. The true value of (β′, θ)′ is (2, −3, 1/2)′and [Π, Γ] is chosen as: [Π, Γ] =√µ2n‖ZC‖Π0, where Π0 is obtained by taking the firstthree columns of the identity matrix of dimension l, C is an l × 1 vector of ones, andµ2 ∈ {0, 10, 1000, 10000} is the concentration parameter that characterizes the strength ofthe instruments. The values µ2 ≤ 613 correspond to weak instruments while µ2 > 613 isfor strong instruments [see Hansen et al. (2008) ]. In particular, when µ2 = 0, the instru-ments are irrelevant and (β′, θ)′ is completely unidentified. The simulations are run withsample sizes n = 100 and 500. The number of replications is N = 10000 and the empiricalrejections of the statistics are computed using the 95% critical value value of χ2(2).Table 1 presents the results. In the first column, we report the statistics while thesecond column reports the number of instruments. In the other columns, for each value ofendogeneity ρυ1 and instrument quality µ2, the rejection frequencies are reported.The first part of the table deals with the setup where the endogeneity of X is ig-4nored. As seen, all tests are seriously size distorted when identification is strong [columnρυ1 = 0, µ2 ∈ {1000, 10000}]. The size distortions persists even for n = 500. When in-struments are irrelevant or weak [column ρυ1 = 0, µ2 ∈ {0, 10}], W1 and W4 are still sizedistorted but W2 and W3 are overly conservative. In all cases, the maximal size distortionof the tests approaches 40% with 3 instruments and around 50% with 20 instruments.The second part of the table corresponds to the setup by Hausman and Taylor(1980,1981), where [Y, Xˆ ] is used as instruments under Hp0. Two observations emergein this case depending on the sample size. Firstly, when n = 100, all tests have correctlevel when identification is very strong [column ρυ1 = 0, µ2 = 10000], but are conservationfor moderate identification strength [column ρυ1 = 0, µ2 = 1000]. However, when identi-fication is weak, W1 and W4 are size distorted when l = 3 but overly conservative whenl = 20. At the same time, W2 and W3 are overly conservative in all cases. Secondly, whenn = 500, all tests are conservative when identification is strong or moderate. With weak in-struments, W1 and W4 are still size distorted when l = 3, with maximal distortion around12%. This suggests that the null asymptotic distributions of the statistics is far from aχ2(2)-distribution, even with moderate identification strength. Moreover, we can see thateven when µ2 = 1000 (moderate identification), the tests have no power when l = 20 andρυ1 = −.2; .3. The sample realizations of the statistics are often close to zero even for arelatively high endogeneity. Overall, the use of DWH tests to assess partial exogeneity hy-potheses, as suggested Hausman and Taylor (1980,1981), may be misleading if identificationis not very strong. The use of C-type tests, as described here, is inappropriate and shouldbe avoided.5Table 1. Size and power at nominal level 5%Rejections when the endogeneity of X is ignoredn = 100ρ1 = −.9 ρ1 = −.2 ρ1 = 0 ρ1 = .3 ρ1 = .8Statistics k2 ↓ µ2 → 0 10 1000 10000 0 10 1000 10000 0 10 1000 10000 0 10 1000 10000 0 10 1000 10000W1 3 31.1 65.8 89.6 87.4 30.0 42.9 79.5 96.7 29.6 38.6 37.6 39.3 29.9 46.1 94.2 99.7 31.6 62.7 93.1 99.4W2 3 1.5 4.9 100.0 100.0 0.8 1.7 74.7 96.0 0.7 1.7 31.2 35.5 1.0 2.2 92.4 99.7 1.6 6.8 100.0 100.0W3 3 1.8 5.9 100.0 100.0 1.0 2.2 76.8 96.5 0.9 2.2 33.7 38.0 1.3 2.6 93.1 99.7 2.0 8.1 100.0 100.0W4 3 40.0 85.5 100.0 100.0 38.1 53.3 79.1 96.6 37.4 49.0 37.1 38.4 37.6 58.5 94.2 99.7 40.3 96.3 100.0 100.0W1 20 9.0 12.7 99.3 100.0 7.2 7.8 38.9 80.2 7.0 7.1 17.1 13.7 7.4 8.1 54.1 95.9 8.7 19.0 99.5 100.0W2 20 3.4 4.7 98.8 100.0 2.0 2.1 29.7 76.3 1.7 1.9 11.3 11.4 2.1 2.4 44.4 94.7 3.1 8.8 99.2 100.0W3 20 4.2 5.9 99.0 100.0 2.7 2.8 32.6 78.7 2.5 2.4 13.2 12.7 2.8 3.2 48.1 95.5 3.9 10.5 99.3 100.0W4 20 8.6 12.1 99.3 100.0 6.9 7.5 38.0 79.4 6.6 6.8 16.6 13.2 7.1 7.7 53.3 95.7 8.4 18.3 99.5 100.0n = 500W1 3 35.6 63.3 99.3 99.8 34.2 40.6 80.3 99.8 33.3 39.3 50.2 32.7 33.6 44.3 95.5 100.0 35.0 80.0 93.1 100.0W2 3 1.4 3.1 99.4 100.0 0.7 1.4 73.0 99.8 0.7 1.1 39.8 31.3 1.0 1.4 92.6 100.0 1.5 4.5 99.5 100.0W3 3 1.5 3.2 99.4 100.0 0.7 1.5 73.4 99.8 0.7 1.1 40.4 31.6 1.0 1.4 92.8 100.0 1.5 4.6 99.5 100.0W4 3 39.5 70.4 100.0 100.0 37.7 44.9 80.3 99.8 36.7 43.3 50.1 32.6 37.1 48.8 95.6 100.0 38.9 90.9 100.0 100.0W1 20 8.0 10.1 95.7 100.0 6.6 6.8 30.7 91.2 6 6.7 20.8 29.6 6.9 7.4 48.9 99.7 7.8 15.0 97.8 100.0W2 20 3.0 3.7 93.9 100.0 1.7 1.9 21.5 90.3 1.4 1.7 13.2 27.2 1.8 1.8 38.4 99.6 2.8 6.7 97.0 100.0W3 20 3.1 3.9 93.9 100.0 1.7 2.0 22.1 90.5 1.6 1.8 13.6 27.8 2.0 1.9 39.1 99.6 2.9 6.9 97.1 100.0W4 20 7.9 10.0 95.7 100.0 6.5 6.7 30.5 91.2 5.9 6.6 20.7 29.5 6.8 7.3 48.8 99.7 7.7 14.8 97.8 100.0Rejections when [Y, Xˆ] is used as instruments under Hp0n = 100ρ1 = −.9 ρ1 = −.2 ρ1 = 0 ρ1 = .3 ρ1 = .8Statistics k2 ↓ µ2 → 0 10 1000 10000 0 10 1000 10000 0 10 1000 10000 0 10 1000 10000 0 10 1000 10000W1 3 7.3 18.3 99.9 100.0 8.1 11.1 39.4 75.4 8.5 11.8 3.0 5.2 7.4 11.2 81.5 98.1 3.5 17.8 100.0 100.0W2 3 0.3 1.3 100.0 100.0 0.3 0.8 29.0 71.3 0.3 0.7 1.7 3.8 0.3 0.7 72.2 97.3 0.2 1.0 100.0 100.0W3 3 0.3 1.7 100.0 100.0 0.4 0.8 33.0 74.1 0.3 0.8 2.2 4.7 0.4 0.9 76.2 97.9 0.2 1.3 100.0 100.0W4 3 9.1 23.1 100.0 100.0 10.8 14.2 38.3 74.4 10.9 15.0 2.9 4.8 9.5 14.8 80.6 98.0 4.3 21.9 100.0 100.0W1 20 17.8 20.2 99.7 100.0 0.3 0.4 13.9 56.9 0.0 0.1 1.5 5.2 0.6 0.9 25.4 90.6 13.3 15.7 99.2 100.0W2 20 6.9 8.1 99.1 100.0 0.1 0.1 7.9 51.4 0 0 0.7 3.71 0.2 0.2 15.9 88.0 4.9 5.7 97.8 100.0W3 20 8.7 10.5 99.3 100.0 0.1 0.1 9.7 54.8 0 0 0.9 4.7 0.3 0.3 19.1 89.6 6.4 7.5 98.3 100.0W4 20 16.8 19.4 99.6 100.0 0.2 0.4 13.3 55.8 0 0.1 1.3 4.9 0.6 0.8 24.5 90.1 12.6 14.8 99.2 100.0n = 500W1 3 7.5 14.2 100.0 100.0 9.0 10.6 18.6 98.5 9.39 10.7 1.6 1.2 8.0 10.9 58.0 100.0 3.7 11.2 100.0 100.0W2 3 0.3 0.7 98.1 100.0 0.2 0.5 8.6 98.2 0.3 0.3 0.7 1.0 0.3 0.5 35.4 100.0 0.2 0.5 98.4 100.0W3 3 0.3 0.7 98.3 100.0 0.2 0.6 9.0 98.4 0.3 0.3 0.8 1.1 0.3 0.5 36.9 100.0 0.2 0.5 98.5 100.0W4 3 8.5 15.8 100.0 100.0 10.1 11.9 18.4 98.5 10.5 12.1 1.6 1.1 9.0 12.2 57.7 100.0 4.2 12.4 100.0 100.0W1 20 5.7 6.9 93.1 100.0 0.0 0.0 0.3 69.3 0 0 0 0.5 0.0 0.0 2.7 99.1 2.3 2.5 89.5 100.0W2 20 1.4 1.8 80.3 100.0 0.0 0.0 0.1 64.8 0 0 0 0.4 0.0 0.0 0.8 98.8 0.5 0.6 75.0 100.0W3 20 1.5 1.9 81.0 100.0 0.0 0.0 0.1 65.6 0 0 0 0.4 0.0 0.0 0.9 98.8 0.6 0.7 75.9 100.0W4 20 5.5 6.8 93.0 100.0 0.0 0.0 0.3 69.0 0 0 0 0.5 0.0 0.0 2.6 99.1 2.3 2.4 89.4 100.064. Concluding remarksIn this paper, we question the validity of the use of standard DWH specification tests forassessing partial exogeneity hypotheses. Our results show that these tests are invalid (levelis not controlled) if the endogeneity of the regressors whose exogeneity is not being testedis ignored within inference. When the fitted values from the first stage regression of theseregressors are used as instruments under the subset null hypothesis of interest, we findthat some versions of the tests are invalid when identification is weak and the number ofinstruments is relatively moderate. However, all tests are overly conservative and have nopower when the number of instruments increases, even for moderate identification strength.So, applying them to assess partial exogeneity hypotheses, as it is often the case in appliedwork, may be misleading. This underscores the importance to develop tests that do notexhibit these problems when testing for partial exogeneity.AcknowledgmentsThe author thanks Mardi Dungey (University of Tasmania), Jean-Marie Dufour (McGillUniversity) and Russell Davidson (McGill University) for helpful comments and suggestions.ReferencesAndrews, D. W. K. , Stock , 2006. Inference with weak instruments. In: R. Blundell,W. Newey , T. Pearson, eds, Advances in Economics and Econometrics, Theory andApplications, 9th Congress of the Econometric Society Vol. 3 . Cambridge UniversityPress, Cambridge, U.K., chapter 6.Baun, C. F., Schaffer, M. E. , Stillman, S. , 2003. Instrumental variable and GMM: Esti-mation and testing. Technical report, Department of Economics, Boston College, USABoston, USA.Chaudhuri, S. , Rose, E. , 2009. Estimating the veteran effect with endogenous schoolingwhen instruments are potentially weak. Technical report, IZA Discussion Papers No.4203.7Doko Tchatoka, F., Dufour, J.-M., 2011a. Exogeneity tests and estimation in IV regressions.Technical report, Department of Economics, McGill University, Canada Montre´al,Canada.Doko Tchatoka, F. , Dufour, J.-M. , 2011b. On the finite-sample theory of exogeneity testswith possibly non-gaussian errors and weak identification. Technical report, Depart-ment of Economics, McGill University, Canada Montre´al, Canada.Dufour, J.-M., 2003. Identification, weak instruments and statistical inference in economet-rics. Canadian Journal of Economics 36(4), 767–808.Durbin, J., 1954. Errors in variables. Review of the International Statistical Institute 22, 23–32.Guggenberger, P. , 2010. The impact of a Hausman pretest on the size of the hypothesistests. Econometric Theory 156, 337–343.Hahn, J., Ham, J. , Moon, H. R. , 2010. The Hausman test and weak instruments. Journalof Econometrics 160, 289–299.Hansen, C., Hausman, J., Newey, W., 2008. Estimation with many instrumental variables.Journal of Business and Economic Statistics 26(4), 398–422.Hausman, J., 1978. Specification tests in econometrics. Econometrica 46, 1251–1272.Hausman, J., Taylor, W., 1981. Panel data and unobservable individual effects. Economet-rica 49(6), 1377–1398.Hausman, J., Taylor, W. E., 1980. Comparing specification tests and clissical tests. Technicalreport, Department of Economics, MIT Boston, MA.Staiger, D. , Stock, J. H. , 1997. Instrumental variables regression with weak instruments.Econometrica 65(3), 557–586.Wu, D.-M., 1973. Alternative tests of independence between stochastic regressors and dis-turbances. Econometrica 41, 733–750.8

English

On the Validity of Durbin-Wu-Hausman Tests for Assessing Partial Exogeneity Hypotheses with Possibly Weak Instruments

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
On the Validity of Durbin-Wu-Hausman
Tests for Assessing Partial Exogeneity
Hypotheses with Possibly Weak
Instruments
Firmin Doko Tchatoka
School of Economics and Finance, University of Tasmania
20 July 2012
Online at https://mpra.ub.uni-muenchen.de/40184/
MPRA Paper No. 40184, posted 20 July 2012 10:58 UTC
On the Validity of Durbin-Wu-Hausman Tests for Assessing
Partial Exogeneity Hypotheses with Possibly Weak
Instruments
Firmin Doko Tchatoka
School of Economics and Finance
University of Tasmania, Private Bag 85, Hobart TAS 7001
Tel: +613 6226 7226; Fax:+61 3 6226 7587; e-mail: Firmin.dokotchatoka@utas.edu.au
July 6, 2012
ABSTRACT
We investigate the validity of the standard specification tests for assessing the exogeneity
of subvectors in the linear IV regression. Our results show that ignoring the endogeneity
of the regressors whose exogeneity is not being tested leads to invalid tests (level is not
controlled). When the fitted values from the first stage regression of these regressors are
used as instruments under the partial null hypothesis of interest, as suggested Hausman
and Taylor (1980, 1981), some versions of these tests are invalid when identification is weak
and the number of instruments is moderate. However, all tests are overly conservative and
have no power when the number of instruments increases, even for moderate identification
strength.
Key words: Partial exogeneity; sized distortions; weak identification.
JEL classification: C12; C15; C3.
1. Introduction
Exogeneity tests of the type proposed by proposed by Durbin (1954), Wu (1973), and
Hausman (1978), henceforth DWH tests, are widely used in applied work to decide whether
ordinary least squares (OLS) or instrumental variable (IV) methods is appropriate. It
is now well known that if the null hypothesis of interest is specified on the whole set of
supposedly endogenous regressors, DWH tests are valid even when instruments are weak
[see Staiger and Stock (1997), Hahn et al. (2010), Guggenberger (2010), and Doko Tchatoka
and Dufour (2011a, 2011b)]. However, their validity when testing for partial exogeneity is
unclear, especially when identification is weak.
Many economic questions often involve more than one supposedly endogenous variable,
but researcher may want to challenge the exogeneity of only a subset of them. This may be
particularly motivated by some economic theory. Furthermore, efficient estimation of model
parameters requires to use available instruments only for regressors that are endogenous.
Thus, assessing individual exogeneity of supposedly endogenous regressors is an important
issue.
In this paper, we focus on the linear IV regression and question the validity of two testing
practices, based on DWH type statistics, for assessing partial exogeneity hypotheses. The
first is the use of Sargan-Hansen C-type tests ignoring the endogeneity of the regressors
whose exogeneity is not being tested [see Chaudhuri and Rose (2009)]. The second is the
extension of DWH tests to partial exogeneity hypotheses, proposed by Hausman and Taylor
(1980, 1981). Theses procedures use the fitted values from the first stage regression of the
regressors whose exogeneity is not being investigated as instruments under the partial null
hypothesis of interest. Both practices are widely used in applied work.
In both cases, we examine the size and power of the corresponding tests through a
Monte Carlo experiment, allowing for the presence of weak instruments. We find that the
use of either C-tests or Hausman and Taylor (1980, 1981) extension may lead to misleading
conclusions even when identification strength is moderate.
The remainder of the paper is organized as follows. Section 2 formulates the model
and presents the statistical problem of interest. Section 3 analyze the behavior of the tests
through a Monte Carlo experiment. Conclusions are drawn in Section 4.
1
2. Model, statistical problem and test statistics
We consider the following linear IV regression model
y = Y β +Xθ + u , (2.1)
Y = ZΠ+ υ, X = ZΓ+ ξ (2.2)
where y ∈ Rn is a vector of observations on a dependent variable, Y ∈ Rn×my and
X ∈ Rn×mx (my + mx = m ≥ 1) are two matrices of (possibly) endogenous explana-
tory variables, Z ∈ Rn×l is a matrix of exogenous instruments, u = (u1, . . . , un)′ ∈ Rn is
the vector of structural disturbances, υ ∈ Rn×my and ξ ∈ Rn×mx are matrices of reduced
form disturbances, β ∈ Rmy and θ ∈ Rmx are unknown structural parameter vectors, while
Π ∈ Rl×my and Γ ∈ Rl×mx are unknown reduced form coefficient matrices. Model (2.1)-
(2.2) can be modified to include exogenous variables Z1. If so, our results do not alter
qualitatively if we replace the variables that are currently in (2.1)-(2.2) by the residuals
that result from their projection onto Z1. We shall assume that the instrument matrix Z
has full-column rank l with probability one and l ≥ m.
The usual necessary and sufficient condition for identification of model (2.1)-(2.2) is
rank([Π, Γ]) = m. If [Π, Γ] is close not to have full rank, (β′, θ′)′ is not identifiable but
some of its linear combinations are ill-determined by the data, a situation often called “weak
identification” in this type of setup [see Andrews and Stock (2006), Dufour (2003)]. When
[Π, Γ] = 0, the instruments Z are irrelevant and (θ′, β′)′ is completely unidentified.
We will study in turn the problem of testing the partial exogeneity of Y , i.e. the
hypothesis
Hp0 : cov(Y, u) = συu = 0 (2.3)
where the regressors X whose exogeneity is not being tested may be endogenous. Haus-
man and Taylor (1980) [see also Hausman and Taylor (1981, p.1389, footnote 10)] pro-
pose to assess Hp0 via the standard DWH specification tests upon comparing the two stage
least squares estimator (βˆ
′
0,2SLS, θˆ
′
0,2SLS)
′ obtained using [Y, Xˆ] as instruments, to those
(βˆ
′
1,2SLS, θˆ
′
1,2SLS)
′ obtained from [Yˆ , Xˆ], where [Yˆ , Xˆ ] = PZ [Y, X] and PZ = Z(Z
′Z)−1Z ′.
2
This is mathematically equivalent1 to test the exogeneity of Y in model
y = Y β + Xˆθ + e , (2.4)
Y = ZΠ+ υ. (2.5)
Which means that the corresponding DWH statistics can be written as a quadratic form
in the subvector of differences of the 2SLS estimates of β alone. An important question is
whether the tests obtained in such a way are still valid, especially when identification is weak.
The problem stems from the fact that when instruments are weak, Xˆ is not independent of
u even in large-sample. So, using Xˆ as instruments under Hp0 may be problematic.
From (2.4)-(2.5), we can express βˆ0,2SLS and βˆ1,2SLS as: βˆ0,2SLS =
(Y ′MXˆY )
−1Y ′MXˆy, βˆ1,2SLS = (Yˆ
′MXˆ Yˆ )
−1Yˆ ′MXˆy, where MXˆ = In − PXˆ . We
consider four alternative DWH type statistics written in the following unified formulation:
Wj = κj(βˆ1,2SLS − βˆ0,2SLS)′Σˆ−1j (βˆ1,2SLS − βˆ0,2SLS), j = 1, 2, 3, 4 (2.6)
Σˆ1 = σ˜
2
2∆ˆ, ∆ˆ = Ωˆ
−1
IV − Ωˆ−1LS , Σˆ2 = σ˜2Ωˆ−1IV − σˆ2Ωˆ−1LS , Σˆ3 = σ˜2∆ˆ, Σˆ4 = σˆ2∆ˆ,
ΩˆIV = Yˆ
′MXˆ Yˆ /n, ΩˆLS = Y
′MXˆY/n, σ˜
2 = (y − Y βˆ1,2SLS)′MXˆ(y − Y βˆ1,2SLS)/n,
σ˜22 = σˆ
2 − (βˆ1,2SLS − βˆ0,2SLS)′∆ˆ−1(βˆ1,2SLS − βˆ0,2SLS), κ1 = (n− 2my)/my
σˆ2 = (y − Y βˆ0,2SLS)′MXˆ(y − Y βˆ0,2SLS)/n, κi = n, j = 2, 3, 4.
The statistics in (2.6) differ only through the variance estimators of the errors in (2.4)-
(2.5) and the scaling factors κj . σˆ
2 and σ˜2 are the usual IV-based estimators (without
correction for degrees of freedom), while σ˜22 can be interpreted as an alternative IV-based
scaling factor. The statistic W1 is an extension of Wu (1973) T2-statistic. Wj (j ≥ 2)
are analogues to alternative Hausman (1978) type-statistics studied by Staiger and Stock
(1997), Guggenberger (2010), and Doko Tchatoka and Dufour (2011a, 2011b). The Sargan-
Hansen C-tests ignoring the endogeneity of X consist of replacing Xˆ by X in (2.6), so that
βˆ0,2SLS collapses to the OLS estimator of β in (2.1).
1See Baum et al. (2003) for further details.
3
Following Hausman and Taylor (1980, 1981), Hp0 is rejected if the corresponding statistic
is greater than the desired critical value of a χ2(my)-distribution. This however requires Xˆ
be independent of the structural error, which is not guaranteed when instruments are weak.
We shall now analyze the size and power of the tests through a Monte Carlo experiment,
in both strong and weak identification setups.
3. Simulation results
In each of the following experiments, data are generated from model (2.1)-(2.2) with three
endogenous regressors (Y : n× 2, W : n× 1) by setting
ui = (1 + ρ
2
υ1 + ρ
2
υ2 + ρ
2
ξ)
−1/2(ε1i + ρυ1ε2i + ρυ2ε3i + ρξε4i),
υi = [(1 + ρ
2
υ1)
−1/2(ρυ1ε1i + ε2i), (1 + ρ
2
υ2)
−1/2(ρυ2ε1i + ε3i)],
ξi = (1 + ρ
2
ξ)
−1/2(ρξε1i + ε4i), (ε1i, ε2i, ε3i, ε4i)
′ ∼ IIDN (0, I4) (3.7)
for all i = 1, . . . , n, ρυ1 ∈ {−.9, −.2, 0, .3, .8} , ρυ2 = ρυ1/
√
3 and ρξ = 0.4. The re-
sults are qualitatively the same for alternative choices of ρξ and ρυ2 . The exogeneity of
Y is then expressed as Hp0 : ρυ1 = 0. Z contains l ∈ {3, 20} instruments each generated
IIDN(0, 1) and is kept fix within experiment. The true value of (β′, θ)′ is (2, −3, 1/2)′
and [Π, Γ] is chosen as: [Π, Γ] =
√
µ2
n‖ZC‖Π0, where Π0 is obtained by taking the first
three columns of the identity matrix of dimension l, C is an l × 1 vector of ones, and
µ2 ∈ {0, 10, 1000, 10000} is the concentration parameter that characterizes the strength of
the instruments. The values µ2 ≤ 613 correspond to weak instruments while µ2 > 613 is
for strong instruments [see Hansen et al. (2008) ]. In particular, when µ2 = 0, the instru-
ments are irrelevant and (β′, θ)′ is completely unidentified. The simulations are run with
sample sizes n = 100 and 500. The number of replications is N = 10000 and the empirical
rejections of the statistics are computed using the 95% critical value value of χ2(2).
Table 1 presents the results. In the first column, we report the statistics while the
second column reports the number of instruments. In the other columns, for each value of
endogeneity ρυ1 and instrument quality µ
2, the rejection frequencies are reported.
The first part of the table deals with the setup where the endogeneity of X is ig-
4
nored. As seen, all tests are seriously size distorted when identification is strong [column
ρυ1 = 0, µ
2 ∈ {1000, 10000}]. The size distortions persists even for n = 500. When in-
struments are irrelevant or weak [column ρυ1 = 0, µ
2 ∈ {0, 10}], W1 and W4 are still size
distorted but W2 and W3 are overly conservative. In all cases, the maximal size distortion
of the tests approaches 40% with 3 instruments and around 50% with 20 instruments.
The second part of the table corresponds to the setup by Hausman and Taylor
(1980,1981), where [Y, Xˆ ] is used as instruments under Hp0. Two observations emerge
in this case depending on the sample size. Firstly, when n = 100, all tests have correct
level when identification is very strong [column ρυ1 = 0, µ
2 = 10000], but are conservation
for moderate identification strength [column ρυ1 = 0, µ
2 = 1000]. However, when identi-
fication is weak, W1 and W4 are size distorted when l = 3 but overly conservative when
l = 20. At the same time, W2 and W3 are overly conservative in all cases. Secondly, when
n = 500, all tests are conservative when identification is strong or moderate. With weak in-
struments, W1 and W4 are still size distorted when l = 3, with maximal distortion around
12%. This suggests that the null asymptotic distributions of the statistics is far from a
χ2(2)-distribution, even with moderate identification strength. Moreover, we can see that
even when µ2 = 1000 (moderate identification), the tests have no power when l = 20 and
ρυ1 = −.2; .3. The sample realizations of the statistics are often close to zero even for a
relatively high endogeneity. Overall, the use of DWH tests to assess partial exogeneity hy-
potheses, as suggested Hausman and Taylor (1980,1981), may be misleading if identification
is not very strong. The use of C-type tests, as described here, is inappropriate and should
be avoided.
5
Table 1. Size and power at nominal level 5%
Rejections when the endogeneity of X is ignored
n = 100
ρ1 = −.9 ρ1 = −.2 ρ1 = 0 ρ1 = .3 ρ1 = .8
Statistics k2 ↓ µ2 → 0 10 1000 10000 0 10 1000 10000 0 10 1000 10000 0 10 1000 10000 0 10 1000 10000
W1 3 31.1 65.8 89.6 87.4 30.0 42.9 79.5 96.7 29.6 38.6 37.6 39.3 29.9 46.1 94.2 99.7 31.6 62.7 93.1 99.4
W2 3 1.5 4.9 100.0 100.0 0.8 1.7 74.7 96.0 0.7 1.7 31.2 35.5 1.0 2.2 92.4 99.7 1.6 6.8 100.0 100.0
W3 3 1.8 5.9 100.0 100.0 1.0 2.2 76.8 96.5 0.9 2.2 33.7 38.0 1.3 2.6 93.1 99.7 2.0 8.1 100.0 100.0
W4 3 40.0 85.5 100.0 100.0 38.1 53.3 79.1 96.6 37.4 49.0 37.1 38.4 37.6 58.5 94.2 99.7 40.3 96.3 100.0 100.0
W1 20 9.0 12.7 99.3 100.0 7.2 7.8 38.9 80.2 7.0 7.1 17.1 13.7 7.4 8.1 54.1 95.9 8.7 19.0 99.5 100.0
W2 20 3.4 4.7 98.8 100.0 2.0 2.1 29.7 76.3 1.7 1.9 11.3 11.4 2.1 2.4 44.4 94.7 3.1 8.8 99.2 100.0
W3 20 4.2 5.9 99.0 100.0 2.7 2.8 32.6 78.7 2.5 2.4 13.2 12.7 2.8 3.2 48.1 95.5 3.9 10.5 99.3 100.0
W4 20 8.6 12.1 99.3 100.0 6.9 7.5 38.0 79.4 6.6 6.8 16.6 13.2 7.1 7.7 53.3 95.7 8.4 18.3 99.5 100.0
n = 500
W1 3 35.6 63.3 99.3 99.8 34.2 40.6 80.3 99.8 33.3 39.3 50.2 32.7 33.6 44.3 95.5 100.0 35.0 80.0 93.1 100.0
W2 3 1.4 3.1 99.4 100.0 0.7 1.4 73.0 99.8 0.7 1.1 39.8 31.3 1.0 1.4 92.6 100.0 1.5 4.5 99.5 100.0
W3 3 1.5 3.2 99.4 100.0 0.7 1.5 73.4 99.8 0.7 1.1 40.4 31.6 1.0 1.4 92.8 100.0 1.5 4.6 99.5 100.0
W4 3 39.5 70.4 100.0 100.0 37.7 44.9 80.3 99.8 36.7 43.3 50.1 32.6 37.1 48.8 95.6 100.0 38.9 90.9 100.0 100.0
W1 20 8.0 10.1 95.7 100.0 6.6 6.8 30.7 91.2 6 6.7 20.8 29.6 6.9 7.4 48.9 99.7 7.8 15.0 97.8 100.0
W2 20 3.0 3.7 93.9 100.0 1.7 1.9 21.5 90.3 1.4 1.7 13.2 27.2 1.8 1.8 38.4 99.6 2.8 6.7 97.0 100.0
W3 20 3.1 3.9 93.9 100.0 1.7 2.0 22.1 90.5 1.6 1.8 13.6 27.8 2.0 1.9 39.1 99.6 2.9 6.9 97.1 100.0
W4 20 7.9 10.0 95.7 100.0 6.5 6.7 30.5 91.2 5.9 6.6 20.7 29.5 6.8 7.3 48.8 99.7 7.7 14.8 97.8 100.0
Rejections when [Y, Xˆ] is used as instruments under Hp
0
n = 100
ρ1 = −.9 ρ1 = −.2 ρ1 = 0 ρ1 = .3 ρ1 = .8
Statistics k2 ↓ µ2 → 0 10 1000 10000 0 10 1000 10000 0 10 1000 10000 0 10 1000 10000 0 10 1000 10000
W1 3 7.3 18.3 99.9 100.0 8.1 11.1 39.4 75.4 8.5 11.8 3.0 5.2 7.4 11.2 81.5 98.1 3.5 17.8 100.0 100.0
W2 3 0.3 1.3 100.0 100.0 0.3 0.8 29.0 71.3 0.3 0.7 1.7 3.8 0.3 0.7 72.2 97.3 0.2 1.0 100.0 100.0
W3 3 0.3 1.7 100.0 100.0 0.4 0.8 33.0 74.1 0.3 0.8 2.2 4.7 0.4 0.9 76.2 97.9 0.2 1.3 100.0 100.0
W4 3 9.1 23.1 100.0 100.0 10.8 14.2 38.3 74.4 10.9 15.0 2.9 4.8 9.5 14.8 80.6 98.0 4.3 21.9 100.0 100.0
W1 20 17.8 20.2 99.7 100.0 0.3 0.4 13.9 56.9 0.0 0.1 1.5 5.2 0.6 0.9 25.4 90.6 13.3 15.7 99.2 100.0
W2 20 6.9 8.1 99.1 100.0 0.1 0.1 7.9 51.4 0 0 0.7 3.71 0.2 0.2 15.9 88.0 4.9 5.7 97.8 100.0
W3 20 8.7 10.5 99.3 100.0 0.1 0.1 9.7 54.8 0 0 0.9 4.7 0.3 0.3 19.1 89.6 6.4 7.5 98.3 100.0
W4 20 16.8 19.4 99.6 100.0 0.2 0.4 13.3 55.8 0 0.1 1.3 4.9 0.6 0.8 24.5 90.1 12.6 14.8 99.2 100.0
n = 500
W1 3 7.5 14.2 100.0 100.0 9.0 10.6 18.6 98.5 9.39 10.7 1.6 1.2 8.0 10.9 58.0 100.0 3.7 11.2 100.0 100.0
W2 3 0.3 0.7 98.1 100.0 0.2 0.5 8.6 98.2 0.3 0.3 0.7 1.0 0.3 0.5 35.4 100.0 0.2 0.5 98.4 100.0
W3 3 0.3 0.7 98.3 100.0 0.2 0.6 9.0 98.4 0.3 0.3 0.8 1.1 0.3 0.5 36.9 100.0 0.2 0.5 98.5 100.0
W4 3 8.5 15.8 100.0 100.0 10.1 11.9 18.4 98.5 10.5 12.1 1.6 1.1 9.0 12.2 57.7 100.0 4.2 12.4 100.0 100.0
W1 20 5.7 6.9 93.1 100.0 0.0 0.0 0.3 69.3 0 0 0 0.5 0.0 0.0 2.7 99.1 2.3 2.5 89.5 100.0
W2 20 1.4 1.8 80.3 100.0 0.0 0.0 0.1 64.8 0 0 0 0.4 0.0 0.0 0.8 98.8 0.5 0.6 75.0 100.0
W3 20 1.5 1.9 81.0 100.0 0.0 0.0 0.1 65.6 0 0 0 0.4 0.0 0.0 0.9 98.8 0.6 0.7 75.9 100.0
W4 20 5.5 6.8 93.0 100.0 0.0 0.0 0.3 69.0 0 0 0 0.5 0.0 0.0 2.6 99.1 2.3 2.4 89.4 100.0
6
4. Concluding remarks
In this paper, we question the validity of the use of standard DWH specification tests for
assessing partial exogeneity hypotheses. Our results show that these tests are invalid (level
is not controlled) if the endogeneity of the regressors whose exogeneity is not being tested
is ignored within inference. When the fitted values from the first stage regression of these
regressors are used as instruments under the subset null hypothesis of interest, we find
that some versions of the tests are invalid when identification is weak and the number of
instruments is relatively moderate. However, all tests are overly conservative and have no
power when the number of instruments increases, even for moderate identification strength.
So, applying them to assess partial exogeneity hypotheses, as it is often the case in applied
work, may be misleading. This underscores the importance to develop tests that do not
exhibit these problems when testing for partial exogeneity.
Acknowledgments
The author thanks Mardi Dungey (University of Tasmania), Jean-Marie Dufour (McGill
University) and Russell Davidson (McGill University) for helpful comments and suggestions.
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http://mpra.ub.uni-muenchen.de/40184/1/MPRA_paper_40184.pdf

On the validity of Durbin-Wu-Hausman tests for assessing partial exogeneity hypotheses with possibly weak

instruments

On the validity of Durbin-Wu-Hausman tests for assessing partial exogeneity hypotheses with possibly weak instruments

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