The Case Against JIVE - Rutgers...

The Case Against JIVE when Instruments are Weak

The Case Against JIVERelated literature, Two comments and One reply

PhD. student Freddy Rojas CamaEconometrics Theory II

Rutgers University

November 14th, 2011


Literature

1 Literature2 Key denitions3 The case against JIVE4 The bias of 2SLS5 Estimators

UJIVE 1UJIVE 2JLS or JIVELIML

6 Experiments7 Results8 Conclusions9 Two comments10 ... and One reply11 Estimation by using STATA12 References13 Appendix


Literature

Instrumental variables performance with weak instruments

Davidson and McKinnon (2004). The case Against JIVE.Journal of Applied Econometrics 21: 827-833.

Blomquist and Dahlberg (1999). Small sample properties ofLIML and Jaccknife IV estimators: Experiments with WeakInstruments. Journal of Applied Econometrics 14: 69-88.

Angrist, J., W. Imbens and A. Krueger (1999). JaccknifeInstrumental Variables Estimation. Journal of AppliedEconometrics 14: 57-67.


Key denitions

Key denitions

Jackknife Instrumental Variables Estimation (JIVE).

Unbiased Jackknife Instrumental Variables Estimation(UJIVE).

Limited Information maximum likelihood (LIML)

Two Stage Linear Squares (2SLS)

Finite-small sample properties

Monte Carlo simulations.


The case against JIVE

JIVE performs very badly when the instruments are weak.

Davidson and McKinnon (2004) [DM] perform montecarloexperiments to compare the performance of "jackknifeinstrumental variables estimator" with the 2SLS and LIMLestimators. They nd NO evidence for using JIVE instead ofLIML; LIML has better nite small sample properties. Interms of size, 2SLS is less dispersed than JIVE.

The results of DMs paper does not support ndings inBlomquist and Dahlberg (1999) [BD] and Angrist, J., W.Imbens and A. Krueger (1999) [AIK].


The case against JIVE

Summarizing

Finite-sample properties Evaluation(According to each study)

UJIVE LIML Comment

DM X LIML the best in reducing biasBD ? ? Hard to nd a winner!AIK X A reduced space of parameters


The bias of 2SLS

The System of Equations

We have the following system of equations (in matricial terms);

Y = X β+ ε (1)

X = Zπ + η (2)

where X , Z and η are matrices of dimension n L, n k andn L respectively. The number of overidentied restricctions canbe calculated as r = k L. Also, there are M common elements inX and Z , then M columns of n L matrix η are zero. Theendogeneity comes from the following expression

E

εiη0i

Z = σεη (3)


The bias of 2SLS

The bias of OLS

We have the following assumptions E [η0i jZ ] = 0 andE [ηiη

0i jZ ] = Ση. The rank of Ση is equal to LM.

The inconsistency feature of OLS estimator

βOLS =X 0X

1 X 0Y is shown in the following steps

βOLS =(Zπ + η)0 (Zπ + η)

1(Zπ + η)0 Y

=

η0 + π0Z 0(Zπ + η)

1(Zπ + η)0 Y


The bias of 2SLS

The bias of OLS

βOLS =

η0 + π0Z 0(Zπ + η)

1 η0 + π0Z 0

Y

=

η0 + π0Z 0(Zπ + η)

1 η0 + π0Z 0

(X β+ ε)

=

η0 + π0Z 0(Zπ + η)

1 η0 + π0Z 0

((Zπ + η) β+ ε)

Then applying law of iterative expectations

EZ[E [βOLS ]jZ ] = E

((η0 + π0Z 0) (Zπ + η))1

(η0 + π0Z 0) ((Zπ + η) β+ ε)

Z


The bias of 2SLS

The bias of OLS

EZ[E [βOLS ]jZ ] = E

β+ ((η0 + π0Z 0) (Zπ + η))1

(η0 + π0Z 0) ε

Z= E

hβ+

η0 + π0Z 0

(Zπ + η)

1η0εZi

= β+ E

"(η0 + π0Z 0) (Zπ + η)

N

1 η0ε

N

Z#

In terms of consistency

limN!∞

βOLS = β+π0ΣZπ + Ση

1σεη


The bias of 2SLS

The bias of OLS

limN!∞

βOLS = β+π0ΣZπ + Ση

1σεη

Where ΣZ = limN!∞

Z 0ZN , Ση = lim

N!∞

η0ηN and η0ε

Np! σεη.


The bias of 2SLS

The bias of 2SLS

The optimal instrumental variable estimation must fulll thefollowing condition

E(Y X βIV )

0 Θ (Z )= 0

an analytical solution is available since

Y 0Θ (Z ) = β0IVX0Θ (Z )

Y 0Θ (Z )X 0Θ (Z )

1= β0IV

orβIV =

Θ (Z )0 X

1Θ (Z )0 Y

Woodridge states that Θ (Z ) = Zπ is the optimal instrument interms of e¢ ciency (instead of only Z ).


The bias of 2SLS

k-class estimator

Note that (Z bπ)0 = X 0PZ = X 0Z (Z 0Z )1Z 0. PZ is theorthogonal projection on to the span of the columns of Z .

Following to Davidson and Mackinnon (2007) we must takenotice that we can nd a matrix A with the property thatAZ = Z will lead us to have another estimator.

Particularly, if we set up the choice A = I κ (I PZ ), wehave a κclass estimator. Thus, quite generally we considerthe estimator β by using Θ (Z ) Θ (Z , κ) .


The bias of 2SLS

The bias of 2SLS

because π is unknown (unfeasible estimation) we approach theestimator with a feasible version of the optimal IV estimation

bβ2SLSIV = (Z bπ)0 X 1 (Z bπ)0 Y (4)

where bπ = (Z 0Z )1 Z 0X .It is value to take notice that expression(4) is not properly the estimator for 2SLS; [AIK] states thatβ2SLSIV has much better small sample properties than β2SLS inthe presence of many instruments (Nagar; 1959).


The bias of 2SLS

The bias of 2SLS

We show the bias of 2SLS estimator in the following lines

E [εiZi bπ] = EZ[E [εiZi bπ]jZ ]

= EZ

hEhεiZi

Z 0Z

1 Z 0X iZi= E

Z

hZiZ 0Z

1 Z 0E [εiX ]Zi= E

Z

hZiZ 0Z

1 Z 0E [εi (Zπ + η)]Zi

= EZ

hZiZ 0Z

1 Z 0E [εiZπ + εiη]Zi

= EZ

hZiZ 0Z

1 Z 0E [εiη]Zi= E

Z

hZiZ 0Z

1 Z 0Ξεi η

Zi


The bias of 2SLS

The bias of 2SLS

where Ξ0εi η is a row vector with zeros and just one element σεη 6= 0in the i-position. Equivalently,

E [εiZi bπ] = EZ

hZiZ 0Z

1 Z 0Ξεi η

Zi= E

Z

hZiZ 0Z

1 Z 0i E [εiηi ]Zi= E

Z

hZiZ 0Z

1 Z 0i Zi σ0εη

=KN

σεη

Thus, for a xed σεη we have that for small samples E [εiZi bπ]increases with the number of instruments. The result is that wehave a bias in the 2SLS estimator.


Estimators

UJIVE 1

UJIVE 1

JIVE removes the dependence of of the constructed instrumentZi bπ on the endogenous regressor for observation i by using thefollowing estimator

eπ (i) = Z (i)0 Z (i)1 Z (i)0 X (i) (5)

the estimate of the optimal instrument is Zi eπ (i) ; then, because εiis independent of Xj if j 6= i we claim that

E [εiZi eπ (i)] = 0this is easily veriable

EEεiZ 0i eπ (i)Z E

hZiZ (i)0 Z (i)

1Z (i)0 E [εiX (i)]

Zi= 0

See Phillips and Hale (1977) for details.


Estimators

UJIVE 1

UJIVE 1

Thus, bXi ,UJIVE = Zi eπ (i) ; then the estimator of β is

bβUJIVE = bX0UJIVEX1 bX0UJIVEY (UJIVE 1)

We require to perform the estimator in (5) by each observation i .[AIK] show a sort of shortcut

Zi eπ (i) = Zi bπ hiXi1 hi

where hi = Zi (Z 0Z )1 Z 0i


Estimators

UJIVE 2

UJIVE 2

An alternative estimator is

Zi eeπ (i) = Zi bπ hiXi1 1

N

the resulting estimator for β is

bβUJIVE 2 = bX0UJIVE 2X1 bX0UJIVE 2Y (UJIVE 2)

where bXi ,UJIVE 2 = Zi eeπ (i) . The projection between UJIVE1 andUJIVE2 are closely similar and both are consistent. The probabilityof the estimator of β and their rst-order asymptotic distributionare therefore the same as those of βIV and β2SLS . DM states thatthese estimators may di¤er noticeably in any particular sample.


Estimators

JLS or JIVE

JIVE

DM consider that an alternative estimator is

bβJIVE = bX0JIVE bXJIVE1 bX0JIVEY (JIVE)

where bXi ,JIVE = Ziπ (i). DM states that JIVE is biased in thedirection of zero, just as OLS estimator when the explanatoryvariable is measured with error. But, This class of estimator isconsistent.


Estimators

LIML

Limited Information Maximum Likelihood

We can estimate the parameters in linear regressions withendogenous regressors by using thelimited-information-maximum-likelihood estimator. The likelihoodis based on normality for the reduced form errors and withcovariance matrix, although consistency and asymptotic normalityof the estimator do not rely on this assumption. The log likelihoodfunction is

L =N

∑i=1 ln (2π) 1

2ln jΩj (LIML)

12

Yi (Ziπ)0 βXi Ziπ

0Ω1

Yi (Ziπ)0 βXi Ziπ

(6)


Experiments

Experiments

DM have the following system of equations (in matricial terms) inorder to do the simualtions;

Y = ιβ1 + xβ2 + ε (Structural eq.)

x = ση (Zπ + η) (Reduced eq.)

where X = [ι x ], Z and η are matrices of dimension n 2, n land n 1 respectively. The number of overidentied restricctionscan be calculated as r = l 2. The elements of ε and η havevariances σ2ε and 1 respectively, and correlation ρ.In order to start with the simulations we need to impose values forparameters. For this, an important guide is the size of the ratiokπk2 to σ2η.


Experiments

The concentration parameter

The ratio kπk2 to σ2η is interpreted as the signal-to-noise ratio inthe reduced-form equation.


Experiments

Setting up parameters

DM x values of the πj to be equal excepting π1 = 0.

The parameter which does varies is denoted by R2∞ =kπk2

kπk2+σ2η

(This is the asymptotic R2)

R2∞ is monotonically increasing function of the the ratio kπk2to σ2η.

A small value of R2∞ implies that the instruments are weak.

In the experiments, DM vary the sample size n, the number ofoveridentifying restrictions (r), the correlation between errors(ρ) and R2∞


Experiments

Parameters set up

500 000 Montecarlo replications.

When R2∞ = 0, β2 is not asymptotically identied.

Absolute value of ρ matters; the sign just a¤ects the directionof bias.

r varies from 0 to 16.

Sample size: 25, 50, 100, 200, 400 and 800.


Experiments

Performance evaluation

Because LIML and JIVE estimators have no moments (seedavison and Mackinnon; 2007). DM reports the median bias,this is

median bias = βd0.5 β

As measure of dispersion DM reports the nine decile range,this is

9d range = βd0.95 βd0.05

Ackberg and Devereux (2006) suggest to consider

Trimmed mean bias =1n ∑

jβj β

where βj 2 [βd0.99 βd0.01] .


Results

Results (I): Median bias evaluation


Results

Results (II): Median bias evaluation


Results

Results (III): Median bias evaluation


Results

Results (IV): Rejection frequencies


Results

Results (V): Dispersion


Conclusions

Conclusions

1 AIK and BD give divergent views of JIVE performance. DMspaper re-examinate this issue.

2 DM conclude that in most regions of the parameters spacethey have studied, JIVE is inferior to LIML regarding tomedian bias, dispersion and reliability of inference.

3 LIML should be preferred whenever you need to deal withestimators which have no moments.

4 DM points out that, however, montecarlo simulations does nosupport unambiguoulsy the usage of LIML; when theinstruments are weak the dispersion is signicant in thisestimator.

5 Chao and Swanson (2004) state that LIML is not a consistentestimator in a context of heteroskedasticity; but UJIVE is(under some conditions).


Two comments

Ackerberg and Devereuxs comments

They disagree with DM conclusion: The LIML estimatorshould almost always be prefered to the JIVE estimators.

Ackerberg and Devereux (2003) show that advantages overLIML are signicantly reduced when using the "improved"JIVE (IJIVE). They extend the unbiased estimator (UIJIVE).What matters is the results in the space of weak instruments;the resulting advantages of LIML in terms of median bias anddispersion are small in that space. But not much di¤erent fromIJIVE estimators.There are important advantages of JUVE estimators over theLIML estimator; particualrly regarding the robustness of JIVEestimators to heteroskedasticity.(see Chao and Swanson; 2004).


Two comments


About the performance of IJIVE estimators.

Particularly, AD shows that in the region where R2 > 0.2 theIJIVE estimator removes almost all the median bias of theJIVE estimator. But, LIML is better in terms of bias in thespace of weak instruments.

IJIVE improves on the 9-decile range of JIVE by about 20%;although it is considerably larger that that of LIML.

If they consider the trimmed mean statistics, UIJIVE bias.issmaller than that of LIML.


Two comments


They conclude the following about advantages of LIML

By using IJIVE or UIJIVE one can either closely match themedian bias of LIML or closely match the dispersion andbetter the trimmed mean bias of LIML.

LIML does better in terms of median bias for values of R2

below 0.2. But, the emprical work is not possible in thatsituation; mainly because

When R2 is below 0.2 the average rst stage F-statistic isbelow 3 and the corresponding p-value is above 0.10; onetipically would not be able to reject the hypothesis that theinstruments are irrelevant.The variance of all estimators is likely to be large.


Two comments


LIML is not the panacea

IJIVE and UIJIVE appear to be very robust estimators. Inparticular, these estimators are consistent in the case ofheteroskedasticity.

Chao and Swanson (2004) states that LIML is not aconsistent estimator in this context of heteroskedasticity.

Chao and Swanson (2004) provide and alternative proof thatJIVE estimators are consistent under heteroskedasticity

Could "a sort of" LIML estimator become consistent underthis context?. Such estimator would probably not have aclosed solution and might be less robust to otherperturbations (non-normality) than standard LIML estimator.


Two comments

Blomquist and Dahlbergs comments

they do nd dicult to understand categorical rejection ofJIVE from DM;

JIVE or LIML do not dominate the 2SLS in terms of RSME.In terms of bias, DMs results shows a good perfomance ofLIML.In terms of variance 2SLS outperforms both LIML and JIVEestimators. LIML outperforms JIVE for some values of theparameter space, in others spaces we nd the oppositeconclusion.

The complexity of the DGP matter for a conclusion. AIK, BDand DM should be more nuanced.


... and One reply

Davidson and Mackinnons reply

DM agree that there is some space of parameters whereUJIVE outperforms LIML.

They do simulations and evaluate the perform of UIJIVEsAckerberg and Devereux (2006); interestingly, UIJIVE tendsto be subtantially less dispersed than other JIVE estimatorsand just when the instruments are weak the this IJIVE is lessdispersed than LIML.

They propose to look at

n1/2bβ2 β2

= t0 + n1/2t1 + op

n1/2

to show the approximate biases for these estimators.


... and One reply

Davidson and Mackinnons reply

They states out that altought UIJIVE si constructed to reducemean bias; it does not do nearly so well as regards medianbias. That implies that estimator is highly skewed.

DM proposes to look more closely at modied LIML estimatorproposed by Fuller (1977) for which the moment exist andthen see how it performs relative to UIJIVE.


Estimation by using STATA

STATA command

In order to perform the UJIVE estimator there is an availableSTATA command for that; the syntaxis is as follows

jive depvar [varlist1 ] (varlist2=varlistiv) [if] [in] [, options]

The options are UJIVE1, UJIVE2 (see AIK), JIVE1 JIVE 2 (seeBD) and calculation of a robust matrix for dealing withheteroskedasticity. The command jive saves information in e(),the most interesting matrices are: e(b) -the coe¢ cient vector,e(V) -the variance and covariance matrix, e(F1) - rst stage Fstat and e(r2_1) -rst stage R2.


References

References

Angrist, J., W. Imbens and A. Krueger (1999). JaccknifeInstrumental Variables Estimation. Journal of AppliedEconometrics 14: 57-67.

Blomquist and Dahlberg (1999). Small sample properties ofLIML and Jaccknife IV estimators: Experiments with WeakInstruments. Journal of Applied Econometrics 14: 69-88.

Davidson, R., and J. McKinnon (2004). The case AgainstJIVE. Journal of Applied Econometrics 21: 827-833.

Davidson, R., and J. McKinnon (2006). Reply to Ackerbergand Devereux and Blomquist and Dhalberg on "The caseAgainst JIVE". Journal of Applied Econometrics 21: 843-844.


References

References

Blomquist and Dahlberg (2006). The case Against JIVE: Acomment. Journal of Applied Econometrics 21: 839-841

Ackerberg and Deverwux (2006). .Comment on "The caseAgainst JIVE". Journal of Applied Econometrics 21: 835-838.

Davidson, R., and J. McKinnon (2007). Moments of IV andJIVE estimators. The Econometrics Journal Vol 10 Number 3.

Stock, J., J. Wright and M. Yogo (2002). A Survey of WeakInstruments and Weak Identication in Generalized Method ofMoments. Journal of Business and Economic Statistics, Vol 20Number 4.

Hausman, J., W. Newey, T. Woutersen, J. Chao and N.Swanson (2006). Instrumental variable Estimation withHeteroskedasticity and Many Instruments. Working paper.


References

References

Chao, J. and N. Swanson. (2004). Estimation and TestingUsing Jackknife IV in heteroskedasticity Regressions Withmany Weak Instruments. Working paper, Rutgers University.

Nagar, A. (1959). The Bias and Moment matrix of the generalk-class estimators of the parameters in simultaneous equation.Econometrics, 27, 575-595.

Phillips, G. D. A., and C. Hale. (1977). The bias ofinstrumental variable estimators of simultaneous equationsystems. International Economic Review 18: 219228.

Fuller, W.A. (1977) Some properties of a modication of theLimited Information estimator. Econometrica 45, 939-953.

References

Chao, J. and N. Swanson. (2004). Estimation and TestingUsing Jackknife IV in heteroskedasticity Regressions Withmany Weak Instruments. Working paper, Rutgers University.

Nagar, A. (1959). The Bias and Moment matrix of the generalk-class estimators of the parameters in simultaneous equation.Econometrics, 27, 575-595.

Phillips, G. D. A., and C. Hale. (1977). The bias ofinstrumental variable estimators of simultaneous equationsystems. International Economic Review 18: 219228.

Fuller, W.A. (1977) Some properties of a modication of theLimited Information estimator. Econometrica 45, 939-953.

2011-11-13


References

Notes

Notes


Appendix

AIK results

Median Absolute ErrorN=100, L=2, M=5000

DGP Estimators2SLS LIML UJIVE1 UJIVE2 OLS

1 0.11 0.12 0.13 0.13 0.592 0.28 0.13 0.17 0.17 0.593 0.16 0.25 0.32 0.15 0.174 0.80 1.01 0.88 0.88 0.805 0.28 0.41 0.20 0.20 0.59


Appendix

AIK results

Coverage rate 95% conf. IntervalN=100, L=2, M=5000


1 0.91 0.96 0.96 0.96 0.002 0.31 0.94 0.94 0.94 0.003 0.57 0.97 0.97 0.95 0.034 0.00 0.71 0.71 0.71 0.005 0.38 0.93 0.93 0.94 0.00


AppendixAIK results

Coverage rate 95% conf. IntervalN=100, L=2, M=5000


1 0.91 0.96 0.96 0.96 0.002 0.31 0.94 0.94 0.94 0.003 0.57 0.97 0.97 0.95 0.034 0.00 0.71 0.71 0.71 0.005 0.38 0.93 0.93 0.94 0.0020

11-11-13


Appendix

Notes

NotesThe DGP processes for each model are as follows1. There is a single overidentifying restriction K=3, L=2 and ρ = 0.252. A large number of instruments relative to the number of regressors;K=21, L=2 and ρ = 0.253. The model is non-linear and heteroscedatic; K=21, L=2 and ρ = 1.04. This model sets the true reduced-form coe¢ cients to zero for allinstruments in an attempt to ascertain how misleading the estimatorsmight be in this non-identied case; K=21, L=2 and ρ = 0.255. AIK introduces a misspecication; one of the instruments hasincorrectly been left out of the main regression; K=21, L=2 and ρ = 0.25


Appendix

BD results

Results of Simulationsρ = 0.2, M=1000 replications

Stats Estimators2SLS LIML UJIVE USSIV OLS

MEAN BIAS54.26.8

-4.3-0.5

-12.8-1.3

-6.5-0.4

241243.9

MEDIAN BIAS0.4140.135

0.4650.136

0.4840.140

0.6740.179

1.1991.223

RMSE0.6040.196

0.6730.198

0.7030.199

0.9870.275

1.231.22

SIZE0.0890.053

0.0620.053

0.0360.046

0.0490.045

0.9991.000

BD results



MEAN BIAS54.26.8

-4.3-0.5

-12.8-1.3

-6.5-0.4

241243.9


0.4650.136

0.4840.140

0.6740.179

1.1991.223

RMSE0.6040.196

0.6730.198

0.7030.199

0.9870.275

1.231.22

SIZE0.0890.053

0.0620.053

0.0360.046

0.0490.045

0.9991.000

2011-11-13


Appendix

Notes

Notes1. The percent bias is dened as (δ-δ)/δ.

2. The rst and second row in each statistic refers to sample size equalto 500 and 4000 respectively.


Appendix

BD results



MEAN BIAS182.525.1

-12.8-1.2

-50-4.2

-25.9-1.6

587.4590


0.4670.137

0.5700.137

0.7350.184

2.3962.951

RMSE1.020.226

0.6940.199

0.9290.205

1.2100.281

2.942.95

SIZE0.5120.108

0.0680.052

0.0540.043

0.0440.037

1.001.00

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