The Case Against JIVE

Transcription

1 The Case Against JIVE Davidson and McKinnon (2004) JAE 21: PhD. student Freddy Rojas Cama Econometrics Theory II Rutgers University November 14th, 2011

2 Literature 1 Literature 2 The case against JIVE 3 The model 4 Estimators UJIVE 1 LIML 5 Experiments 6 Results 7 Conclusions 8 References

3 Literature Instrumental variables performance with weak instruments Davidson and McKinnon (2004). The case Against JIVE. Journal of Applied Econometrics 21: Blomquist and Dahlberg (1999). Small sample properties of LIML and Jaccknife IV estimators: Experiments with Weak Instruments. Journal of Applied Econometrics 14: Angrist, J., W. Imbens and A. Krueger (1999). Jaccknife Instrumental Variables Estimation. Journal of Applied Econometrics 14:

4 The case against JIVE Summarizing Finite-sample properties Evaluation (According to each study) UJIVE LIML Comment DM X LIML the best in reducing bias BD?? Hard to nd a winner! AIK X A reduced space of parameters

5 The model The System of Equations We have the following system of equations (in matricial terms); Y = X β + ε (1) X = Z π + η (2) where X, Z and η are matrices of dimension n L, n k and n L respectively. The number of overidenti ed restricctions can be calculated as r = k L. Also, there are M common elements in X and Z, then M columns of n L matrix η are zero. The endogeneity comes from the following expression E ε i η 0 i Z = σ εη (3)

6 Estimators UJIVE 1 UJIVE 1 JIVE removes the dependence of of the constructed instrument Z i bπ on the endogenous regressor for observation i by using the following estimator eπ (i) = Z (i) 0 Z (i) 1 Z (i) 0 X (i) (4) the estimate of the optimal instrument is Z i eπ (i) ; then, because ε i is independent of X j if j 6= i we claim that E [ε i Z i eπ (i)] = 0 this is easily veri able E E ε i Z 0 i eπ (i) Z E = 0 h Z i Z (i) 0 Z (i) i 1 Z (i) 0 E [εi X (i)] Z See Phillips and Hale (1977) for details.

7 Estimators UJIVE 1 UJIVE 1 Thus, bx i,ujive = Z i eπ (i) ; then the estimator of β is bβ UJIVE = 1 X b UJIVE 0 X X b UJIVE 0 Y (UJIVE 1) We require to perform the estimator in (4) by each observation i. [AIK] show a sort of shortcut where h i = Z i (Z 0 Z ) 1 Z 0 i Z i eπ (i) = Z i bπ h i X i 1 h i

8 Estimators LIML Limited Information Maximum Likelihood We can estimate the parameters in linear regressions with endogenous regressors by using the limited-information-maximum-likelihood estimator. The likelihood is based on normality for the reduced form errors and with covariance matrix, although consistency and asymptotic normality of the estimator do not rely on this assumption. The log likelihood function is L = N i= ln (2π) ln jωj (LIML) 2 (Zi π) 0 0 β Ω 1 Yi (Zi π) 0 β (5) Z i π Z i π Yi X i X i

9 Experiments Experiments DM have the following system of equations (in matricial terms) in order to do the simualtions; Y = ιβ 1 + x β 2 + ε (Structural eq.) x = σ η (Z π + η) (Reduced eq.) where X = [ι x], Z and η are matrices of dimension n 2, n l and n 1 respectively. The number of overidenti ed restricctions can be calculated as r = l 2. The elements of ε and η have variances σ 2 ε and 1 respectively, and correlation ρ. In order to start with the simulations we need to impose values for parameters. For this, an important guide is the size of the ratio kπk 2 to σ 2 η.

10 Experiments Setting up parameters DM x values of the π j to be equal excepting π 1 = 0. The parameter which does varies is denoted by R 2 = kπk2 kπk 2 +σ 2 η (This is the asymptotic R 2 ) R 2 is monotonically increasing function of the the ratio kπk 2 to σ 2 η. A small value of R 2 implies that the instruments are weak. In the experiments, DM vary the sample size n, the number of overidentifying restrictions (r), the correlation between errors (ρ) and R 2

11 Experiments Performance evaluation Because LIML and JIVE estimators have no moments (see davison and Mackinnon; 2007). DM reports the median bias, this is median bias = β d 0.5 β As measure of dispersion DM reports the nine decile range, this is 9d range = β d 0.95 β d 0.05 Ackberg and Devereux (2006) suggest to consider Trimmed mean bias = 1 n j β j β where β j 2 [β d 0.99 β d 0.01 ].

12 Results Results (I): Median bias evaluation

13 Results Results (II): Median bias evaluation

14 Results Results (III): Median bias evaluation

15 Results Results (V): Dispersion

16 Conclusions Conclusions 1 AIK and BD give divergent views of JIVE performance. DM s paper re-examinate this issue. 2 DM conclude that in most regions of the parameters space they have studied, JIVE is inferior to LIML regarding to median bias, dispersion and reliability of inference. 3 LIML should be preferred whenever you need to deal with estimators which have no moments. 4 DM points out that, however, montecarlo simulations does no support unambiguoulsy the usage of LIML; when the instruments are weak the dispersion is signi cant in this estimator.

17 References References Angrist, J., W. Imbens and A. Krueger (1999). Jaccknife Instrumental Variables Estimation. Journal of Applied Econometrics 14: Blomquist and Dahlberg (1999). Small sample properties of LIML and Jaccknife IV estimators: Experiments with Weak Instruments. Journal of Applied Econometrics 14: Davidson, R., and J. McKinnon (2004). The case Against JIVE. Journal of Applied Econometrics 21: Davidson, R., and J. McKinnon (2006). Reply to Ackerberg and Devereux and Blomquist and Dhalberg on "The case Against JIVE". Journal of Applied Econometrics 21: