Ultra High Dimensional Variable Selection with Endogenous Variables

Transcription

1 1 / 39 Ultra High Dimensional Variable Selection with Endogenous Variables Yuan Liao Princeton University Joint work with Jianqing Fan Job Market Talk January, 2012

2 2 / 39 Outline 1 Examples of Ultra High Dimensional Econometric Model 2 Problem I: Ultra-high dim. covariates selection with endogeneity Inconsistency of Penalized OLS Penalized GMM and Penalized empirical likelihood 3 Problem II: Ultra-high dimensional instrumental selection Linear model: 2SLS with many weak instruments Nonlinear model: ultra-high dimensional sieve approximation 4 Implementation and Simulation

3 Examples of Ultra High Dimensional Econometric Model 3 / 39 Examples of Ultra High Dimensional Econometric Model

4 4 / 39 Examples of Ultra High Dimensional Econometric Model Cross-country Growth Regression Estimating the effect of an initial GDP per capita on the growth rates of GDP per capita. Solow-Swan-Ramsey model: poorer countries should grow faster, and catch up with richer countries. effect of initial GDP on growth rate should be negative Rejected using a simple bivariate regression ( Barro and Sala-i-Martin 1995) Conditional effects: For countries with similar characteristics, the effect of initial GDP on growth rate is negative.

5 Examples of Ultra High Dimensional Econometric Model 5 / 39 Cross-country Growth Regression y i = a 0 + a 1 log G i + x T i β + ɛ i y: growth rate; G: initial GDP x: country s char.: measures of edu, policies, trade openness, saving rates, investment rate, etc. H 0 : a 1 < 0. Barro and Lee (1994): p = 62, n = 90. Severe criticism of literature for relying on ad hoc covariate selection (Levine and Renelt 92) Development of a data-driven procedure for covariates selection is essential.

6 Examples of Ultra High Dimensional Econometric Model 6 / 39 Home price prediction Housing market based on state-level panel data can capture state-specific dynamics and variations If focus on local levels, including only macroeconomic variable cannot capture the cross-sectional correlation among local levels. y i,t+s = p y kt β ik + x T t θ i + ɛ i,t+s. k=1 x: macroeconomic variables p 1000; n < 200 for monthly sales data in ten years. Only a few county-level info. should be useful conditioning on national factors.

7 7 / 39 Examples of Ultra High Dimensional Econometric Model Home price prediction Fan, Lv and Qi (2011): monthly repeated sales of 352 largest counties in US from January 2000 to December 2009(n = 120) Testing periods: black: historical data blue: OLS with national house-price appreciation only red: penalized variable selection

8 Examples of Ultra High Dimensional Econometric Model 8 / 39 Labor Economics: Wage regression Effect of education on future income Response: log-wage y = E(y x) + ɛ. Nonparametric sieve approx. E(y x) = p i=1 β ip i (x) + r, P 1 (x),..., P p (x) are either polynomials or spline transformations. No guarantee that r is small using low-order polynomials. Possible oscillatory behavior associated with advanced degrees higher order

9 9 / 39 Examples of Ultra High Dimensional Econometric Model Labor Economics: Wage regression Belloni and Chernozhukov 11

10 Examples of Ultra High Dimensional Econometric Model 10 / 39 Instrumental Selection y = θ 0 + θ 1 z + w T γ + u 1 z = x T β + w T δ + u 2 y : wage; z : education. Angrist and Krueger: 180 IV s Two approaches in classical literature: 1 use 3 leading IV s: large variance 2 use 180 IV s: large bias 37 Lasso selected IV s. (Belloni and Chernozhukov 11)

11 Examples of Ultra High Dimensional Econometric Model 11 / 39 Model setting Consider y i = x T i β 0 + ɛ i, i = 1,..., n. dim(x) = p >> n. Allow p = exp(n α ), for some α (0, 1). Assume β 0 to be sparse. β 0 = (β 0S, 0) where dim(β 0S ) = s << n. Accordingly, x = (x S, x N ): important and unimportant

12 Examples of Ultra High Dimensional Econometric Model 12 / 39 Two Problems in This Talk

13 13 / 39 Examples of Ultra High Dimensional Econometric Model Problem I: Ultra-high dim. covariates selection y = x T β 0 + ɛ, β 0 = (β 0S, 0) x may contain many endogenous components. How to achieve oracle property? 1 ˆβ S β 0S = O p ( s/n log s). 2 P( ˆβ N = 0) 1. 3 ˆβ S has asymptotic normality. Solution: penalized GMM and penalized EL.

14 14 / 39 Examples of Ultra High Dimensional Econometric Model Problem II: Ultra high dim. instrumental selection dim(w) can be ultra high. y = x T S β 0S + ɛ x S = Θ 0 w + v dim(w) = O(exp(n α )), α (0, 1). Many instruments are weak. Solution: Penalized LS in 2SLS.

15 Problem I: Ultra-high dim. covariates selection with endogeneity 15 / 39 Problem I: Ultra-high dimensional covariates selection

16 16 / 39 Problem I: Ultra-high dim. covariates selection with endogeneity Penalized OLS Inconsistency of Penalized OLS Find ˆβ as: 1 ˆβ = arg min β n n (y i x i β)2 + i=1 P n : penalty function. Lasso: P n ( β j ) = λ n β j, where λ n 0. SCAD (Fan and Li 2001), etc. Key assumption: E(ɛ x S, x N ) = 0 p P n ( β j ). Unimportant predictors are artificially added; more desirable to assume only E(ɛ x S ) = 0 In many cases, important covariates are also endogenous. Instead, E(ɛ w) = 0. j=1

17 17 / 39 Problem I: Ultra-high dim. covariates selection with endogeneity A simulation example Inconsistency of Penalized OLS β 0 = (2.5, 4, 7, 1.5, 0,..., 0), p = 10, n = 100. (x 1,..., x 4 ) = (z 1,..., z 4 ), x j = (z j + 2)(ɛ + 1) where z N p (0, Σ), Σ ij = 0.5 i j. Penalized OLS +SCAD λ = 0.2 λ = 0.7 λ = 1.2 λ = 1.7 TP-Mean FP-Mean FP-Median (0.901) (0.799) (0.912) (0.83)

18 18 / 39 Problem I: Ultra-high dim. covariates selection with endogeneity Inconsistency of POLS Inconsistency of Penalized OLS Theorem 1 Suppose Ex l ɛ >> 0 for some x l. If β = ( β T S, β T N )T is POLS estimator, then either β S β 0S p 0, or lim sup P( β N = 0) < 1. n The inconsistency of POLS comes from the fact that, when x l is endogenous, E(y x T β 0 )x l = 0 is misspecified.

19 Problem I: Ultra-high dim. covariates selection with endogeneity Penalized GMM and Penalized empirical likelihood 19 / 39 Ultra-high dim. covariates selection with endogeneity Consider more general E[g(y, x T β 0 ) w] = 0, β 0 = (β 0S, 0). linear model: g = y x T β 0 logit model: g = y exp(x T β 0 )/(1 + exp(x T β 0 )) probit model: g = y Φ(x T β 0 ) Both important and unimportant covariates are possibly endogenous w: a set of valid instrumental variables.

20 Problem I: Ultra-high dim. covariates selection with endogeneity Penalized GMM and Penalized empirical likelihood 20 / 39 Penalized GMM Let v be p-dim. technical instruments. If dim(w) dim(x), v w. v = (f 1 (w),..., f p (w)). For fixed β R p, let v(β) contain only components {v l : β l 0} e.g., p = 3, β = (1, 0, 2), then v(β) = (v 1, v 3 ). Define L GMM (β) = [ 1 n n g(y i, x T i β)v i (β)] T W [ 1 n i=1 n g(y i, x T i β)v i (β)] i=1 p Q GMM (β) = L GMM (β) + P n ( β i ). i=1

21 Problem I: Ultra-high dim. covariates selection with endogeneity Penalized GMM and Penalized empirical likelihood 21 / 39 P n : penalty function. 1 P n is concave, increasing on [0, ), differentiable 2 P n(0 + ) > n 1/2 ; P n(t) = o(1) when t > c > 0. 3 max β0j 0 P n (β 0j ) = o(1). Examples 1 Lasso (Tibshirani 1986): P n(t) = λ n t 2 SCAD (Fan and Li 2001): P n(t) = λ n[λ n + λ n (aλ n t) + (a 1)λ n dt] 3 MCP (Zhang 2009): P n(t) = 1 (aλn t)+dt. a 4 Hard thresholding (Antoniadis 1996): a = 1.

22 Problem I: Ultra-high dim. covariates selection with endogeneity Penalized GMM and Penalized empirical likelihood 22 / 39 x 10-3 SCAD (Fan and Li 2001) Lasso (Tibshirani 1986)

23 23 / 39 Problem I: Ultra-high dim. covariates selection with endogeneity Oracle properties of PGMM Penalized GMM and Penalized empirical likelihood Theorem 1 Either E(g(y, x T β 0 ) x S ) = 0 or E(g(y, x T β 0 ) w) = 0 0 < c < λ min (Ex S v(β 0S ) T ) λ max (Ex S v(β 0S ) T ) < M. s 3 log s = o(n). Under regularity conditions, there exists a strictly local minimizer of Q GMM : 1 2 P( ˆβ N = 0) 1. s log s ˆβ S β 0S = O p ( + sp n n(min( β 0S ))). 3 Asymptotic normality of ˆβ S.

24 24 / 39 Problem I: Ultra-high dim. covariates selection with endogeneity Penalized empirical likelihood Penalized GMM and Penalized empirical likelihood Theorem 2 L EL (β) = 1 max λ R k β 0 n n log{1 + λ T [(y i x T i β)v i (β)]}. (2.1) i=1 Q EL (β) = L EL (β) + p P n ( β j ). (2.2) s 4 log s = o(n), there exists a strictly local minimizer of Q EL : 1 2 P( ˆβ N = 0) 1. j=1 s log s ˆβ S β 0S = O p ( + sp n n(min( β 0S ))). 3 Asymptotic normality of ˆβ S.

25 Problem II: Ultra-high dimensional instrumental selection Linear model: 2SLS with many weak instruments 25 / 39 Problem II: Ultra-high dimensional instrumental selection

26 26 / 39 Problem II: Ultra-high dimensional instrumental selection Linear model: 2SLS with many weak instruments Ultra high dim. instrumental selection Suppose oracle property is achieved, w.p.a.1, we identity: E[g(y, x T S β 0S) w] = 0. Optimal IV: A(w) = D(w) T Ω(w) 1, (Newey 01) D(w) = E( g(β 0S) β S w), Ω(w) = E(g(y, x T β 0S )g(y, x T β 0S ) T w). Ω(w): homoskedasticity. dim(w) can be ultra high. y = x T S β 0S + ɛ x S = Θ 0 w + v D(w) = Θ 0 w. But many instruments are weak. Including many weak IV s in 2SLS is severely biased.

27 27 / 39 Problem II: Ultra-high dimensional instrumental selection Linear model Linear model: 2SLS with many weak instruments Method based on MSE: (Donald&Newey 01, Kuersteiner&Okui 10) dim(w) n. requires natural ordering of IV s. In general, computationally infeasible: NP-hard. Proposed method: on the first stage, 1 ˆθ l = arg min θ n n (x Sl w T i θ) 2 + i=1 p P n ( θ j ). j=1 x = ˆΘv, ˆΘ = (ˆθ1,..., ˆθ s ) T. We allow dim(w) = o(exp( T )). s LASSO: ˆθ l θ 0l = O p ( 1 log s 1 n + s 1 λ n ), s SCAD: ˆθ l θ 0l = O p ( 1 log s 1 n ).

28 Problem II: Ultra-high dimensional instrumental selection Linear model: 2SLS with many weak instruments 28 / 39 Recent literature proposed methods based on l 1 penalty: (Belloni et al. 10, Garcia 11, Can&Fan 11) computationally efficient Lasso: choice of λ n is very restrictive. λ n large miss many important IVs. λ n small include too many weak IVs, complicated model Adaptive lasso: P n ( β j ) = β j 1 λ n β j. requires initial estimator, which is hard to obtain when w is ultra high dimensional. iterative algorithm may permanently remove important IVs. Proposed method allows more adaptive penalties.

29 Problem II: Ultra-high dimensional instrumental selection Linear model: 2SLS with many weak instruments 29 / 39 Nonlinear model Optimal IV: D(w) = E( g(β 0S) w) β S Estimate based on sieve approx. (Newey 01) p 1 D(w) = θ i f i (w) + r, p 1 n. i=1 No guarantee r is small if p 1 is small. Goal: allow for higher order polynomials

30 30 / 39 Problem II: Ultra-high dimensional instrumental selection Ultra-high dim. sieve approximation Nonlinear model: ultra-high dimensional sieve approximation Assumption: 1 There is a large set of technical IV s v = (f 1 (w),..., f p1 (w)) T (possibly p n): D(w) = Θ 0 v + a(w), max l s ( 1 n n a l (w i ) 2 ) = O p (cn) 2 i=1 2 max l s i / T l θ 0l,i < n α 1, minl s,i Tl θ 0l,i = h n > n α 2 max l s #{i : i T l } = s 1 = o(n). Penalized estimator: 1 ˆθ l = arg min θ n n ( g(y i, ˆx T i ˆβ S ) v T i θ) 2 + β Sl i=1 ˆD(w) = ˆΘv, ˆΘ = (ˆθ 1,..., ˆθ s ) T. p P n ( θ j ). j=1

31 Problem II: Ultra-high dimensional instrumental selection Nonlinear model: ultra-high dimensional sieve approximation 31 / 39 Theorem 2 There exists a strictly local minimizer ˆθ l = (ˆθ ls, ˆθ ls ), s.t. s log s s1 log s 1 ˆθ ls θ 0l,S = O p ( + + s 1 n α 1 + s 1 c n n n 1 n n i=1 + s 1 P n(h n )). lim P(ˆθ ln = 0) = 1. n ˆD l (W i ) D l (W i ) 2 = O p ( s 1s log s 1 n +s 2 1 n 2α 1 + s 2 1 c2 n + s 2 1 P n(h n ) 2 ). + s2 1 log s 1 n

32 Implementation and Simulation 32 / 39 Implementation

33 33 / 39 Smoothing Implementation and Simulation L GMM is not continuous. = p j=1 L GMM (β) = w j [ 1 n n i=1 [ 1 n n i=1 (y i x T i β)x i (β)] T W (y i x T i β)x ij I(β j 0) ] T [ 1 n n i=1 [ 1 n n i=1 (y i x T i β)x i (β) (y i x T i β)x ij I(β j 0) ] ] L EL (β) = max λ 1 n n i=1 1 = max λ j R k,j=1,...,p n log(1 + λ T (y i x T i β)x i (β) n log(1 + i=1 p j=1 λ T j (y i x T i β)x ij I(β j 0)),

34 Implementation and Simulation Replace I(β j 0) with K (β 2 j /σ n ), σ n 0 K (0) = 0, K (+ ) = 1, lim t K (t)t = 0, lim t K (t)t <. K (.) < M. Kernel K is similar to a cdf, as in smoothed maximum score. Horowitz (1992) Example: K (t) = 0.5(Φ(t) 0.5). Theorem 3 Under regularity conditions of P n, K n, and Theorems 1-4, smoothed PGMM and PEL achieve oracle properties. 34 / 39

35 35 / 39 Simulation Implementation and Simulation E(ɛ x S ) = 0, without knowing x S. y = x T β 0 + ɛ β 0 = (2.5, 4, 7, 1.5, 0,..., 0), ɛ N(0, 1). z N p (0, Σ), Σ ij = 0.5 i j. (x 1,..., x 4 ) = (z 1,..., z 4 ), x j = (z j + 2)(ɛ + 1)

36 Implementation and Simulation 36 / 39 Table: POLS and PGMM when p = 50, n = 200 POLS PGMM λ = 0.05 λ = 0.1 λ = 0.5 λ = 1 λ = 0.05 λ = 0.1 λ = 0.2 λ = 0.4 MSE S (0.053) (0.043) (0.301) (0.329) (0.094) (0.069) (0.076) (0.245) MSE N (0.035) (0.016) (0.016) (0.019) (0.010) (0) (0.009) (0.014) TP-Mean Median (0) (0) (0.385) (0.504) (0) (0) (0) (0.503) FP-Mean Median (2.902) (3.045) (3.334) (1.557) (0.337) (0) (0.141) (0.569)

37 37 / 39 Implementation and Simulation Sensitivity to minimal signal β 4 = 1.5 β 4 = 0.5 Table: Penalized GMM when p = 20, β 4 = 0.5 λ MSE S (0.090) (0.117) (0.102) (0.117) (0.083) (0.158) TP-Mean Median (0.197) (0.273) (0.239) (0.288) (0.197) (0.435) FP-Mean Median (1.545) (2.447) (2.139) (0.141) (0) (0)

38 Implementation and Simulation 38 / 39 Conclusion Many applications in economics contains ultra. high dim. regressors Careful about POLS for variable selection PGMM/ PEL allow endogeneity in ultra high dim. estimation and selection. Allow ultra high dim. instruments for 2SLS Allow ultra high dim. sieve approx. for optimal IV.

39 39 / 39 Some references Implementation and Simulation HANSEN, L. (1982). Large sample properties of generalized method of moments estimators. Econometrica, OWEN, A. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75, ANDREWS, D. (1999). Consistent moment selection procedures for generalized method of moments estimation. Econometrica, CANER, M. (2009). Lasso-type GMM estimator. Econometric Theory, BELLONI, A. and CHERNOZHUKOV, V. (2011). High-Dimensional Sparse Econometric Models, an Introduction. In Ill-Posed Inverse Problems and High-Dimensional Estimation, Springer Lecture Notes in Statistics. SALIA-I-MARTIN, X. (1997). I Just Ran Two Million Regressions. The American Economic Review 87,