Limited Information Econometrics

Transcription

1 Limited Information Econometrics Walras-Bowley Lecture NASM 2013 at USC Andrew Chesher CeMMAP & UCL June 14th 2013 AC (CeMMAP & UCL) LIE 6/14/13 1 / 32

2 Limited information econometrics Limited information econometric models leave the determination of some endogenous variables unspeci ed. Single equation or systems? Computation, e ciency, robustness - identi cation? AC (CeMMAP & UCL) LIE 6/14/13 2 / 32

3 Limited information econometrics Incomplete models may not be point identifying when there are: discrete outcomes (binary, ordered, counts, multiple discrete choice) rich speci cations of heterogeneity (random coe cients, individual e ects, frailties in duration analysis). The common feature is that unobservables are not single valued functions of observed variables. When unobservables are set valued functions of observable variables incomplete models are generically partially identifying. AC (CeMMAP & UCL) LIE 6/14/13 3 / 32

4 Limited information econometrics Reactions? ignore endogeneity, work up enthusiasm for a point identi ed parameter, beef up the restrictions and use point identifying models. AC (CeMMAP & UCL) LIE 6/14/13 4 / 32

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6 Limited information econometrics Use the model which embodies your beliefs about the economic process being studied. If that model is partially identifying then so be it. Keep Calm and Carry On. Barriers to the use of partially identifying models? Methods for conducting inference? Recent advances. Developing usable characterizations of sharp identi ed sets? We provide a simple characterization of the identifying power of a wide class of limited information models. I explain, and illustrate, drawing on research done with my colleagues Konrad Smolinski and Adam Rosen. AC (CeMMAP & UCL) LIE 6/14/13 6 / 32

7 Example: binary outcome IV model This linear model for endogenous Y = (Y 1, Y 2 ), exogenous Z, unobserved U 1 : Y 1 = α 0 + α 1 Y 2 + U 1 U 1? Z can point identify α 1. But this binary model set identi es α 1 Y 1 = 1[α 0 + α 1 Y 2 + U 1 > 0] U 1? Z even in a parametric probit case. This IV probit model is studied in Chesher (2010, 2013). Here is a picture of the identi ed set for (α 0, α 1 ). AC (CeMMAP & UCL) LIE 6/14/13 7 / 32

8 Probit IV model: Iden4fied set for α 0 and α 1 α α 1 = α 0

9 Complete triangular model A complete recursive, triangular model is often employed: Y 1 = 1[α 0 + α 1 Y 2 + U 1 > 0] Y 2 = g(z, U 2 ) (U 1, U 2 )? Z Can be point identifying because it implies conditional exogeneity of Y 2 : U 1? Y 2 ju 2 Only useful if U 2 is a single valued, identi able (control) function of (Y 2, Z ). so, only scalar U 2 allowed, no discrete Y 2 (but see e.g. Chesher (2005), Jun, Pinkse Xu (2012), for partial identi cation) AC (CeMMAP & UCL) LIE 6/14/13 8 / 32

10 Complete triangular model This control function solution is the industry standard Y 1 = 1[α 0 + α 1 Y 2 + U 1 > 0] Y 2 = g(z, U 2 ) (U 1, U 2 )? Z see e.g. STATA s ivprobit command, Heckman (1978), Smith and Blundell (1986), Rivers and Vuong (1988), Altonji and Ichimura (2003), Blundell and Powell (2003, 2004), Chesher (2003), Vytlacil and Yildiz (2007), Florens, Heckman, Meghir, and Vytlacil (2007), Imbens and Newey (2009), Jun, Pinkse and Xu (2009), Petrin and Train (2010), Shaikh and Vytlacil (2010), Blundell, Chen and Kristenssen (2013), and many more. And a lot of applications. AC (CeMMAP & UCL) LIE 6/14/13 9 / 32

11 Plausible restrictions Many other complete point identifying models can be entertained. But can we believe the additional restrictions embodied in these models? AC (CeMMAP & UCL) LIE 6/14/13 10 / 32

12 "I can't believe that!" said Alice. "Can't you?" the queen said in a pitying tone. "Try again, draw a long breath, and shut your eyes." Alice laughed. "There's no use trying," she said. "One can't believe impossible things.

13 "I dare say you haven't had much praccce," said the queen. "When I was your age, I always did it for half an hour a day. Why, somecmes I've believed as many as six impossible things before breakfast. Lewis Carroll (1871)

14 Exploring partially identifying power of models Choosing a point-identifying completion of the incomplete model is selecting a point in the incomplete model s identi ed set. it s worth at least a look at that set... AC (CeMMAP & UCL) LIE 6/14/13 12 / 32

15 Characterizing identi ed sets Our models restrict a function h : R YZU! R that delivers sets Y level sets: Y h (u, z) fy : h(y, z, u) = 0g U level sets: U h (y, z) fu : h(y, z, u) = 0g Example: a linear model, y = (y 1, y 2 ) with z excluded y 1 = α 0 + α 1 y 2 + u has: h(y, z, u) = y 1 (α 0 + α 1 y 2 + u) Y level sets: Y h (u, z) = f(α 0 + α 1 y 2 + u, y 2 ) : y 2 2 support of Y 2 g U level sets: U h (y, z) = fy 1 α 0 α 1 y 2 g AC (CeMMAP & UCL) LIE 6/14/13 13 / 32

16 Identi ed sets in incomplete models Example: a binary model, again y = (y 1, y 2 ) with z excluded y 1 = 1[α 0 + α 1 y 2 + u > 0] has h(y, z, u) = y 1 1[α 0 + α 1 y 2 + u > 0] non-singleton U level sets 8 < (, (α 0 + α 1 y 2 )], y 1 = 0 U h (y, z) = : ( (α 0 + α 1 y 2 ), + ], y 1 = 1 AC (CeMMAP & UCL) LIE 6/14/13 14 / 32

17 Distributions of unobservables and structures Our models restrict a function h : R YZU! R that delivers sets Y level sets: Y h (u, z) fy : h(y, z, u) = 0g U level sets: U h (y, z) fu : h(y, z, u) = 0g Our models also restrict a collection of probability distributions: G U jz fg U jz =z : z 2 R Z g where (notation) for a set S R U : G U jz =z (S) = P[U 2 SjZ = z] So models de ne admissible structures (h, G U jz ). AC (CeMMAP & UCL) LIE 6/14/13 15 / 32

18 What data tell us Data informs about distributions of observable variables: F Y jz ff Y jz =z : z 2 R Z g where for a set T R Y : F Y jz =z (T ) = P[Y 2 T jz = z] The collection of structures admitted by a model that can deliver a family of distributions F Y jz is that model s identi ed set. In our papers we use concepts and methods drawn from the theory of random sets, introduced into econometrics by Beresteanu, Molchanov and Molinari (2011). AC (CeMMAP & UCL) LIE 6/14/13 16 / 32

19 Characterizing sharp identi ed sets For sets S R U de ne a set: J h (S, z) fy : U h (y, z) Sg where U h (y, z) fu : h(y, z, u) = 0g J h (S, z) contains all values of Y which structural function h says eventuate (when Z = z) if and only if U 2 S. AC (CeMMAP & UCL) LIE 6/14/13 17 / 32

20 Characterizing sharp identi ed sets For sets S R U de ne a set: J h (S, z) fy : U h (y, z) Sg U h (y, z) fu : h(y, z, u) = 0g The identi ed set delivered by F Y jz and a model M, comprises structures (h, G U jz ) admitted by M that satisfy: G U jz (Sjz) F Y jz =z (J h (S, z)), a.e. z 2 R Z for a su ciently rich collection of test sets S R U. proof uses results in Artstein (1983), Molchanov (2005), Norberg (1992). sharp identi ed set, characterized by moment inequalities. simpli es under independence U? Z. G U (S) sup z2r Z F Y jz =z (J h (S, z)) AC (CeMMAP & UCL) LIE 6/14/13 18 / 32

21 Example: Interval censored endogenous variable The model: Y 1 = g(y 2, U) U? Z 2 R Z P[ Y 2 Y 2 Y 2 ] = 1 g is strictly increasing in U and Y 2. g is normalized so that U Unif (0, 1). Observed: Y (Y 1, Y 2, Y 2 ) and Z. Manski and Tamer (2002) meets Chernozhukov and Hansen (2005). Identi ed set comprises admissible functions g such that, for all [t, t] [0, 1]: where t t sup z 2R Z F Y jz =z (J h ([t, t], z)) J h ([t, t], z) fy : g(y 2, t) y 1 g(y 2, t)g AC (CeMMAP & UCL) LIE 6/14/13 19 / 32

22 Numerical example Generate data : (probability distributions) using a Gaussian triangular model. U V Y 1 = γ 0 + γ 1 Y 2 + U Y2 = δ 1 Z + V 0 σuu? Z N, 0 σ uv σ uv σ vv Y 2 = Y 2 W 2 Y 2 = Y 2 + W 3 W i Exp(λ i ) Data generating parameter values, ( E [W i ] = 1/λ i ): γ 0 γ 1 δ 1 σ uu σ uv σ vv λ 2 λ f1, 2g f5, 10g f5, 10g Z 2f 1, 1g AC (CeMMAP & UCL) LIE 6/14/13 20 / 32

23 Numerical example The model is parametric Gaussian: Y 1 = g 0 + g 1 Y 2 + su U? Z U N(0, 1) Calculate identi ed set for (g 0, g 1, s). P[Y 2 Y 2 Y 2] = 1 Use a selection of intervals [t, t]: for m = 1/M 2 (0, 1): 2 3 [0, m] [0, 2m] [0, 3m] [0, 1] [m, 2m] [m, 3m] [m, 1] 6 4 [2m, 3m] [2m, 1] and M 2 f4, 5, 6, 7, 8, 9, 11g, involves from 9 to 133 inequalities. AC (CeMMAP & UCL) LIE 6/14/13 21 / 32

24 1.0 g g s

25 1.0 g g s

26 1.0 g g s

27 1.0 g g s

28 1.0 g g s

29 1.0 g g s

30 1.2 g s g 0 0.1

31 1.2 g s g 0 0.1

32 s g g 1

33 s g

34 The e ect of family size on female employment Angrist and Evans (1998), Angrist (2001), Angrist and Pishke (2009). Sample: 1980 US Census Public Use Micro Samples 254,654 married mothers aged with at least 2 children, oldest < 18. Binary outcome: Y 1 = 1 if worked for pay in 1979, Y 1 = 0 otherwise. Explanatory variable: Y 2 = 1 if 3 or more children, Y 2 = 0 if 2 children. Instrumental variables: Z 1 = 1 if rst two children are same-sex, 0 otherwise. Z 2 = 1 if at 2nd birth there are twins, 0 otherwise AC (CeMMAP & UCL) LIE 6/14/13 23 / 32

35 Parameters of interest I consider two models both with the structural equation: Y 1 = 1[ a by 2 < U] U? Z Focus on two parameters. The counterfactual probability: that Y 1 = 1 (work) when Y 2 = 0 (only 2 children) ρ 0 = Y 1 (0) P[ a < U] delivered by the marginal distribution of U. The Average Treatment E ect (ATE), the di erence in counterfactual probabilities: ρ 1 = Y 1 (1) Y 1 (0) P[ a b < U] P[ a < U] AC (CeMMAP & UCL) LIE 6/14/13 24 / 32

36 A complete model Heckman s (Econometrica 1978), point identifying, triangular model Y 1 = 1[ a by 2 < U] U V Y 2 = 1[ c dz < V ] 0 jz N 0 which has (Φ is the standard normal CDF) 1 r, r 1 ρ 0 = Φ(a) ρ 1 = Φ(b + a) Φ(a) ML estimates using the same-sex instrument: ˆρ 0 = (0.011) ˆρ 1 = (0.029) AC (CeMMAP & UCL) LIE 6/14/13 25 / 32

37 Incomplete model The incomplete model has no equation for Y 2, only this: Y 1 = 1[ a by 2 < U] U? Z The same-sex instrument has low predictive power for advancing beyond 2 children. P[Y 2 = 1jsame-sex] = 0.41 P[Y 2 = 1jnot same-sex] = 0.35 AC (CeMMAP & UCL) LIE 6/14/13 26 / 32

38 Incomplete model: identified set. Complete model: MLE 1.0 Same sex instrument 0.5 ρ ρ 0

39 Twins instrument The twins instrument is a much better predictor of advancing beyond 2 children (Y 2 = 1) P[Y 2 = 1jtwins] = 1.00 P[Y 2 = 1jnot twins] = 0.38 I amend Heckman s model to allow for perfect prediction. ML estimates using the twins instrument: ˆρ 0 = (0.007) ˆρ 1 = (0.017) AC (CeMMAP & UCL) LIE 6/14/13 28 / 32

40 Twins instrument The twins instrument is a much better predictor of advancing beyond 2 children (Y 2 = 1) P[Y 2 = 1jtwins] = 1.00 P[Y 2 = 1jnot twins] = 0.38 I amend Heckman s model to allow for perfect prediction. ML estimates using the twins instrument: ˆρ 0 = (0.007) ˆρ 1 = (0.017) The incomplete model point identi es ρ 0 + ρ 1 = Y 1 (1), but neither ρ 0 nor ρ 1 individually. AC (CeMMAP & UCL) LIE 6/14/13 28 / 32

41 Twins instrument The twins instrument is a much better predictor of advancing beyond 2 children (Y 2 = 1) P[Y 2 = 1jtwins] = 1.00 P[Y 2 = 1jnot twins] = 0.38 I amend Heckman s model to allow for perfect prediction. ML estimates using the twins instrument: ˆρ 0 = (0.007) ˆρ 1 = (0.017) The incomplete model point identi es ρ 0 + ρ 1 = Y 1 (1), but neither ρ 0 nor ρ 1 individually. The identi ed set for (ρ 0, ρ 1 ) is a manifold - a one-dimensional line with slope 1 and intercept equal to ρ 0 + ρ 1. AC (CeMMAP & UCL) LIE 6/14/13 28 / 32

42 Incomplete model: identified set. Complete model: MLE 1.0 Twins instrument 0.5 ρ ρ 0

43 Incomplete model: identified set. Complete model: MLE 1.0 Identified set: same sex instrument Identified set: twins instrument 0.5 ρ MLE: same sex instrument MLE: twins instrument ρ 0

44 Remarks Incomplete single equation models are generically partially identifying when unobserved variables are not single valued functions of observed variables. discrete outcomes non-scalar heterogeneity some models characterized by inequality restrictions, some models admitting multiple equilibria Most of the applied work on these sorts of problems uses point identifying complete models, but there are many ways to complete these models and data cannot distinguish one from another. Now we can characterize the sharp identi ed sets identi ed by these incomplete models. Identi ed sets are characterized by systems of moment inequalities. AC (CeMMAP & UCL) LIE 6/14/13 30 / 32

45 Remarks We have particular results for discrete outcome cases. binary and ordered outcome IV models, Chesher (2010, 2013), Chesher and Smolinski (2011, 2012), multiple discrete choice with endogenous explanatory variables. Chesher, Rosen and Smolinski (2013), Research proceeds on cases with continuous endogenous variables. How to select from an uncountable in nity of moment inequalities? Estimation and inference procedures are available - Andrews and Shi (2013), Chernozhukov, Lee and Rosen (2013). Computational challenges remain. AC (CeMMAP & UCL) LIE 6/14/13 31 / 32

46 Remarks An incomplete model s identi ed set contains all the points identi ed by its various completions. So our results provide a tool to: conduct systematic studies of fragility of inference to failure of whimsical assumptions. taking (some of) the con out of econometrics? AC (CeMMAP & UCL) LIE 6/14/13 32 / 32