Estimation in the Fixed Effects Ordered Logit Model. Chris Muris (SFU)

Transcription

1 Estimation in the Fixed Effects Ordered Logit Model Chris Muris (SFU)

2 Outline Introduction Model and main result Cut points Estimation Simulations and illustration Conclusion

3 Setting 1. Fixed-T panel. A random sample {(y it, X it ), i = 1,, N, t = 1,, T }, with N 2. Ordered logit. y it is an ordered response in {1, 2,, J}, y it = α i + X it β + u it, 1 if yit < γ 1, 2 if γ 1 yit y it = < γ 2,.. J if γ J 1 yit, for cut points γ j. Errors are logistic. 3. Fixed effects. Joint distribution of α i and X i is unrestricted.

4 Contribution This paper: Estimation of differences of the cut points More efficient estimation of the regression coefficient Why does this matter? Cut points: bounds on partial effects Model is heavily used (BSW, 2015: >150 cites)

5 Application (1): Allen and Arnutt (WP, 2013) Effect of Teach First program on educational outcomes. y it : letter grade student i for subject-year t D it {0, 1}: school enrolled in Teach First? Latent variable model: where y it = α i + β 1 D it + X it β 2 + u it, α i is unobserved student ability X it are controls

6 Application (1): Allen and Arnutt (WP, 2013) All three model ingredients are present 1. Fixed-T : number of subjects per student is much smaller than the number of students 2. Ordered: letter grade is an ordered outcome 3. Fixed effects: schools with results in the bottom 30% are eligible

7 Application (2): Frijters et al. (AER, 2004): Effect of income on life satisfaction y it : life satisfaction on scale {0,, 10} completely dissatisfied to completely satisfied. X it : real household income Latent variable model:y it = α i + β 1 X it + Z it β 2 + u it α i : unobserved student ability X it may correlated with α i Z it : other controls.

8 More applications Health Labor Khanam et al. (JHE, 2014): income and child health Carman (AER, 2013): intergenerational transfers and health Frijters et al. (JHE, 2005): income on health Hamermesh (JHR, 2001): earnings shocks and job satisfaction Das and van Soest (JEBO, 1999): expectations about future income

9 More applications (2) Happiness Frijters et al. (AER, 2004): income and life satisfaction Blanchflower and Oswald (JPE, 2004): trends in US life satisfaction Credit / debt ratings Amato, Furfine (JBF 2003): credit ratings are not procyclical Afonso et al. (IJOFE, 2013): determinants of sovereign debt ratings Education Allen and Alnutt (2013): effect of Teach First program on student achievement

10 Literature Chamberlain (RES, 1980): binary choice and unordered choice Das and van Soest (JEBO, 1999): all cutoffs Ferrer-i-Carbonell and Frijters (EJ, 2004): individual-specific cutoffs Baetschmann et al. (JRSS-A, 2015): small-sample improvements None of these papers estimate the cut point differences.

11 Outline Introduction Model and main result Cut points Estimation Simulations and illustration Conclusion

12 Model Random sample of size n, T fixed: {(y i1,, y it, X i1,, X it ), i = 1,, n} y it is an ordered outcome in {1,, J} X it = (X it,1,, X it,k ) are covariates Unobserved heterogeneity in the latent variable: y it = α i + X it β + u it Serially independent, exogenous logistic errors u i1,, u it (X i1,, X it ), α i iidlog (0, 1) Link between latent and observed by cut points 1 if yit < γ 1 2 if γ 1 yit y it = < γ 2.. J if γ J 1 yit.

13 Incidental parameters For each category j, P (y it = j X it, α i ) = Λ (γ j α i X it β) Λ (γ j 1 α i X it β), where Λ = exp (x) / (1 + exp (x)). Likelihood is n T J [Λ (γ j α i X it β) Λ (γ j 1 α i X it β)] 1{yit=j}. i=1 t=1 j=1 Fixed T : maximum likelihood estimator (MLE) is inconsistent

14 Incidental parameters (logit) ˆβ ML : maximum likelihood estimator for T = J = 2 Inconsistent (Abrevaya, 1997) ˆβML p 2β as n Solution (Chamberlain, 1980) y i1 + y i2 is a sufficient statistic for α i conditional MLE (CMLE) with P (y i = (1, 0) y i1 + y i2 = 1, X i, α i ) = is consistent Drawback: CMLE uses only switchers exp ((X i2 X i1 ) β)

15 Incidental parameters (Ordered logit) Solution for incidental parameters problem is model-specific No sufficient statistic (yet?) for ordered logit No exponential form: P (y it = j X it, α i ) = Λ (γ j α i X it β) Λ (γ j 1 α i X it β)

16 Incidental parameters (Takeaway) Unobserved heterogeneity can cause inconsistency Solution exists for the case of binary logit Solution uses only switchers Does not extend to ordered logit model

17 Ordered choice Consider ordered choice with y it {1,, J} Dichotomization: Pick some { j {1,, J 1} and define the binary variable 1 if y it j, d it,j = 0 otherwise. Apply Chamberlain s CMLE to y it,j Consistent but inefficient: Information is lost by discarding more precise measurement y it Winkelmann and Winkelmann (1998): {0,, 10} collapsed to {0, 1} by cutting at 7 Out of observations, only 2523 are switchers

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19 Non-switcher: not informative

20 Switcher: informative

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25 Das and van Soest: multiple cutoffs

26 Time-invariant transformations do not catch flat patterns

27 Time-varying transformations catch flat patterns

28 There are (J 1) T (J 1) time-varying transformations

29 Main result (notation) Cutoff categories π t J 1 π = (π 1,, π T ) is a transformation d it,π = 1 {y it π t } is the π transformed dependent variable time series for unit i: d i,π {0, 1} T di,π = t d it,π: number of times below cutoff F d is the set of all binary T vectors f with sum d

30 Main result Theorem If the random vector (y i, X i ) follows the fixed effects ordered logit model, then for any transformation π, the conditional probability distribution of the π-transformed dependent variable d i,π is given by p i,π (d β, γ) P ( d i,π = d d i,π = d, ) X i, α i (1) 1 = f F exp { d t (f t d t ) ( )} (2) γ π(t) X it β for any d {0, 1} T.

31 Main result (remarks) 1. Conditional probability does not depend on α i 2. Sufficient statistic exists for (J 1) T transformations of y i 3. Existing approaches use at most (J 1) of those transformations

32 Main result (T = 2) Evaluate the conditional probability for d = (1, 0) For any time-invariant transformation: exp { (X i2 X i1 ) β} For time-varying transformation π = (j, k), j k exp {(γ k γ j ) (X i2 X i1 ) β} Identification of γ k γ j. Intuition: subpopulation with X i2 = X i1

33 Outline Introduction Model and main result Cut points Estimation Simulations and illustration Conclusion

34 Cut points: binary Panel data binary choice (J = 2): no interpretation of the magnitude of β evaluation of partial effects requires value/distribution α i Existing estimators for ordered choice inherit this problem by eliminating thresholds Marginal effect of a ceteris paribus change in regressor m with coefficient β m : P (y it j X it, α i ) X it,m = β m Λ (α i + X it β γ j ) [1 Λ (α i + X it β γ j )]

35 Change in y it for unit change in X it,m?

36 If y it = α i + X it β + u it < β m, then y it is unchanged.

37 No marginal effects without info on α i or α i X it.

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39 Bounds (notation) Consider a ceteris paribus change in X it of x The counterfactual latent dependent variable is ỹ it = y it + ( x) β; ỹ it : the counterfactual ordered outcome.

40 Bounds Conditional probability for the observed counterfactual outcome: 1 if ( x) β > γ j γ j 1, P (ỹ it > j y it = j, X it ) = 0 if ( x) β < 0, else F v(γ j X it β) F v(γ j (X it + x)β) F v(γ j X it β) F v(γ j 1 X it β) Paper presents a more general result along the same lines. Note: intermediate category.

41 Bounds (2) Using the first component: Minimum required change in X itm to move everybody with y it = j up: δ j m γ j γ j 1 β m Let x m be the ceteris paribus change in X it,m, then x m > δ j m P (ỹ it > j y it = j, X it ) = 1

42 Outline Introduction Model and main result Cut points Estimation Simulations and illustration Conclusion

43 Estimation (one transformation) γ π, are the n π cut point differences that show up θ π,0 = (β 0, γ π,,0 ): true parameter value The CMLE for transformation π is ( ) ˆθ π = ˆβ, ˆγπ, 1 = arg max R K R nπ n n 1 {d i = d} ln p iπ (d θ π ) i=1

44 Estimation (one transformation) Assumption The variance matrix of the regressors,var is positive definite. X i1. X it, exists and Theorem Let ({y i, X i }, i = 1,, n) be a random sample from the fixed effects ordered logit model, and let π be an arbitrary transformation. If the above assumption holds, then ˆθ π is consistent and n (ˆθ π,n θ π,0 ) d N ( 0, H 1 π Σ π H 1 ) as n, (3) where H π and Σ π are the variance and expected derivative of (??). π

45 Estimation (one transformation) The score s i,π (d θ π ) = = [ ln pi,π ( d θ π) β ln p i,π ( d θ π) γ π, ] [ si,π,β (β, γ π, ) s i,π,γ (β, γ π, ) ] can be used to show global concavity Identification is guaranteed by condition on var (vec (X )) Assumption are as for linear panel model

46 Estimation (more transformations) CMLE is equivalent to solving the moment conditions [ ] si,π,β (β E 0, γ π,,0 ) = 0. s i,π,γ (β 0, γ π,,0 ) GMM provides framework for combining information from multiple transformations.

47 Estimation (more transformations) For time-invariant transformations, γ π, is empty These are transformations used by existing procedures

48 Estimation (more transformations) Time-invariant π. Combine moment conditions for β 0 : s i,(1,,1) (β 0 ) E [s i,1,β (β 0 )] = E. = 0 (4) s i,(j 1,,J 1) (β 0 ) GMM estimator based on (4) is β W1,n = arg min s 1,n (β) W 1,n s 1,n (β) where s 1,n (β) = 1 n n i=1 s i,1,β (β) Existing procedure corresponds to choice for W 1,n β is optimal estimator in this class

49 Estimation (even more transformations) Main result: (J 1) T (J 1) additional, time-varying transformations Scores involve n γ cut point differences γ = (γ π, ) π Collect the scores for γ in the n γ 1 vector s i,2,γ (β, γ ) = (s i,π,γ (β, γ π, ), π : n π 1) Scores for β from time-varying π: s i,2,β (β, γ ) = (s i,π,β (β, γ π, ), π : n π 1)

50 Estimation (even more πs) Proposal: estimation using s i,1,β (β 0 ) E [s i (β 0, γ,0 )] = E s i,2,β (β 0, γ,0 ) = 0 s i,2,γ (β 0, γ,0 ) Optimal estimator in this class: ( ˆβ, ˆγ ) Question: Is ˆβ more efficient than β?

51 Efficiency (result) Theorem Let ({y i, X i }, i = 1,, n) be a random sample from [...] Then, as n, n ( β β 0 ) d N (0, V1 ), n (( ˆβ ˆγ ) ( β0 γ,0 )) d N (0, V ), where [...]. Furthermore, let V β be the top-left K K block of V. Then V 1 V β is positive semidefinite.

52 Efficiency (proof sketch) ( ˆβ, ˆγ ) is based on s i,1,β (β 0 ) E s i,2,β (β 0, γ,0 ) s i,2,γ (β 0, γ,0 ) Information from s i,2,γ (β 0, γ,0 ) exactly identifies γ,0 β estimation based on s i,1,β (β 0 ) is unaffected by adding s i,2,γ (β 0, γ,0 ) s i,2,β (β 0, γ,0 ) yields efficiency gains for β 0

53 Efficiency (OMD) The efficient minimum distance estimator based on all ˆβ π is asymptotically equivalent to the optimal GMM estimator ˆβ

54 Outline Introduction Model and main result Cut points Estimation Simulations and illustration Conclusion

55 Implementation Stata s clogit command implements ˆθ π augment regressors with indicator for π Apply clogit for each π Optimally combine the results using suest

56 Simulations J = 3, K = 1, T = 2, N = 5000 β γ 2 γ 1 Estimator %Bias RelSD %Bias RelSD Oracle CSLogit π = (1, 1) π = (2, 2) π = (1, 2) π = (2, 1) DvS OMD

57 Simulations (sensitivity) (J = 3, K = 3) (J = 3, K = 5) (J = 5, K = 5) Estimator %Bias RelSD %Bias RelSD %Bias RelSD Coefficient β π = (1, 1) π = (1, 2) DvS OMD Cut point γ 2 γ 1 π = (1, 2) π = (2, 1) OMD

58 Simulations: many-π bias Estimation of weight matrix affects small samples Proposal: composite likelihood estimator (CLE) ˆθ CLE = arg max π n 1 {d i = d} ln p iπ (d θ π ) i=1 sacrifices efficiency robust finite-sample performance (large J, T ) even easier to implement in Stata (expand + clogit)

59 Simulations: many π results Other parameters unchanged T = 4 T = 6 T = 8 Estimator %Bias RelSD %Bias RelSD %Bias RelSD Oracle π = (1, 1) BUC DvS CLE OMD

60 Family income and children s health Relationship between reported (subjective) children s health status and total household income Seminal paper: Case et al. (2002) 1. children s health is positively related to household income 2. relationship is stronger for older children. Currie and Stabile (2003) replicate using Canadian panel Murasko (2008) replicates using Medical Expenditure Panel Survey (MEPS) Currie et al. (2007) use British data: confirm finding #1 no evidence for #2 Khanam et al. (2014) use Australian data first to control for unobserved heterogeneity no evidence for #2

61 Illustration (data) Data: Panel 16 of the Medical Expenditure Panel Survey (MEPS). US data. MEPS is a rotating panel (Agency for Healthcare Research Quality, 1996) Demographic and socioeconomic variables (survey) Data on health and healthcare usage (admin) 4131 children in 2011 and Dependent variable: self-reported health status (RTHLTH) 1 = Poor - 5 = Excellent. Explanatory variables (de-meaned) total household income interaction age and income. year dummies, family size

62 Illustration: results RE CRE BUC CLE log(income) it (0.03) (0.06) (0.07) (0.08) Age log(income) it (0.007) (0.013) (0.015) (0.016) Family size (0.04) (0.14) (0.15) (0.16) γ 2 γ (0.05) (0.05) (0.05) γ 3 γ (0.09) (0.09) (0.12) γ 4 γ (0.23) (0.23) (0.29)

63 Illustration: discussion (1) Controlling for unobserved heterogeneity is important Not enough evidence for income effect at average age Sufficient evidence for age-dependent income-health effect CLE is the only estimator to detect it.

64 Illustration: discussion (2) BUC: no cut points CLE: income increase > 900% to move a 15-year old from Fair to Good or higher. Correlated random effects (CRE): close to correctly specified standard errors are only slightly smaller 100% income increase changes probability of Good or above from to

65 Outline Introduction Model and main result Cut points Estimation Simulations and illustration Conclusion

66 Conclusion Estimation for fixed effects ordered logit model Using (J 1) T fixed effects binary choice logit models Cut point differences for bounds on partial effects Regression coefficient: increased efficiency

67 Extensions 1. Better bounds 2. Other panel data models 2.1 transformation model 2.2 interval-censored model 3. Semiparametric ordered choice 4. Dynamic ordered choice 5. Time-varying cut points