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Lecture Advanced Econometrics (Part II) - Chapter 5: Limited dependent variable models - Truncation, censoring (tobit) and sample selection

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Lecture "Advanced Econometrics (Part II) - Chapter 5: Limited dependent variable models - Truncation, censoring (tobit) and sample selection" presentation of content: Truncation, censored data, some issues in specification, sample selection model, regression analysis of treatment effects.

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Nội dung Text: Lecture Advanced Econometrics (Part II) - Chapter 5: Limited dependent variable models - Truncation, censoring (tobit) and sample selection

  1. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models Chapter 5 LIMITED - DEPENDENT VARIABLE MODELS: TRUNCATION, CENSORING (TOBIT) AND SAMPLE SELECTION. I. TRUNCATION: The effect of truncation occurs when sample data are drawn from a subset of a larger population of interest. 1. Truncated distributions: Is the part of an untruncated distribution that is above or below some specified value • Density of a truncated random variable: If a continuous random variable x has pdf f (x) and a is a constant then: f ( x) f ( x x > a) = Prob( x > a ) If x ~ N ( µ , σ 2 ) a−µ a−µ  → P ( x > a ) = 1 − Φ  = 1 − Φ (α ) , α =    σ   σ  − ( x− µ )2 1 e 2σ 2 f ( x) 2πσ 2 f ( x x > a) = = 1 − Φ (α ) 1 − Φ (α ) 1 x−µ φ  σ  σ  = (φ = Φ ' ) 1 − Φ (α ) o Truncated standard normal distribution: 2. Moments of truncated distributions: ∞ E[ x x = > a] ∫ xf ( x x > a= a )dx µ V = ∫ a (x − µ ) 2 Nam T. Hoang UNE Business School 1 University of New England
  2. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models o Truncated mean and truncated variance If x ~ N ( µ , σ 2 ) and a is a constant E[ x x > a ] = µ + σλ (α ) < Var[ x x > a] = σ 2 [1 − δ (α ) < a−µ  Where α =   , φ (.) is this standard normal density  σ  And λ (α ) = φ (α ) [1 − Φ (α )] if x > a λ (α ) = − φ (α ) Φ (α ) if x < a And δ (α ) = λ (α )[λ (α ) − α ] 0 < δ (α ) < 1 for all values of α σ truncated 2 a ] = X i β + σ 1 − Φ[(a − X i β ) / σ ] ∂E[Yi Yi > a ] ∂α = β + σ (dλi / dα i ) i ∂X i ∂X i = β + σ (λ2i − α i λi )(− β ) σ = β (1 − λ2i + α i λi ) = β (1 − δ i ) a − X i βi Where: α i = , λi = λ (α i ) , δ i = δ (α i ) σ 1 − δ i is between zero and 1  for every element of Xi , the marginal effect is less than the corresponding coefficient Nam T. Hoang UNE Business School 2 University of New England
  3. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models Var[Yi Yi > a] = σ 2 (1 − δ ) o Estimate: Yi Yi > a = E[Yi Yi > a] + ui = X i β + σλi + ui Var[ui ] = σ 2 (1 − δ i ) If we use OLS on (Yi,Xi)  we omit λi  all the biases that arise because of an omitted variable can be expected. o If E ( X Y ) in the full population is a linear function of Y then b = βτ for some τ II. CENSORED DATA • A very common problem in micro economic data is censoring of dependent variable. • When the dependent variable is censored, value in a certain range are all transferred to (or reported as) a single value. 4. The censored normal distribution: When data is censored the distribution that applies to the sample data is a mixture of discrete and continuous distribution. Define a new random variable Y transformed from the original one, Y * by:  Y = 0 if Y* ≤ 0  * Y = Y if Y* > 0 If Y * ~ N ( µ ,σ 2 ) −µ µ Prob( y = 0) = Prob(Y * ≤ 0) = Φ ( ) = 1 − Φ( ) σ σ If Y * > 0 then Y has the density of Y * This is the mixture of discrete and continuous parts. Moments: Y * ~ N ( µ ,σ 2 ) and Y = a if Y * ≤ a or else Y = Y * then: E[Y ] = Φ.a + (1 − Φ )( µ + σλ ) Var[Y ] = σ 2 (1 − Φ )[(1 − δ ) + (α − λ ) 2 Φ ] Where: Nam T. Hoang UNE Business School 3 University of New England
  4. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models  a−µ  Φ  σ  = Φ (α ) = Prob(Y * ≤ a) = Φ     ϕ  λ ϕ / (1 − Φ ) =( = )  1 − Φ  δ λ 2 − λα =   µ For a=0  E[Y a = 0] = Φ ( )( µ + σλ ) σ µ φ( ) λ= σ µ Φ( ) σ 5. The censored Regression Model: (Tobit Model) a. Model: Yi* = X i β + ε i  Yi = 0 if * Yi ≤ 0  * * Yi = Yi if Yi > 0 We only know Yi − Xiβ E[Yi X i ] = Φ + ( X i β + σλi ) σ Note: µ = E[Yi* X i ] = X i β φ[(0 − X i β ) / σ ] φ(Xiβ /σ ) Where: λi = = 1 − Φ[(0 − X i β ) / σ ] Φ ( X i β / σ ) * For the Y variable [ ∂E Y * X i ] = β but Y * is unobservable ∂X i b. Marginal Effects: Yi* = X i β + ε i  Yi = a if Yi* ≤ a  * Yi = Y if a < Yi* < b  Y = b if b < Yi*  i Let f (ε ) & F (ε ) denote the density and cdf of ε assume ε ~ iid (0,σ 2 ) and f (ε X ) = f (ε ) Then Nam T. Hoang UNE Business School 4 University of New England
  5. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models ∂E (Y X ) = β * Prob[a < Y * < b] ∂X This result does not assume ε is normally distributed. For the standard case with censoring at zero and normally distributed disturbances; ε ~ N (0,σ 2 ) ∂E (Yi X i ) X β = β .Φ i  ∂X i  σ  OLS estimates usually = MLE estimate times the proportion of non-limit observations in the sample ∂E (Yi X i ) o A useful decomposition of ∂X i ∂E (Yi X i ) = β .{Φ i [1 −λ i (α i + λi )] + φiα i + λi )} ∂X i Xiβ φi Where: α i = , Φ i = Φ (α i ) and λi = σ Φi Taking two parts separately ∂E (Yi X i ) ∂E[Yi X i , Yi > 0] = Prob[Yi > 0]. ∂X i ∂X i ∂Prob[Yi > 0] + E[Yi X i , Yi > 0]. ∂X i Thus, a change in Xi has two effects: It affects the conditional mean of Yi * in the positive part of the distribution and it affects the probability that the observation will fall in that part of the distribution. 6. Estimation and Inference with Censored Tobit: Estimation of Tobit model and the truncated regression is similar using MLE. The log-likelihood for the censored regression model is   X i β  1   Yi − X i β   = ∏ 1 − Φ   ∏  ϕ  σ   yi > 0 σ   σ   =yi 0  1  (Yi − X i β ) 2    X i β  L ln= →=  ∑ (− 2 ) ln(2π ) + ln σ 2 + σ 2  + ∑ ln 1 − Φ   σ   yi > 0   yi = 0  The two parts correspond to the classical regression for the non-limit observations and the relevant probabilities for the limit observation. This likelihood is a mixture of discrete and Nam T. Hoang UNE Business School 5 University of New England
  6. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models continuous distribution  MLE produce an estimator with all the familiar desirable properties attained by MLEs. β 1 o With γ = and θ = → σ σ 1 L →= ∑ (− 2 ) ln(2π ) − ln θ 2 + (θ yi − X i γ ) 2  + ∑ ln [1 − Φ ( X i γ ) ] Yi > 0 Yi = 0  The Hessian is always negative definite. Newton-Raphson method is simple to use and usually converges quickly. o By contrast, for the truncated model 1 Yi − X i β  ϕ  n σ σ  = ∏ yi > 0 f (Yi Y= i > a ) ∏ i= 1  a − Xiβ  1− Φ    σ  n  −1  (Yi − X i β ) 2    a − Xiβ    L ln= =  ∑   ln(2π ) + ln(σ 2 ) + σ 2  − ln 1 − Φ   σ   i =1 2       1 γ After convergence, the original parameters can be uncovered using σ = and β = θ θ Asymptotic covariance matrix of ( β ,σ ) ai X i' X i bi X i  '   = A( X i )  bi X i ci  Where { ai = −σ −2 X i γφi − [φi2 (1 − Φ i )] − Φ i } { bi = σ −3 ( X i γ ) 2 φi + φi − [( X i γ ) φi2 (1 − Φ i )] / 2 } { ci = −σ −4 ( X i γ ) 3 φi + ( X i γ )φi − [( X i γ ) φi2 (1 − Φ i )] − 2Φ i / 4 } β γ= φi and Φ i are evaluated at X iγ σ −1  n  VarCov( β ,σ ) = ∑ A( X i )  i =1  Where:     A( X i ) =         o Researchers often compute least squares estimates despite their inconsistency. Nam T. Hoang UNE Business School 6 University of New England
  7. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models o Empirical regularity: MLE estimates can be approximated by dividing OLS estimates by the propotion of non-limit observation in the sample: ε − Xiβ  X β Pr ob(Yi* > 0) = Pr ob( X i β + ε > 0) = Pr ob(ε i > − X i β ) = Pr ob i >  = Φ i  σ σ   σ   Xiβ  β OLS = β MLE Φ   σ  o Another strategy is to discard the limit observations, that just trades the censoring problem for the truncation problem. III. SOME ISSUES IN SPECIFICATION Heteroscedascticity and Non-normality: o Both heteroskedasticity & non-normality result in the Tobit estimator βˆ being inconsistent for β . o Note that in OLS we don’t need normality, consistency based on the CLT and we only need E (ε X ) = 0 (exogeneity)  data censoring can be costly. o Presence of hetero or non-normality in Tobit on truncated model entirely changes the functional forms for E (Y X , Y > 0) and E (Y X ) . IV. SAMPLE SELECTION MODEL: 7. Incidental Truncation in a Bivariate Distribution: o Suppose that y & Z have a bivariate distribution with correlation ρ . o We are interested in the distribution of y give that Z exceeds a particular value  If y & Z are positively correlated, then the truncation of Z should push the distribution of Y to the right. o The truncated joint density of y and Z is f ( y, Z ) f ( y, Z Z > a ) = Prob( Z > a ) For the bivariate normal distribution: Theorem: If y and Z have a bivariate normal distribution with mean µ y and µ Z , standard deviations σ y and σ Z and correlation ρ , then: Nam T. Hoang UNE Business School 7 University of New England
  8. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models  E ( y Z > a ) = µ y + ρσ y λ (α Z )  Var ( y Z > a )= σ y [1 − ρ δ (α Z )] 2 2 Where:  α z = (a − µ Z ) σ Z  φ (α z )  λ (α z ) =  [1 − Φ (α z )] δ (α z ) = λ (α z )[λ (α z ) − α z ] − φ (α Z ) If the truncation is Z < a → λ (α Z ) = Φ (α Z ) For the standard bivariate normal: ( y, Z ) ~ N (0,0,1,1, ρ ) E ( y Z = a ) = ρa V ( y Z = a) = 1 − ρ 2 φ (a) E ( y Z < a) = − ρ Φ(a) φ (a ) E ( y Z > a) = ρ 1 − Φ (a ) Var ( y Z > a ) = 1 − ρ 2δ (a ) General case: Let y ~ N ( µ , ∑) and partition y, µ and ∑ into: y   µ1  ∑ ∑12  y =  1  µ =   , ∑ =  11   y2   µ2   ∑ 21 ∑ 22  Then the marginal distribution of y1 is N ( µ1 , ∑11 ) , y2 ~ N ( µ 2 , ∑ 22 ) . Conditional distribution of y1 y2 is: y1 y2 ~ N [ µ1 + ∑12 ∑ 22 −1 ( y2 − µ 2 ), ∑11 − ∑12 ∑ 22 −1 ∑ 21 ] 8. The Sample Selection Model: a) Wage equation: Z i* = Wiγ + ui Z i* : difference between a person’s market wage and her reservation wage, the wage rate necessary to make her choose to participate in the labour for Z i* > 0 participate Nam T. Hoang UNE Business School 8 University of New England
  9. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models Z i ≤ 0 do not participate Wi : education, age,… b) Hours equation Yi X i β + ε i = Yi : number of hours supplied X i : wage # children, marital status.  Yi is observed only when Z i* > 0 . Suppose ε i & ui have a bivariate normal distribution with zero mean and correlation ρ . E (Yi Yi is observed ) = E (Yi Z i* > 0) = E (Yi ui > −Wiγ ) = X i β + E (ε i ui > −Wiγ ) = X i β + ρσ ε λi (α u ) = X i β + β λ λi (α u ) φ (Wiγ σ u ) Where: α u = − Wiγ σ u λ (α u ) = Φ (Wi γ σ u ) Yi Z i* > 0 = E (Yi Z i* > 0) + vi = X i β + β λ λi (uu ) + vi β λ = ρσ ε OLS estimation produces inconsistent estimates of β because of the omitting of relevant variable λi (α u ) . Even if λi were observed, the OLS would be inefficient. The disturbance vi is heteroskedasticity. We reformulate the model as follow: Z i* Wi γ + ui =   1 if Z i* > 0  biary choice model Zi =   0 otherwise  → Prob( Z i = Φ (W i γ ) 1W i) = Prob( Z i = 0 W i ) = 1 − Φ (W i γ ) Regression model: Yi = X i β + ε , observed only if Z i = 1 [0, 0 , 1, σε , ρ] (ui , ε i ) ~ bivariate normal µ u , µε σ uu , σ ε , σ uε Nam T. Hoang UNE Business School 9 University of New England
  10. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models Suppose that, as in many of these studies, Z i & Wi are observed for a random sample of individuals but Yi is observed when only Z i = 1 . → E[Yi Z i = 1, X i ,Wi ] = X i β + ρσ ε λ (Wiγ ) c) Estimation The parameters of the sample selection model can be estimated by maximum likelihood estimation. However Heckman’s (1979) two-step estimation procedure is usually used instead: o Estimate the probit equation by MLE to obtain estimates of γ . For each observation in the selected sample, compute λˆi = φ (Wiγˆ ) Φ (Wiγˆ ) and δˆi = λˆi (λˆi + Wiγˆ ) o 2. Estimate β and β λ = ρσ ε by least-squares regression of Y and X & λˆ . o Asymptotic covariance matrix of [ βˆ , βˆλ ] : (Yi Z i = 1, X i ,Wi ) = X i β + ρσ ε λi + vi Heteroskedasticity: Var[v1 Z i = 1, X i ,Wi ] = σ ε2 (1 − ρ 2δ i ) Let β * = [ βˆ , βˆλ ] , X i* = [ X i , λi ] VarCov( β * ) = σ ε2 [ X *' X ' ]−1[ X *' ( I − ρ 2 ∆) X * ][ X *' X ' ]−1 Where I − ρ 2 ∆ is a diagonal matrix with ( I − ρ 2δ i ) on the diagonal.  ˆ2 1 ' ˆ 2 1   σ ε = e e + δ βˆλ ;δ = p lim ∑ δˆi   n n  βˆλ* ρˆ * = σˆ ε* d) Model:  X β + ε Y > 0 * Xi =[ X i , Wi ] Y1 =  1 1 2  0 Y2 < 0 Z= Y2* Y2* = Xβ 2 + ε 2 ε  Assume ε =  1  ~ N (0, ∑) ε2  σ σ 12  With ∑ =  11   σ 12 1  Nam T. Hoang UNE Business School 10 University of New England
  11. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models Heckman’s two steps estimation: In the subsample for which Y1 ≠ 0 we have E (Yi1 Yi*2 > 0)= X i β1 + E (ε i1 Yi*2 > 0) = X i β1 + E (ε i1 ε i 2 > − X i β 2 ) = X i β + ρσ 1λi (α i ) X i β2 Where α i = − σ2 Therefore, in the subsample for which Y1 ≠ 0 * Yi1 Yi= 2 > 0 E (Yi1 Yi*2 > 0) + vi = X i β + ρσ 1λi (α i ) + vi = X i β + β λ λi + vi This is a proper regression equation in the sense that: E (vi xi , λi , Yi*2 > 0) = 0 σ 12 Note that: ρ = σ1 Regression of Y1 on X is subject to the omitted variable bias. o Heckman’s two steps estimation: (Heckit) procedure 1. Estimate the probit equation by MLE to get βˆ2 . Use this estimates to construct: φ (− X i βˆ2 ) λˆi = 1 − Φ (− X i βˆ2 ) 2. Regress Yi1 on X i and λˆi o Maximum likelihood: There are two data regimes: Y2 = 0 and Y2 = 1 . Construct the Likelihood Function: Regime What is known about ε 1 Y1 not observed , Y2 = 0 ε 2 < − Xβ 2 2 Y1 observed , Y2 = 1 ε1 = Y1 − Xβ1 , ε 2 > − Xβ 2 Regime 1: likelihood element: Nam T. Hoang UNE Business School 11 University of New England
  12. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models − Xβ 2 ∫ f (ε 2 )dε 2 (Φ(− Xβ 2 )) −∞ Regime 2: +∞ −X ∫β f (Y − Xβ , ε 1 1 2 ) dε 2 2 − X β2 +∞ =(β1 , β 2 , ∑) ∏ ∫ 1 f (ε 2 )d ε 2 ∏ 2 ∫ f (Y1 − X β1 , ε 2 )d ε 2 −∞ − X β2 V. THE DOUBLE SELECTION MODEL:  Xβ1 + ε1 Y2* > 0 or (and ) Y3* > 0 or both Y1 =   0 otherwise Y2* = Xβ 2 + ε 2  * Y3 = Xβ 3 + ε 3 VI. REGRESSION ANALYSIS OF TREATMENT EFFECTS: Ei = X i β + δCi + ε i Ci is a dummy variable indicating whether or not the individual attended college. Does δ measure the value of a college education? (Assume the rest of the regression model is correctly specified) The answer is no If the typical individual who chooses to go to college would have relatively high earnings whether or not he or she went to college  The problem is one of seft-selection (sample selection).  δ will overestimate the treatment effect.  Other settings in which the individuals themselves decide whether or not they will receive the treatment. Ci* = Wiγ + ui Ci = 1 if Ci* > 0   Ci = 1 otherwise E (Yi Ci = 1, X i , Z i ) = X i β + δ + E (ε i Ci = 1, X i , Z i ) = X i β + δ + ρσ ε λ (−Wiγ )  estimate this model using the two-step estimator. For non-paticipate: Nam T. Hoang UNE Business School 12 University of New England
  13. Advanced Econometrics - Part II Chapter 5: Limited - Dependent Variable Models  − φ (Wiγ )  E (Yi Ci = 0, X i , Z i ) = X i β + ρσ ε   1 − Φ (Wiγ )  The difference in expected earings between participants and non-participant is then:  φ  E (Yi Ci = 1, X i , Z i ) − E (Yi Ci = 0, X i , Z i ) = δ + ρσ ε    Φ i (1 − Φ i )  δ least square overestimate the effect. Nam T. Hoang UNE Business School 13 University of New England
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