Econometrics Master in Business and Quantitative Methods

Transcription

1 Econometrics Master in Business and Quantitative Methods Helena Veiga Universidad Carlos III de Madrid

2 This chapter deals with truncation and censoring. Truncation occurs when the sample data are drawn from a subset of a larger population. Example: Studies of income based on incomes above or below some poverty line may be of limited usefulness for inference about the population. This is essentially a characteristic of the distribution from which the sample is drawn.

3 Truncation In this part of the chapter, we are interested in inferring the characteristics of a population from a sample drawn from a restricted part of the population. A truncated distribution is the part of an untruncated distribution that is above or below some value. Theorem Density of a truncated random variable: If a continuous variable x has pdf f(x) and a is a constant, f (x x > a) = f (x) P(x > a)

4 The truncated normal distribution If x has a normal distribution with mean µ and standard deviation σ, ( ) a µ P(x > a) = 1 Φ = 1 Φ(α), σ where Φ(.) is the standard normal distribution and α = a µ σ. The truncated normal distribution is f (x x > a) = f (x) 1 Φ(α) where φ(.) is the standard normal density. 1/σφ((x µ)/σ) =, 1 Φ(α)

5 Figure: Truncated normal distribution.

6 Moments of the truncated normal distribution Theorem If x N(µ,σ 2 ) and a is a constant, where α = (a µ)/σ and E[x truncation] = µ + σλ(α) Var[x truncation] = σ 2 (1 δ(α)), λ(α) = φ(α) 1 Φ(α) if truncation is x > a, λ(α) = φ(α) Φ(α) if truncation is x < a, δ(α) = λ(α)(λ(α) α).

7 An important result is 0 < δ(α) < 1, for all values of α. The function λ(α) is called the inverse Mills ratio and when x > a it is also the Hazard function for the distribution. Summing up: If the truncation is from below, the mean of the truncated variable is greater then the mean of the original one. If the truncation is from above, the mean of the truncated variable is smaller than the mean of the original one. Truncation reduces the variance compared to the variance of the original variable.

8 The truncated regression model Here we assume that the µ = β X i then, where Therefore, y i = µ + u i, u i X i N(0,σ 2 ). y i X i N(µ,σ 2 ). We are interested in the distribution of y i given that it is greater than a certain truncation point a.

9 So, or φ((a µ)/σ) E(y i y i > a,x i ) = µ + σ 1 Φ((a µ)/σ). E(y i y i > a,x i ) = β X i + σλ(α i ), where α i = (a β X i )/σ. This conditional mean depends on X and β. The marginal effects in this model in the subpopulation can be obtained by: ( ) = β + σ dλi δα i dα i δx i δe(y i y i >a,x i ) δx i = β + σ(λ 2 i α i λ i ) = β(1 δ(α i )) ( ) β σ

10 The marginal effect of any variable X i is less than the corresponding coefficient since δ(α i ) ]0,1[. The conditional variance is: that it is less than σ 2. Var(y i y i > a,x i ) = σ 2 (1 δ(α i )), The marginal effects are of interest if the analysis is to be confined to the subpopulation. Otherwise, β is actually more interesting.

11 Estimation OLS The regression model is: y i y i > a = E(y i y i > a,x i ) + ε i = β X i + σλ i + ε i. By construction, ε i has mean zero but it is heteroscedastic Var(ε i X i ) = σ 2 (1 δ(α i )), which is a function of X i. The estimation of the previous model by OLS with a regression of y on X leads to biases and consequently to the inconsistency of the OLS estimator because we have ommited the nonlinear term λ i.

12 Maximum likelihood estimation In fact, it has been found that the OLS estimates are biased towards zero. f (y i y i > a,x i ) = 1 σ φ((y i µ)/σ) 1 Φ((a µ)/σ) The log likelihood is the sum of logs of these densities, lnl = n (ln(2π) + 2 lnσ2 ) 1 i (yi β X i ) 2 )) i (1 ln Φ. ( (a β Xi ) σ 2σ 2

13 The necessary conditions for maximization are: δlnl δβ = ( yi β X i σ 2 λ ) i X i = 0, σ i δlnl δσ 2 = i ( 1 2σ 2 + (y i β X i ) 2 2σ 4 α ) iλ i 2σ 2 = 0, where and α i = a β X i σ λ i = φ(α i) 1 Φ(α i ). The Hessian is quite involved. Hausman and Wise suggest using the Berndt et al. estimator during the iteration and to obtain the standard errors instead.

14 Censoring is a more common problem in recent studies. Examples: Censored Data 1. In the studies of income, suppose that instead of being unobserved, incomes below the poverty line are reported as if they are at the poverty line. 2. The number of hours worked by a woman in the labor force. 3. The number of extramarital affairs. Some of these studies analyzes a dependent variable that is zero for a significant fraction of the observations. The censoring of a range of values of the variable of interest introduces a distortion into conventional statistical results similar to that of truncation. Unlike truncation, censoring is essentially a defect in the sample data.

15 We assume that the censoring point is zero, though this is only a convenient normalization. When data are censored, the distribution that applies to the sample data is a mixture of discrete and continuous distributions. In fact, y = 0 if y 0, Therefore, if y N(µ,σ 2 ) the y = y if y > 0. P(y = 0) = P(y 0) = Φ ( µ ) σ = 1 Φ (µ/σ) and if y > 0, y has the density of y. The distribution is a mixture of discrete and continuous parts.

16 Figure: Partially censored distribution.

17 Moments of the censored normal variable Theorem If y N(µ,σ 2 ) and y = a if y a else y = y then and E(y) = Φa + (1 Φ)(µ + σλ) Var(y) = σ 2 (1 Φ)[(1 δ) + (α λ) 2 Φ], where Φ((a µ)/σ) = Φ(α) = P(y a) = Φ, λ = φ/(1 Φ) and δ = λ 2 λα. For the special case of a = 0, E(y a = 0) = Φ where λ = φ(µ/σ) Φ(µ/σ). ( µ σ) (µ + σλ),

18 The censored regression model- Tobit The Tobit model is: y i = β X i + u i y i = 0 if y i 0, y i = y i The conditional expected value is: E(y i X i ) = P(y i 0 X i )0+P(y i where λ i = φ(β X i /σ) Φ(β X i /σ). if y i > 0. > 0 X i )E(y i > 0 X i ) = Φ ( ) β X i (β X i +σλ i), σ There is a discussion about which expected value is the correct one, the E(y i X i ) or the E(y i X i).

19 When we are interested in predicting an upcoming event, the censored mean is the relevant quantity. If the objective is to study the need for a new facility, the mean of the latent variable, yi, would be more interesting. There are differences in the marginal effects in the models as well. δe(y i X i ) δx i = β. Given the censoring, the marginal effect is only ( E(y i X i ) β ) X i = βφ. δx i σ Once again, which one is relevant depends on the purpose of the estimates.

20 Estimation Estimation, nowadays, is on the level of ordinary regression. The log likelihood for the censored regression model is: lnl = 0.5 (ln(2π) + ln(σ 2 ) + (yi ) β X i ) 2 + ( ( )) β X i ln 1 Φ. σ 2 σ y i >0 y i =0 Doing the following reparameterization we can simplify the previous log likelihood. Let γ = β/σ and θ = 1/σ, then lnl = ( 0.5 ln(2π) lnθ 2 + (θy i γ X i ) 2) + ln(1 Φ(γ X i )). y i >0 y i =0 This form has the virtue that the Hessian is always negative definite. After the estimation, the original parameters can be recovered using σ = 1/θ and β = γ/θ.

21 The asymptotic covariance matrix for these estimates can be obtained from that for the estimates of [γ,θ] using: Asy.Var[β,σ] = JAsy.Var[γ,θ]J where ( ) δβ/δγ δβ/δθ J = δσ/δγ = δσ/δθ ( (1/θ)I ( 1/θ 2 )γ 0 1/θ 2 Some studies estimate this model by OLS despite its inconsistency. These estimators are smaller in absolute value than the MLEs and these latter can often be approximated by dividing the OLS estimates by the proportion of nonlimit observations in the sample. )

22 Application The file data.txt includes data from 753 households. The variables are expenditure (the expenditure made in the purchase of a car), income (the income of the household in the last year), children (the number of kids under 18 years old) and age (the age of the head of the household). The following model specifies the relation of interest: Expenditure i = β 0 + β 1 income i + β 2 children i + β 3 age i + u i Note that during the inquiry some households did not purchase any car in that period. To this households were assigned a null expenditure. a. Estimate the tobit model b. Test if the variables are statistically significant c. Determine the marginal effect of income on the conditional expected values, E(expenditure i X i) and E(expenditure i X i ) d. Calculate the elasticity demand income at the average point.

23 Some issues in specification Two issues that commonly arise in microeconometric data, heteroscedasticity and nonnormality, are going to be analyzed below: Nonnormality: It has been shown that if the underlying disturbances are not normally distributed, the usual estimator is inconsistent. The recent research focus on alternative estimators and on methods for testing this type of misspecification. Testing: 1. we can employ a Hausman test. Recent applications [e.g., Melenberg and van Soest (1996)] have used the Hausman test to compare the tobit/normal estimator with Powells consistent, but inefficient (robust), estimator. 2. Chesher and Irish (1987) have devised an LM test of normality in the tobit model based on generalized residuals.

24 Heteroscedasticity Heteroscedasticity emerges in this setup as a serious problem because can cause the MLE to be inconsistent. Petersen and Waldman (1981) presented an alternative to estimate a tobit model with heteroscedasticity of several types. The solution is based on replacing σ with σ i in the log likelihood function and including σ 2 i in the summations. So, it is necessary to specify a particular model for σ i and to test for the possibility of heteroscedasticity of this type. A Lagrange multiplier test can be used for this purpose. Consider the heteroscedastic tobit model in which we specify that: σ 2 i = σ 2 exp(α w i ). This a general model that includes many familiar ones as special cases.

25 The null hypothesis of homoscedasticity is α = 0. After some algebra, the necessary conditions for maximizing the log likelihood under the null hypothesis reduces to: δlnl δβ = i a ix i, δlnl δσ 2 = i b i, δlnl δα = i σ2 b i w i where z i is 1 if y i is positive and 0 otherwise, ( ui ) a i = z i σ 2 + (1 z i)( φ/(1 Φ i )) σ and b i = z i(u 2 i /σ2 1) 2σ 2 + (1 z i)(β X i φ i /(1 Φ i )) 2σ 3.

26 The sums are taken over all observations and all functions involving unknown parameters are evaluated at the restricted (homoscedastic) maximum likelihood estimates. To construct the Lagranger multiplier statistic, we use the Berndt et al. estimator for the information matrix. Under the null, at the maximum likelihood estimates, δlnl δlnl δβ and are both zero. δσ 2 LM = ( δlnl δα ) ( ) δlnl Q αα, δα

27 with A = i and Q αα = A 1. ai 2X ix i a i b i X i σ 2 a i b i X i w i a i b i X i bi 2 σ 2 bi 2wi σ 2 a i b i wix i σ 2 bi 2w i σ 4 bi 2w iw i The statistic is asymptotically distributed as chi-squared with degrees of freedom equal to the number of variables in w i.

28 Sample selection A selected sample is a general term that describes a nonrandom sample. There are a variety of selection mechanisms that result in nonrandom samples: 1. sample design, 2. the behavior of the units being sampled, including nonresponse on survey questions and attrition from social programs. Sample selection can only be an issue once the population of interest has been carefully specified. If we are interested in a subset of a larger population, then the proper approach is to specify a model for that part of the population, obtain a random sample from that part of the population, and proceed with standard econometric methods.

29 Some examples: Example 1: Suppose we wish to estimate a saving function for all families in a given country, and the population saving function is: saving = β 0 + β 1 income + β 2 age + β 3 married + β 4 kids + u, where age is the age of the household head and the other variables are self-explanatory. However, we only have access to a survey that included families whose household head was 45 years of age or older. This limitation raises a sample selection issue because we are interested in the saving function for all families, but we can obtain a random sample only for a subset of the population.

30 Example 2: We are interested in estimating the effect of worker eligibility in a particular pension plan on family wealth. Let the population model be: wealth = β 0 + β 1 plan + β 2 educ + β 3 age + β 4 income + u where plan is a binary indicator for eligibility in the pension plan. However, we can only sample people with a net wealth less than $200,000, so the sample is selected on the basis of wealth. In these two examples data were missing on all variables for a subset of the population as a result of survey design.

31 Regression in a model of selection In most cases the dependent variable of interest z is not observed. Rather we observe only its sign. Consider the next example of female labor supply: 1. Wage equation: the wage rate necessary to make female choose to participate in the labor market is function of characteristics such as age, education, number of children and where they live. 2. Hours equation: the desired number of labor hours supplied depends on the wage, home characteristics such as whether there are small children present, marital status and so on. The second equation describes desired hours, but it is observed only if the individual is working. We infer from this that the market wage exceeds the reservation wage. Thus, the hours variable in the second equation is incidentally truncated.

32 We can infer the sign of z but not its magnitude, from this information. Since there is no information on the scale of z, the variance of the error in the selection equation cannot be estimated. Theorem (Moments of the Incidentally Truncated Bivariate Normal Distribution) If y and z have a bivariate normal distribution with means µ y and µ z, standard deviations σ y and σ z, and correlation ρ, then E(y z > a) = µ y + ρσ y λ(α z ) Var(y z > a) = σ 2 y(1 ρ 2 δ(α z )) where α z = (a µ z )/σ z, λ(α z ) = φ(α z )/(1 Φ(α z )) and δ(α z ) = λ(α z )(λ(α z ) α z )

33 Selection Mechanism zi = γw i + u i, z i = 1 if zi > 0, z i = 0 if zi 0, P(z i = 1 w i ) = Φ(γ W i ), P(z i = 0 w i ) = 1 Φ(γ W i ).

34 Regression Model y i = β X i + ε i, observed only if z i = 1, (u i,ε i ) bivariate normal [0,0,1,σ ε,ρ]. Suppose that, as in many of these studies, z i and W i are observed for a random sample of individuals, but y i is observed only when z i = 1. Therefore, E(y i X i,z i = 1) = E(y i X i,z i > 0) = E(y i X i,u i > γ W i ) = β X i + E(ε i u i > γ W i ) = β X i + ρσ ε λ i (α u ) where α u = γ W i and λ(α) = φ(γ W i ) Φ(γ W i ).

35 Therefore, y i X i,z i > 0 = β X i + ρσ ε λ i (α u ) + v i. Least squares regression using the observed data, for instance, OLS regression of hours on its determinants, using only data for women who are working, produces inconsistent estimates of β. We can view this problem as an omitted variable. The marginal effect of the regressors on y i in the observed sample consists of two components: 1. There is a direct effect on the mean of y i, which is β. 2. For a particular explanatory variable, if it appears in the probability that z i is positive, it will influence y i through its presence in λ i.

36 The full effect of a unit change in a regressor that appears in both X i and W i on y is: where δ i = λ 2 i α i λ i. δe(y i X i,z i > 0) δx ik = β k γ k (ρσ ε )δ i (α u ), Suppose that ρ is positive and E(y i X i ) is greater when z i is positive than when it is negative. The additional term serves to reduce the marginal effect because 0 < δ i < 1.

37 Estimation The parameters of the sample selection model can be estimated by ML, but this is not so easy. An alternative consists in using the procedure by Heckman (1979). Heckman s two-step estimation procedure is as follows: 1. Estimate the probit equation by ML to obtain estimates of γ. For each observation in the selected sample compute ˆλ i = φ(ˆγ W i ) Φ(ˆγ W i ). and ˆδ i = ˆλ i (ˆλ i ˆγ W i ). 2. Estimate β and β λ = ρσ ε by least squares regression of y on X and ˆλ.

38 It is also possible to obtain consistent estimators of the individual parameters ρ and σ ε. At each observation, the true variance of the disturbance would be: σ 2 i = σ 2 ε(1 ρ 2 δ i ). The average variance for the sample would converge to plim 1 σi 2 = σε 2 n σ2 ε ρ2 δ i and this is estimated by the least squares residual variance, e e/n. We have also that plimb 2 λ = ρ2 σ 2 ε,

39 while based on the probit results, we have plim 1 ˆδ i = n δ. Then we can obtain a consistent estimator of σ 2 ε using: i ˆσ 2 ε = e e n + δb 2 λ. Finally, an estimate of ρ 2 is obtained as ˆρ 2 = b2 λ. ˆσ 2 ε In order to test hypotheses, we need an asymptotic covariance matrix of (b,b λ ).

40 We have two cases: case 1. Suppose for the moment that λ and δ are known, this means that we do not have to estimate γ. Let X i = (X i,λ i ) and let b be the least squares coefficient vector in the regression of y on X in the selected data. Then, using the appropriate form of the variance of OLS when there is heteroscedasticity, we have to estimate: Var(b X ) = σ 2 ε(x X ) 1 [ i (1 ρ2 δ i )X i X i ] (X X ) 1 = σ 2 ε(x X ) 1 (X (I ρ 2 )X )(X X ) 1, where I ρ 2 is a diagonal matrix with (1 ρ 2 δ i ) on the diagonal.

41 case 2. The parameters in γ do have to be estimated using the probit equation. Heckman has shown that the earlier variance matrix can be appropriately corrected by adding a term inside the brackets, Q = ˆρ 2 (X ˆ W)Est.Asy.Var(ˆγ)(W ˆ X ), where ˆV = Est.Asy.Var(ˆγ), the estimator of the asymptotic covariance of the probit coefficients. Finally, the complete expression is: Est.Asy.Var(b,b λ ) = ˆσ 2 ε (X X ) 1 (X (I ˆρ2 ˆ )X +Q)(X X ) 1.