M2R “Development Economics” Empirical Methods in Development Economics Universit´e Paris 1 Panth´eon Sorbonne

Selection and Heckman procedure R´emi Bazillier [email protected]

Semester 1, Academic year 2016-2017

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Introduction

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The purpose of the class today is to present the potential problem (and its solution) induced by the violation of Condition 2.

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Remember that Condition 2 requires that one works on a random sample of n observations from the population:

{(xi1 , ..., xik , yi )} with i = 1, ..., n

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Introduction I

To better analyze the consequences of the violation of Condition 2, let’s introduce some basic notations.

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Assume that we can estimate the following population model if we get {(xi1 , ..., xik , yi )} (i = 1, ..., n) from a random draw: yi = xi β + ui , where xi β = β0 + β1 xi1 + ... + βk xik .

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Assume that, instead of getting the random sample {(xi1 , ..., xik , yi )} (i = 1, ..., n), we only get a subsample of it that is called the selected sample (i.e. for some reason, yi is not observed for certain i).

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Introduction

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This means that we can define a selection indicator si for each i by si = 1 if we observe all of (yi , xi ), and si = 0 otherwise

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Thus, si = 1 indicates that we will use the observation in the analysis, while si = 0 means the observation will not be used.

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In other words, we can only estimate the following model: si yi = si xi β + si ui .

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Introduction

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Clearly, if we start with a random sample and randomly drop observations, OLS estimates will still be unbiased and consistent (since Condition 2 still holds).

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But selected samples that stem from random selection are clearly not the rule.

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Many selected samples result instead from incidental truncation.

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Yet, incidental truncation usually violates Condition 2.

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Introduction

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Outline of the class today: 1. Incidental truncation: what is it? 2. Incidental truncation: why is it a problem? 3. Incidental truncation: what can we do about it?

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2. Incidental truncation: what is it?

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Let’s consider the following equation: y = xβ + u, where xβ = β0 + β1 x1 + ... + βk xk .

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Suppose y = log (wage ), where wage is the hourly wage that an individual could receive in the labor market.

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2. Incidental truncation: what is it?

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If the person is actually working at the time of the survey, then the wage she could receive in the labour market is the wage we observe.

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But for people out of the work force, we cannot observe wage.

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Therefore, the truncation of wage offer is incidental because it depends on another variable, namely, labor force participation.

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2. Incidental truncation: what is it? I

The usual approach to incidental truncation is to add an explicit selection equation to the model of interest.

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This model is given by: y = xβ + u,

(1)

where u ∼ Normal (0, 1). I

The explicit selection equation is given by: s = 1[zγ + v ≥ 0],

(2)

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s = 1 if we observe y , and s = 0 otherwise v ∼ Normal (0, 1) xβ = β0 + β1 x1 + ... + βk xk zγ = γ0 + γ1 z1 + ... + γm zm 9 / 35

2. Incidental truncation: what is it?

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We assume that z is exogenous in Equation (2).

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This means that E (v |z) = 0.

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In addition, for the correction of the bias derived from incidental truncation to work well, it is better that x be a strict subset of z: any xj is also an element of z, and we have some elements of z that are not also in x.

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2. Incidental truncation: why is it a problem?

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Intuitively, the OLS estimate of β in Equation (1), that we b will be biased if there are omitted variables that denote by β determine both: I

the probability to participate in the labour force

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the wage.

Put differently, incidental truncation generates biased OLS estimates if u and v are correlated.

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3. Incidental truncation: what can we do about it?

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A two step procedure is needed.

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This procedure was introduced by Heckman in 1976.

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Hence its name: Heckman procedure or Heckit method.

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3. Incidental truncation: what can we do about it?

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First, using all n observations, one estimates the following probit model: P (s = 1|z) = Φ(zγ). (3) ch (h = 1, ..., m) and therefore allows to compute This gives γ the inverse Mills ratio λbi = λ(zi γ b ) for each i such as si = 1.

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3. Incidental truncation: what can we do about it?

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Second, using the selected sample, that is, the observations for which si = 1 (say, n1 of them), one computes the OLS estimates of the following equation: b + u. y = xβ + ρλ

(4)

b which is unbiased and consistent. This gives β

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3. Incidental truncation: what can we do about it? I

b is Note that is suffices to check whether the coefficient of λ statistically significant to know whether there is a sample selection problem or not.

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The STATA command is given by heckman varlist1, select (varlist2).

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For instance, one can have heckman lw education age children, select(education age children married).

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The twostep option allows to generate estimates for both Equation (4) and Equation (5).

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3. Incidental truncation: what can we do about it? I

We mentioned earlier that it is better if x is a strict subset of z.

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This means that we have at least one element of z that is not also in x.

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Put differently, one needs to rely on a kind of instrument in the first step of the Heckman procedure: this variable affects the selection but has no direct impact on y .

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The results are usually less than convincing if z = x.

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Do you see why?

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3. Incidental truncation: what can we do about it?

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The reason for this is that while the inverse Mills ratio is a nonlinear function of z, it is often well-approximated by a linear function.

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b can be highly correlated with x. If z = x, λ

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Such multi-collinearity can lead to very high standard errors for the OLS estimates of Equation (3) and therefore artificially lower the chance of rejecting the null hypothesis according to which these estimates are equal to 0.

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4. Selection and earning in South Africa I

South African data

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Building of a model which allows for the selection into employment Focus on black male South Africans

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The residual of the selection equation may contain personal motivation I

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Major differences by both gender and race

u and v can be positively correlated

Reservation wages I

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People with high reservation wages wait longer before taking a job and are less likely to have a job But conditional on having a job, they are likely to have a higher wage Since we do not observe reservation wage, it will go into the residuals of the model u and v will be negatively correlated 18 / 35

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Earnings with selection

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Selection variables: I

Other income / Married

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These variables should have an effect on the probability to work but should not influence earning directly

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No convincing ways to test the validity of the exclusion restrictions: you need to argue for their plausibility

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4. Selection and earning in South Africa

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Earnings with selection

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The probability of having a wage job: I

It increases with age at a decreasing rate

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It increases with education at an increasing rate

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→ young people with little education are relatively unlikely to have a job

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We can calculate the Inverse Mills Ratio manually I

Ratio of the standard normal density and the standard normal cumulative distribution function

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4. Selection and earning in South Africa

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Interest of the ‘manual’ procedure: we can see how the Mills ratio is correlated with age and education

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However: standard errors are not correct in the second stage I

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Because we have not taken into account the fact that the IMR is a generated regressor

To get the right standards errors, use the heckman command

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4. Selection and earning in South Africa

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4. Selection and earning in South Africa

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IMR is significant: selection bias

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IMR is negative: negative correlation between the residuals: consistent with the hypothesis that individuals differ in their willingness to wait for a well-paid jobs (reservation wage)

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Correcting for sample reduction reduces the estimated age effect on earnings I

Higher age has a positive effect on earnings and a positive effect on the likelihood of having a job

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The type of individual least likely to be selected into the sample is one with a high reservation wage and little experience

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4. Selection and earning in South Africa

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5. The occurence of zeros I

Corner response variables: I

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The dependent variable can take a continuous range of values but where there are a large number of zeros value

Example: How firms in Africa invest? I

Zero is a quite common outcome - some firms invest nothing in any particular year

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We need to ask why those zeros may have arisen

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Here, there is evidence that physical capital is ’irreversible’ (selling out fixed capital is difficult)

The issue here is not that we cannot observe the outcome

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Investment in Ghana’s manufacturing sector

invratei = β0 + β1 anyfori + ui

(5)

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4. Selection and earning in South Africa

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Zeros investment appears to be a much more common choice among domestically owned firms

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OLS estimates are biased

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The problem of zeros I

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Zeros or missing values? I

If missing values are random, not necessarily a problem (no selection bias)

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It is the case if values are missing for reasons correlated with other variables (eg. firm survival rate in a study on firms productivity)

If “real” zeros: I

Think about the reasons explaining the large share of zeros

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We can use Heckman procedure I I

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We need exclusion restrictions Without exclusion restrictions, a model is technically identified because of the nonlinearity of the IMR, but often results in imprecise estimates

See chapter on gravity 34 / 35

Conclusion

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Instruments do not only allow to solve the biases induced by I

the standard omitted variables problem;

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the reverse causality problem;

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the measurement error problem.

They also help run the Heckman procedure and therefore correct the (omitted variables) bias induced by incidental truncation.

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