Sample Selection: Heckman Model

We are interested in estimating the model

\[\begin{equation} \mathbf{y^*=xb + \epsilon} \end{equation}\]

but for a subset of our data, the dependent variable is either missing or coded to some arbitrary values (e.g. 0 or -999). Of concern however, is that the pattern of this missingness is non-random in a way that could induce bias in our estimated \(\beta\)’s if we apply OLS to observed values from the above equation.

Selection Mechanism

Suppose that the pattern of missingness (I’ll refer to this as censored hereafter) is related to the latent (unobserved) process

\[\begin{equation} \mathbf{z}^* = \mathbf{w}\gamma + \mathbf{u} \end{equation}\]

From this process, the researcher can observe

\[\begin{split}\begin{align} z_i =& 1 \text{ if } z^*_i > 0 \\ =&0 \text{ if } z^*_i \le 0 \end{align}\end{split}\]

or \(z_i=1\) (\(y_i\) not censored) when

\[\begin{equation} u_i \ge - \mathbf{w}_i\gamma \end{equation}\]

The probability of \(y_i\) not censored is

\[\begin{split}\begin{align} Pr(u_i \ge - \mathbf{w}_i\gamma) =& 1- \Phi(-\mathbf{w}_i\gamma)\\ &=\Phi(\mathbf{w}_i\gamma) \end{align}\end{split}\]

if we are willing to assume that \(\mathbf{u}\sim N(\mathbf{0},\mathbf{I})\). Note for identification purposes in the Heckman Model we restrict \(Var(u_i) = 1\). Also note that \(1- \Phi(-\mathbf{w}_i\gamma)=\Phi(\mathbf{w}_i\gamma)\) by symmetry of the standard normal distribution.

To visualize this, consider the following figure. The probability of \(y_i\) not being censored (\(Pr(u_i \ge - \mathbf{w}_i\gamma)\)) is the set of errors greater than \(-\mathbf{w}_i\gamma\). The probability that an error draw satisfies this condition is the darker shaded area to the right of \(-\mathbf{w}_i\gamma\).

../_images/heckman_selection.png

Fig. 15 Probabilities in the Selection Mechamism

Amounts Mechanism and Sample Selection Bias

Having constructed a model a model for censoring, we can construct “amounts” equation as follows. Denoting \(\mathbf{y}\) as the not censored (observed) dependent variable, the censoring model defines what is in the estimation sample as

\[\begin{align} y_i = y^*_i = \mathbf{x}_i \beta + \epsilon_i \text{ observed, if $z_i$ = 1} \end{align}\]

Finally, the joint distribution of the errors in the selection (\(u_i\)) and amounts equation (\(\epsilon\)) is distributed iid as

\[\begin{split}\begin{equation} \begin{bmatrix} u_i \\ \epsilon_i \end{bmatrix} \sim Normal\left( \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} 1 & \rho \\ \rho & \sigma^2_\epsilon \end{bmatrix} \right) \end{equation}\end{split}\]

To see how the selection and amounts model are related, consider

\[\begin{split}\begin{align} E(y_i | y_i \text{ observed}) = & E(y_i | z^* > 0 ) \\ =&E(y_i | u_i > -\mathbf{w}_i \gamma) \\ =&\mathbf{x}_i \beta + E(\epsilon_i | u_i > -\mathbf{w}_i \gamma) \\ =&\mathbf{x}_i \beta + \rho \sigma_\epsilon \frac{\phi(\mathbf{w}_i \gamma)}{\Phi(\mathbf{w}_i \gamma)} \end{align}\end{split}\]

What is immediately apparent is that the conditional mean (\(E(y_i | y_i \text{ observed})\)) differs from the unconditional mean (\(\mathbf{x}_i\beta\)) only if \(\rho \neq 0\) since all the other elements in the far right hand term (i.e., the variance of the error in the amounts equation, \(\sigma_\epsilon\), and the Inverse Mills Ratio, \(\frac{\phi(\mathbf{w}_i \gamma)}{\Phi(\mathbf{w}_i \gamma)}\)) in the preceding equation are strictly positive. So if the errors in the amounts and selection equations are uncorrelated (\(\rho=0\)) we can safely apply ordinary least squares to uncover unbiased estimates for \(\beta\) and can ignore endogenous selection effects and the selection equation portion of the model.

However, there are some further nuances to consider. Denoting the correlation amongst the two sets of independent variables in the model \(\mathbf{x},\mathbf{w}\) as \(\rho_{\mathbf{x},\mathbf{w}}\), we examine the following four cases of correlation patterns amongst data and errors. As the final equation above shows, when \(\rho\) is zero (Case 1 and Case 2) below, OLS will always give unbiased \(\beta\) even if \(\rho_{\mathbf{x},\mathbf{w}}\) is non-zero. This is happening because the missingness patterns in \(\mathbf{y}\) are essentially random or select cases in ways that are symmetric around the regression line that preserves slope and intercept coefficients (we have plots below of these two cases later for providing more intuition about this statement).

../_images/heckman_example_rho_0_rhoxy_0.png

Fig. 16 Case 1

../_images/heckman_example_rho_0_rhoxy_8.png

Fig. 17 Case 2

Case 3 and Case 4 are more interesting and represent instances where OLS will be inconsistent and we should apply the Heckman Model. For gaining intuition about how cases 3 and 4 differ consider again the conditional mean

\[\begin{equation} E(y_i | y_i \text{ observed}) = \mathbf{x}_i \beta + \rho \sigma_\epsilon \frac{\phi(\mathbf{w}_i \gamma)}{\Phi(\mathbf{w}_i \gamma)} \end{equation}\]

We can think of the preceding equation as guiding the regression equation we ought to be estimating if the sample selection process as outlined above is happening. If instead, we ignore the final term (\(\rho \sigma_\epsilon \frac{\phi(\mathbf{w}_i \gamma)}{\Phi(\mathbf{w}_i \gamma)}\)) when in fact \(\rho\neq 0\) then we are putting this information into the error term of the misppecified OLS model regressing \(\mathbf{y}\) on \(\mathbf{x}\) making the estimation equation

\[\begin{equation} \mathbf{y = x \tilde{\beta} + \psi} \end{equation}\]

where \(\psi = \rho \sigma_\epsilon \mathbf{IMR} + \epsilon\), where \(\mathbf{IMR}\) is the \(N \times 1\) vector

\[\begin{split}\begin{equation} \begin{bmatrix} \frac{\phi(\mathbf{w}_1 \gamma)}{\Phi(\mathbf{w}_1 \gamma)}\\ \vdots \\ \frac{\phi(\mathbf{w}_i \gamma)}{\Phi(\mathbf{w}_i \gamma)} \\ \vdots \\ \frac{\phi(\mathbf{w}_N \gamma)}{\Phi(\mathbf{w}_N \gamma)} \end{bmatrix} \end{equation}\end{split}\]

For the OLS estimator to be unbiased in this context, we need \(E[\mathbf{b}] = \beta\), so that (without showing steps in proof which is analogous to the OLS proof of unbiasedness):

\[\begin{split}\begin{align} E[\mathbf{b}] =& \beta +E[(\mathbf{x'x})^{-1}\rho \sigma_\epsilon \mathbf{x}'\mathbf{IMR} + (\mathbf{x'x})^{-1}\mathbf{x}'\epsilon] \\ =& \beta + \rho \sigma (\mathbf{x'x})^{-1}E[\mathbf{x}'\mathbf{IMR}] \end{align}\end{split}\]

since \(E[\mathbf{x}'\epsilon]=0\) and \(\rho\) and \(\sigma_\epsilon\) are unknown constants. If \(\rho \neq 0\), this term won’t be zero. Simplifying the last term from above further, we have

\[\begin{align} \rho \sigma (\mathbf{x'x})^{-1}E[\mathbf{x}'\mathbf{IMR}] = \rho \sigma (\mathbf{x'x})^{-1}(\mathbf{x}'\mathbf{IMR} + Cov(\mathbf{x}',\mathbf{IMR})) \end{align}\]
../_images/heckman_example_rho_8_rhoxy_0.png

Fig. 18 Case 3

For Case 3, the independent variables in the model \(\mathbf{x}\) and \(\mathbf{w}\) are uncorrelated so it is likely that \(Cov(\mathbf{x}',\mathbf{IMR})\) is very close to zero, since all variation in the relationship is being driven by the two sets of independent variables in the model (\(\mathbf{x}\) and \(\mathbf{w}\)). Setting \(Cov(\mathbf{x}',\mathbf{IMR})=0\), shows that there will still be bias since \(\mathbf{x}'\mathbf{IMR}\) won’t be zero. Here we are adding (or subtracting depending on the sign of \(\rho\)) a constant value to the model with mean \(\rho \sigma (\mathbf{x'x})^{-1} \mathbf{x}'\mathbf{IMR}\). So this miss-specified model’s constant term will need to depart from the true value \(\beta_0\) to account for this positive or negative shift in the mean value of all observation’s errors. However, since there is no correlation amongst \(\mathbf{x}\) and \(\mathbf{w}\) in our miss-specified OLS model we won’t impart missing variable bias on our slope coefficients. We see this in Case 3. Cases that are selected tend to have higher draws for \(\epsilon\). So the points that are censored tend to be below (for our case here) in a way that would only affect the intercept coefficient.

For Case 4, we have the bias imparted on our intercept coefficient as discussed above and bias induced by correlation between our model error \(\psi\) and \(\mathbf{x}\) (since \(\mathbf{x}\) and \(\mathbf{w}\) are correlated). This has the effect of also biasing our slope coefficients. Case 4 shows how this is happening visually. As in Case 3, the observations that tend to be included in the amounts equation (the dark points in the right panel) on average tend to be above the True Regression line. This is happening because in the selection equation we tend select observations having higher draws of \(u_i\). Since (for this example) \(u_i\) and \(\epsilon_i\) are positively correlated, we also tend to be selecting cases above the True regression line. Additionally, when \(\mathbf{w}\) and \(\mathbf{x}\) are correlated the cases we select tend to be (for this example) higher values of our independent variable \(\mathbf{x}\) (since higher values of \(\mathbf{z}\) tend to be selected). This has the effect of shifting the some of points that are censored (compared to Case 3) below the regression line which can lead to bias in slope and intercept if we apply OLS.

../_images/heckman_example_rho_8_rhoxy_8.png

Fig. 19 Case 4

Model Log-Likelihood

Defining person i’s contribution to the log-likelihood function (\(C_i\)) as

\[\begin{split}\begin{equation} \small C_i = \left\{ \begin{aligned} &ln\Phi \left(\frac{\mathbf{w_i \gamma} + \rho \left (\frac{y_i - \mathbf{x}_i \beta}{\sigma_\epsilon}\right)} {\sqrt{1 - \rho^2}}\right) - \frac{1}{2} \left( \frac{y_i - \mathbf{x}_i\beta}{\sigma_{\epsilon}} \right)^2-ln\left(\sigma_{\epsilon}\sqrt{2\pi}\right) \text{, $z_i$ = 1}\\ &ln\left( 1 - \Phi(\mathbf{w}_i\gamma) \right) \text{, $z_i$ = 0} \end{aligned} \right. \end{equation}\end{split}\]

The log-likelihood function is

\[\begin{equation} LogL = \sum_{i = 1}^N C_i \end{equation}\]

Source: Stata pdf manual for Heckman Model.

Two-step Estimation

There are two estimators one can employ. The first method (known as the two-step method) was the only practical way to estimate the model when the paper was first published in 1979. This method follows these steps:

  1. Run Probit on the Selection Model

  2. Recover Estimated Inverse Mills Ratio

  3. Using Odinary Least Squares, run the regression

    \[\begin{equation} y_i = \mathbf{x}_i \beta + \rho \sigma_\epsilon \frac{\phi(\mathbf{w}_i \hat{\gamma})}{\Phi(\mathbf{w}_i \hat{\gamma})} \end{equation}\]

    where \(\rho \sigma_\epsilon\) is treated as a single parameter to be estimated.

  4. “Back Out” separate estimates for \(\rho\) and \(\sigma_\epsilon\)

  5. Adjust standard errors to account for the fact that the Inverse Mills Ratio is an estimate (and hence random) covariate in the above model.

The key two steps are to first run a probit and using information from the results from that model estimate a corrected form of the OLS model. This is the only estimation method available in a beta branch of Python Statsmodels as of November 2018.

The second and preferred method is to use Maximum Likelihood over the full parameter set \(\beta, \gamma, \rho\), and \(\sigma\) in the log-likelihood function above. This is the default method in Stata.

Code for this Lecture

The python code generating the toy data for the figures above is given below. This version examines Case 4.1

# or ["remove_output", "remove_input"]
# start a connected stata17 session
from pystata import config
config.init('be')
config.set_streaming_output_mode('off')
import numpy as np
import pandas as pd

# true parameters
rho_t = np.array([0.8])
rho_x_w_t = np.array([0.8])
gamma_t = np.array([.5,1.0])
beta_t = np.array([-2.0,0.5])
sigma_e_t = np.array([1.0])
N =5000

# generate toy data consistent with heckman:
# generate potentially correlated x,w data
mean_x_w = np.array([0,0])
cov_x_w = np.array([[1,rho_x_w_t[0]],[rho_x_w_t[0], 1]])
w, x = np.random.multivariate_normal(mean_x_w, cov_x_w, N).T

# add constant to first position and convert to DataFrame
w_ = pd.DataFrame(np.c_[np.ones(N),w],columns=['Constant (Selection)','Slope (Selection)'])
x_ = pd.DataFrame(np.c_[np.ones(N),x], columns=['Constant','Slope'])

# generate errors 
mean_u_eps = np.array([0,0])
cov_u_eps = np.array([[1,rho_t[0]],[rho_t[0],sigma_e_t[0]]])
u, epsilon = np.random.multivariate_normal(mean_u_eps, cov_u_eps, N).T

# generate latent zstar
zstar = w_.dot(gamma_t) + u
# generate observed z (indicator=1 if zstar is positive)
z = zstar > 0  

# generate latent ystar
ystar = x_.dot(beta_t) + epsilon
y=ystar.copy()
# generate observed y [if z=0, set y to NaN]
y[~z] = np.nan

stata_data = pd.DataFrame(np.c_[y,z,x,w], columns=['y','z','x','w'])

We can estimate a Two-Step Heckman Model in Python using an unmerged branch from StatsModels (this replicates the Stata two-step results).

import heckman as heckman
res = heckman.Heckman(y, x_, w_).fit(method='twostep')
print(res.summary())
       Heckman Regression Results      
=======================================
Dep. Variable:                        y
Model:                          Heckman
Method:                Heckman Two-Step
Date:                  Mon, 17 May 2021
Time:                          12:54:35
No. Total Obs.:                    5000
No. Censored Obs.:                 1887
No. Uncensored Obs.:               3113
==============================================================================
                 coef    std err          z      P>|z|      [0.025      0.975]
------------------------------------------------------------------------------
Constant      -1.9901      0.038    -52.076      0.000      -2.065      -1.915
Slope          0.4937      0.024     20.312      0.000       0.446       0.541
========================================================================================
                           coef    std err          z      P>|z|      [0.025      0.975]
----------------------------------------------------------------------------------------
Constant (Selection)     0.4636      0.022     21.427      0.000       0.421       0.506
Slope (Selection)        0.9794      0.027     35.859      0.000       0.926       1.033
================================================================================
                   coef    std err          z      P>|z|      [0.025      0.975]
--------------------------------------------------------------------------------
IMR (Lambda)     0.8066      0.062     13.066      0.000       0.686       0.928
=====================================
rho:                            0.797
sigma:                          1.013
=====================================

First table are the estimates for the regression (response) equation.
Second table are the estimates for the selection equation.
Third table is the estimate for the coef of the inverse Mills ratio (Heckman's Lambda).

And in Stata, we can estimate the Full Information Maximum Likelihood model over the toy dataset as

%%stata -d stata_data
heckman y x, select(z=w)
Iteration 0:   log likelihood = -6342.6713  
Iteration 1:   log likelihood = -6341.5671  
Iteration 2:   log likelihood =  -6341.565  
Iteration 3:   log likelihood =  -6341.565  

Heckman selection model                         Number of obs     =      5,000
(regression model with sample selection)              Selected    =      3,113
                                                      Nonselected =      1,887

                                                Wald chi2(1)      =     669.86
Log likelihood = -6341.565                      Prob > chi2       =     0.0000

------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
y            |
           x |   .5086451   .0196527    25.88   0.000     .4701264    .5471637
       _cons |  -2.013987   .0250826   -80.29   0.000    -2.063148   -1.964826
-------------+----------------------------------------------------------------
z            |
           w |   .9853258   .0254098    38.78   0.000     .9355236    1.035128
       _cons |   .4686382   .0213757    21.92   0.000     .4267426    .5105338
-------------+----------------------------------------------------------------
     /athrho |   1.177133   .0628499    18.73   0.000     1.053949    1.300317
    /lnsigma |   .0257877   .0168219     1.53   0.125    -.0071826     .058758
-------------+----------------------------------------------------------------
         rho |   .8265454   .0199123                       .783337    .8618046
       sigma |   1.026123   .0172613                      .9928431    1.060519
      lambda |   .8481373   .0316568                      .7860911    .9101836
------------------------------------------------------------------------------
LR test of indep. eqns. (rho = 0): chi2(1) = 259.62       Prob > chi2 = 0.0000

As discussed in class, the command bundles the likelihood ratio test statistic (at the bottom) for testing whether \(\rho=0\) (the OLS Model could be applied) or not (the Heckman Model should be applied).

Outputs from the Heckman Model

Once we have run the model, we can use the results for creating parameter values that we might want to use for verifying our understanding of the Heckman Model.

Parameter

Stata Command

\(\sigma\)

e(sigma)

\(\rho\)

e(rho)

\(log(L)\)

e(ll)

\(\beta_{name}\)

_b[name]

\(\gamma_{name}\)

_b[z:name] (where z is referring to your selection variables)

\(\mathbf{x}\hat{\beta}\)

predict xb, xb

\(\mathbf{w}\hat{\gamma}\)

predict zg, xbsel

One thing worth noting is that we have two sets of parameters being maintained in the background by stata. To extract individual ones, given our variable names use syntax like this

%%stata
gen beta_x = _b[x]
di "beta x coefficient: "beta_x
gen gamma_w = _b[z:w]
di "gamma w coefficient:", gamma_w
. gen beta_x = _b[x]

. di "beta x coefficient: "beta_x
beta x coefficient: .50864506

. gen gamma_w = _b[z:w]

. di "gamma w coefficient:", gamma_w
gamma w coefficient: .98532575

. 

Usually for this class, you will be working with the linear predictor, so you’ll be calculating \(\mathbf{w}\gamma\) and \(\mathbf{x}\beta\) using

%%stata
predict zg, xbsel
predict xb, xb
. predict zg, xbsel

. predict xb, xb

. 

Expected Value

Formula

Stata Command

\(E[y \shortmid y\hspace{.03in} obs]\)

\(\mathbf{x}_i \beta^{H} + \rho \sigma_{\epsilon} \frac{\phi(\mathbf{w}_i \gamma)}{\Phi(\mathbf{w}_i \gamma)}\)

margins, predict(ycond)

predict ycond, ycond

\(Prob(Not Censored)\)

\(\Phi(\mathbf{w}_i \gamma)\)

margins, pr(psel)

predict probobs, pr(psel)

\(E[y]\)

\(\Phi(\mathbf{w}_i \gamma)\left[\mathbf{x}_i \beta^H + \rho \sigma_{\epsilon} \frac{\phi(\mathbf{w}_i \gamma)}{\Phi(\mathbf{w}_i \gamma)}\right]\)

margins, predict(yexpected)

predict y, yexpected

\(E[y^*]\)

\(\mathbf{x}_i \beta^{H}\)

margins, xb

predict xb, xb

Noting that \(\beta^H\) are the estimates of \(\beta\) from the Heckman Model.

For example, if we want to predict \(E[y \shortmid y\hspace{.03in} obs]\) for each observation we can just issue this command:

%%stata
predict ycond, ycond
list y ycond in 1/10
. predict ycond, ycond

. list y ycond in 1/10

     +------------------------+
     |          y       ycond |
     |------------------------|
  1. | -1.6150838   -1.488948 |
  2. |          .   -1.601148 |
  3. |  -2.743624   -1.757029 |
  4. | -1.4453615    -1.15765 |
  5. | -1.9769977   -1.660356 |
     |------------------------|
  6. | -2.0649865   -1.152555 |
  7. |          .   -1.264049 |
  8. | -1.8709723   -1.669293 |
  9. |  -1.467229   -1.609621 |
 10. |          .   -1.407274 |
     +------------------------+

. 

1

Note, we are using Stata 17 Jupyter Notebook integration using the %%stata magic.