Chapter 8 Limited Dependent Variables

8.1 Introduction

Limited response variables: Mixed discrete and continuous features of the dependent variable.

Typical sources: Censoring and truncation.

Economic explanations:

  • Corner solutions: Utility-maximizing choice of individuals is at the corner of the budget set, typically zero.
  • Sample selection: Data deficiency causes response to be unknown (or different) for subsample.
  • Treatment effects: Response for each individual is only observable for one level of a “treatment” variable.

Examples: Typical economic examples for limited responses.

  • Labor supply.
    \(y_i\): Number of hours worked by person \(i\).
    Potential covariates: Education, non-work income, …

  • Expenditures for health services.
    \(y_i\): Expenditures of person \(i\) for health services last month.
    Potential covariates: Health status, income, gender, …

  • Wage equations.
    \(y_i\): Earnings of person \(i\) derived from tax returns.
    Potential covariates: Education, previous experience, …

  • Unemployment programs.
    \(y_i\): Wage of person \(i\) after unemployment.
    Treatment: Training program (potentially non-random assignment).
    Potential covariates: Occupation, duration of unemployment, …

8.1.1 Example: PSID 1976 (Mroz data)

Cross-section d from the 1976 Panel Study of Income Dynamics (PSID), based on data for the previous year, 1975.

A data frame containing 753 observations on 21 variables, including:

Variable Description
participation Did the individual participate in the labor force in 1975? (equivalent to wage \(> 0\) or hours \(> 0\))
hours Wife’s hours of work in 1975.
youngkids Number of children less than 6 years old.
oldkids Number of children between ages 6 and 18.
age Wife’s age in years.
education Wife’s education in years.
wage Wife’s average hourly wage, in 1975 USD.
fincome Family income, in 1975 USD.

Example: Corner solution

Example: Sample selection

8.2 Tobin’s Corner Solution Model

Problem: As motivated above, many microeconomic variables have

  • non-negative values,
  • a cluster of observations at zero.

Note: OLS cannot be used sensibly due to

  • possibly negative predictions,
  • imposed constant marginal effects (which cannot be remedied by logs as \(\log(0)\) is not defined).

Quantities of interest:

  • \(P(y = 0 ~|~ x) = 1 - P(y > 0 ~|~ x)\), probability of zero.
  • \(E(y ~|~ y > 0,~ x)\) expectation conditional on positive \(y\).
  • \(E(y ~|~ x) = P(y > 0 ~|~ x) \cdot E(y ~|~ y > 0,~ x)\) expectation.

Ideas:

  • Two-part model with binary selection and truncated outcome.
  • Latent variable driving both selection and outcome.

Tobin’s solution: Employ the latter approach with latent Gaussian variable \(y^*\) and observed discrete-continuous response \(y\):

\[\begin{eqnarray*} y^* & = & x^\top \beta ~+~ \varepsilon, \qquad \varepsilon ~|~ x \sim \mathcal{N}(0, \sigma^2),\\ y\phantom{^*} & = & \max(0, y^*), \end{eqnarray*}\]

i.e., \(y\) is a censored version of \(y^*\).

Likelihood:

\[\begin{equation*} L(\beta, \sigma; y, x) ~=~ \prod_{i = 1}^n f(y_i ~|~ x_i, \beta, \sigma)^{I(y_i > 0)} \cdot P(y_i = 0 ~|~ x_i)^{I(y_i = 0)} \end{equation*}\]

where

\[\begin{eqnarray*} P(y_i = 0 ~|~ x_i) & = & P(y_i^* \le 0 ~|~ x_i) \\ & = & \Phi \left(\frac{0 - x_i^\top \beta}{\sigma} \right) ~=~ 1 - \Phi \left(\frac{x_i^\top \beta}{\sigma} \right) \\ f(y_i ~|~ x_i, \beta, \sigma) & = & \frac{1}{\sigma} \cdot \phi \left( \frac{y_i - x_i^\top \beta}{\sigma} \right) \end{eqnarray*}\]

Remarks:

  • Model is known as tobit model.
  • Despite the name it does not belong to the same model class as the binary logit and probit models.
  • Log-likelihood can be shown to be globally concave.
  • MLE is well-behaved (asymptotically normal etc.).

In R: tobit() from package AER. Convenience interface to survreg() from package survival which provides much more general regression tools for censored responses.

Alternatively: crch(..., left = 0) from package crch for censored regression with conditional heteroscedasticity. Dedicated interface for tobit-type models with either censoring or truncation, different response distributions (beyond normal).

Example: Regression for annual hours of work in PSID 1976 data, using reduced-form specification. Wage is missing as a regressor (as unobserved for non-working subsample).

Model equation of interest:

hours_f <- hours ~ nwincome + education +
  experience + I(experience^2) + age + youngkids + oldkids

Fitting tobit model and naive OLS models:

hours_tobit <- tobit(hours_f, data = PSID1976)
hours_ols1 <- lm(hours_f, data = PSID1976)
hours_ols2 <- lm(hours_f, data = PSID1976,
  subset = participation == "yes")

Compare coefficients:

sapply(list("Tobit" = hours_tobit, "OLS (all)" = hours_ols1,
  "OLS (positive)" = hours_ols2), coef)
##                    Tobit OLS (all) OLS (positive)
## (Intercept)      965.305 1330.4824      2056.6428
## nwincome          -8.814   -3.4466         0.4439
## education         80.646   28.7611       -22.7884
## experience       131.564   65.6725        47.0051
## I(experience^2)   -1.864   -0.7005        -0.5136
## age              -54.405  -30.5116       -19.6635
## youngkids       -894.022 -442.0899      -305.7209
## oldkids          -16.218  -32.7792       -72.3667

8.2.1 Truncated normal distribution

Question: How can these models be interpreted?

Needed: Better understanding of truncated (normal) distributions.

Probability density function: Random variable \(y\) with density function \(f(y)\), truncated from below at \(c\). General form and \(\mathcal{N}(\mu, \sigma^2)\) distribution.

\[\begin{eqnarray*} f(y ~|~ y > c) & = & \frac{f(y)}{P(y > c)} ~=~ \frac{f(y)}{1 - F(c)} \\ & = & \frac{1}{\sigma} \cdot \phi\left( \frac{y - \mu}{\sigma} \right) \left/ \left\{ 1 - \Phi \left(\frac{c - \mu}{\sigma} \right) \right\} \right. \end{eqnarray*}\]

Example: Standard normal distribution truncated at \(c = -1\) and \(c = 0\).

Expectation: For \(\varepsilon \sim \mathcal{N}(0, 1)\).

\[\begin{eqnarray*} E(\varepsilon ~|~ \varepsilon > c) & = & \int_c^\infty \varepsilon \cdot f(\varepsilon ~|~ \varepsilon > c) ~d \varepsilon \\ & = & \frac{1}{1 - \Phi(c)} ~ \int_c^\infty \varepsilon \cdot \phi(\varepsilon) ~d \varepsilon \\ & = & \frac{1}{1 - \Phi(c)} ~ \left\{ \left. - \phi(\varepsilon) \phantom{\frac{.}{.}} \! \right|_c^\infty \right\} \\ & = & \frac{\phi(c)}{1 - \Phi(c)} \end{eqnarray*}\]

Because \(\phi'(x) = -x \cdot \phi(x)\).

Note: This simple solution does not hold in general.

Inverse Mills ratio: Defined as

\[\begin{equation*} \lambda(x) ~=~ \frac{\phi(x)}{\Phi(x)}. \end{equation*}\]

Hence, due to symmetry of the normal distribution, the following holds for \(\varepsilon \sim \mathcal{N}(0, 1)\) and \(y \sim \mathcal{N}(\mu, \sigma^2)\), respectively:

\[\begin{eqnarray*} E(\varepsilon ~|~ \varepsilon > c) & = & \lambda(-c), \\ E(y ~|~ y > c) & = & \mu ~+~ \sigma \cdot \lambda\left(\frac{\mu - c}{\sigma}\right). \end{eqnarray*}\]

Example: \(y \sim \mathcal{N}(\mu, 1)\) with truncation at \(c = 0\).

8.2.2 Interpretation of the tobit model

Uninteresting: Expectation of latent variable is straightforward but lacks substantive interpretation.

\[\begin{equation*} E(y^* ~|~ x) ~=~ x^\top \beta \end{equation*}\]

Of interest: Depending on substantive question, consider

\[\begin{eqnarray*} P(y > 0 ~|~ x) & = & \Phi(x^\top \beta / \sigma),\\ E(y ~|~ y > 0,~ x) & = & x^\top \beta ~+~ \sigma \cdot \lambda(x^\top \beta / \sigma),\\ E(y ~|~ x) & = & P(y > 0) \cdot E(y ~|~ y > 0,~ x) \\ & = & \Phi(x^\top \beta / \sigma) \cdot \left\{ x^\top \beta ~+~ \sigma \cdot \lambda(x^\top \beta / \sigma) \right\}. \end{eqnarray*}\]

Example: For \(\sigma = 1\).

Marginal effects:

\[\begin{eqnarray*} \frac{\partial P(y > 0 ~|~ x)}{\partial x_l} & = & \phi(x^\top \beta / \sigma) \cdot \beta_l / \sigma\\[0.5cm] \frac{\partial E(y ~|~ y > 0,~ x)}{\partial x_l} & = & \beta_l \cdot \left[ 1 - \lambda(x^\top \beta / \sigma) ~ \{ x^\top \beta / \sigma ~+~ \lambda(x^\top \beta / \sigma) \}\right] \\[0.5cm] \frac{\partial E(y ~|~ x)}{\partial x_l} & = & \frac{\partial P(y > 0 ~|~ x)}{\partial x_l} \cdot E(y ~|~ y > 0,~ x) ~+~ \\ & & P(y > 0 ~|~ x) \cdot \frac{\partial E(y ~|~ y > 0,~ x)}{\partial x_l} \\ & = & \beta_l \cdot \Phi(x^\top \beta / \sigma) \\ \end{eqnarray*}\]

Remarks:

  • Overall effect on \(E(y ~|~ x)\) is sum of an effect at the
    • extensive margin (e.g., how much more likely a person is to join the labor force as education increases times the expected hours) and an effect at the
    • intensive margin (e.g., how much the expected hours of work increase for workers as education increases times the probability of participation).
  • OLS estimation (both for full and positive sample) is biased towards zero due to inability to capture non-constant marginal effects.
  • Effects (e.g., at mean regressors) can again be considered instead of marginal effects.

Example: Effects in tobit model for PSID 1976 data (hand-crafted).

Specification issues:

  • The classical (uncensored) OLS regression model is rather robust to misspecifications of distributional form and even non-constant variances. Estimates remain consistent.
  • In the tobit model, this is not the case. Misspecification of the likelihood (including constant variances) will lead to misspecification of the score. Hence, \(E(y ~ |~ y > 0,~ x)\) and \(E(y ~|~ x)\) are misspecified and estimates inconsistent.
  • Crucial assumption: The same latent process drives the probability of a corner solution and the expectation of positive outcomes.
  • Example: A single factor, such as number of old kids, can not increase the probability to work but decrease the expected number of hours conditional on participation in the labor force.
  • Check for the latter: Cragg two-part model.

8.3 Cragg Two-Part Model

Idea: Similar to hurdle model for count data, employ two parts.

  • Is \(y\) equal to zero or positive?
  • If \(y > 0\), how large is \(y\)?

Formally: Likelihood has two separate parts.

  • \(P(y > 0 ~|~ x)\): Binomial model (typically probit) for \(I(y > 0)\).
  • \(E(y ~|~ y > 0,~ x)\): Truncated normal model for \(y\) given \(y > 0\).

Remarks:

  • The tobit model is nested in the two-part model with probit link. Thus, LR tests etc. can be easily performed.
  • If the tobit model is correctly specified, estimates from the probit model \(\hat \gamma\) would be consistent (but less efficient) for the scaled coefficients from the tobit model. Thus, \(\hat \beta / \hat \sigma\) should be similar to \(\hat \gamma\) in empirical samples.

In R: Employ separate modeling function. glm() with "probit" link can be used for selection part.

part_f <- update(hours_f, participation ~ .)
part_probit <- glm(part_f, data = PSID1976,
  family = binomial(link = "probit"))
coeftest(part_probit)
## 
## z test of coefficients:
## 
##                 Estimate Std. Error z value Pr(>|z|)
## (Intercept)      0.27007    0.50808    0.53   0.5950
## nwincome        -0.01202    0.00494   -2.43   0.0149
## education        0.13090    0.02540    5.15  2.6e-07
## experience       0.12335    0.01876    6.58  4.8e-11
## I(experience^2) -0.00189    0.00060   -3.15   0.0017
## age             -0.05285    0.00846   -6.25  4.2e-10
## youngkids       -0.86833    0.11838   -7.34  2.2e-13
## oldkids          0.03601    0.04403    0.82   0.4135

truncreg() from package truncreg (or trch() from package crch) can be employed for estimating the truncated normal regression model.

library("truncreg")
hours_trunc <- truncreg(hours_f, data = PSID1976,
  subset = participation == "yes")
coeftest(hours_trunc)
## 
## z test of coefficients:
## 
##                 Estimate Std. Error z value Pr(>|z|)
## (Intercept)     2055.713    461.265    4.46  8.3e-06
## nwincome          -0.501      4.935   -0.10  0.91912
## education        -31.270     21.796   -1.43  0.15139
## experience        73.007     20.229    3.61  0.00031
## I(experience^2)   -0.970      0.581   -1.67  0.09521
## age              -25.336      7.885   -3.21  0.00131
## youngkids       -318.852    138.073   -2.31  0.02093
## oldkids          -91.620     41.202   -2.22  0.02617
## sigma            822.479     39.320   20.92  < 2e-16

Comparison of (scaled) parameter estimates from tobit, probit, and truncated model.

cbind("Censored"  = coef(hours_tobit) / hours_tobit$scale,
      "Threshold" = coef(part_probit),
      "Truncated" = coef(hours_trunc)[1:8] / coef(hours_trunc)[9])
##                  Censored Threshold  Truncated
## (Intercept)      0.860327  0.270074  2.4994098
## nwincome        -0.007856 -0.012024 -0.0006093
## education        0.071875  0.130904 -0.0380188
## experience       0.117256  0.123347  0.0887641
## I(experience^2) -0.001661 -0.001887 -0.0011788
## age             -0.048488 -0.052852 -0.0308044
## youngkids       -0.796795 -0.868325 -0.3876719
## oldkids         -0.014454  0.036006 -0.1113943

LR test and information criteria:

loglik <- c("Tobit" = logLik(hours_tobit),
  "Two-Part" = logLik(hours_trunc) + logLik(part_probit))
df <- c(9, 9 + 8)
-2 * loglik + 2 * df
##    Tobit Two-Part 
##     7656     7620
-2 * loglik + log(nrow(PSID1976)) * df
##    Tobit Two-Part 
##     7698     7698
pchisq(2 * diff(loglik), diff(df), lower.tail = FALSE)
##  Two-Part 
## 1.273e-08

8.4 Sample Selection Models

Idea: Employ censored regression model for observations \(y ~|~ y > c\) but model censoring threshold \(c\) as random variable.

Name: Incidental censoring or self selection.

Example: Distribution of wages.

  • Wages are only observed for workers.
  • Thus, the wages that non-workers would receive (if they decided to work) are unknown/latent.
  • Individuals decide to work if their (possibly latent) offered wage \(w_0\) exceeds their individual reservation wage \(w_r\).
  • The distribution of wages for workers are thus \(f(w_o ~|~ w_o > w_r)\) with random \(w_r\).

Formally: With \(y\) and \(c\) stochastic rewrite

\[\begin{equation*} f(y ~|~ y > c) ~=~ f(y ~|~ y - c > 0) ~=~ f(y_1 ~|~ y_2 > 0). \end{equation*}\]

Special cases:

  • \(y_1 = y_2\): Standard censoring regression (i.e., tobit).
  • \(y_1\) and \(y_2\) independent: \(f(y_1 ~|~ y_2 > 0) = f(y_1)\) (i.e., no selection).

Simple approach: Bivariate normal distribution for \((y_1, y_2)\) with correlation \(\varrho\), including special cases \(|\varrho| = 1\) (identity) and \(\varrho = 0\) (independence).

Alternatively: Employ different bivariate distribution or semiparametric approach.

Regression model: Employ (potentially overlapping) regressors for mean of both components.

\[\begin{eqnarray*} \mbox{Outcome equation:} & y_1 ~= & x^\top \beta ~+~ \varepsilon_1, \\ \mbox{Selection equation:} & y_2 ~= & z^\top \gamma ~+~ \varepsilon_2, \end{eqnarray*}\]

where

\[\begin{equation*} \left( \begin{array}{cc} \varepsilon_1 \\ \varepsilon_2 \end{array} \right) ~\sim~ \mathcal{N}\left( \left( \begin{array}{cc} 0 \\ 0 \end{array} \right), \left( \begin{array}{cc} \sigma^2 & \varrho \\ \varrho & 1 \end{array} \right) \right). \end{equation*}\]

Remarks:

  • Observation rule: \(y_1\) is non-censored, observed for \(y_2 > 0\).
  • In selection equation, only \(y_2 > 0\) vs. \(y_2 \le 0\) is observed. Hence, standardize \(Var(y_2) = 1\) for identifiability (as in probit model).
  • Terminology: Heckman model, heckit model, sample selection model, self-selection model, tobit-2 model.

Conditional expectation:

\[\begin{equation*} E(y_1 ~|~ y_2 > 0,~ x) ~=~ x^\top \beta ~+~ \sigma \cdot \varrho \cdot \lambda(z^\top \gamma). \end{equation*}\]

Selection effect: Sample is

  • positively selected for \(\varrho > 0\): \(E(y_1 ~|~ y_2 > 0,~ x) > E(y_1 ~|~ x)\),
  • negatively selected for \(\varrho < 0\),
  • and no selection occurs for \(\varrho = 0\).

Estimation:

  • Efficient: Full maximum likelihood.
  • Biased: OLS for non-censored observations.
  • Consistent (and employed only for historical reasons): Two-step estimator. Probit model yielding \(\hat \gamma\) followed by OLS regressing \(y\) on \(x\) and \(\lambda(z^\top \hat \gamma)\).

Classical example: Estimation of mean wage offers (or worker’s productivity) from a sample of workers and wages.

In this context: Sample selection model called Gronau-Heckman-Roy model (by Winkelmann & Boes 2009) due to the authors that first recognized this problem and established its inference, respectively.

Identification: Economic considerations guide selection of variables \(x\) that affect the outcome and variables \(z\) that affect the selection.

  • Typically, all variables \(x\) for the outcome also affect selection, especially in the case of “self selection”.
  • Additional variables might affect the selection but not the outcome. Example: Presence of small children will affect selection but probably not the reservation wage.
  • Such variables play a similar role as “instruments”.

Example: Female wages in PSID 1976 data. Employ semi-logarithmic wage equation with education and quadratic polynomial in experience.

wage_f <- log(wage) ~ education + experience + I(experience^2)

OLS estimation: For selected subsample.

wage_ols <- lm(wage_f, data = PSID1976, subset = participation == "yes")

Sample selection model: Additionally employ age, number of old and young children, and other income as regressors in selection equation (corresponding to previously used part_f).

In R available in selection() from package sampleSelection:

library("sampleSelection")
wage_ghr <- selection(part_f, wage_f, data = PSID1976)

Sample selection models

coeftest(wage_ghr)
## 
## z test of coefficients:
## 
##                  Estimate Std. Error z value Pr(>|z|)
## (Intercept)      0.266449   0.508958    0.52   0.6006
## nwincome        -0.012132   0.004877   -2.49   0.0129
## education        0.131341   0.025382    5.17  2.3e-07
## experience       0.123282   0.018724    6.58  4.6e-11
## I(experience^2) -0.001886   0.000600   -3.14   0.0017
## age             -0.052829   0.008479   -6.23  4.7e-10
## youngkids       -0.867399   0.118651   -7.31  2.7e-13
## oldkids          0.035872   0.043475    0.83   0.4093
## (Intercept)     -0.552696   0.260379   -2.12   0.0338
## education        0.108350   0.014861    7.29  3.1e-13
## experience       0.042837   0.014879    2.88   0.0040
## I(experience^2) -0.000837   0.000417   -2.01   0.0449
## sigma            0.663398   0.022707   29.21  < 2e-16
## rho              0.026607   0.147078    0.18   0.8564

Interpretation: \(\hat \varrho\) is essentially zero signaling that there are no significant selection effects. Outcome part of Gronau-Heckman-Roy model and OLS for selected subsample are thus virtually identical.

cbind("GHR (outcome)" = coef(wage_ghr, part = "outcome"),
  "OLS (positive)" = coef(wage_ols))
##                 GHR (outcome) OLS (positive)
## (Intercept)        -0.5526963     -0.5220406
## education           0.1083502      0.1074896
## experience          0.0428368      0.0415665
## I(experience^2)    -0.0008374     -0.0008112

Similarly: Selection part of Gronau-Heckman-Roy model and probit for selection are thus virtually identical.

cbind("GHR (selection)" = coef(wage_ghr)[1:8],
  "Probit (0 vs. positive)" = coef(part_probit))
##                 GHR (selection) Probit (0 vs. positive)
## (Intercept)            0.266449                0.270074
## nwincome              -0.012132               -0.012024
## education              0.131341                0.130904
## experience             0.123282                0.123347
## I(experience^2)       -0.001886               -0.001887
## age                   -0.052829               -0.052852
## youngkids             -0.867399               -0.868325
## oldkids                0.035872                0.036006

Effects: Compare education effects graphically.

Set up auxiliary data with average regressor and varying education.

X <- model.matrix(hours_tobit)[, -c(1, 5)]
psid <- lapply(colnames(X), function(i)
  if(i != "education") mean(X[,i]) else
    seq(from = min(X[,i]), to = max(X[,i]), length = 100))
names(psid) <- colnames(X)
psid <- do.call("data.frame", psid)

Predictions for OLS:

wage_ols_out <- predict(wage_ols, newdata = psid)

Compute predictions for sample selection model by hand:

xb <- model.matrix(delete.response(terms(wage_f)),
  data = psid) %*% coef(wage_ghr, part = "outcome")
zg <- model.matrix(delete.response(terms(part_f)), data = psid) %*%
    head(coef(wage_ghr), 8)    
wage_ghr_out <- xb + coef(wage_ghr)["sigma"] *
  coef(wage_ghr)["rho"] * dnorm(zg)/pnorm(zg)