Question

For a sample of 600 married females, we are interested in explaining participation in market employment from exogenous characteristics in \(x_{i}\) (age, family composition, education). Let \(y_{i}=1\) if person \(i\) has a paid job and 0 otherwise. Suppose we estimate a linear regression model $$ y_{i}=x_{i}^{\prime} \beta+\varepsilon_{i} $$ by ordinary least squares. (a) Give two reasons why this is not really an appropriate model. As an alternative, we could model the participation decision by a probit model. (b) Explain the probit model. (c) Give an expression for the loglikelihood function of the probit model. (d) How would you interpret a positive \(\beta\) coefficient for education in the probit model? (e) Suppose you have a person with \(N\). What is your prediction for her labour market status \(y_{i}\) ? Why? (f) To what extent is a logit model different from a probit model? Now assume that we have a sample of women who are not working \(\left(y_{i}=0\right)\), part-time working \(\left(y_{i}=1\right)\) or full-time working \(\left(y_{i}=2\right)\). (g) Is it appropriate, in this case, to specify a linear model as \(y_{i}=x_{i}^{\prime} \beta+\varepsilon_{i} ?\) (h) What alternative model could be used instead that exploits the information contained in part-time versus full-time working? (i) How would you interpret a positive \(\beta\) coefficient for education in this latter model? (j) Would it be appropriate to pool the two outcomes \(y_{i}=1\) and \(y_{i}=2\) and estimate a binary choice model? Why or why not?

Short answer

A linear model is not suitable for binary or categorical outcomes due to range and variance issues. The Probit model uses the normal CDF for probabilities and can be expressed with a specific loglikelihood function. An ordered model is better for distinctions between part-time and full-time work.

Step-by-step solution

Identifying Inappropriateness of a Linear Model

The linear regression model for the binary outcome variable, market employment, is not appropriate because:1. The model can predict probabilities outside the [0, 1] range.2. It assumes homoscedasticity (constant variance across observations), which is often not true for binary outcomes.

Explaining the Probit Model

In a Probit model, the probability that the binary outcome variable equals 1 is modeled using the standard normal cumulative distribution function (CDF). This can be expressed as: \[P(y_i = 1|x_i) = \text{CDF}(x_i' \beta)\]

Expressing the Log-Likelihood Function

The loglikelihood function for the Probit model can be written as:\[\text{LL}(\beta) = \text{log}\bigg(\text{CDF}(x_i' \beta)\bigg)^{y_i}\bigg(1 - \text{CDF}(x_i' \beta)\bigg)^{1 - y_i}\bigg)\]

Interpreting Positive \(\beta\) for Education

A positive \(\beta\) coefficient for education in the Probit model indicates that higher education is associated with a higher probability of having a paid job.

Prediction for a Hypothetical Person

For a person with a given set of characteristics \(x_i\), the prediction for her labor market status \(\hat{y}_i\) would be:\[\text{Predicted probability} = \text{CDF}(x_i' \beta)\]The predicted status \(y_i\) would be 1 if this probability is above 0.5 and 0 otherwise.

Logit vs Probit Model Differences

The Logit model uses the logistic function for the probability that \(y_i = 1\), while the Probit model uses the standard normal CDF. Though their outputs are often similar, the choice depends on theoretical considerations.

Suitability of a Linear Model

For cases where \(y_i\) can take values 0, 1, or 2, a linear model is not suitable because it does not capture the ordering of the outcomes or the differences between part-time and full-time work.

An Alternative Model for Ordered Outcomes

An ordered Probit or Logit model should be used to properly exploit the information contained in part-time versus full-time working due to the natural ordering of the outcomes.

Interpreting Positive \(\beta\) for Education in Ordered Model

In an ordered model, a positive \(\beta\) coefficient for education suggests that higher education increases the likelihood of transitioning to a higher category (e.g., from not working to part-time, or part-time to full-time).

Pooling Outcomes

Pooling the outcomes \(y_i = 1\) and \(y_i = 2\) into a binary choice model ignores the distinct difference between part-time and full-time work and can lead to loss of information and biased estimates.

Key concepts

Binary Outcome Variables

Binary outcome variables are variables that can take on only one of two possible outcomes. In econometrics, they are often used to model decisions or classifications, such as whether a person is employed or not.
For example, a variable can take a value of 1 if a person has a paid job (employed) and 0 otherwise (not employed).

• These types of variables are common in many areas of social sciences where binary outcomes (yes/no decisions) are analyzed.
• Common examples include voting behavior, disease prevalence, or any on/off condition.

Since binary outcomes are limited to two states, specific statistical models like the Probit and Logit models are more appropriate than linear regression models.

Log-Likelihood Function

The log-likelihood function in econometrics is used to estimate the parameters (coefficients) of a model.
For binary outcome models like the Probit model, the log-likelihood function compares the predicted probabilities to the actual outcomes.

• The general form of the log-likelihood function for the Probit model is:

\[\text{LL}(\beta) = \text{log}\bigg(\text{CDF}(x_i' \beta)\bigg)^{y_i}\bigg(1 - \text{CDF}(x_i' \beta)\bigg)^{1 - y_i}\bigg)\bigg\text{LL(\beta)} \]
This function is maximized to find the parameter values (\beta) that make the observed data most probable.
In simpler terms, it helps in getting the best fitting model by checking how well the predicted probabilities match the actual outcomes.

Ordered Probit Model

The ordered Probit model is an extension of the regular Probit model that is used for ordinal dependent variables.
In cases where the outcome variable is not just binary but ordered (e.g., not working, part-time working, full-time working), the ordered Probit model is appropriate.

• It can handle more than two categories and respects the inherent order of these categories.
• Thus, it provides more accurate results compared to treating the categories as purely nominal.

For example, in the case where women are classified into not working, part-time working, or full-time working, the ordered Probit model would account for the progression from part-time to full-time work.

Linear Regression Limitations

Linear regression has limitations when used for binary outcome variables.

• First, it can predict probabilities outside the [0, 1] range, which is not meaningful for probabilities.
• Second, it assumes homoscedasticity (constant variance), which often does not hold for binary outcomes where the variance varies with the mean.

Using linear regression for binary outcomes also overlooks the fact that these types of data are better suited to models that can address the binary nature of the dependent variable, such as the Probit or Logit models.

Interpretation of Coefficients in Econometrics

Interpreting coefficients in econometrics, especially for models like Probit or Logit, can be nuanced.

• In the Probit model, a positive coefficient \(\beta\) for a variable (e.g., education) means that higher values of this variable are associated with a higher probability of the dependent variable being 1 (e.g., being employed).
• The magnitude of \(\beta\) provides information on the strength and direction of this relationship.

For example, a positive \(\beta\) for education would imply that more education leads to a higher probability of employment.
However, unlike linear regression, the actual effect size on the probability isn't directly given by the coefficient but involves the standard normal cumulative distribution function.