Question

A top university requires all students that apply to do an entry exam. Students who obtain a score of less than 100 are not admitted. For students who score above 100 , the scores are registered, after which the university selects students from this group for admittance. We have a sample of 500 potential students who did their entry exam in 2010 . For each student, we observe the result of the exam being: \- 'rejected', if the score is less than 100 , or \- the score, if it is 100 or more. In addition, we observe background characteristics of each candidate, including parents' education, gender and the average grade at high school. The dean is interested in the relationship between these background characteristics and the score for the entry exam. He specifies the following model \(y_{i}^{*}=\beta_{0}+x_{i}^{\prime} \beta_{1}+\varepsilon_{i}\), \(y_{i}=y_{i}^{*}\) \(\quad\) if \(y_{i}^{*} \geq 100\) \(\quad=\) 'rejected' \(\quad\) if \(y_{i}^{*}<100\), where \(y_{i}\) is the observed score of student \(i\) and \(x_{i}\) is the vector background characteristics (excluding an intercept). (a) Show that the above model can be written as the standard tobit model (tobit I). (b) First, the dean does a regression of \(y_{i}\) upon \(x_{i}\) and a constant (by OLS), using the observed scores of 100 and more \(\left(y_{i} \geq 100\right)\). Show that this approach does not lead to consistent or unbiased estimators for \(\beta_{1}\). (c) Explain in detail how the parameter vector \(\beta=\left(\beta_{0}, \beta_{1}^{\prime}\right)\) ' can be estimated consistently, using the observed scores only. (d) Explain how you would estimate this model using all observations. Why is this estimator preferable to the one of \(\mathbf{c} ?\) (No proof or derivations are required.) (e) The dean considers specifying a tobit II model (a sample selection \(\mathrm{~ m o d e l ) . ~ D e s c r i b e ~ t h i s ~ m o d e l .}\)

Short answer

The Tobit I model uses censored regression for latent exam scores. OLS is biased because it ignores censored data. Consistent estimates require Tobit modeling with MLE, using all observations including censored ones.

Step-by-step solution

- Understanding the Tobit Model

The Tobit model, also known as censored regression model, is used when the dependent variable is not always fully observed. In this case, the exam scores are censored at 100.

- Rewriting the Given Model

Rewrite the given model in the context of the Tobit I model. The Tobit I model can be specified as: \[ y_{i}^{*} = \beta_{0} + x_{i}^{\prime} \beta_{1} + \varepsilon_{i} \] \[ y_{i} = y_{i}^{*}, \quad if \quad y_{i}^{*} \geq 100 \quad otherwise \quad y_{i} = 100 \] In this case, the observed score is the actual latent score when it is above 100 but is censored at 100 otherwise.

- Inconsistency of OLS Estimation

OLS regression on the observed scores does not provide consistent estimators because it only uses data where scores are 100 or more, ignoring censored data. This leads to biased estimates since the censoring threshold affects the distribution of the observed dependent variable.

- Consistent Estimation with Tobit Model

To estimate the parameter vector \(\beta = (\beta_{0}, \beta_{1}^{\prime})'\) consistently, we must use the Maximum Likelihood Estimation (MLE). The Tobit model accounts for the censored nature of the data, allowing us to estimate parameters considering both censored and uncensored observations.

- Using All Observations

To use all observations for estimation, employ the full Tobit model (Tobit II). This model incorporates both censored and uncensored data during estimation. It gives more efficient and unbiased estimations compared to only using uncensored observations.

- Describing Tobit II Model

The Tobit II model, or sample selection model, separates the selection process from the outcome process. It explains both whether a score is observed (selection equation) and, when observed, what the score is (outcome equation). This two-part model allows for better handling of selection bias in the data.

Key concepts

Censored Data

Censored data occurs when the dependent variable in a dataset is only partially observed. For example, in our exercise, exam scores below 100 are recorded simply as 'rejected.' This kind of censoring means we don't have full information on the actual scores for those students.
This poses a challenge for statistical analysis because standard methods like OLS (Ordinary Least Squares) assume that all values of the dependent variable are fully observed.
Ignoring censored data can lead to biased results and improper inferences about the relationship between independent variables and the dependent variable.

• The Tobit model is designed specifically to handle censored data situations.
• It includes adjustments so that estimates account for the censored nature of the data.

Maximum Likelihood Estimation (MLE)

MLE is a method used for estimating the parameters of a statistical model. It finds the values of the parameters that make the observed data most probable.
In the context of our Tobit model, we use MLE to estimate the parameters even when some of the data is censored.
MLE works by constructing a likelihood function, which represents the probability of observing the given data as a function of the model parameters.

• This likelihood function is maximized to find the parameter estimates that are most consistent with the observed data.
• In the Tobit model, the likelihood function incorporates both the probability of observing a censored score and the probability of observing an uncensored score.

Using MLE in Tobit models accounts for both types of observations, leading to consistent and unbiased parameter estimates.

Sample Selection

Sample selection refers to the process of only observing a subset of the population, based on some selection criteria.
In our example, the university only records exam scores for students who score 100 or above, so we have incomplete data on all potential applicants.
This selective recording can bias the analysis if not properly accounted for.

• The Tobit II model, or sample selection model, addresses this issue by modeling the selection process separately from the outcome process.
• It includes a selection equation that determines whether a student's score is observed and an outcome equation that models the observed scores.

This two-part approach helps correct for the bias introduced by selective recording and provides more accurate estimates of the relationship between background characteristics and exam scores.