Tutorial 7 - Censored Variables

Question 1

What is censored data?

Answer 1

• Random sample
• But partial information about the value of a variable—we know it is beyond some boundary, but not how far above or below it (or 0)

Question 2

What is truncated data?

Answer 2

Data are truncated when the data set does not include observations in the analysis that are beyond a boundary value. Having a value beyond the boundary eliminates that individual from being in the analysis.

• -> no random sample!
• -> no information on censored variables!

Question 3

What is the Tobit model used for?

Answer 3

The Tobit model is used in situations in which the dependent variable y is censored

Question 4

What are corner solutions?

Answer 4

The dependent variable y has bunches at certain points (typically zero) due to individual behaviour (corner solution)

• Hours worked for women
• Monthly spending on cigarettes
• Amount invested in a project

Here, the term “censoring” is also used for corner solutions.

Question 5

What are the Tobit model assumptions for the dependent variable?

Answer 5

The latent variable y‡* = xβ + ϵ ϵ | x ~ N(0; σ²) iid
is assumed to :

• be a linear function of the covariates,
• with normally distributed, homoscedastic errors

Question 6

How can you express “left-censoring” or “censoring from below” (at zero) for observed variable y as a function of latent variable y*?

Answer 6

Question 7

What is the expected value of the observed variable y in a Tobit model?

Answer 7

Question 8

What is the probability that y is positive in censored data?

Answer 8

with

• Φ_σ= cdf of normal distribution with a standard deviation of σ and a mean of zero
• Φ (.) = cdf of standard normal distribution (a standard deviation of one and a mean of zero)

Question 9

In the tobit model, what is the expected value of y, given that y is positive?

Answer 9

Question 10

In the tobit model, what is the expected value of y?

Answer 10

Question 11

What is the partial effect of continuous regressor x_K on the observed variable y in the Tobit model?

Answer 11

Question 12

What is the partial effect of continuous regressor x_K on the latent variable y* in the Tobit model?

Answer 12

Question 13

Which partial effect is more interesting for which case of censored data?

Answer 13

• for “real” censoring model, it is in most cases the effect on the latent variable
• for corner solution model, it is the observed variable

Question 14

What is the average partial effect (APE) of a continuous variable x_Kon y?

Answer 14

Question 15

What is the average partial effect (APE) of a discrete variable x_Kon y?

Answer 15

Question 16

How would you interpret this average partial effect of the variable ‘kids05 = number of kids below 6’ on ‘hours = hours worked per year’?

Answer 16

The effect of an additional kid<6 ranges from -855.7 to -13.78, and the Average Partial Effect (APE) is -526.2 One additional child (younger of six) reduces averagely the yearly working time by approximately 526 hours.

Question 17

Can you compare these coefficients?

Answer 17

Coefficients of OLS vs. Tobit are only comparable in
signs, not in magnitudes! To compare magnitudes, we have to compare e.g. APEs (for OLS: the coefficient, for Tobit: the APE on observed value)

Question 18

How can you solve the problem of taking the logarithm of censored data (where 0 is a frequent value)?

Answer 18

• For these observations, impute an artifical value of 1 -> Log(Investment + 1) is the dependent variable
• This is a trick sometimes used in models with corner solutions at zero