Tobit Regression

Do households spend more on arts and cultural activities as their income rises? Suppose a researcher surveyed 1,000 households on their annual spending on arts and cultural activities. Some households spent a considerable amount on things like performances and exhibitions, but others spent nothing at all.

When many households spend $0 like this, what goes wrong if you apply ordinary linear regression? The ordinary linear regression model doesn’t directly reflect a structure in which the dependent variable is limited at a certain lower or upper bound. So, depending on how the data were generated, the regression coefficients and predictions can be hard to interpret properly.

A method to consider in this situation is the Tobit regression model.

The Tobit model is a regression model for a structure in which there is an unobserved, latent continuous outcome variable and the actual observations are censored at a certain threshold. For example, we can assume a structure in which actual spending is observed as $0 whenever the latent spending propensity is at or below a certain threshold.

But having spending of $0 doesn’t by itself make the Tobit model appropriate. The process of deciding not to spend and the process of deciding how much to spend may be different.

In this lesson we’ll look at the basic principle of the Tobit model, when it applies, how it is computed, a worked example, the difference between regression coefficients and marginal effects, comparisons with other models, and how to interpret the results.

The best-known case is the left-censored Tobit model, in which the lower limit of the dependent variable is set to 0.

For example, suppose a household’s latent propensity to spend on arts and cultural activities exists as a continuous value. This latent variable can vary with things like the household’s income and level of interest, but the spending we actually observe can’t be less than $0. The Tobit model expresses this situation as follows.

Here Y*_i is the unobserved latent outcome variable and Y_i is the spending actually observed. X_i are predictors such as income and household size, and β are the regression coefficients of the predictors.

This model assumes that whenever the latent variable is 0 or below, the actual spending is recorded as 0. So observations of $0 are also used as important information in the model.

However, you need to check whether the $0 values that actually occur came about through this kind of censoring of a latent variable. If whether to spend and how much to spend are determined by different decision processes, a different model may be more appropriate.

In both situations the observable range of the data is limited, but the way information is lost differs. For example, suppose that in a survey of arts and culture spending, households that didn’t spend were also included and their spending was recorded as $0. In this case we know how many households are at $0.

If instead only households that spent on arts and culture were surveyed, households that didn’t spend are left out of the data entirely. In this case we can’t even observe the number of households at $0.

The same regression analysis shouldn’t be applied to both cases. In particular, the fact that $0 households are included in the survey and the model assumption that those $0 values arose from censoring of a latent variable are two separate issues.

With censored data, however, the observed outcomes don’t reflect all values of the underlying latent variable as they are. For example, suppose the latent propensity to spend on arts and culture falls as household income gets lower. An ordinary linear regression model may extend this relationship and predict negative spending for some households.

The Tobit model builds into the model the structure that actual observations are limited at $0. It also estimates the parameters using not only the amounts for households with positive spending but also the share of households observed at $0.

However, the Tobit model isn’t always better than OLS. If the way $0 values actually arise differs from the Tobit assumptions, you can end up with inappropriate estimates.

The researcher wants to find out how household income, education level, household size and so on relate to arts and culture spending. Of all households, 280 spent nothing on arts and cultural activities, and the remaining 720 recorded some spending (more than $0).

Composition of the analysis data

In this example we’ll assume that a latent variable model that explains the $0 observations as left censoring is appropriate. The research question is:

In actual research you need to examine whether this model assumption is valid. In particular, if households first decide whether to spend and only those who spend then decide the amount, you can consider a two-part model or similar.

In the left-censored Tobit model, observations that are positive and observations that are 0 contribute differently to the likelihood. Positive observations use the continuous probability density of the latent variable. Households observed at $0, on the other hand, use the probability that the latent variable is 0 or below.

If we assume the error term follows a normal distribution with mean 0 and variance σ², the probability of a $0 observation is:

Here Φ is the cumulative distribution function of the standard normal distribution. The Tobit model uses the information from positive observations and from $0 observations together to estimate β and σ. So $0 observations aren’t arbitrarily dropped or replaced with a small positive number.

But this estimation relies on assumptions such as the linear structure of the latent variable, normality and homoscedasticity. If these assumptions aren’t appropriate, the standard Tobit maximum likelihood estimates and inferences can be distorted.

Tobit regression results

The dependent variable is in dollars.

The coefficient for household income was 84. This means that, other model conditions being equal, a household income $20,000 higher is associated with a latent arts and culture spending variable that is on average $84 higher.

The coefficient for household size was 50, meaning the latent spending variable was $50 higher for each additional household member. Households with a college degree or higher also showed a latent spending variable $172 higher than households with a high school education or less.

A regression coefficient shows how the mean of the latent variable changes with a predictor. A change in actually observed spending, on the other hand, also reflects the structure that spending is limited at $0.

In the left-censored Tobit model, under the standard normal Tobit assumptions, the marginal effect of a continuous predictor X_k on the conditional mean of the observed outcome is:

That is, the marginal effect on actually observed spending reflects both the probability of observing positive spending under the given conditions and the regression coefficient of the latent variable. For example, if the household income coefficient is 84 and the probability of positive spending under particular covariate values is 0.70, we get:

So, under these conditions, when household income increases by $20,000, the conditional mean of actually observed spending is estimated to increase locally by about $58.80.

This is an illustrative calculation; in actual research you can compute the marginal effect for each observation and report their average, the average marginal effect (AME). For a categorical predictor, it’s appropriate to compute how much the predicted mean changes when the category is switched, instead of using a derivative.

The first is the probability that spending is greater than $0. The second is the expected value of actually observed spending across all households. The third is the conditional mean spending of households with positive spending.

For example, if higher-income households spend more on arts and culture, that could be because a larger share of households spend, or because households that spend spend larger amounts. Both can happen at the same time.

Results you can get from a Tobit analysis

These results are connected to one another but are not the same numbers. For example, as income rises, the share of households that spend may go up a little while households already spending also spend more. In that case, the change in overall mean spending reflects both components.

However, the standard Tobit model assumes that whether to spend and how much to spend are determined by the same latent variable process. If the two decision processes are fundamentally different, this assumption may not be appropriate.

A two-part model analyzes whether to spend and the actual amount spent as separate processes. The first part analyzes whether a household spent on arts and cultural activities, using logistic regression or similar. The second part analyzes the amount spent, only for households with positive spending.

For example, suppose some households don’t spend because they have no interest at all in arts and cultural activities, while others are interested but don’t spend because their budget is tight. If these households follow different decision processes, a Tobit model that explains them with one latent variable may be too restrictive.

So you shouldn’t automatically choose the Tobit model just because there are many $0 values. The researcher first needs to understand why the $0 values occurred and how the data were generated.

In particular, the classic normal Tobit model assumes that, conditional on the predictors, the error term follows a normal distribution with mean 0 and constant variance. The predictors also need to be exogenous.

For example, if arts and culture spending is much more variable in higher-income households, the homoscedasticity assumption may be inappropriate. If the spending distribution is very skewed or some households show extremely large amounts, the standard latent normal assumption also needs to be examined.

The relationship between the predictors and the outcome may also be nonlinear. Spending may rise quickly with income at low incomes but increase by less at high incomes. In such cases you can consider nonlinear terms, more flexible censored regression models, two-part models and so on.

The key point is that checking whether the Tobit coefficients are statistically significant isn’t enough. You also need to judge whether the latent variable structure and the censoring process fit the actual data.

The Tobit model assumes a latent variable structure in which observations are censored at a certain boundary. If the actual $0 values arose from a separate choice or participation decision, another model may fit better.

Distinguish observed spending from the latent variable.

The regression coefficients fundamentally refer to effects on the latent variable. To describe the average change in actual spending or the probability of spending, you need to compute separate predictions and marginal effects.

Set the censoring threshold correctly.

In some studies the lower limit isn’t 0, and there may be both an upper and a lower limit. If a survey recorded all spending above a certain amount at the same top-coded value, you may need to consider right censoring or two-sided censoring.

Distinguish it from sample selection problems.

If information on households that didn’t spend is left out of the sample entirely, the data structure differs from the simple Tobit censoring model. If the process by which subjects were selected is the issue, you may need to consider a Heckman selection model or similar.

Don’t treat regression coefficients from observational data as causal effects.

Even if higher-income households spend more on arts and culture, we can’t conclude that income alone creates the difference. Unobserved factors such as cultural preferences or the living environment may be involved.

In particular, distinguishing the Tobit regression coefficients from the average marginal effects on the actually observed outcome helps readers understand the results more accurately.

In actual research, it’s good to also state the model assumptions about the error distribution and homoscedasticity, what the marginal effects were computed for and at what reference values, and the estimation method used.

For example, with data in which arts and culture spending is limited at $0, we can model the relationship between the latent spending variable and actually observed spending. In doing so, information from households observed at $0 is used along with households with positive spending.

However, because Tobit regression coefficients describe relationships with the latent variable, they need to be distinguished from changes in actual mean spending. If you want to know the average effect on actual spending, you need to compute marginal effects.

Also, a Tobit model isn’t appropriate for every dataset with many $0 values. When whether to spend and how much arise from separate decision processes, other methods such as a two-part model should be considered.