# Limited dependent variable

A limited dependent variable is a type of variable in regression analysis that is constrained in some way, such as being categorical, binary, or bounded within a specific range.
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Updated: Jun 21, 2024

## 3 key takeaways

• Limited dependent variables are variables that have restricted values, such as binary outcomes, categorical data, or truncated ranges.
• Analyzing limited dependent variables requires specialized statistical techniques, such as logistic regression, probit models, or Tobit models.
• These models help to properly interpret relationships and make accurate predictions when dealing with constrained dependent variables.

## What is a limited dependent variable?

A limited dependent variable is a variable whose values are restricted in some manner. Unlike continuous variables that can take any value within a range, limited dependent variables are constrained by their nature. Examples include binary variables (yes/no), categorical variables (different categories or groups), and truncated or censored variables (where values below or above certain thresholds are not observed or are grouped).

### Example

An example of a limited dependent variable is a binary variable representing whether a person has purchased a product (1) or not (0). Traditional linear regression models are not suitable for such data, as they do not accommodate the binary nature of the outcome.

## Types of limited dependent variables

### Binary variables

Binary variables take on two possible values, often coded as 0 and 1. Examples include whether an individual votes in an election (yes/no) or whether a loan application is approved (approved/rejected).

### Categorical variables

Categorical variables have more than two categories, but no inherent order among the categories. Examples include the type of car owned (sedan, SUV, truck) or a person’s blood type (A, B, AB, O).

### Ordinal variables

Ordinal variables have categories with a meaningful order but not necessarily equal intervals between them. Examples include survey responses rated on a scale from “strongly disagree” to “strongly agree” or education levels (high school, bachelor’s, master’s, PhD).

### Truncated or censored variables

Truncated variables are observed only within a certain range, while censored variables have values that fall outside a range but are recorded as the boundary value. For example, income data might be censored if values above a certain amount are recorded as “over \$100,000.”

## Statistical methods for limited dependent variables

### Logistic regression

Logistic regression is used for binary dependent variables. It models the probability that the dependent variable equals 1 as a function of the independent variables. The logistic function ensures that predicted probabilities are between 0 and 1.

### Probit regression

Probit regression is similar to logistic regression but uses the cumulative normal distribution to model the probability of the binary outcome. It is often used when the underlying data is assumed to follow a normal distribution.

### Multinomial logistic regression

Multinomial logistic regression is used for categorical dependent variables with more than two categories. It generalizes logistic regression by modeling the probabilities of the different categories.

### Ordinal logistic regression

Ordinal logistic regression, also known as ordered logit, is used for ordinal dependent variables. It models the cumulative probabilities of the categories, taking into account the order of the categories.

### Tobit model

The Tobit model is used for truncated or censored dependent variables. It accounts for the fact that the observed values are only a subset of the true underlying distribution and models the relationship accordingly.

• Appropriate modeling: Specialized models for limited dependent variables provide more accurate and meaningful results than traditional linear regression.
• Interpretation: These models help in interpreting probabilities, odds, or likelihoods, which are often more intuitive for binary and categorical outcomes.
• Handling constraints: Models for limited dependent variables properly account for the constraints and boundaries in the data, ensuring valid inferences.

• Complexity: Specialized models for limited dependent variables can be more complex to implement and interpret compared to traditional regression models.
• Assumptions: These models often rely on specific assumptions about the distribution of the dependent variable or the nature of the relationship with independent variables.
• Data requirements: Some models, like the Tobit model, require a large sample size to accurately estimate the parameters and provide reliable results.
• Binary logistic regression: Explore the use of logistic regression for modeling binary outcomes and interpreting odds ratios.
• Categorical data analysis: Learn about methods for analyzing categorical data, including chi-square tests and log-linear models.
• Econometrics: Understand the broader field of econometrics, which includes techniques for modeling and analyzing economic data, including limited dependent variables.

Analyzing limited dependent variables requires specialized statistical techniques to handle the constraints and provide accurate, interpretable results. By using appropriate models, researchers and analysts can gain valuable insights and make reliable predictions when dealing with binary, categorical, ordinal, or truncated data.

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