Why do we need to discard one dummy variable?

Answer: We discard one dummy variable to avoid multicollinearity in regression analysis.

Explanation:

When dealing with categorical variables in regression analysis, such as in linear regression or logistic regression, it’s common practice to use dummy variables to represent categorical data numerically. A dummy variable is a binary variable that takes on the value of 1 if an observation falls into a particular category and 0 otherwise.

Now, why do we need to discard one dummy variable? This practice is necessary to prevent multicollinearity, which occurs when two or more predictor variables are highly correlated. In the context of dummy variables, if we include dummy variables for all categories of a categorical variable, it can lead to perfect multicollinearity because the values of these variables will sum up to a constant.

Here’s a detailed explanation:

1. Perfect Multicollinearity: If we include dummy variables for all categories of a categorical variable in a regression model, the sum of these dummy variables across all categories will always equal 1. This is because an observation can only belong to one category. As a result, the dummy variables are perfectly multicollinear, meaning that their values can be perfectly predicted from one another. This causes issues in the regression analysis, such as making it impossible to estimate the coefficients of the dummy variables accurately.
2. Reducing Redundancy: By discarding one dummy variable, we effectively remove the redundancy in the model caused by perfect multicollinearity. This allows the regression model to estimate the coefficients of the remaining dummy variables accurately and interpret their effects on the outcome variable independently.
3. Reference Category: The dummy variable that is dropped is often referred to as the reference category. The coefficients of the remaining dummy variables represent the difference between each category and the reference category in terms of the effect on the outcome variable.
4. Interpretability: Dropping one dummy variable also makes the interpretation of the regression coefficients more straightforward. The coefficients of the remaining dummy variables indicate the change in the outcome variable relative to the reference category.
5. Avoiding the Dummy Variable Trap: Including all dummy variables without discarding one can lead to the “dummy variable trap,” where the regression model becomes overparameterized and unable to produce meaningful results.

In summary, discarding one dummy variable when representing categorical variables in regression analysis helps to avoid multicollinearity, reduce redundancy, improve model interpretability, and avoid potential issues such as the dummy variable trap. It is a common practice in regression modeling when dealing with categorical data.