Question

Consider the dependent variable \(y=\) fuel efficiency of a car (mpg). a. Suppose that you want to incorporate size class of car, with four categories (subcompact, compact, midsize, and large), into a regression model that also includes \(x_{1}=\) age of car and \(x_{2}=\) engine size. Define the necessary dummy variables, and write out the complete model equation. b. Suppose that you want to incorporate interaction between age and size class. What additional predictors would be needed to accomplish this?

Short answer

The dummy variables for size class (subcompact, compact, midsize, large) would be \(D1, D2, D3\) respectively. The complete model equation would be: \(y = β0 + β1x_{1} + β2x_{2} + β3D1 + β4D2 + β5D3 + ε\). To incorporate interactions between age and size class, the additional predictors will be \(x_{1}*D1, x_{1}*D2, x_{1}*D3\). With these variables, the fully expanded regression model becomes: \(y = β0 + β1x_{1} + β2x_{2} + β3D1 + β4D2 + β5D3 + β6(x_{1}*D1) + β7(x_{1}*D2) + β8(x_{1}*D3) + ε\).

Step-by-step solution

Define Dummy Variables

Define dummy variables for the categorical variable 'size class of the car'. Since there are four categories: subcompact, compact, midsize, large. Let's assign the following dummy variables: Let \(D1\) represent subcompact, \(D2\) represent compact, \(D3\) represent midsize. If \(D1, D2, D3\) are all 0, then it's large size.

Write Out the Complete Model Equation Using Dummy Variables

The regression model that includes the variable age of car, engine size and size class of car is written as:\(y = β0 + β1x_{1} + β2x_{2} + β3D1 + β4D2 + β5D3 + ε\) where \(β0\) is the intercept, \(β1\) is the coefficient for age of car, \(β2\) is the coefficient for engine size, \(β3\), \(β4\), \(β5\) are coefficients for the respective size class.

Incorporate Interaction Between Age and Size Class

To incorporate interaction between age and size class, additional predictors are needed. These are the product of age ( \(x_{1}\) ) and each of the dummy variables:The interaction variables would be \(x_{1}*D1\), \(x_{1}*D2\), \(x_{1}*D3\).The resulting model is:\(y = β0 + β1x_{1} + β2x_{2} + β3D1 + β4D2 + β5D3 + β6(x_{1}*D1) + β7(x_{1}*D2) + β8(x_{1}*D3) + ε\)where \(β6\), \(β7\), \(β8\) are coefficients for the respective interaction terms.

Key concepts

Fuel Efficiency

Fuel efficiency is a critical measure for any vehicle, indicating the distance a car can travel per unit of fuel, commonly miles per gallon (mpg) in the U.S. When analyzing factors that influence fuel efficiency, regression models are highly useful. They allow us to quantify the relationship between several variables, like a car's age and engine size, and its fuel efficiency.

Understanding how different characteristics affect fuel efficiency helps manufacturers and consumers make informed decisions. For instance, a newer, smaller engine might be associated with higher fuel efficiency, which could influence both design choices and consumer preferences. By using regression analysis, we can predict fuel efficiency based on a set of vehicle features, providing valuable insights into performance and environmental impact.

Categorical Variables in Regression

Categorical variables represent types or categories of data, such as the size class of a car. These variables can take on a limited, fixed number of possible values, which don't have a natural numeric representation. To include such variables in a regression model, we transform them into a series of dummy variables.

Dummy variables are binary (\(0\text{ or }1\)) indicators, representing the presence or absence of a category. In our example, we create dummy variables for each size class of car (except for the reference category). The introduction of dummy variables allows the regression model to distinguish between different car sizes and assess their individual impact on fuel efficiency. Each dummy coefficient tells us the difference in fuel efficiency compared to the reference category, while controlling for other factors in the model.

Interaction Terms in Regression

Interaction terms in regression are crucial when the effect of one variable on the dependent variable depends on the level of another variable. By including interaction terms, we can capture the combined effect of two variables working together. In our car example, this means examining how the size class's impact on fuel efficiency changes with the age of the car.

To create interaction terms, we multiply the dummy variables by the continuous variable they interact with. Adding these terms to our regression model allows for differential slopes, suggesting that the relationship between age and fuel efficiency can vary across different car sizes. These terms enhance the model's flexibility and give us a more nuanced understanding of how combined factors influence the dependent variable, like fuel efficiency in various car types.