Business Statistics: Communicating with Numbers By Sanjiv Jaggia and Alison Kelly
Chapter 16 Learning Objectives (LOs) LO 16.1:Use and evaluate polynomial regression models. LO 16.2:Use and evaluate log-transformed models. LO 16.3:Describe the method used to compare linear with log-transformed models.
The Rental Market in Ann Arbor, Michigan • Marcela Treisman is analyzing the rental market in Ann Arbor, the home of the University of Michigan. • She has data on monthly rent, along with several property characteristics. • Marcela will evaluate models that predict rental income from home characteristics. She will select the most appropriate model and make predictions for rental income for specific property characteristics. 16-3
16.1Polynomial Regression Models LO 16.1 Use and evaluate polynomial regression models. 16-4
The Quadratic Regression Model LO 16.1
The “Flexible” Quadratic Model LO 16.1
Example 16.1 LO 16.1 • Suppose we want to estimate the relationship between average cost and output. We gather data for 20 manufacturing firms on output and average cost. • When using a scatterplot to display the relationship, notice that a quadratic curve seems to better fit the data. 16-7
Example 16.1 LO 16.1 • We then estimate both a linear and quadratic model, determining which is more appropriate from our goodness-of-fit measures. • In order to estimate a quadratic regression model, we first need to compute the squared output: 16-8
Estimation Results LO 16.1 16-9
The Marginal Effect LO 16.1 16-10
Maximum or Minimum? LO 16.1 16-11
Example 16.2 LO 16.1 16-12
Higher Order Models LO 16.1 16-13
Example 16.4 LO 16.1 • We often use a cubic model to estimate a firm’s total costs. • Let’s work through an example using the data labeled Total Cost on the text website. Here we are given the total cost for various levels of output. • We first must compute output squared and output cubed before estimating the model: 16-14
Example 16.4 LO 16.1 16-15
16.2 Regression Models with Logarithms LO 16.2 Use and evaluate log-transformed models.
The Log-Log Model LO 16.2
The Slope as an Elasticity LO 16.2
Prediction with the Model LO 16.2
Example 16.5 LO 16.2
The Logarithmic Model LO 16.2 16-21
Example 16.6 LO 16.2
The Exponential Model LO 16.2 16-24
Prediction with the Exponential LO 16.2
Example 16.7 LO 16.2
Summary of Logarithmic Models LO 16.2 16-27
Ann Arbor Rentals LO 16.2 • In the introductory case, Marcela Treisman wants to examine what factors influence rental prices in Ann Arbor, Michigan. • She has data on 40 properties, including the number of bedrooms and baths, the floor space, and the rental price. • As a first look, she plots each of the explanatory variables against the rental price.
Plotting the Data LO 16.2 • When plotting the number of bedrooms and number of bathrooms against Rent, the relationship appears to be roughly linear.
Plotting the Data LO 16.2 • Since the relationship between rent and floor space seems to change as the floor space increases, we should consider possible logarithmic models. 16-30
Potential Models LO 16.2
Estimation Results LO 16.2 • Since all three models have three explanatory variables, we need not present the adjusted R2. • Yet, we cannot simply compare the standard error and R2 of all four models since the response variable differs. 16-32
Predicting the Rental Price LO 16.2
Comparing Linear and Log-Transformed Models LO 16.3 Describe the method used to compare linear with log-transformed models.
Computing R2 LO 16.3
Ann Arbor Rentals LO 16.3 • Let’s return to model selection for the introductory case. • Recall that of the models that used an untransformed response variable, the logarithmic model performed best. • Of the models that used the transformed response variable, the log-log model outperformed the exponential model. • But we now have the tools to select the most appropriate model overall.
Computing R2 LO 16.3
Conclusion LO 16.3 16-38