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Business Statistics: Communicating with Numbers By Sanjiv Jaggia and Alison Kelly

Business Statistics: Communicating with Numbers By Sanjiv Jaggia and Alison Kelly

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## Business Statistics: Communicating with Numbers By Sanjiv Jaggia and Alison Kelly

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**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