Find your first car in San Jose
Author(s): Ren Zhi-yi, Chen Yu-kai, Xu Jie-mo
Class: 2nd year of SJSU-FCU 2+2 Bachelor's Program in Business Analytics
Student ID: D0571956, D0565634, D0571990
Instructor: Dr. Cathy W. S. Chen Course: Introduction to Data Analytics
Department: International School of Technology and Management Academic Year: Semester 1, 2017-2018
This paper examines the relationship between the price of used cars in San Jose on December 23, 2017 from Carfax website. We use the multiple regression analysis to investigate 9 explanatory variables and use Stepwise Selection approach to select the best fitted model. The collected data shows the mileage which has been driven has negative correlation with second-hand cars’ price. The drive wheel types such as All Wheel Drive (AWD), Front-Wheel Drive (FWD) and Rear-Wheel Drive (RWD) indicate the negative correlation as well. From the perspective of customers, they suggested that they pay more attention to the year of model and the number of images that sellers upload to the website. Second-hand cars, which are new models and have more pictures may have higher price. The quantity of cylinders that the engine has, engine displacement, the amount of miles when consuming a gallon of gasoline (MPG), whether it is for personal or business use and gearbox have strong positive correlation among our 196 randomly selected observations.
Keyword: Influential Point, Model Selection, Multicollinearity, Regression Analysis,
2 FCU e-Paper (2017-2018) Table of Contents
1. Introduction ... 3
2. Methods ... 4
2.1 Flow chart ... 4
2.2 Explanatory variable and response variables ... 4
2.3 Descriptive statistics ... 5
2.4 Scatter plot ... 6
2.5 Correlation plot ... 10
2.6 Full Model and diagnose multicollinearity ... 11
2.7 Discuss the outliers and influential points ... 12
2.8 Model Selection ... 12
2.9 Verify for assumption ... 15
3. Results ... 17
4. Discussion ... 17
5. Conclusion... 18
6. Appendix ... 19
Our department cooperates with San Jose State University and we students have the opportunity of studying abroad in California to receive a dual degree. However, the area of San Jose State University lacks public transport, such as MRT (Mass Rapid Transit) system. Residents usually drive cars for their commute. Our motivation is to help our classmates find their first cars that are high quality but low price in San Jose State. In addition, we hope to provide this resource as one available reference for those marketing researchers to assess second-handed cars’ price. Purchasing used cars can save money and we will not lose too much once we would like to sell it again. Moreover, used cars do not have plastic smell. Thus, purchasing second-hand cars is a wise choice. In this study, we will be considering what factors affect the price of used-cars.
The regression analysis is a very advanced statistical tool to learn more about the relationship between several independent or predictor variables and a dependent or criterion variable. For instance, personnel professionals customarily use multiple regression procedures to determine equitable compensation nowadays. With the maturity of people's concept of purchasing cars as well as more and more products that American automotive market can provide, car demanders can choose more models of cars in second-hand car markets. Purchasing used car is more suitable for young consumers' consumption ability. At the same time, according to its high hedging rate, affordable features, it is also very suitable for students who study abroad. Of course, our report has its limitations. We did not take the tax implications of buying a car into consideration.
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2. Method
2.1 flow chart
Our data resource came from the website of US biggest used car chain store--Carmax. We used the “Web Crawler” by Python to collect predictor variables that we are interested in such as mileages, number of photos, etc. Then we use Excel to delete several data with missing value. After rearranging data, we got 7724 pieces of data in our dataset. Finally, we randomly picked up 196 observations by SAS as our research sample.
2.2 Explanatory variable and response variables
Table 1 presents the explanatory variable and response variables.
Explanatory Variable Description
Price Used car’s price in San Jose. We use the dollar as the unit.
Table 1. Explanatory variable description
Find the data
Collect the data
Randomly select the data Clean the data
Response Variable Description
Mileage A number of how many miles a vehicle has traveled or covered.
Year The model year of a vehicle.
imgcount Numbers of photos that the sellers upload to the website.
engine How many cylinders a vehicle has.
displacement The swept volume of all the pistons inside the cylinders of a reciprocating
engine in a single movement.
mpg The number of miles that the car can drive when consuming a gallon of
gasoline. Usually, it can test the efficiency of vehicles. Person 1 if a vehicle is for personal use; otherwise 0.
Automatic 1 if a vehicle’s transmission is automatic; otherwise 0.
AWD 1 if a vehicle’s transmission system is All-Wheel Drive; otherwise 0.
FWD 1 if a vehicle’s transmission system is Front-Wheel Drive; otherwise 0.
RWD 1 if a vehicle’s transmission system is Rear-Wheel Drive; otherwise 0.
Table 2. Response variable description
2.3 Descriptive statistics
Table 3 provides the basic summary statistics analysis including mean, standard deviation, minimum, maximum and others. We notice that the ranges of “Price” and “Mileage” are quite large. Therefore we consider to transform these two variables by taking the natural logarithm so that we can figure out the result more significant.
Variable Minimum Q1 Median Q3 Maximum Range
Price 22346.84 12149.14 4899.00 13992.00 19410.00 29988.00 71984.00 67085.00
Mileage 49204.56 42114.30 827.00 23948.50 35791.00 64512.00 282584.00 281757.00
Year 2013.71 3.20 2000.00 2012.00 2015.00 2016.00 2018.00 18.00
imgcount 22.63 13.02 0.00 15.50 21.50 30.00 63.00 63.00
6 FCU e-Paper (2017-2018) displacement 2.87 1.12 1.40 2.00 2.50 3.50 6.40 5.00 mpg 24.61 6.18 14.00 20.00 24.00 28.00 52.00 38.00 Person 0.68 0.47 0.00 0.00 1.00 1.00 1.00 1.00 Automatic 0.96 0.20 0.00 1.00 1.00 1.00 1.00 1.00 AWD 0.17 0.38 0.00 0.00 0.00 0.00 1.00 1.00 FWD 0.46 0.50 0.00 0.00 0.00 1.00 1.00 1.00 RWD 0.30 0.46 0.00 0.00 0.00 1.00 1.00 1.00
Table 3. Summary statistics before taking natural log
Table 4 shows the results after we log the two independent variables
Variable Minimum Q1 Median Q3 Maximum Range
Price 9.87 0.54 8.50 9.55 9.87 10.31 11.18 2.69 Mileage 10.46 0.89 6.72 10.08 10.49 11.07 12.55 5.83 Year 2013.71 3.20 2000.00 2012.00 2015.00 2016.00 2018.00 18.00 imgcount 22.63 13.02 0.00 15.50 21.50 30.00 63.00 63.00 engine 5.15 1.42 3.00 4.00 4.00 6.00 8.00 5.00 displacement 2.87 1.12 1.40 2.00 2.50 3.50 6.40 5.00 mpg 24.61 6.18 14.00 20.00 24.00 28.00 52.00 38.00 Person 0.68 0.47 0.00 0.00 1.00 1.00 1.00 1.00 Automatic 0.96 0.20 0.00 1.00 1.00 1.00 1.00 1.00 AWD 0.17 0.38 0.00 0.00 0.00 0.00 1.00 1.00 FWD 0.46 0.50 0.00 0.00 0.00 1.00 1.00 1.00 RWD 0.30 0.46 0.00 0.00 0.00 1.00 1.00 1.00
Table 4. Summary statistics before taking natural log
Figure 1.1 illustrates a negative correlation between mileage and price , which indicates more miles the second-hand cars have been driven may cause lower price. As the plots in figure 1.2 show, there is a positive correlation between used cars’ price and the number of photos uploaded in description pages .
Figure 1.3 shows a positive correlation between the model year of cars and sales price. The newer types of cars they are, the higher price they will cost. A positive correlation can be observed in Figure 1.4, which means more cylinders may lead to higher price.
Figure 1.4 engine versus price Figure 1.2 imgCount versus
price
Figure 1.3 year versus price Figure 1.1 mileage versus price
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The scatter plot shown in Figure 1.5 has positive correlation relationship which indicates that the larger displacement is, the higher price they have. Figure 1.6 renders a positive correlation in mileage versus engines, which demonstrates the more cylinders a vehicle has, the higher price the seller may sell it out.
Figure 1.7 Person versus price
Figure 1.6 mpg versus price
In the figures 1.7 and 1.8, the scatter plots of 2 dummy variables test whether personal use or automatic transmission have effect. In these patterns, the price will not be affected by the 2 dummy variables.
Figure 1.5 displacement versus price
Figure 1.11 RWD versus price
Figure 1.10 FWD versus price Figure 1.9 AWD versus price
In Figures 1.9, 1.10 and 1.11, the scatter plots of 3 dummy variables show the price is not affected by the wheel drive type, whether it is all-wheel-drive or front-drive-type or rear-wheel-drive.
10 FCU e-Paper (2017-2018) 2.5 Correlation plot
The correlation matrix table reflects price and mileage; mpg; FWD have positive correlations. That the price has significant positive correlation with mileage shows the less miles they drive, the higher price they may sell.
First, between the engine and displacement, there is a significant positive relationship. In normal situation, when the engine has more cylinders, the displacement will be larger.
Secondly, the relationship between displacement and mile per gallon shows a negative relationship. The larger the displacement is, the more gasoline is needed to mix with air.
Thirdly, since engine and displacement have highly positive correlations, that the more engines the car has means the displacement is larger as well, hence, the engine and MPG also show a negative relationship.
In the end, the meaning of “year” in our variables is the model year of the car, but not
produced. So, they have negative correlation between year and mileage.
2.6 Full Model and diagnose multicollinearity
Table 5. Parameter Estimate of the full model
From table 5, the full model is:
Adjusted R² for this model is 0.7888, and is 0.06221.
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Multicollinearity is a problem that we can run into when we’re fitting a regression model. As usual, we use 10 as our standard.
In our dataset, the outputs above show that the VIF don’t go over 10.
Hence, it refers to predictor variables that are not correlated with other predictors in the model.
If VIF>10, we can assume that the regression coefficients are poorly estimated due to multicollinearity, which means it has high multicollinearity.
And we can solve Multicollinearity by redefining variables, principal components, use biased estimation – ridge regression, standardized coefficients.
2.7 Discuss the outliers and influential points
Output Statistics
Obs Residual Std Error
Residual Student Residual Cook’s D 159 0.7607 0.235 3.242 0.127 187 0.7566 0.244 3.100 0.035
Table 6. detect the outliers and influential points
We consider Residual and Student Residual test to detect outliers and use Cook’s D to identify any influential points. There are 2 observations(No.159 and No.187) whose Student Residual are greater than 3 and Residual is greater than 3 =0.748.Therefore, it has 2 observations which can be considered as outliers.For influential point, every observation’s Cook’s D is less than 0.5, there is no influential point in the dataset. Since these 2 pieces of data do not affect the estimated model too much in general, we do not remove the outliers.
2.8 Model Selection
According to Freund, Wilson, and Sa (2006), we can use 5 approaches to fit the regression model. Table 7 provides the selected factors in different selection
of Adjusted R-Square Selection and Mallow’s Cp we can notice is that they can not recognize the group of dummy variables. Finally, we choose the result of Stepwise Regression selection as our best fitted model.
Selection procedures Slentry/Slstay Mileage Year ImgCount Engine Displacement mpg Person Automatic AWD FWD RWD
Backward Elimination Slstay=0.15 √ √ √ √ √ √ √ √ √
Forward Selection Slentry=0.15 √ √ √ √ √ √ √ √ √
Stepwise Regression
Slstay=0.15 Slentry=0.15
√ √ √ √ √ √ √ √ √
Adjusted R-Square Selection √ √ √ √ √ √ √ √
Mallow’s Cp √ √ √ √ √ √ √ √
Table 7. Model Selection
Backward Elimination
Backward elimination starts with all variables, tests the deletion of each variable if p-value is greater than 0.15 and deletes the variable whose loss gives the most statistically insignificant deterioration of the model fit, then repeats this process until no variables can be deleted without a statistically significant loss of fit.
Forward selection
Forward selection starts with no variables in the model, tests the addition of each variable if p-value is smaller than 0.15 and adds the variable whose inclusion gives the most statistically significant improvement of the fit, then repeats this process until no variables can improve the model.
Stepwise Regression
Stepwise Regression is one approach to fit the model with all potential variables, which combines “Backward Elimination” and “Forward Selection”. The first selection procedure is putting only one variable, and select the smallest p-value which is also smaller than 0.15, however, it would be delete if p-value is greater than 0.15 after adding other variables.
14 FCU e-Paper (2017-2018) Adjusted R-Square Selection
Adjusted R-Square selection will calculate the adjusted R-Square of all the possible models, and the benefit of this method is that it would not be affected by increasing the number of variables.
Mallow’s Cp
In the Mallows’ Cp Selection, Cp value of all the possible models will be calculated. The selection criterion is that the Cp value is as small as possible, and the Cp value
closer to the number of parameters will be considered as the best model.
After we do the model selection, the variance inflation factor decreases significantly.
From table 8, the best selection model is:
Adjusted R² for this model is 0.7904, and is 0.06173.
2.9 Verify for assumption
Linear regression is an analysis that assesses whether one or more predictor variables explain the dependent variable. The regression has four key assumptions:
• The mean of error term is equal to zero. • Linearity (The variance of residual is ). • No autocorrelation
• Normality of the error distribution Assumption1:
Tests for Location: =0
Test Statistic p Value
Student’s t t -0.01106 Pr > |t| 0.9912
Sign M -1 Pr >= |M| 0.9431
Signed Rank S -82 Pr >= |S| 0.9182
Table 9. Assumption for The mean of error term is equal to zero.
We use Student's t, Sign, Signed Rank, three methods to verify the assumption. If the p-value is larger than = 0.05, it will not reject the null hypothesis. From table 9, the p-value is greater than ----fail to reject the null hypothesis, which is . Therefore, the mean of error term is equal to zero.
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Figure 3. Studentized Residual of Predicted Value
To check the second assumptions, we should use residuals versus fitted values plot. Above is the plot from the regression analysis we did. The errors have constant variance, with the residuals scattered randomly around zero. Besides, the residuals plots do not have any patterns, therefore, the errors have constant variance.
Assumption3:
Durbin-Watson Statistics
Order DW Pr < DW Pr > DW
1 1.9326 0.3142 0.6858
Table 10. Assumption for autocorrelation.
As usual, we use Durbin-Watson Statistics to test whether having autocorrelation. From table 10, both p-values for testing positive and negative autocorrelation are
greater than . So we fail to reject The hypothesis is valid.
Tests for Normality
Test Statistic p Value
Shapiro-Wilk M 0.9913 Pr < W 0.2883
Kolmogorov-Smirnov D 0.0460 Pr > D >0.1500
Cramer-von Mises W-Sq 0.0667 Pr > W-Sq >0.2500
Anderson-Darling A-Sq 0.4257 Pr > A-Sq >0.2500
Table 11. Assumption for Normality of the error distribution.
We use above 4 tests to verify the assumption. From table 11, we can notice that all the p-value is greater than 0.05. We fail to reject the null hypothesis. Therefore, the error distribution is normal.
3. Results
This dataset includes 2 outliers and no influential point. All the variables in the best fitted model are significant except one wheel drive type called all-wheel-drive.
The main findings suggest that the model of year, the automatic transmission, for personal use, 4 wheel drive types, the number of description pages’photos and cylinders have positive correlation between the price and these factors. Moreover, there is a negative relationship between price and mileage; other wheel drive types.
4. Discussion
After verifying the assumptions, our hypothesis was supported.
In the past, people would have the new car experiences as the basis for the pricing of second-hand cars. Our defect is that the sample observations are not enough, so it cannot represent a very comprehensive information. But the advantage is that our data
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is very timely and real. It is possible to reflect the more authentic conditions of purchasing of used cars in San Jose. And we have studied other used cars in a similar way, for example: The automatic transmission will be relatively expensive.
5. Conclusion
In conclusion, if people want to buy a used car in San Jose State and they have a budget, they should choose the one with the mileage as low as possible, and that which has the latest year. They may ignore how many images are in the description pages, because they should contact the sellers, then go and see the seller in person. If individuals would like to enjoy freeway in U.S., you should choose the second-hand car which has big engines, like the muscle car, though the price of the car will be higher as well.
Even though the car for personal use is expensive, do not risk buying the car which is not for personal use like rent cars, taxi cars, etc., you would never know how last driver drove the car, what it was used for, and what had happened to the car.
Nowadays most of the cars are automatic, and manual car is much cheaper. But we are going to San José, which is a busy metropolis, so driving a manual car will become a disaster .
There are various transmission systems you can choose, and all of them have benefits and drawbacks. From our perspectives, FWD is the best choice, thanks to its lower price and fuel economy. Also it does not snow in San José in winter, so it’s quite safe to drive. But, if you want something for fun, RWD is the option for you.
Carfax. (2017). Used cars for sale in San Jose, CA. Retrieved November 23, 2017, from https://www.carfax.com/Used-Cars-in-San-Jose-CA_c1023
Freund, R. J., Wilson, W. J., & Sa, P. (2006). Regression analysis: Statistical modeling
