Introduction

• As the population increases, the need for transportation increases.
• It takes longer to travel from one place to another in the city due to congestion, resulting increased carbon emission by cars.
• Demand for car use is inelastic regardless of the fuel price (Borger, & Rouwendal, 2014).
• Increasing the tax on fuel does not help much due to cars with higher fuel efficiency being more expensive compared to cars with lower fuel efficiency.
• To solve the increased carbon emission problem, people need to use cars with better fuel efficiency.

Problem Statement

• However, it is hard to achieve because of the perception of people that fuel efficient cars are more expensive (Alberini, Bareit, & FIlippini, 2016).
• The aim of this paper is to find out if there really is a relationship between fuel efficiency and the price of the car using regression analysis.
• To find out how well the fuel efficiency of a car can predict the price of the car, Automobile data set from UCI is used (Schlimmer, 1987).

Data

• The data contains 26 different attributes of automobiles.
• However, we’re only interested in the two attributes - price and fuel efficiency.
• Car price ranging from $5,118 to $45,400 (continuous variable).
• City fuel efficiency 13 to 49 miles per gallon (continuous variable).
• A total of 205 entries of cars were collected.
  
• The data is also independent, i.e. each entry is unique and does not affect other entries.

Descriptive Statistics and Visualisation

• Descriptive statistics
• Rows with NA values are dropped resulting in 201 rows without NA values

cars.all <- read.csv('cars.csv', header=FALSE, 
                 dec=".", 
                 na.strings='?', 
                 col.names = c("symboling","normalized-losses","make","fuel-type","aspiration", "num-of-doors", "body-styles","drive-wheels","engine-location","wheel-base","length","width", "height","curb-weight","engine-type","num-of-cylinders","engine-size","fuel-system","bore","stroke","compression-ratio","horsepower","peak-rpm","city-mpg","highway-mpg","price"))

cars <- cars.all[,c("city.mpg", "price")]
cars <- cars %>% na.omit()

stats <- c( min(cars$price,na.rm = TRUE), quantile(cars$price,probs = .25,na.rm = TRUE), median(cars$price, na.rm = TRUE), quantile(cars$price,probs = .75,na.rm = TRUE), max(cars$price,na.rm = TRUE), mean(cars$price, na.rm = TRUE), sd(cars$price, na.rm = TRUE), length(cars$price), sum(is.na(cars$price)), min(cars$city.mpg,na.rm = TRUE), quantile(cars$city.mpg,probs = .25,na.rm = TRUE), median(cars$city.mpg, na.rm = TRUE), quantile(cars$city.mpg,probs = .75,na.rm = TRUE), max(cars$city.mpg,na.rm = TRUE), mean(cars$city.mpg, na.rm = TRUE), sd(cars$city.mpg, na.rm = TRUE), length(cars$city.mpg), sum(is.na(cars$city.mpg))
)
summary <- matrix(stats, ncol = 9, byrow = TRUE)
colnames(summary) <- c("Min", "Q1", "Median",   "Q3",   "Max",  "Mean", "SD",   "n",    "Missing")
rownames(summary) <- c("Price", "City MPG")

knitr::kable(summary)

Descriptive Statistics and Visualisation cont.

ggplot(cars, aes(x=price, y=city.mpg)) + geom_point()

• City MPG by Price has a negative non-linear relationship.
• We can explore the frequencies for both columns to understand why.

Descriptive Statistics and Visualisation cont.

a <- ggplot(cars, aes(x=price)) + geom_histogram(aes(y=..density..)) + geom_density()
b <- ggplot(cars, aes(x=city.mpg)) + geom_histogram(aes(y=..density..)) + geom_density()
grid.arrange(a, b, nrow = 1)

• Both plots are skewed to right and are not normally distributed.

Descriptive Statistics and Visualisation cont.

• After performing logarithmic transforms on both columns:

a <- ggplot(cars, aes(x=log(price))) + geom_histogram(aes(y=..density..)) + geom_density()
b <- ggplot(cars, aes(x=log(city.mpg))) + geom_histogram(aes(y=..density..)) + geom_density()
grid.arrange(a, b, nrow = 1)

• Both are now somewhat more normally distributed.

Descriptive Statistics and Visualisation cont.

ggplot(cars, aes(x=log(price), y=log(city.mpg))) + geom_point()

• Negative linear relationship.

Descriptive Statistics and Visualisation cont.

model1 <- lm(log(city.mpg) ~ log(price), data = cars)

model1 %>% summary()

## 
## Call:
## lm(formula = log(city.mpg) ~ log(price), data = cars)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.38139 -0.09863 -0.01045  0.06525  0.46221 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  7.02840    0.19350   36.32   <2e-16 ***
## log(price)  -0.41006    0.02067  -19.84   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1468 on 199 degrees of freedom
## Multiple R-squared:  0.6643, Adjusted R-squared:  0.6626 
## F-statistic: 393.7 on 1 and 199 DF,  p-value: < 2.2e-16

• We apply the linear regression model and test assumptions after summarising the model.

Descriptive Statistics and Visualisation cont.

• Visualising line of best fit over the transformed scatter plot:

alpha <- model1$coefficients[["(Intercept)"]]
beta <- model1$coefficients[["log(price)"]]
model.function <- function(x) alpha + beta*x

ggplot(cars, aes(x=log(price), y=log(city.mpg))) + geom_point() + stat_function(fun = model.function) + xlim(8, 11)

• Line of best fit: \(log(economy) = 7.03 - 0.41 * log(price)\)

Hypothesis Testing

• Hypotheses for the overall linear regression model
  • \(H_0\): The data does not fit the linear regression model
  • \(H_A\): The data fits the linear regression model

model1 %>% summary

## 
## Call:
## lm(formula = log(city.mpg) ~ log(price), data = cars)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.38139 -0.09863 -0.01045  0.06525  0.46221 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  7.02840    0.19350   36.32   <2e-16 ***
## log(price)  -0.41006    0.02067  -19.84   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1468 on 199 degrees of freedom
## Multiple R-squared:  0.6643, Adjusted R-squared:  0.6626 
## F-statistic: 393.7 on 1 and 199 DF,  p-value: < 2.2e-16

• \(P\) value for F-test is very small \(F(1,199) = 393.7,p<.001\). City fuel economy explained 66.43% (Multiple R-squared) of the variability in the price of a car. The model is statistically significant. We can now interpret the model coefficients.

Hypothesis Testing Cont.

• Hypotheses for linear regression model parameters:
  • Intercept:
    • \(H_0: \alpha= 0\)
    • \(H_A: \alpha \neq0\)
  • Slope:
    • \(H_0: \beta= 0\)
    • \(H_A: \beta \neq0\)

model1 %>% summary() %>% coef()

##               Estimate Std. Error   t value     Pr(>|t|)
## (Intercept)  7.0283979 0.19350366  36.32178 9.413514e-90
## log(price)  -0.4100568 0.02066568 -19.84240 4.753214e-49

model1 %>% confint()

##                  2.5 %    97.5 %
## (Intercept)  6.6468170  7.409979
## log(price)  -0.4508086 -0.369305

• \(H_0: \alpha/\beta=0\), \(H_A: \alpha/\beta \neq0\)
  • \(H_0: \alpha = 0\) is not captured by the 95% CI \([6.65,7.41]\), \(p < .001\). Test is statistically significant.
  • \(H_0: \beta = 0\) is not captured by the 95% CI \([-0.45,-0.37]\), \(p < 0.001\). Test is statistically significant.
• We reject \(H_0\) for model parameters.

Testing Assumptions - Normality of Residuals

cars$predicted <- predict(model1)
cars$residuals <- residuals(model1)

a <- ggplot(cars, aes(x = log(price), y = log(city.mpg))) +
  geom_smooth(method = "lm", se = FALSE, color = "lightgrey") +
  geom_segment(aes(xend = log(price), yend = predicted), alpha = .2) +
  geom_point(aes(color = residuals)) +  
  scale_color_gradient2(low = "blue", mid = "white", high = "red") +  
  guides(color = FALSE) +
  geom_point(aes(y = predicted), shape = 1) +
  theme_bw()
b <- ggplot(cars, aes(x = predicted, y = residuals)) + geom_point() +  geom_smooth(aes(colour = "red"), se= FALSE) + 
 geom_hline(yintercept=0) + theme(legend.position = "none")

grid.arrange(a, b, nrow = 1)

• Residuals vs Fitted sum lie close to 0.
• This indicates normality of residuals and homoscedasticity.

Testing Assumptions - Normality of Residuals, Homoscedasticity and Influential Cases

grid.arrange(a, b, c, nrow = 1)

• No obvious departures from normality for most residuals
• No influential cases
• Variance in square root of residuals is mostly consistent (no pattern)
• Assume homoscedasticity

Correlation coefficient

r.squared <- model1 %>% summary()
r.squared <- r.squared$r.squared

#Correlation Coefficient r
r <- cor(log(cars$city.mpg), log(cars$price), use = "complete.obs")

#r 95%CI
library(psychometric)
n <- cars$city.mpg %>% length()
print(CIr(r, n = n, level = .95))

## [1] -0.8567765 -0.7626498

detach("package:psychometric", unload = TRUE)

• \(R^2\)
  • \(R^2 = 0.6643\)
  • Log city fuel economy explained 66.43% of the variability in the price of the car.
• \(r\)
  • \(r = -0.82\)
  • Indicates strong negative linear relationship
  • 95%CI [-0.86, -0.76] does not capture \(r =\) -0.82. Statistically significant evidence of negative correlation between the two variables.

Results

• Simple linear regression summary:
  • Linearity was assumed, normality of residuals OK, homoscedasticity OK, no influential cases.
  • \(r\) = -0.82, \(R^2\) = 0.66.
  • Model ANOVA, \(F(1,199) = 393.7, p < .001\).
  • \(\alpha = 7.03, p < .001, CI(6.65, 7.41)\)
  • \(\beta = −0.41, p < .001 CI(-0.55, -0.37)\)
• Decision:
  • Overall model: Reject H0.
  • Intercept: Reject H0.
  • Slope: Reject H0.
  • \(Log(Efficiency) = 7.03 − 0.41 × log(Price)\)
• Conclusion:
  • There was a statistically significant negative linear relationship between a car’s city fuel economy and its price.
  • For every one unit increase in log(price), log(city mpg) was estimated to decrease on average by -0.41.

Discussion

• Results are contradictory from the findings of Borger & Rouwendal et.al (2014).
• There is statisitically significant evidence of a negative linear relationship showing the prices of cars actually increases as the fuel efficiency of the car decreases.
• This shows that the previous findings showing fuel efficient cars are more expensive is incorrect.
• This means that being eco friendly and driving fuel efficient cars is actually cheaper as people will be saving on both car and fuel costs.
• However, the findings can be due to possible presence of confounding variables such as curb weight of the car.
• There is a possible positive relationship between curb weight and price of the car meaning heavier cars are more expensive and also less fuel efficient.
• Therefore, curb weight needs to be controlled in the future researches to find out if there is a real negative relationship between the fuel efficiency and price of the car.

References

• Schlimmer, J. (1987). Automobile data set. Retrieved from https://archive.ics.uci.edu/ml/datasets/automobile

• Borger, B. D., & Rouwendal, J. (2014). Car user taxes, quality characteristics, and fuel efficiency: Household behaviour and market adjustment. Journal of Transport Economics and Policy. 48(3), 345-366

• Alberini, A., Bareit, M., & FIlippini M., (2016). What is the effect of fuel efficiency information on car prices? Evidence from Switzerland. The Energy Journal, 37(3). 2016.