Influence of Various Options on Asking Price for Used Mustangs - Case Study Example

Influence of various options on asking price for used Mustangs – a report Executive Summary This report analyses the impacts of various options of vintage cars on the asking price of the cars. Aspects like color, number of miles travelled, presence of air-conditioning, number of cylinders in the engine, nature of transmission and whether the car is privately owned or owned by the dealer are found to have no impact on asking price of the vehicles. The age of the vehicle is inversely related to its price. A GT model fetches a substantially higher price compared to a non-GT model. A convertible also is associated with higher price than a non-convertible. I. Introduction The objective of this report is to identify the effects of various aspects of used cars on the price of the car in order to assist the shop owner Tom come up with a price list that matches the preferences of the customers in the market. Using data from 35 different sellers of used cars, influences of characteristics on prices are estimated. The structure of the report is as follows: section II presents the data and methodology of estimation, section III presents the results and analysis of the estimation. Section IV extends the analysis to consider the potential threat of multicollinearity. Finally, the report concludes in section V with the final recommendations for Tom. II. Data and methodology The dataset includes 35 observations on the price (PRICE), age of the car (AGE), approximated odometer reading on miles (MILES) and the number of cylinders in the engine (CYL). Additionally, dummy or indicator variables reflecting whether the car is a convertible or not (CONVERT), whether the car has an automatic or manual transmission (TRANS), whether the car has air-conditioning or not (AIR), whether the model is GT or not (GT), whether the car is owned privately or by the dealer (OWNER) and finally, what the color is of the car (COLOR) are also included. To identify the degree of influence of these traits on the prices of vintage cars, a simple OLS regression model is used. The equation estimated is the following: III. Results and analysis This section presents the results of running the regressions. The results of running an OLS regression in accordance with the specification above are given below1 in table 1. From the table, the first point to note is that the estimated equation is: PRICE = 10789 + 3119 CONVERT - 1277 AGE - 0.00032 MILES + 494 TRANS - 512 AIR + 227 CYL - 108 COLOR + 2791 GT - 490 OWNER Before proceeding to interpret this equation, it is critically important to gauge the significance of the coefficients in order to avoid mistaken inferences. However, from the column of p-values, the constant, convertible, age and GT emerge as predictors which are significant at the 1% level (p-values less than 0.01). The other predictors are insignificant implying that there is no statistical evidence of these predictors having any impact on the prices of vintage cars. Table 1: Predicting car prices of vintage cars Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Constant 10788.98 1354.30 7.97 0.00 7999.75 13578.21 convertible 3119.34 456.12 6.84 0.00 2179.94 4058.73 age -1277.21 133.09 -9.60 0.00 -1551.30 -1003.11 miles 0.00 0.01 -0.03 0.97 -0.02 0.02 transmission 493.65 422.31 1.17 0.25 -376.11 1363.42 Air-conditioning -512.39 798.02 -0.64 0.53 -2155.94 1131.16 Number of cylinders 227.35 143.70 1.58 0.13 -68.61 523.31 Color -108.17 90.26 -1.20 0.24 -294.06 77.72 GT 2790.52 505.05 5.53 0.00 1750.35 3830.69 Private/dealer owner -490.33 420.67 -1.17 0.25 -1356.71 376.05 Multiple R R Square Adjusted R Square Standard Error Observations F Significance F 53.35 0.00 0.974942 0.950512 0.932696 981.5446 35 The probability associated with the F-test evaluating the joint significance of the predictors is 0.00 implying that at-least one of the included predictors has a statistically significant impact on the price. This corroborates the results of the tests of individual significance (t-test) where the constant, the dummy variables convertible and GT and finally, the age of the vehicle were found to be significant. Noting these results, the regression is run again with only the significant predictors included. The results are presented in table 2. Table 2: Predicting prices of vintage cars (only significant independent variables included)   Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Constant 11516.27 414.9701 27.75206 0.00 10669.94 12362.61 Convertible 3162.348 389.0679 8.128012 0.00 2368.839 3955.858 Age -1333.4 79.71972 -16.7261 0.00 -1495.99 -1170.81 GT 2879.64 363.6642 7.918404 0.00 2137.942 3621.338 Multiple R R Square Adjusted R Square Standard Error Observations F Significance F 0.965365 0.93193 0.925343 1033.778 35 141.4713 0.00 The estimated equation is: And from the column of p-values all estimated coefficients are significant. Therefore, this equation can be used to analyze the impacts the predictors have on the dependent variable, i.e., price. a) Impact of age on price The estimated coefficient of age is: -1333, and the coefficient is significant. This implies for each additional year of age, the asking price of a vintage car is reduced by $1333.00. The confidence interval is (-1495.99, -1170.81). This provides the range between which the true coefficient would lie 95 times if the entire experiment was repeated a 100 times. b) Impact of whether car is a convertible on price The estimated coefficient of the dummy variable convertible is 3162, and the coefficient is significant. This implies that if the vehicle is a convertible, the asking price is hiked by an additional $3162. The confidence interval in this case is (2368.84, 3955.86). c) Impact of whether car is a GT model on price The estimated coefficient of the indicator variable GT is 2880 and it is significant. Therefore, if a model is GT, then its price is raised by $2880. The confidence interval for the true coefficient is (2137.94, 3621.34). d) The implication of the intercept The intercept or the constant is estimated as being equal to 11516. Since the intercept is the value of the predicted variable when the predictors are all zero, $11516 is the price of a vehicle that is not a convertible, neither is a GT model, but is brand new (age =0 corresponds to a new car). The confidence interval for the true intercept is (10669.94, 12362.61) Comparing tables 1 and 2, the following additional points also emerge: The R-squared and adjusted R-squared values in both specifications are very high implying a very good fit of the model to the data. Although, these values are slightly lower for the 2nd specification, the magnitude of the change is very low although a large number of variables are dropped. This very small change implies that the variations in variables which were found to be insignificant indeed explain very little of the observed variations in the prices. The value of the F-statistic for the 2nd specification increases almost threefold. This is expected since some of the included variables in the 1st specification are insignificant but all variables included in the 2nd specification are significant. Thus the null hypothesis of joint insignificance of the predictors is rejected more strongly in the latter specification. The standard error of the regression in the first specification is 981.54, while it is 1033.778 for the 2nd specification. It should be noted that the standard error is essentially the standard deviation of the predicted variable conditional upon the values of the predictors. Thus, this rise in the standard errors essentially reflects increased variability in the predicted variable, i.e., price of vintage cars in the second model. Although the standard errors may seem to be very large in absolute terms (implying possible multi-collinearity), it should be noted that these are actually quite small when one considers the values of the dependent variable. In particular the mean price of the cars is 8193.88 and relative to this, the standard errors of the two models are not very high. Figure 1 below presents the scatter plots of the significant predictors and the predicted variable. The left segment shows the scatter plot of price and convertible, the middle segment shows the scatter plots of price and age and finally the rightmost segment shows the scatter plot of the price and GT. An evident downward sloping trend is evident in the scatter plot of price and age. This reinforces the negative relationship identified earlier between price and the age of a vehicle. No such trends are evident in the other two segments of the graph. However since the predictors in these cases are indicator variables that assume values of only 0 and 1, no continuous relationship should be expected. Instead cluster of values are located at the values 0 and 1 for the predictor. But these plots do indicate that higher values of price occur when the vehicle is a convertible or when it is a GT model. The effect is more pronounced in case of GT. Figure 1: Scatter plot of price against significant independent variables IV. Extension: Multicollinearity In this section, the possibility of multicollinearity is evaluated. This is done simply by looking at the Variance Inflating Factors (VIFs). Since the inferences are based on the 2nd specification, the VIFs for the estimates of the 2nd specification are reported below2 in table 3. Table 3: Testing for multicollinearity Variable VIF CONVERT 1.012 AGE 1.005 GT 1.011 Mean VIF 1.00933333 From the table above it is found that all VIFs are very close to 1. Therefore, there is no cause for concern regarding multicollinearity. Even if it is present, it is not present to any degree that would create problems for inferences. Therefore, the inferences drawn in the earlier segment are valid and the recommendations can be based on these results. V. Recommendations On the basis of the results obtained above, the following should be noted. The distance travelled by the car, the nature of transmission, presence or lack of air-conditioning, number of cylinders, color and finally whether the car is owned privately or by the dealer has no impact on prices. Therefore, these options should be ignored in determining the asking price of the car. The age of the car is very important and it varies inversely with asking price. Therefore, in the very least, a higher price should be charged for newer cars, and lower prices should be charged for older cars. Specifically, each additional year should imply a reduction in the asking price by some value lying within the range of $1171 to $1495. If a car is a convertible, this feature, by itself should warrant an additional price within the interval of $2368.84 to $ 3955.86. Finally, if the car is a GT model, an additional price lying within the interval of $2137.94 to $3621.34 should be charged. However, although these ranges include the exact impact, Tom’s best bet would be to use the sample estimates of the coefficients reported earlier. One important shortcoming of the analysis here is that the sample contains only 35 observations. The estimates therefore may be affected by a small sample bias. In order to be confident about these results a research should be carried out where the sample is adequately large. APPENDIX: MINITAB OUTPUT 1. Result of OLS estimation with all variables Regression Analysis: PRICE versus CONVERT, AGE, ... The regression equation is PRICE = 10789 + 3119 CONVERT - 1277 AGE - 0.00032 MILES + 494 TRANS - 512 AIR + 227 CYL - 108 COLOR + 2791 GT - 490 OWNER Predictor Coef SE Coef T P VIF Constant 10789 1354 7.97 0.000 CONVERT 3119.3 456.1 6.84 0.000 1.542 AGE -1277.2 133.1 -9.60 0.000 3.106 MILES -0.000323 0.009258 -0.03 0.972 2.891 TRANS 493.7 422.3 1.17 0.253 1.555 AIR -512.4 798.0 -0.64 0.527 1.246 CYL 227.3 143.7 1.58 0.126 2.435 COLOR -108.17 90.26 -1.20 0.242 1.207 GT 2790.5 505.1 5.53 0.000 2.163 OWNER -490.3 420.7 -1.17 0.255 1.501 S = 981.545 R-Sq = 95.1% R-Sq(adj) = 93.3% Analysis of Variance Source DF SS MS F P Regression 9 462613202 51401467 53.35 0.000 Residual Error 25 24085747 963430 Total 34 486698950 Regression Analysis: PRICE versus CONVERT, AGE, GT The regression equation is PRICE = 11516 + 3162 CONVERT - 1333 AGE + 2880 GT Predictor Coef SE Coef T P VIF Constant 11516.3 415.0 27.75 0.000 CONVERT 3162.3 389.1 8.13 0.000 1.012 AGE -1333.40 79.72 -16.73 0.000 1.005 GT 2879.6 363.7 7.92 0.000 1.011 S = 1033.78 R-Sq = 93.2% R-Sq(adj) = 92.5% Analysis of Variance Source DF SS MS F P Regression 3 453569373 151189791 141.47 0.000 Residual Error 31 33129577 1068696 Total 34 486698950 Read More