Question

Using OLS estimation methodology, the study of Morelli and Smith (2015) uses a cross sectional data of 2490 cars for the year 2013 to estimate the factors affecting the price of automobiles in the state of California. The estimation results of regressing the price variable on a set of explanatory variables are shown in Model (1), where the numbers in parentheses are the robust standard errors of the coefficients.

????? = 5647.02 + 5.77 ????ℎ? + 23.64 ??? + 3573.09 ???????      (1)

              (1042.20)      (1.50)              (13.74)            (1230)

???_?^2= 0.65, ? = 2490

Where price is in U.S. dollars, weight is in pounds, mpg is the number of miles per gallon, and foreign is a dummy variable that takes 1 if the ith car is foreign and 0 if domestic.

1. Interpret the above results and discuss whether the signs and statistical significance of the coefficients are as expected.
2. What is the predicted price for a Buick Regal car that weights 3,450 lbs. and with 22 mpg? And what is the predicted price for a Toyota Corolla that weights 2,310 lbs.and with 18 mpg?
3. If in the year 2013 the actual price of Buick Regal was $25,500 and that of Toyota Corolla was $23,640, use the results of point (b) to determine whether your model over or under predicting the price of each car?
4. In one specification, the authors have included the length of the ith car, length, as an additional regressor to Model (1), where length is measured in inches. The resulting regression results are shown in Model (2) as follows,

              ????? = 5524.02 + 6.54 ????ℎ? + 22.73 ??? + 3568.11 ??????? − 93.48 ?????ℎ       (2)

   (1033.10) (4.85)               (13.68)             (1232)                      (32.87)

???_?^2 = 0.92, ? = 2490

If the F-statistic of the coefficients of the four included variables in Model (2) is equal to 54.32, does the inclusion of the variable length in Model (2) creates an econometric problem? Explain in details.

5. In one last specification of the model, the authors have included the size of the trunk of the ith car, trunk, as an additional regressor to Model (1), where trunk is measured in cubic feet. The resulting regression results are shown in Model (3) as follows,

               ????? = 5631.24 + 4.95 ????ℎ? + 25.99 ??? + 3650.22 ??????? + 88.31 ????? (3)

                         (1144.67)    (1.62)              (13.54)             (1285.29)                 (44.38)

???_?^2 = 0.75, ? = 2490

Suppose that the correlations between the variable trunk and the variables price, weight, mpg, and foreign are equal to 0.25, 0.49, -0.38, and -0.36, respectively. Based on these correlations, refer to Model (1) and discuss the direction of the bias of each coefficient of the three included variables. What is your opinion about including the variable trunk as an additional regressor in Model (3)? Does the inclusion of the variable trunk violate any of the OLS assumptions? Explain in details.

Answer 1

a) price is expected to increase with weight. This is obvious given that we can imagine that with more weight better and heavy material is used.

price is expected to increase with mpg. The more a car is expected to go, this is a symbol of better performance, hence more price

price of a foreign car will be more than that of a domestic one. This distinction can be thought to be caused due to the taxes levied on import of car. Hence more expensive.

b) assuming buick regal is a foreign car:

price=5647.02+5.77(3450)+23.64(22)+3573.09

price=$29646.69

assuming toyota corolla is domestic car:

price=5647.02+5.77(2310)+23.64(18)

Price= $19401.24

c)

Car	actual price	Predicted price
Buick regal	25500	29646.69
Toyota corolla	23640	19401.24

It can be said that model is over predicting price of domestic cars and underpredcting price of foreign cars.

d) More info needed

e) weight: it is expected that bigger the trunk of a car, more it will weigh. Hence, by including trunk size as a variable, some of the effect of variable weight is captured by this variable. The direction of bias is negative.

mpg: bigger the trunk, heavier the car, hence more miles per gallon needed to cover same distance. This is why it can be seen that coefficient of mpg has increased. Now, the same amount of mpg will drive the price of car upward because of this additional variable. Direction of bias is positive

foreign: bigger the trunk, heavier the car, hence more taxes are expected to be applied in order to get a foreign car imported. This is why the coefficient on foreign has increased. Direction of bias is positive