Across sectional study in 2017 uses a sampleof 1678 auto dealers to estimate the price elasticity of demand for new compact cars in thestate of New York for the year 2017. Using OLS estimation methodology, the estimationresults of regressing the sales of cars on a set of explanatory variables is as follows,
?????????? = 19.66 - 2.87?????????? + 0.64???????????? + 0.47??????????hModel (1)
(5.46) (0.98) (0.27) (0.24)
?? = 1,678,??????_??!= 0.67
wheredcarsis the value of sales for auto dealeri(in thousands of dollars);priceis the average
price of cars sold by auto dealeri(in thousands of dollars);incomeis the average income of
car owners (in thousands of dollars); andwealthis the average wealth of car owners (in
thousands of dollars). Numbers in parentheses are the robust standard errors of the
a. Interpret the above results. What are the names of the possible econometric problems in model (1)? Explain.
b. Given that the variablepriceis endogenous and an assumption that the
variables incomeand wealthare exogenous, the researchers re-estimated the above
model. They used Instrumental Variable (IV) regression model withstate sales tax on
cars“stax” as an instrumental variable. The estimation results are as follows:
?????????? = 18.63 - 2.01?????????? + 0.62???????????? + 0.45??????????hModel (2)
(6.42) (1.15) (0.30) (0.25)
?? = 1,678,??????_??!= 0.71
What are the important changes between model (1) and model (2)? What is the logic
behind the endogeneity of the variableprice?And what is the logic behind the
exogeneity of the variablesincomeand wealth?What is the logic behind the choice of the
c. Based on Models (1) and (2), what do the results imply about the bias of IV
versus OLS estimation methodologies? What do the results imply about the correlation
between thestaxand the error term, the correlation betweenstax andprice, and the
correlation between thepriceand the error term?Hint: Discuss the expected direction of
the correlations and the relation between the three correlations.
d. Based on Models (1) and (2), what do the results imply about theR2 of the first
stage regression model? Explain.
e. In another specification, the authors add the variablelocal sales tax on cars
“ltax” as another instrument for Model (2) and the model is re-estimated. Write down the
first stage regression. How many instruments are included? What are they?
f. Write down the null and alternative hypothesis for the instrument “relevance
test.” Assuming that the F-statistic of the test is equal to 12.33, what is your conclusion
about the relevance of the instruments? Why?
g. Write down the regression for the “overidentification test.” Write down the
null and alternative hypothesis of the test. Assume that the J-statistic is equal to 4.30,
what is the F-statistic of the overidentification test? Using any significance level, what is
the critical value of the J-statistic? And what is your conclusion on the exogeneity of the
h. Are the instruments used in the study “valid”? Why?
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