Please i need your help with this question
A cross sectional study in 2017 uses a sample of 1678 auto dealers to estimate the price elasticity of demand for new compact cars in the state of New York for the year 2017. Using OLS estimation methodology, the estimation results of regressing the sales of cars on a set of explanatory variables is as follows,
𝑑𝑐𝑎𝑟𝑠 = 19.66 − 2.87 𝑝𝑟𝑖𝑐𝑒 + 0.64 𝑖𝑛𝑐𝑜𝑚𝑒 + 0.47 𝑤𝑒𝑎𝑙𝑡ℎ Model (1)
(5.46) (0.98) (0.27) (0.24)
𝑁 = 1,678, 𝐴𝑑𝑗_𝑅! = 0.67
where dcars is the value of sales for auto dealer i (in thousands of dollars); price is the average
price of cars sold by auto dealer i (in thousands of dollars); income is the average income of
car owners (in thousands of dollars); and wealth is the average wealth of car owners (in
thousands of dollars). Numbers in parentheses are the robust standard errors of the
coefficients.
a. Interpret the above results. What are the names of the possible econometric problems in model (1)? Explain.
b. Given that the variable price is endogenous and an assumption that the
variables income and wealth are exogenous, the researchers re-estimated the above
model. They used Instrumental Variable (IV) regression model with state sales tax on
cars “stax” as an instrumental variable. The estimation results are as follows:
𝑑𝑐𝑎𝑟𝑠 = 18.63 − 2.01 𝑝𝑟𝑖𝑐𝑒 + 0.62 𝑖𝑛𝑐𝑜𝑚𝑒 + 0.45 𝑤𝑒𝑎𝑙𝑡ℎ Model (2)
(6.42) (1.15) (0.30) (0.25)
𝑁 = 1,678, 𝐴𝑑𝑗_𝑅! = 0.71
What are the important changes between model (1) and model (2)? What is the logic
behind the endogeneity of the variable price? And what is the logic behind the
exogeneity of the variables income and wealth? What is the logic behind the choice of the
instrument?
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 the stax and the error term, the correlation between stax and price, and the
correlation between the price and 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 the R2 of the first
stage regression model? Explain.
e. In another specification, the authors add the variable local 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
instruments? Why?
h. Are the instruments used in the study “valid”? Why?
