Introduction to Econometrics 3rd Edition by James Stock, Mark Watson

Exercise 21

The demand for a commodity is given by Q = 0 + 1 P + u, where Q denotes quantity, P denotes price, and u denotes factors other than price that determine demand. Supply for the commodity is given by Q = 0 + 1 P + v , where v denotes factors other than price that determine supply. Suppose that u and v both have a mean of zero, have variances
and
, and are mutually uncorrelated.
a. Solve the two simultaneous equations to show how Q and P depend on u and v.
b. Derive the means of P and Q.
c. Derive the variance of P , the variance of Q , and the covariance between Q and P.
d. A random sample of observations of (Q i , P i ) is collected, and Q i is regressed on P i. (That is, Q i is the regressand and P i is the regressor.) Suppose that the sample is very large.
i. Use your answers to (b) and (c) to derive values of the regression coefficients.
ii. A researcher uses the slope of this regression as an estimate of the slope of the demand function ( ß 1 ). Is the estimated slope too large or too small