Please solve this accurately about introduction to Econometrics: Instrumental Variable.

Graddy (2006) reports the operation of the Fultan Fish Market in Manhattan, New York City. Among other things, Graddy considers a simple regression model: ln????=??0+??1ln????+????, ??=1,…,??,

where ???? is the quantity (the total amount of fish sold on a day) and ???? is the price (the average price on that day), ??0 and ??1 are unknown parameters, and ???? is a mean-zero, unobserved random variable. The econometric problem is to estimate the demand curve for fish.

Suppose that the log price follows the following equation: ln????=??0+??1ln????+????, ??=1,…,??,

where ??0 and ??1 are unknown parameters, and ???? is a mean-zero, unobserved random variable. This equation can be viewed as the supply equation. Assume that cov(????,????)=0, that is, demand and supply shocks are uncorrelated. Further, assume that var(????)>0, var(????)>0, ??1?0, ??1?0, and ??1×??1?1.

(a) [7 points] Show that cov(ln????,????)=??1var(????)1???1??1.

What is the implication of this result?

(b) [7 points] Show that cov(ln????,????)=??1var(????)1???1??1.

What is the implication of this result?

(c) [7 points] Graddy (2006) argues that variations in weather can provide an instrument for ln????. Let ???? denote the storminess of the weather as an instrument. Then cov(????,????)=0 is reasonable since the weather condition in the sea is unlikely to affect demand for fish. Consider the following regression ln????=??0+??1????+????,

where ??0 and ??1 are unknown parameters, and ???? is a mean-zero, unobserved random variable such that cov(????,????)=0. Write a null hypothesis that the instrument is irrelevant in this setup. Explain briefly.

(d) [7 points] Suppose that the TSLS estimate of ?? is -1.25 with the standard error of 0.50. A friend of yours claims that one should reject the null hypothesis that demand is unit elastic at the 5% level. Would you agree? Justify your answer.