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.