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This is about the basic Econometrics stat question. The R part is ok but I need help with the calculation parts. Especially how to find function Q and the variance of that. THANK YOU.



Question 1 - Modelling demand in a simple setting. Consider the following model for the measured price, P, of walnuts. Assume that Q varies only because of weather conditions, and producers
always create as much output as possible. P=a0—a1Q+e where e is simply measurement error (true price is not observed) that is independent of all other variables with IE[E] : 0. Assume: ' V[Q] : 0:29, HQ] : #Q-
- V[6] : 0'? Now: Calculate IE[P] IE[P|Q : q], V[P], and (C(P, Q) Based on our answers above, come up with an expression for (11 based on these population features. Based on (2), propose an estimator of al, 6:], given iid data (P1, Q1), . . . , (PN, QN). Set a0 = 2, a1 = 0.1, pg 2 2, 0Q = 0.1, and 05 = 0.1. Write some code to simulate the data for a sample size of N = 500, assuming
all exogenous variables are normally distributed. 5. Draw 500 different samples of the data above, and for each sample compute your estimator Eel. Draw a picture to show how 6:1 is
distributed. What do you notice? 99.”?


Question 2 - Modelling demand in a structural setting Now instead assume that in addition to the measurement error, 6, there is a structural shock to demand, 1], Le. a random variable that shifts the
demand curve up and down: P*:ao—oqQ—l—q and we observe this P“ with the same error as before: PIClo—G1Q+T]+€ Let EM] : 0 and Vhfl : 0%. Now assume that the market for walnuts is a monopoly, and the monopolist faces costs: C(Q) : Co +le2 and chooses Q to maximize profit: «(62) : P* x Q , C(Ql- Solve the monopolist‘s problem (i.e. solve for Q). What do you notice about the relationship between Q and 7]? Use your answer to the above to make a prediction about whether your estimator, 651 from Question (1) will work. Calculate C(P, Q) and V[Q] in terms of the structural parameters of the model. Now, derive an expression for the estimator 5:1 in terms of the structural parameters. Use this to argue that in this new setting, the estimator
won’t in general be consistent. . Set co : 0, c1 : 0.1, and 0% : 0.1, and retain the other parameters from the previous question. Simulate 500 samples of size 500,
compute the estimator, and show its distribution. Does the simulation prove your point in part (5)? .U‘PPJN." cn

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