Econ 829. Homework 1: Estimation of Single-Agent Dynamic Models. Due date: Monday, April 15, 2013

Description You can download the data from the course website (http://www.u.arizona.edu/~nosal/empio.html). The data is in the CSV format and has three columns: (1) mileage at period t (xt ), (2) observed replacement decision at period t (it ), and (3) mileage next period (xt+1 ). The data generating process is as follows: xt can take on 11 values, i.e. xt 2 f0; 1; 2; : : : ; 10g. At any time period, xt+1 Note that

( minfxt + 1; 10g; = xt ;

with probability ; with probability 1

does not depend on the replacement decision it .

Replacement decision is made in the beginning of each period and is e¤ective immediately (i.e. if it = 1, the agent uses machine with mileage 0 and the next period state xt+1 is 1 with probability and 0 with probability 1 ). The per-period maintenance cost for a bus with mileage x is C(x; ) = The …xed cost of replacement is

0t ; 1t )

iid

+

2x

2

.

3.

In other words, one-period utility function is given by ( 2 1 xt 2 xt + u(xt ; it ; 1t ; 2t ; ) = 3 + 1t ; where (

1x

0t ;

if it = 0; if it = 1

Type 1 Extreme Value.

Assume that the decision-maker has discount factor

= 0:95.

* Together with your answers, please submit your computer code for the questions that follow.

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Problems 1. Write down ML estimator for and solve for it (note that this does not require any nonlinear optimization). Report your estimate. 2. Set 1 = 0:3; 2 = 0:0; 3 = 4:0 and solve the DP problem using your estimate of . Report the probability of replacement for every state. 3. Estimate parameters of the model using MLE with a nested …xed point algorithm: (a) Guess initial parameter values (you can use initial guess);

1

= 0:3;

2

= 0:0;

3

= 4:0 as your

(b) Solve DP problem; (c) Calculate the probability of replacement at each state; (d) Use model predictions (the probability calculated in the previous step) and data to derive a likelihood; (e) Search over parameter values by repeating steps (b)

(d).

Calculate standard errors for your parameters (at this stage assume that Report your results.

is known).

4. Conduct the following policy experiment: Using the estimated parameters, calculate the expected long-run replacement probability for bus engines (you will need to integrate over the states at any time period using simulation). Suppose that the government reduces replacement cost 3 by 10 percent with an investment subsidy. Predict the e¤ect of this subsidy on the long-run replacement probability. 5. Re-estimate parameters of the model using Hotz and Miller approach. To do this in addition to your estimate of you need to obtain initial conditional choice probabilities P (it jxt ) (report your estimates of the CCP’s). Report your parameter estimates. 6. Extend your estimation procedure in the last problem by iterating in the probability space as suggested by Aguirregabiria and Mira. Report your results for the second, third, and …fth iteration (skip calculation of standard errors).

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