Spring 1997 John Rust
Economics 551b 37 Hillhouse, Rm. 27

PROBLEM SET 3

Classical Methods(I)

(Due: February 24, 1997)

QUESTION 1 Let where Y is random vector and is positive definite matrix. Prove .

QUESTION 2 Suppose regression model

(where y, are , X is , and is ) with normality assumption on disturbance, . Derive the maximum likelihood estimator (MLE) of when is known, and show that MLE is identical to the GLS estimator.

QUESTION 3 Show that the Cramer-Rao lower bound, , corresponds to the GLS covariance matrix, , in Question 2.

QUESTION 4 Suppose following SUR model,

where is , , is , and . Also suppose covariance matrix of is . Derive the covariance matix of OLS estimator , and show this covariance matrix is larger than that of GLS estimator, .

QUESTION 5 Derive the maximum likelihood estimator for the model

where are IID double exponential random variables with mean 0 and scale parameter :

A.

Derive a formula for K so that f is a valid probability density (i.e. so ).

B.

Derive the maximum likelihood estimator for .

Hint: Show that the MLE for is identical to the Least Absolute Deviations estimator defined by:

QUESTION 6 (Bayesian) Suppose in SUR model of Question 4. Suppose the prior for the parameters is given by the product of two independent priors for the vector and the inverse covariance matrix :

Suppose i.e. the prior for is a multivariate normal distribution with hyperparameters (the mean of the prior) and (the covariance matrix of the prior). Suppose , i.e. the prior for is a Wishart distribution with hyperparameters ( ) and ( ). Recall that the density of a Wishart is given by:

Show that the posterior distribution of is conditionally conjugate i.e. it can be decomposed as a product of a Normal and Wishart as follows:

where

Hint: Show that the likelihood function can be expressed as:

where is and is . Also use the fact that so we can write:

QUESTION 7 Consider the random utility model:

where is a decision-maker's payoff or utility for selecting alternative d from a set containing D possible alternatives (we assume that the individual only chooses one item). The term is known as the deterministic or strict utility from alternative d and the error term is the random component of utility. In empirical applications is often specified as

where is a vector of observed covariates and is a vector of coefficients determining the agent's utility to be estimated. The interpretation is that represents a vector of characteristics of the decision-maker and alternative d that are observable by the econometrician and represents characteristics of the agent and alternative d that affect the utility of choosing alternative d which are unobserved by the econometrician. Define the agent's decision rule by:

i.e. is the optimal choice for an agent whose unobserved utility components are . Then the agent's choice probability is given by:

where is the vector of observed characteristics of the agent and the D alternatives and is the conditional density function of the random components of utility given the values of observed components X, and is the indicator function given by if and 0 otherwise. Note that the integral above is actually a multivariate integral over the D components of , and simply represents the probability that the values of the vector of unobserved utilities lead the agent to choose alternative d.

Definition: The Social Surplus Function is given by:

The Social Surplus function is the expected maximized utility of the agent.

Problem: Prove the Williams-Daly-Zachary Theorem:

and discuss its relationship to Roy's Identity.

Hint: Interchange the differentiation and expectation operations when computing :

and show that

Question 8 Consider the special case of the random utility model when has a multivariate (Type I) extreme value distribution:

Show that the conditional choice probability is given by the multinomial logit formula:

Hint 1: Use the Williams-Daly-Zachary Theorem, showing that in the case of the extreme value distribution (8) the Social Surplus function is given by

where is Euler's constant.
Hint 2: To derive equation (9) show that the extreme value family is max-stable: i.e. if are IID extreme value random variables, then also has an extreme value distribution. Also use the fact that the expectation of a single extreme value random variable with location parameter and scale parameter is given by:

and the CDF is given by

Hint 3: Let be INID (independent, non-identically distributed) extreme value random variables with location parameters and common scale parameter . Show that this family is max-stable by proving that is an extreme value random variable with scale parameter and location parameter

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