Nonlinear Econometric Analysis (ECO 722) : Homework 1
Due Thursday, February 25
You must turn in this assignment before class begins on the due date. No photocopies, no
loose pages, no paper clips (use a staple).
1. Consider a binary random variable yi that describes a Bernoulli trial in which the probability of observing yi = 1 in any draw is given by θ. Then the probability mass function
(pmf) for an observation is given by
(
θ
if yi = 1
f (yi ; θ) =
(1 − θ) if yi = 0
which can be written in an analytically more convenient way as
f (yi ; θ) = θyi (1 − θ)(1−yi ) .
Consider a random sample of n observations drawn from a Bernoulli distribution with unknown θ.
a. Write down the log likelihood function.
b. What is the first order condition for maximization of the log likelihood with respect to
θ? Solve it. Note that for all such problems, it is often helpful to recognize that
n
n
X ∂f (yi ; θ)
∂ X
f (yi ; θ) =
.
∂θ i=1
∂θ
i=1
c. Calculate the second derivative of the log likelihood w.r.t. the parameter. Show that the
log likelihood function is globally concave.
2. When independent Bernoulli trials are repeated, each with probability θ of success, the
number of trials, yi , it takes to get the first success has a geometric distribution which has
a pmf given by
f (yi ; θ) = (1 − θ)(yi −1) θ for yi = 1, 2, 3, ...
Consider a random sample of n observations drawn from a geometric distribution with unknown θ.
a. Write down the log likelihood function.
b. What is the first order condition for maximization of the log likelihood with respect to
θ? Solve it.
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c. Calculate the second derivative of the log likelihood w.r.t. the parameter. Show that the
log likelihood function is globally concave.
3. Consider an integer valued random variable yi that follows a Poisson distribution with
unknown parameter λ and pmf given by
f (yi ; λ) =
e−λ λyi
for yi = 0, 1, 2, ...
yi !
Consider a random sample of n observations drawn from a Poisson distribution with unknown
λ.
a. Write down the log likelihood function.
b. What is the first order condition for maximization of the log likelihood with respect to
λ? Solve it.
c. Calculate the second derivative of the log likelihood w.r.t. the parameter. Show that the
log likelihood function is globally concave.
4. The Pareto distribution has is as a model for a density function with a slowly decaying
tail and has a pdf given by
f (yi ; Y θ) = θY θ yi (−θ−1) for yi ≥ Y and θ > 1
where Y is a known constant. Consider a random sample of n observations drawn from a
Pareto distribution with unknown θ.
a. Write down the log likelihood function.
b. What is the first order condition for maximization of the log likelihood with respect to
θ? Solve it.
c. Calculate the second derivative of the log likelihood w.r.t. the parameter. Show that the
log likelihood function is globally concave.
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