🔬 Tutorial problems eta \eta

🔬 Tutorial problems eta \(\eta\)#

Note

This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.

\(\eta\).1#

Find the largest domain \(S \subset \mathbb{R}^2\) on which

\[ f(x, y) = x^2 - y^2 - xy - x^3 \]

is concave.

How about strictly concave?

It is useful to review the Hessian based conditions for concavity and the conditions for definiteness of a Hessian of \(\mathbb{R}^2 \to \mathbb{R}\) functions.

\(\eta\).2#

Show that the function \(f(x) = - |x|\) from \(\mathbb{R}\) to \(\mathbb{R}\) is concave.

Because the function is not differentiable everywhere in its domain, using the definition of concavity could be an easier way.

\(\eta\).3#

Consider the function \(f\) from \(\mathbb{R}\) to \(\mathbb{R}\) defined by

\[ f(x) = (c x)^2 + z \]

Give a necessary and sufficient (if and only if) condition on \(c\) under which \(f\) has a unique minimizer.

\(\eta\).4#

Let \(C\) be an \(N \times K\) matrix, let \(z \in \mathbb{R}\) and consider the function \(f\) from \(\mathbb{R}^K\) to \(\mathbb{R}\) defined by

\[ f(x) = x' C' C x + z \]

Show that \(f\) has a unique minimizer on \(\mathbb{R}^K\) if and only if \(C\) has linearly independent columns.

Obviously, you should draw intuition from the preceding question.

Also, what does linear independence of the columns of \(C\) say about the vector \(C x\) for different choices of \(x\)?

\(\eta\).5#

This exercise takes you on a tour of a binary logit model and its properties.

Consider a model when a decision maker is making a choice between \(J=2\) two alternatives, each of which has a scalar characteristic \(x_j \in \mathbb{R}\), \(j=1,2\). Econometrician observes data on these characteristics, the choice made by the decision maker \(y_i \in \{0,1\}\) and an attribute of the decision maker, \(z_i \in \mathbb{R}\). The positive value of \(y_i\) denotes that the first alternative was chosen. The data is indexed with \(i\) and has \(N\) observations, i.e. \(i \in \{1,\dots,N\}\).

To rationalize the data the econometrician assumes that the utility of each alternative is given by a scalar product of a vector of parameters \(\beta \in \mathbb{R}^2\) and a vector function \(h \colon \mathbb{R}^2 \to \mathbb{R}^2\) of alternative and decision maker attributes. Let

\[\begin{split} h \colon \left( \begin{array}{c} x \\ z \end{array} \right) \mapsto \left( \begin{array}{l} x \\ xz \end{array} \right) \end{split}\]

In line with the random utility model, the econometrician also assumes that the utility of each alternative contains the additively separable random component which has an appropriately centered type I extreme value distribution, such that the choice probabilities for the two alternatives are given by a vector function \(p \colon \mathbb{R}^2 \to (0,1) \subset \mathbb{R}^2\)

\[\begin{split} p \colon \left( \begin{array}{c} u_1 \\ u_2 \end{array} \right) \mapsto \left( \begin{array}{c} \frac{\exp(u_1)}{\exp(u_1) + \exp(u_2)}\\ \frac{\exp(u_2)}{\exp(u_1) + \exp(u_2)} \end{array} \right) \end{split}\]

In order to estimate the vector of parameters of the model \(\beta\), the econometrician maximizes the likelihood of observing the data \(D = \big(\{x_j\}_{j \in \{1,2\}},\{z_i,y_i\}_{i \in \{1,\dots,N\}}\big)\). The log-likelihood function \(logL \colon \mathbb{R}^{2+J+2N} \to \mathbb{R}\) is given by

\[ logL(\beta,D) = \sum_{i=1}^N \ell_i(\beta,x_1,x_2,z_i,y_i), \]

where the individual log-likelihood contribution is given by a scalar product function \(\ell_i \colon \mathbb{R}^6 \to \mathbb{R}\)

\[\begin{split} \ell_i(\beta,x_1,x_2,z_i,y_i) = \left( \begin{array}{l} y_i \\ 1-y_i \end{array} \right) \cdot \log\left(p \left( \begin{array}{l} \beta \cdot h(x_1,z_i) \\ \beta \cdot h(x_2,z_i) \end{array} \right) \right) \end{split}\]

Assignments:

  1. Write down the optimization problem the econometrician is solving. Explain the meaning of each part.

    • What are the variables the econometrician has control over in the estimation exercise?

    • What variables should be treated as parameters of the optimization problem?

  2. Elaborate on whether the solution can be guaranteed to exist.

    • What theorem should be applied?

    • What conditions of the theorem are met?

    • What conditions of the theorem are not met?

  3. Derive the gradient and Hessian of the log-likelihood function. Make sure that all multiplied vectors and matrices are conformable.

  4. Derive conditions under which the Hessian of the log-likelihood function is negative definite.

  5. Derive conditions under which the likelihood function has a unique maximizer (and thus the logit model has a unique maximum likelihood estimator).

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