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🔬 Tutorial problems theta

🔬 Tutorial problems theta#

\(\theta\).1#

Determine definiteness of the quadratic forms defined with the following matrixes either by Silvester’s criterion or eigenvalue criterion. For the asymmetric matrices use their symmetric part \(\frac{1}{2}(A+A^{T})\) when constructing a quadratic form (see exercise \(\epsilon\).1)

\[\begin{split} A_1 = \begin{pmatrix} 5 & 0 & 1 \\ 1 & 1 & 0 \\ -7 & 1 & 0 \end{pmatrix} \end{split}\]
\[\begin{split} A_2 = \begin{pmatrix} 5 & -2 & 3 \\ 0 & 4 & 0 \\ 0 & -1 & 3 \end{pmatrix} \end{split}\]
\[\begin{split} A_3 = \begin{pmatrix} 1 & 0 & 12 \\ 2 & -5 & 0 \\ 1 & 0 & 2 \end{pmatrix} \end{split}\]
\[\begin{split} A_4 = \begin{pmatrix} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{pmatrix} \end{split}\]
\[\begin{split} A_5 = \begin{pmatrix} -4 & 2 & -6 \\ 2 & -1 & 3 \\ -6 & 3 & -9 \end{pmatrix} \end{split}\]

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\(\theta\).2#

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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