docs/2-1-econometrics.tex.md

In edxu96/MatrixTSA: Tidy Multivariate (Non)Linear Dynamic Systems

L2

1. We view each single observation (Y_{i}) as a realization of a r.v. also denoted (Y_{i} .) Suppose this r.v. is Bernoulli distributed with success parameter (\theta .) State the density of (Y_{i} .) Show that (E\left[Y_{i}\right] = \theta) and (\operatorname{Var}\left[Y_{i}\right] = \theta(1-\theta))

$$ P(Y_{i}) = \begin{cases} \theta \ 1 - \theta \end{cases} $$

$$ E\left[Y_{i}\right] = \ $$

1. Given data (Y_{1}, Y_{2}, ., Y_{n},) write the Bernoulli model as a set of joint data densities, i.e. like our general formulation of the statistical model, (\mathcal{M}=\left{\mathrm{f}{\psi}\left(y{1}, y_{2}, ., y_{n}\right), \psi \in \Psi\right}, \Psi \subseteq \mathbb{R}^{k} .) State a condition that the model is correctly specified. This will be assumed in the following.

2. Show furthermore that these joint data densities (i.e. the elements of (\mathcal{M}) ) can be written in terms of the sample mean, (\bar{y}) Hintt: use that, for numbers (a_{i}) and (b_{i}) it holds that, (\prod_{i=1}^{n} a_{i} b_{i}=\prod_{i=1}^{n} a_{i} \Pi_{i}^{n} \Pi_{i=1}^{n} b_{i}) Hint2: use eq. 1.3 .1 in Hint 3: use the definition of the sample meanl.

3. State the log-likelihood function for the Bernoulli model and go through the derivations leading to the (\mathrm{MLE}, \widehat{\theta})

4. Show that (\widehat{\theta}) is unbiased and that the variance of it is (\frac{V a r\left[Y_{1}\right]}{n} .) What is the standard error of (\widehat{\theta} .)

5. Consider the distributional statement (\left.\widehat{\theta} \stackrel{P}{\rightarrow} \theta_{0}, \text { where } \theta_{0} \in\right] 0 ; 1[) is the population value/true value. Explain what it means and what assumptions are needed.

6. What does it mean that (\widehat{\theta}) is asymptotically normally (Gaussian) distributed? Explain why it holds.

CI

In 5% of the cases, hypothaticcaly repeated samples are out in the ends, CI will not include theta-node

In 95% of the cases, CI will include the true value.

CI is random, not the true value.

consistent

edxu96/MatrixTSA documentation built on Feb. 5, 2021, 11:30 p.m.