STA512: Final Examination

Question 1

Consider the linear regression model estimated from a sample indexed by $i = 1,..., N < \infty$:

$y _i = x _i^{'} \gamma + \epsilon _i$

with $k$ regressors denoted by $N \times k$ matrix $X$ satisfying:

\textbf{A1:} $rank(X) = k < \infty$ \textbf{A2:} $ y = X\gamma + \epsilon$, where $E\left[\epsilon\right] = 0$ \textbf{A3Rmi:} $E\left[\epsilon |X\right] = 0$ \textbf{A4:}$\epsilon_i$ independently distributed over $i$ \textbf{A5:} $\epsilon _t$ distrbuted Gaussian $N(0, \sigma _{\epsilon _i}^2)$ \noindent Suppose that two thirds of our sample corresponds to observations on women, while the rest are observations on men.

  1. We suspect that the error variance differs across observations depending on whether individual $i$ is a man (error variance $\sigma _m^2$) or a woman (error variance $\sigma _w^2$) with a priori reasoning suggesting that $\sigma _m^2 < \sigma _w^2$. Explain the properties of Ordinary Least Squares (OLS) and Generalized Least Squares (GLS) for estimating $\gamma$. (5 marks)

2.How would you statistically test the differences in variance? Among Wald, Likelihood Ratio, and Lagrangian Multiplier tests, which one do you propose and why? Does your answer change, if $N \rightarrow \infty$? (3 marks)

3.Suppose it is believed that the coefficient themselves differ by gender, i.e., we have $\gamma _m$ for men and (possibly different) $\gamma _w$ for women, as well as $\sigma _m^2 < \sigma _w^2$. An investigator proposes three estimation strategies:

A. Apply OLS to each of the two gender sub-samples, one for men and another for women. B. Maximize w.r.t. parameters $\gamma _m$, $\gamma _w$, $\sigma _m^2$, and $\sigma _w^2$ the log-likelihood function:

$L \equiv c - \frac{N _m}{2} \text{ln} \sigma _m^2 - \frac{N _w}{2} \text{ln} \sigma _w^2 - \sum _{i \in men}^{N _{men}} \frac{2(y _i - x _i^{'} \gamma _m)^2}{2 \sigma _m} - \sum _{i \in women}^{N _{women}} \frac{2(y _i - x _i^{'} \gamma _w)^2}{2 \sigma _w}$

C. Define the $x$ variables interacted with a male dummy and denote them by $x _m$ and the $x$ variables interacted with a female dummy by $x _w$ and apply OLS to the model

$y _i = x _{mi} \gamma _m + x _{wi} \gamma _w + u _i$ and $i= 1,...,N$

Discuss the finite sample properties of the three proposed procedures. (4 marks) Argue if there is any advantage in extended sample size. (2 marks)

  1. Given (3), how does your answer in (2) change? If unaffected, why not so? (3 marks)

  2. Can you argue this setting in the context of Chow Test? (2 marks)

  3. Suppose your team member claims that deep neural network (DNN) allows us to disregard GLS type transformation. In what context is the claim valid? (3 marks)

  4. Another collegue argues that DNN is superior to every proposed estimation method in (3). Discuss validity of the argument. (3 marks)

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