这是一篇来自英国的关于集团项目从回归模型中生成数据，以了解各种估计器和程序的性质的经济代写

Experiments with sparse regression models

We will be generating data from regression models in order to understand properties of various estimators and procedures. Our basic framework requires to generate p predictors in matrix x and a target variable y as follows:

xi ∼ Np (0,S) (1)

yi = β1xi1 + … + βpxip + εi , εi ∼ N(0, σ2 ) (2)

for i = 1, …, n, where S{jk} = ρ |j−k| for some correlation level −1 ≤ ρ ≤ 1 and for elements j, k ∈ {1, …, p}.

1. Use the code Monte Carlo bias.m and do various experiments in order to demonstrate the importance of omitted variable bias on econometric estimates (see Appendix A for more details and guidance).

2. Write code that explores the opposite issue, i.e. what happens if we generate from a regression with three significant predictors but we estimate a regression with p − 3 (p > 3) additional predictors that are irrelevant? Which is more hurtful for regression, omitting an important predictor or including an irrelevant one?

(Hint: make sure you are thorough enough and explore the effect of various choices n, p, σ2 , ρ)

3. Variable selection for small p Generate 10 predictors in x and perform an information-theoretic model averaging approach similar to Pesaran and Timmerman (1995, Journal of Finance) and Kapetanios, Labhard and Price (2008, Journal of Business & Economic Statistics). Write a short MATLAB code that scans through all 210 possible model specifications, estimates each one using OLS, and calculates some measure of fit of your preference (e.g. BIC, AIC, adjusted R2 etc). Find the model with the highest probability of being the “best” model. Notes on the procedure are in Appendix B.

4. Variable selection for large p Use the lasso and elastic net to perform highdimensional variable selection using 5-fold cross validation. Set p large and explore cases where p ≫ n. Alongside the other choices (σ 2 , ρ) explain in which cases the lasso/elastic net choose the correct variables.

HEALTH WARNINGS:

❼ I won’t accept a sloppy copy-paste of a million tables without structure, motivation and scientific structure. Your main task is to build a story and explain what works and what doesn’t, in a structured and thorough way. Your report should be scientific and evidence based, and not opinion or intuition-based like a newspaper article or a blog piece.

❼ You should submit all your code in clear and reproducible form. I won’t accept use of build-in functions (other than the functions for lasso/elastic net).

❼ You can use MATLAB, Python or R. I can read other languages, but it will be harder for me to run your code and replicate things, so you are advised NOT to work in C++, Java, Stata etc.

References

[1] Kapetanios, G., Labhard, V. and Price, S. (2008) Forecasting Using Bayesian and Information-Theoretic Model Averaging, Journal of Business & Economic Statistics,26(1), 33-41.

[2] Pesaran, M.H. and Timmermann, A. (1995), Predictability of Stock Returns:

Robustness and Economic Significance. The Journal of Finance, 50, 1201-1228.

[3] Tibshirani, R. (1996). Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society. Series B (Methodological), 58(1), 267-288.