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MATH0062
Mathematics and Statistics of Algorithmic Trading
MSc Examination
2023
Problem 1.
(a) Consider a multi-variable linear regression model
where 
is the vector of model parameters. Note that in the above model there is no intercept (bias) term.
(i) Show that the minimiser to the ridge regression problem
where 
is
Here I denotes an M × M identity matrix, and
[10 marks]
(ii) Explain why, in contrast to the traditional ordinary least square (OLS) regression, ridge regression is applicable in situations where the number of features is larger than the number of data points. [6 marks]
(b) Under the same model as in (1), write down the mathematical minimisation problem for the LASSO regression. You are not required to solve for the optimal β. [3 marks]
(c) Suppose ridge regression and LASSO regression are applied to the same data set. In the figures below, we plot all the non-zero estimated slope coefficients ˆβi under each regression method against the feature indices i. One of the plots corresponds to the results under ridge regression, and the other one corresponds to the results under LASSO regression. Which plot ((a) or (b)) is more likely to be the one corresponding to LASSO regression? Explain your reasoning briefly. How is your answer related to the idea of “feature selection”?
[6 marks]