Data Science and Machine Learning in Finance (ACCFIN 5246)
Problem Set 1 – Spring 2025
Question 1
Consider two investment assets X and Y . The observed asset returns generated by X have been historically low but always non-negative and equal to 0%, +1% and +2%, whereas observed returns generated by Y have been more volatile and equal to −5%, +2% and +8%. Furthermore, the following table provides further information about their joint investment perfor-mances with a summary that informs the likelihoods associated with each observed return outcomes across each other:
Table 1: Each value within the table is a probability and values across the top row and left column are returns in percentage points.
Each entry shows the probability of the corresponding X and Y values occurring jointly. For example: P(Y = 5, X = 1) = 0.2,
(1.1) Compute the marginal distributions for X and Y . In other words, compute the unconditional probabilities for each value of X: P(X = 0), P(X = 1), P(X = 2), and do the same for Y : With these in hand, compute E[X] and E[Y ].
(1.2) Compute the conditional distribution of Y given X, that is, P(Y = y|X = x) for all possible values of X and Y . Recall that:
(1.3) Compute the conditional expectation of E(Y|X = x) for all three values of X.
(1.4) Compute E[E[Y|X]] and compare with E[Y].
Question 2
(2.1) “Failure to reject H0 means the null hypothesis is true”, true or false? If true, explain why? If false, explain why.
(2.2) Is the statement, “A matrix is a projection matrix iff it is an idempotent matrix”, true? If so, explain why? If not, explain when this can be true.
(2.3) “An idempotent matrix is always invertible”, true or false?
(2.4) “A projection matrix is always invertible”, true or false?
(2.5) In March 1994, Michigan voters approved a sales tax increase from 4% to 6%. In political advertisements, supporters of the measure referred to this as a two percentage point increase, or an increase of two cents on the dollar. Opponents of the tax increase called it a 50% increase in the sales tax rate. Explain which way of measuring the increase in the sales tax is more accurate.
Question 3
Let yi represent the share price of a stock in the S&P, and xi be a dummy variable equal to 1 if stock i is classified in the financial sector and 0 otherwise. Suppose we see N stocks total, and Nx of these stocks are in the financial sector (which of course means that N − Nx are in other sectors).
(3.1) Write (the average of xi) in terms of Nx and N.
(3.2) Suppose we run an OLS regression of yi on a constant and xi
:
yi = β0 + β1xi + νi (2)
Show that is quantitatively equal to the difference in means between the two sectors:
Question 4
Categorical variables d1 and d2, and 1n each is a vector of size n × 1, and that d2 = 1n − d1 with n = n1 + n2 (n1: number of men and n2: number of women) such that:
suppose:
(4.1) Show that
(4.2) Compare and from two OLS regressions:
Question 5
Consider the regression model (y and u each is N × 1, X is N × k and β is k × 1):
y = Xβ + u
and that we additionally wish to examine Rβ = r where R is q × k and r is q × 1. Let RSSU and RSSR denote the unrestricted and restricted sum of squared residuals, respectively.
(5.1) Write a formal expression for the null and alternative hypotheses.
(5.2) Write the problem in terms of a constrained problem (Lagrange problem).
(5.3) Derive the first order conditions and solve.
(5.4) What the value of Lagrange multiplier? Interpret the Lagrange multiplier. What is the sign?
(5.5) What are the equations for RSSU and RSSR?
(5.6) Derive an expression in terms of regression residuals for, RSSR − RSSU .
(5.7) Interpret the term RSSR − RSSU . What is the sign and why?
Question 6
Consider the model yi = α + exp(xiβ) + ui
. Derive the NLS estimators for α and β.