Individual Assignment

**Requirements:**

• Complete your entire assignment in Jupyter Notebook, including your code and

markdown sections for your written answers. Use Latex in markdown sections

where needed.

• Submit the resulting downloaded html file as your entire assignment. Care must

be taken with presentation in this file, however unavoidable error messages and

page formatting issues will be ignored in marking.

• Only relevant analysis outputs (graphs, tables, etc) should appear in the assignment file and all output should appear together with the discussion of that

output, in the file.

**Task 1 (30 marks). Business problem:**

This assignment follows the analysis conducted in the lectures regarding the dependence

between earnings and asset returns for companies listed on the NYSE. You will assess

whether earnings in one year (say t−1) affect asset returns in the subsequent year (say

t), and in particular whether returns are typically higher following positive, compared

to negative, earnings years and also assess whether there may be a linear relationship

between returns and lagged earnings.

Data: The data file for the analysis is “SampleData from US 90 08 wk3.csv” which

was sampled from “US 90 08 wk3.csv”.

2

Questions:

(a) Conduct an appropriate exploratory analysis on the asset returns, both individually

and in terms of one of the primary questions being considered in this assignment:

are returns in the subsequent year t typically higher following positive, compared to

negative, earnings years in year t − 1? Discuss any cleaning of the data you did,

including why and how you did it, or why you did not do it. (3 marks)

(b) Conduct the appropriate t-test (with α = 0.05), median and Mann-Whitney tests,

to assess whether returns are typically higher following positive, compared to negative,

earnings years. For median tests, use two-sided testing. Assess all assumptions made.

(10 marks)

(c) Which test’s result do you believe the most in part (b)? Discuss and explain. (2

marks)

(d) Conduct an appropriate exploratory analysis to assess whether there may be a

linear relationship between returns and lagged earnings. (3 marks)

(e) Conduct a simple linear regression analysis, using OLS estimation, for returns

on lagged earnings. Fully assess all assumptions of OLS. Also list and assess the

assumptions of LAD (no need to obtain the LAD estimates). Discuss any cleaning of

the data you did, including why and how you did it, or why you didn’t do it. (9 marks)

(f) Write a brief (< 0.5 page) report summarising and discussing your findings and

conclusions in layman’s terms. Include a discussion of whether you would recommend

an investment strategy based on your findings. (3 marks)

Task 2 (20 marks). Theoretical derivations:

Consider the population SLR model:

Yi = β0 + β1Xi + εi

and an observed, random sample of data (y1, x1), . . . ,(yn, xn) from that model. An

OLS regression is run on this data.

3

**Questions:**

(a) Show that the mean of the estimated residuals from the OLS regression exactly

equals 0, i.e. ¯e = 0. Hint: look at the first equation found when differentiating the

residual sum of squares with respect to β0. (2 marks)

(b) Show that the correlation between the estimated residuals and the observed x’s

exactly equals 0. How does this result relate to the 2nd LSA? Hint: look at the second

equation found when differentiating the residual sum of squares with respect to β1. (5

marks)

(c) Show that the equality T SS = RegSS + RSS holds, i.e. show that:

Xn

i=1

(yi − y¯)

2 =

Xn

i=1

(ˆyi − y¯)

2 +

Xn

i=1

(yi − yˆi)

2

Hint: add and subtract ˆyi

inside the square on the left side of the equation. (6 marks)

(d) Explain and show why SER2 = Var( d ) = Var( d Y |X). (7 marks)