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BACKGROUND
Following on from the equity research you carried out in Assessment 1 (Report), you are asked by your division to carry out a range of time series analyses to augment your earlier report. Specifically, the firm would like to know if alternative time series models (ARMA and ES) are better at pricing and predicting stock return. In addition, you are also expected to estimate volatility using the ARCH/GARCH family of models.
The terms of reference are detailed below.
PART ONE: STOCK RETURN PREDICTABILITY
UNIVARIATE TIME SERIES MODELS
1. Using the equal-weighted portfolio of the twenty (20) equities from Assessment One, update your portfolio this time to a daily return for the period 1 January 2018 to 30 June 2024.
Using the fund's daily returns
1. Estimate an appropriate Autoregressive Moving Average (ARMA) using the monthly return of your portfolio for the period 1 January 2018 to 31 December 2022.
2. Discuss the diagnostic results (residual analysis) of the ARMA model results.
3. Produce an out-of-sample forecast for 1 February 2023 to 30 June 2024.
4. Compare your forecasts to the return series for the forecast window, using the following measures:
· Mean Square Error (MSE)
· Root Mean Square Error (RMSE) and
· Mean Absolute Percentage Error (MAPE). [20 MARKS]
EXPONENTIAL SMOOTHING & TECHNICAL ANALYSIS
(i) Estimate an appropriate exponential smoothing (ETS) model for the daily price series of one stock in your portfolio from 1 January 2018 to 30 June 2022. Your model should incorporate a trend term and a dampening factor if required.
(ii) Produce an out-of-sample forecast from 1 July 2022 to 30 June 2024 and discuss the forecasts’ accuracy measures. [20 MARKS]
DISCUSSION OF RESULTS
Referring to your ARMA and MACD, discuss the predictability of stock prices. Relate your discussion to the notion that financial markets are informationally efficient. [10 MARKS]
[TOTAL 50 MARKS]
PART THREE: VOLATILITY MODELLING WITH GARCH
In addition to the return on the fund, it is important to understand how volatile the fund is relative to the market.
(i) Using the daily returns (in percentages) from 1 January 2018 to 29 February 2024, estimate the GARCH (1, 1) and the GJR GARCH (1, 1) process for;
· The S&P 500 Index
· The equity fund you formed in Part 1
· Any one (1) stock from the equity fund in Part 1.
(ii) Plot your volatility estimates and summarise the coefficients from your GARCH (1, 1) and GJR GARCH (1,1) models in an appropriate table and comment upon the results. [6 MARKS]
(iii) What do the results for GARCH (1, 1) suggest about the riskiness of your fund and stocks relative to the market portfolio? Discuss. [7 MARKS]
(iv) Discuss whether your volatility estimates are symmetrical using the GJR GARCH (1,1) results. Is this result in line with the economic theory of investor reaction to good and bad news? [7 MARKS]
[TOTAL 20 MARKS]
PART FOUR: VECTOR AUTOREGRESSIVE MODELS
Using the relevant quarterly series from January 2010 to December 2023:
(i) Estimate a vector autoregressive model (VAR) using the following series.
· The equity fund’s return series (from Part One)
· Real Gross Domestic Product [GDPC1]
· Industrial Output [INDPRO]
· Consumer price index [USACPIALLMINMEI]
· Federal fund effective rate [FEDFUNDS]
· Yield spreads (difference between the US 3 months T-Bill and 10-Year Treasury Constant Maturity Rate) [T10Y3M]
· SP 500 Index [^GSPC]
Note: the labels in the square bracket [ ] are the variable names in the Federal Reserve Economic Database (FRED), except for SP500, the variable in Yahoo Finance.
(ii) Report the following results using appropriate tables and charts:
· Granger causality between your fund’s return and all the variables in the model [ignoring all other causality results]
· Impulse response function from all other factors to your fund’s return series [ignoring all other impulse response function results].
(iii) Discuss only the Granger causality results and impulse response function on the fund return series. [10 MARKS]
(iv) Estimate a structural VAR (SVAR) using the following ordering and restrictions.
· Variable order Real Gross Domestic Product [GDPC1], SP 500 Index [^GSPC], The equity fund’s return series (from Part One).
· Restriction
Note: In the estimation, the ordering above Real GDP, SP 500 Index, and the fund’s return series must be maintained.
(v) Estimate the Granger Causality between the fund’s returns, Real GDP, and SP 500 Index.
(vi) Conduct an impulse response from Real GDP and SP 500 and the lag of the fund’s return series. [10 MARKS]
[TOTAL 20 MARKS]
FORMAT OF THE REPORT
An R script showing the codes used in estimating your results should be submitted for all estimations.
[Link for submitting the R Script. is provided below]
A well-structured report with clarity of discussion, logical presentation, and proficient level of depth (references).
[TOTAL 10 MARKS]
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