DEGREE OF MASTER OF SCIENCE IN FINANCIAL ECONOMICS FINANCIAL ECONOMETRICS HILARY TERM 2019 COMPUTATIONAL ASSIGNMENT 1 PRACTICAL WORK 3

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1 DEGREE OF MASTER OF SCIENCE IN FINANCIAL ECONOMICS FINANCIAL ECONOMETRICS HILARY TERM 2019 COMPUTATIONAL ASSIGNMENT 1 PRACTICAL WORK 3 Thursday 31 January Assignment must be submitted before noon (12:00) Friday 15 February 2019 (5th Week) by uploading to SAMS. This is group work. Groups of 3 or 4 are permitted. Groups with less than 3 or more than 4 are not permitted without explicit permission. All solutions must be submitted by the due date and time. Do not write the names of members of your group on your submission. Candidates should answer all questions. Suggested Length: 7 pages; limit 10 pages. Material on pages 11+ will not be assessed. Limit does not include the cover sheet, academic honesty declaration or submitted code files. All material, including figures, equations, and explanatory text must fit within 10 pages.

2 Assessment This assignment is assessed in 3 parts: 50% - Report. This report should focus on the analysis of the results and not code or the numerical values of the problems. 25% - Autograder. 5 functions must be submitted to compute the required outputs using the inputs. The signature of each function is provided as part of the problem. You must exactly match the function name. 25% - Presentation. There is a 5-minute presentation. The presentation covers one of problems 1 or 2 (not both). However, you do not find out until the time of presentation which problem your group needs to present and so you must be prepared for both. 2 turn over

3 Problem 1 The ID of your group members determines which data series you should analyze. The formula to determine your data set ID is: n Data Set ID = I D i mod i =1 where n is the number of members of your group. Your Data Set ID should be in {1, 2,..., 6}. The data sets are: Data Set ID FRED Code Transformation 1 T10Y2Y End-of-month 2 INDPRO None 3 PAYEMS None 4 BAMLH0A0HYM2 Biweekly, Wednesday, ending price 5 PCE None 6 T10YIE Weekly, Wednesday, ending price 1. Report your Data Set ID and describe the data (quantitative, graphical, qualitative as your group finds useful). 2. Download data for your assigned data set from FRED. If the data set needs to be transformed, the transformation can be performed in FRED before downloading. 3. Ignoring seasonality, is the data stationary? If not, transform it to be stationary. 4. Using the transformed data (if any was needed), build ARMA models for each series using the first 50% of the data. Assess the accuracy of this model using the second half. 5. Compare your models at horizons 1, 3 and 12 against random walk forecasts. Code Problems Mincer-Zarnowitz Implement Mincer-Zarnowitz regression. [alpha, beta, alpha_tstat, beta_tstat, joint_stat] = mincer_zarnowitz(realization, forecast) alpha - scalar, ˆα from MZ regression beta - scalar, ˆβ from MZ regression alpha_tstat - scalar, the t-stat for the null H 0 : α = 0 in a MZ regression 3 turn over

4 beta_tstat - scalar, the t-stat for the null H 0 : β = 1 in a MZ regression joint_stat - scalar, Wald statistic testing the joint null of correct specification realization - R by 1 vector of realizations of variable being forecast forecast - R by 1 vector of 1-step ahead forecasts variable being forecast. The forecast in position i is the time i 1 forecast of the value in i so that the forecast errors are realization - forecast. Diebold-Mariano Implement a Diebold-Mariano test. [avg_diff, std_err, dm_stat, concl] = diebold_mariano(loss_a, loss_b, nw_bandwidth) avg_diff - Mean loss difference where δ t = L A t L B t std_err - Estimated standard error of the difference dm_stat - The Diebold-Mariano test statistic concl - Conclusion. Should be -1 if null is rejected in favor of model A, 0 if null is not rejected and 1 if null is rejected in favor of model B. loss_a - R by 1 vector of losses from model A loss_b - R by 1 vector of losses from model B nw_bandwidth - Bandwidth (number of lags) to use in the Newey-West estimator 4 turn over

5 Problem 2 The ID of your group members determines which data series you should analyze. The formula to determine your data set ID is: n Data Set ID = I D i mod i =1 where n is the number of members of your group. Your Data Set ID should be in {1, 2,..., 6}. The data sets are: Data Set ID Portfolio 1 Portfolio 2 1 Excess Market (Mkt-RF) Momentum (MOM) 2 Size (SMB) Robust-Minus-Weak (RMW) 3 Value (HML) Conservative-Minus-Aggressive (CMA) All factors come from the 5-factor daily data set available on Ken French s site. The Momentum factor is stored in a separate file. This data set is longer than the other so only use data from July 1, Download daily data on your groups assigned portfolios from Ken French s website. 2. Construct the time series of monthly realized variance by squaring and summing the returns within each month. 3. Using this data and the transformation discussed in the notes and lectures, build an ARCH model for each series using the first half (50%) of the available data. 4. Assess the accuracy of your model using the second 50% of the data. 5. Build a HAR for the same data using the log of the monthly model. 1,2 6. Compare the HAR to the ARCH-type model selected in (3) at horizons of 1 month and 3-months based on cumulative volatility. 7. Suppose r t has an annualized volatility of x %. A portfolio that holds 20/x units of r t has an expected annualized volatility of 20%. Volatility-targeting can be implemented dynamically using the modelbased forecasts and rescaling returns monthly. Assess the ability of the models to volatility target a return with an annualized volatility of 20%. 8. Compare the performance in terms of Sharpe ratio of the dynamically targeted portfolio with one that uses only the historical average volatility to scale the portfolio. 1 When forecasting with the ln RV, if residuals are conditionally normal so that ln RV t +1 N ( µ t, σ 2),then using the properties of the log-normal, E t [RV t +1] = exp ( µ + 1 σ2). 2 2 Since this is monthly data, you should think about which lags are needed in the HAR 1, 5 and 22 are probably not the correct lags to use here. 5 turn over

6 Code Problems HAR Forecast HAR forecasts from a daily model that uses 1-, 5- and 22-day lagged terms. forecasts = recursive_har_forecast(data, in_sample) forecasts - T by 1 vector of 1-step ahead forecasts computed where the value in position t uses observations up-to-and-including time t. data - T by 1 vector of realized variances in_sample - scalar integer value indicating the in-sample period. The first forecast should be produced using observations 1, 2, 3,...,in_sample. Sharpe Ratio Compute the Sharpe ratio from a vector of returns. sr = sharpe_ratio(returns) sr - The Shape Ratio returns - T by 1 vector of returns MZ-GLS Implement the GLS version of a Mincer-Zarnowitz regression that can be used with variance forecasts. [alpha, beta, alpha_tstat, beta_tstat, joint_stat] = mincer_zarnowitz_gls(realization, forecast) alpha - scalar, ˆα from MZ regression 6 turn over

7 beta - scalar, ˆβ from MZ regression alpha_tstat - scalar, the t-stat for the null H 0 : α = 0 in a MZ regression beta_tstat - scalar, the t-stat for the null H 0 : β = 1 in a MZ regression joint_stat - scalar, Wald statistic testing the joint null of correct specification realization - R by 1 vector of realizations of variable being forecast forecast - R by 1 vector of 1-step ahead forecasts variable being forecast. The forecast in position i is the time i 1 forecast of the value in i so that the forecast errors are realization - forecast. 7 last page