IB9Y60: Empirical Finance Group Project
Ganesh Viswanath-Natraj Andrea De Polis∗ University of Warwick, Warwick Business School
Guidelines
All questions can be solved using any language of your choice. It is recommended you stick to Matlab given the seminar material, however the questions can also be done via Python or R if that is your preferred language. Soft copies of your written answers and code must be submitted via myWBS by 8pm Thursday March 25th. Code should be commented and be able to execute to generate the results. The Group Project is worth a total of 20 marks. Each question is worth 5 marks.
1 Cointegration
1.1 Australia’s Real Exchange Rate and the Terms of Trade
In this question we use a simple VECM model to study the long-term cointegration of Australia’s real exchange rate with the terms of trade, which measures the ratio of export to import prices.
1. Load Q1FXreal.xlsx and plot the time series of the Australian real exchange rate and the terms-of-trade over the full sample.
2. Conduct a Dickey Fuller (constant, no trend) test on the log(price) of the real exchange rate and the log of the terms of trade. Are they stationary series? Discuss.
∗ganesh.viswanath-natraj@wbs.ac.uk and phd17ad@mail.wbs.ac.uk respectively.
3. Conduct the first step of the Engle-Granger procedure, by regressing the log price of the real exchange rate rert on the terms of trade tott
rert = α0 + α1tott + zt
Plot the residuals zˆ of this regression. Using the Dickey-Fuller (constant, no trend)
test, are the residuals stationary?
4. Estimate the bivariate VECM model for the real exchange rate and terms of trade below,
∆rert = β11∆rert−1 + β12∆tott−1 + δ1z1,t−1 + ε1,t ∆tott = β21∆rert−1 + β22∆tott−1 + δ2z2,t−1 + ε2,t
5. Interpret the coefficient δ1 in the above regression. Do the estimates line up with the findings of Reserve Bank of Australia economists who estimate a similar VECM specification in https://www.rba.gov.au/publications/rdp/2015/2015-12/the-baseline-ecm. html? 1
6. Re-estimate the models using data until December 2010 and forecast 1-quarter ahead for the log(price) of the real exchange rate within an expanding window scheme. Plot the 1-step forecasts and the actual values of the real exchange rate, and evaluate the root mean square error (RMSE) of your 1-quarter ahead forecasts.
7. Repeat the previous question, but now compute a 2 quarter ahead and 4 quarter ahead forecast. Report the RMSE of these forecasts.
8. Compute the RMSE of 1, 2 and 4 quarter ahead forecasts using a random walk model. (hint: under a random walk, the real exchange rate follows the process rert = rert−1 + εt)
9. Compute the ratio of the RMSE of the VECM model to the RMSE of the random walk for the 1, 2 and 4 quarter-ahead forecasts. How do your results support the
1The VECM specification by economists at the Reserve Bank of Australia have additional variables such as the real interest rate differential and the VIX, therefore your error correction estimates may differ from the RBA model.
t
2
hypothesis in Meese and Rogoff (1983)?
2 Volatility Modeling
In this question we conduct volatility modeling of a nominal exchange rate pair. All exchange rates are quoted as foreign currency units per US dollar. You will analyse volatility results for only one pair, which is randomly assigned based on your group student ID. Please refer to the second spreadsheet ”README” in FXnominal.xlsx for details on which pair to select.
1. Load Q2FXnominal.xlsx, and select the exchange rate series matched to your group number in the README spreadsheet. Estimate an AR(1) model on exchange rate returns Rer,t as follows:
Rer,t = φ0 + φ1Rer,t1 + εt
2. Take the residuals and estimate an ARCH(1) model defined as
σ t2 = α 0 + α 1 ε 2t − 1
Test for ARCH effects using the Engle Lagrange Multiplier test. What does this
test tell you about the presence of conditional heteroscedasticity?
3. Using the AR(1) model from before, take the residuals and estimate a GARCH(1,1) model defined as:
σ2 = α + α ε2 + βσ2
t 0 1t−1 t−1
4. Now take the residuals from the AR(1), and estimate a GJR GARCH(1,1), defined by the following equation:
σ2 = α + α ε2 + βσ2 + γε2 I
t 0 1 t−1 t−1
t−1 t−1
It−1 = 1,if
εt−1 < 0 0, otherwise
What is the estimate of γ? Is there a statistically significant leverage effect in exchange rate returns?
3
3
5.
6.
Now take the residuals from the AR(1), and estimate a t-GARCH(1,1), which assumes the errors are t-distributed.
Compute the likelihood ratio test statistic comparing a standard GARCH(1,1) with (i) a GJR-GARCH, and (ii) a t-GARCH(1,1). Which model you prefer and why?
Factor Modeling and PCA
In this question we conduct a factor analysis of excess currency returns for six portfolios sorted on interest rates, available from Lustig, Roussanov and Verdelhan (2011).2
1. Load Q3CurrencyPortfolios.xlsx, and conduct a PCA analysis on the six portfolio excess returns. Document the variance decomposition of the PCA. What per cent of variation in asset returns is explained by the first two factors respectively?
2. Plot the covariance between the first principal component with all of the 6 portfolio returns. Do the same for the second principal component.
3. Based on Lustig, Roussanov and Verdelhan (2011), compute the dollar factor as the average dollar returns across all 6 portfolios, Dol = 1 6 Ri and the high-minus-
6 i=1
low factor as the difference between the high interest rate portfolio returns minus
the low interest rate portfolio returns, HML = R6 − R1 (portfolio 6-portfolio 1). Plot the dollar factor against the first principal component, and the HML factor against the second principal component. What does this tell you about what the first two principal components correspond to?
4. Compute the betas of the dollar and HML factor, as calulated in the previous part, in the following regression for each portfolio i = 1, 2, ..., 6. Tabulate βi,dol and βi,HML for the 6 portfolios. In addition, plot the average (over full sample) of realised returns for the six portfolios against the average predicted returns for each portfolio.
Ri,t = αi + βi,dolDolt + βi,HMLHMLt + εt
2Data is taken from http://web.mit.edu/adrienv/www/Data.html. The full paper is at- tached along with the group project. A useful set of slides summarising their paper is found at https://www.snb.ch/n/mmr/reference/sem_2008_09_22_pres_verdelhan/source/sem_2008_09_ 22_pres_verdelhan.n.pdf.
4
5. Tabulate the pricing errors αi, standard errors and t-statistics for each portfolio. Are the pricing errors individually significant? Conduct a joint test of significance of the pricing errors using the Gibbons, Ross and Shanken (GRS) test statistic. Are the pricing errors jointly significant?
6. Using the Fama-Macbeth procedure, what is the price of risk (λ) for the dollar and HML factor? Tabulate the price of risk, and the betas of the dollar and HML factor.3
7. Conduct the Fama Macbeth procedure as before, but now account for time-varying beta. Set the initial sample at 60 periods.4What is the price of risk (λ) for the dollar and HML factor using this method? Compare your results to the method used in the previous question.
8. What have we learned from this exercise? Explain the economic intuition behind the pricing of each factor, and what explanations do the authors in Lustig, Roussanov and Verdelhan (2011) offer to explain their findings?
3In Lustig, Roussanov and Verdelhan (2011), the authors conduct the following Fama Macbeth proce-
dure. In the first step, they use the full sample to estimate the betas of the dollar and HML factors for all
6 portfolios. In the second step, they run a cross-sectional regression at each point in time and estimate
the lambdas, using the betas estimated in step 1. Finally, they take averages of lambda, λ ̄ = 1 T λi T i=1
to estimate the price of risk for the dollar and HML factor. Please refer to the paper for more details. 4Following the seminar, construct λ for the cross-sectional regression in period 61. Repeat the pro- cedure by rolling the initial sample of 60 periods 1 period forward, to then estimate a lambda for the
cross-sectional regression in period 62. Finally, produce an estimate of λ ̄ = 1 T λi that is the T −60 i=61
mean lambda across the estimated cross-sectional regressions.
5
4 Research Proposal
Write a research proposal on a topic in empirical finance. Where possible, use methods discussed in class. The proposal is to be two pages maximum (excluding references). Here are some tips on how to frame the proposal.
Introduction and Motivation
(1-2 paragraphs)
• What is the question you are answering? Why is it important? Were there any particular economic circumstances leading to the emergence of your topic?
• Is your question filling a gap in a particular line of research? Where does it fit in the literature?
• Is there a potential policy decision that may be informed by your research?5 Con- vince the reader that your question matters
Research Hypothesis
(1-2 paragraphs)
• Clearly state your specific, empirically answerable research question. If possible, state it in a way that can be empirically tested using the framework for hypothesis testing.
• Yourresearchhypothesisshouldbeaclearoutcomeofyourmotivation/introduction.
• What is it you want to find out and using what measurable variables?
Data and Methodology
(2-4 paragraphs)
• What data will you use? What are the independent and dependent variables you need to collect? What is the appropriate temporal and geographic unit for each variable? It will be a great bonus if you can show us that you already have access to the data, for instance by providing a link to your source.
5Not all questions can be motivated by a policy. However, relating your question to the real world and how it enhances understanding of financial markets is important
6
• What econometric techniques will you use to analyze your data? For example, are you using a VECM, a GARCH, or PCA analysis? You are open to use other methods if they are more relevant in your context, however we recommend using one of the methods discussed in class.
• Wherepossible,stateclearlytheequationsoftheregressionspecification/methodology. Relate the steps of the method to the research hypothesis.
Hypothesized results
(1-2 paragraphs)
• What results of interest do you expect your analysis to give? Do you think they would have external validity i.e. would they hold up in other geographic, temporal, etc. conditions?
• What are some proposed modifications to your hypothesis and/or methods to better answer the question?
• How broadly applicable do you expect your results to be?
• If you do have results, then you are welcome to briefly state them. However, addi- tional material (Tables/Figures) may be relegated to an appendix after references. This is NOT required for the project and you will be graded on the 2 page research proposal.
7