Solutions of Final Exam 2020, MGMTMFE 407 Empirical Methods in Finance
Prof. . Lochstoer
You are only free to consult your lecture notes, homeworks, and the textbook (Tsay) when answering this exam. You are not allowed to discuss the exam with anyone else. Please be clear and concise. Good luck!
1. Time: 180 minutes, 11:30am to 2:30pm. Emails must be sent by 2:45pm, which gives you 15 minutes extra for scanning.
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2. There are a total of 4 longer questions (100 points in total). Please answer all questions. To get credit you must show your work.
3. Write your anwers on blank sheets of paper, number each page.
4. When done, you have to scan your pages and email them back to the MFE office as a pdf file with title: EmpiricalYOURFULLNAME.pdf. YOURFULLNAME = last name and first name as in LochstoerLars or ZhangDanyu. The first page should have your name, student id, and the sentence ”I acknowledge my obligations under the UCLA Honor Code” with your signature following.
1. Autocorrelation and ARMA models (30 pts)
(a) (10 points) Consider the following ARMA-model:
yt+1 = 0.5 + 0.9yt − 0.25εt + εt+1, (1)
where εt is i.i.d. standard Normal for each t.
i.What are the unconditional mean and variance of this process?
ii.Assume yt = 0.3 and εt = 1. What are Et [yt+1] and Et [yt+2]?
i.E(yt) = φ0 = 5 1−φ1
V ar(y ) = σ2 1+θ12−2φ1θ1 = 3.22 t ε 1−φ21
ii.Et(yt+1) = 0.52
Et(yt+2) = 0.5 + 0.9Et(yt+1) = 0.968
(b) (10 points) You have a sample of the return on equity for Google. The un- conditional mean of roe is 0.1. The unconditional variance is 0.22. The first, second, and third order autocorrelations are (rounded to two decimal places) 0.9, 0.81, and 0.66.
i. What is the most parsimonious ARMA process that captures this pattern? (Example: AR(1), AR(2), MA(1), MA(2), ARMA(1,1) or other?) Explain how you got to your answer.
ii. Write down the ARMA model you choose, including the values of all coef- ficients in the model (similar to Equation 1 given above).
The most parsimonious process is AR(3), MA(3), or ARMA(1,2). Besides φ0 and σ2 , three parameters are needed here to match with the first, second, and third-order covariances. AR(3) is used as an example of how to get the coeffi- cients of the model. We could plug the numbers into the following equations to solve the coefficients.
rt = φ0 + φ1rt−1 + φ2rt−2 + φ3rt−3 + εt, εt ∼ WN(0,σε2) E= φ0
1−φ1 −φ2 −φ3
V ar = φ21V ar+φ2V ar+φ23V ar+σ2+2φ1φ2Cov(1)+2φ1φ3Cov(2)+2φ2φ3Cov(1)
Cov(1) = φ1V ar + φ2Cov(1) + φ3Cov(2) Cov(2) = φ1Cov(1) + φ2V ar + φ3Cov(1) Cov(3) = φ1Cov(2) + φ2Cov(1) + φ3V ar
(c) (10 points) Assume monthly inflation follows an AR(1) process with autocorre- lation 0.99. Due to seasonalities you want to estimate a model using 12-month sums of inflation overlapping monthly. That is, if the first observation is the sum of monthly inflation January through December, the second observation is the sum from February through January, etc.
Ignoring the intercept term, give the ARMA process this data follows, includ- ing the value of the coefficients. Remember to show your work.
rt is the monthly inflation and yt is the 12-month sums of inflation.
2. VAR models, return predictability and the present-value restriction(30 pts)
(a) (6 points) Write down a VAR(1) that has two state-variables: log market re- turns (rt) and the log price-dividend ratio (pdt). Clearly define all variables and parameters.
Zt+1 − μ = φ(Zt − μ) + εt+1 where Zt = (rt, pdt)′ and μ = (μ1, μ2)′ φ φ
rt+1 = φ1rt + εt+1(φ1 = 0.99)
11 rt−i = φ1 11 rt−i−1 + 11 εt−i
i=0 i=0 yt=φ1yt−1+11 εt−i
This process is captured by ARMA(1,11) with coefficients shown above.
φ= 11 12 φ21 φ22
The residual vector εt = (εrt , εpdt ) has a 2×2 covariance matrix Σ. We could assume the residuals are jointly normal and i.i.d., but this is not needed for consistency. You do need your residuals to be stationary. The could be implied from the assumptions you state. Your residual assumption is important for the discussion of the estimation below.
(b) (6 points) Explain in words how you would estimate the parameters of this VAR.
Estimate the model via OLS. That is, run the following two regressions: rt+1 = φ01 + φ11rt + φ12pdt + εr,t+1
pdt+1 = φ02 + φ21rt + φ22pdt + εpd,t+1
OLS is consistent even if the errors are non-normal and/or not i.i.d. In the case of heteroskedasticity and autocorrelation, the standard errors of the regression coeffcients must be estimated accordingly using, e.g., Newey-West standard errors.
(c) (6 points) Using the VAR you wrote down in 2.a, derive the formulas for the following expectations:
i. Et (rt+1), Et (rt+2) .
ii. E ∞ ρjr ,where|ρ|<1.
i.The eigenvalues of φ need to all be less than 1.
Define the vector er = (1, 0). Thus, er Zt+k = rt+k . The k-period
forecast at time t is: Et(erZt+k) = er(μ+φk(Zt −μ)). Where μ = (I2 −φ1)−1φ0
Therefore, we could get the cases for k=1 and k=2, which are,
Et (rt+1) = er(μ + φ(Zt − μ)).
Et (rt+2) = er(μ + φ2(Zt − μ)).
ii.E ∞ ρjr t j=1 t+j
= er(∞j=1 ρjμ + ∞j=1 ρjφj(Zt − μ)) =er( ρ μ+ρφ(I2 −ρφ)−1(Zt −μ))
(d) (6 points) Define DR = E ∞ ρjr . Derive a formula that uses your
t t j=1 t+j
VAR to get an expression for the cash flow component of the pd-ratio: CFt =
E ∞ ρj∆d . t j=1 t+j
Denote the vector epd = (0, 1). Thus, epdZt = pdt.
Campbell- gives us,
pd =constant+E(∞ ρj−1∆d )−E ∞ ρj−1r t t j=1 t+j t j=1 t+j
So we could have the following formula, CFt = ρepdZt + DR − ρ × constant
(e) (6 points) Now, you want to instead estimate a VAR(2) using the same vari- ables (returns and pd-ratio). Write this 2-variable VAR(2) in the form of a 4-variable VAR(1).
r φ φ φ φ φ r ε t+1 01 11 12 13 14 t 1t+1
pdt+1 φ02 φ21 φ22 φ23 φ24 pdt ε2t+1 r =0+1 0 0 0r +0
t pdt
t − 1 0 1 0 0 pdt−1 0
3. Volatility models(20 pts)
(a) (5 points)Give the three main stylized facts about market return volatility? Solution:
1) Volatility clusters. 2) Volatility is stationary. 3) Leverage effect: negative returns are followed by larger increase in volatility than the positive returns.
(b) (5 points) Let σt2 ≡ Et−1 [ε2t ] . Explain why an AR(1) process for ε2t is not
appropriate for modeling σt2.
Since ε2 = σ2 + κ , what we could observe is conditional variance plus t+1 t+1 t+1
noise. This means that AR(1) process for ε2t is not appropriate for modeling σt2. Also, the shock in AR(1) model could be both positive and negative, which may cause ε2t to turn negative.
(c) (5 points) Consider the GARCH(1,1) process σ2 = 0.1 + 0.08ε2 + 0.9σ2. If t+1 t t
σ2 =0.22 andε2 =0.12,whatisE σ2 ? t t tt+2
E σ2 = 0.1368
E σ2 = 0.1 + (0.08 + 0.9) × 0.1368 = 0.234
(d) (5 points) What differentiates an EGARCH(1,1) from a GARCH(1,1)? Solution:
EGARCH could capture leverage effect.
4. Factor models(20 pts) You are evaluating a long-short equity hedge fund and are given the below regression results:
Re =0.05+0.5×MKT −0.8×HML +0.8×SMB +0.1×ε, (2) fund,t t t t t
where the factors are the FF3 factors and where ε is a standard Normal error term. Assume all coefficients are significant.
(a) (5 points) What investment ’styles’ would you say characterizes this fund? Solution:
The fund invests in growth companies and small companies.
(b) (5 points) Describe in detail how you would construct a factor-neutral version of this fund. Give the expression for the return on this factor-neutral fund (denote the ensuing return Rtα).
I would long 1 unit fund, short 0.5 unit Mkt, long 0.8 unit HML, and short 0.8 unit SMB. Rtα = 0.05 + 0.1εt
(c) (5 points) What is the expected return and Sharpe ratio of this factor-neutral fund?
E(Rα) = 0.05, std(Rα) = 0.1, Sharp ratio= E(Rtα) = 0.5
= 0.5 × MKTt −0.8×HMLt +0.8×SMBt. The expected excess return on this port- folio is 6% and the standard deviation is 10%. Assuming you could buy the factor-neutral version of the hedge fund with returns Rtα, what is the mean- variance efficient combination of Re and Rα that achieves an unconditional
standard deviation of 15%? Solution:
t t std(Rtα )
(d) (5 points) Assume your existing portfolio has excess returns Re
0.12 0 −1 0.05
5 wMVE =kΩ−1Re =k 0 0.12 0.06 =k 6
The standard deviation is 15%. So we have, 0.12 05
0.15=k56 0 0.12 6
k = 0.19. The MVE weights are (0.96, 1.15)′.
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