Financial Econometrics
Open Book Final Examination WS 2020/21, February 17th, 2021
Chair of Statistics and Econometrics Institute of Economics
University of Freiburg
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This exam contains 3 questions, each of which is worth 15 points. You have to answer all 3 questions within 90 minutes.
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Problem 1: Univariate Volatility Model
Consider the following conditional variance model for a daily log-return series rt of a financial asset:
rt = c + εt (1) 2 iid
εt = zt σt zt ∼ N (0, 1) (2) σt2 = ω + αε2t−1 + γε2t−11{εt−1<0}, (3)
where ω ≥ 0, α > 0, α + γ > 0. 1{εt−1<0} denotes the indicator function, which is equal to 1, if εt−1 < 0 and 0 otherwise.
a) What is the idea behind the model specified in (3)? How is it different from an ARCH(1) model?
b) Describe shortly the concept of the leverage effect in the context of condi- tional volatility modeling.
c) Is the model defined in (3) able to capture the leverage effect? Provide
both a mathematical and an intuitive explanation. For the mathematical
explanation derive cov(σ2 , εt|Ft−1), where Ft is the information set avail- t+1
able up to time t. Under which parameter restrictions is cov(σ2 ,εt|Ft−1) t+1
Hint: Note that 0 z3φ(z)dz = −2 and ∞ z3φ(z)dz = 2 with φ(·)
be the pdf of the standard normal distribution. Moreover, you can use the
fact that cov[ε2t , 1{εt<0}|Ft−1] = 0.
d) Derive the conditional variance forecast over the next two periods E[σ2 |Ft].6 P t:t+2
Problem 2: Model and Value-at-Risk
Assume you want to invest 10 000e into a long position of a financial asset with a log-return rt. Table 2.1 summarises the outputs from estimating various GARCH models for rt such that:
rt = c + εt (1) 2 iid
εt = zt σt zt ∼ N , (0, 1) (2) where σt2 is modelled by ARCH(1), GARCH(1,1), TGARCH(1,1) and EGARCH(1,1).
Table 2.1: Estimated parameters, p-value of the ARCH LM test on standardized residuals and information criteria.
ARCH GARCH
TGARCH EGARCH
−0.03∗∗∗ −0.03∗∗∗ 0.09∗∗∗ 0.03∗∗∗ 0.12∗∗∗ 0.24∗∗∗ 0.86∗∗∗ 0.97∗∗∗
−0.02∗∗∗ −0.03∗∗∗
ωˆ 0.65∗∗∗ 0.07∗∗∗
αˆ 0.29∗∗∗ 0.12∗∗∗
βˆ 0.87∗∗∗
γˆ 0.01 −0.01
AIC 3.92 3.80 BIC 3.93 3.82
p-value ARCH LM 0.03 0.71
3.81 3.82 3.83 3.84
∗∗∗ indicates significance of the parameter at the 1% level. no star indicates no significance at 1,5 or 10% level.
a) Based on the results of Table 2.1, which model is the best choice for σt2? Explain your answer based on the results of the table.
b) Please write down the equation for σt2 of the model you choose to be best at point a). Please define correctly the parameters and their restrictions.
c) Compute the Value-at-Risk at the p = 1% level for the next two periods
VaR(p,2) for your initial investment of 10 000e and by using the best
model chosen at point a). You have that εˆ = −0.015 and σˆ = 0.05. tt
Hint: The 1% quantile of the standard normal distribution is -2.3263.
Problem 3: Transition-GARCH Processes
Consider the following process:
Yt = c+εt (1)
εt = σt2 zt (2)
σ2 = w+αε2 G(−ε )+γε2 G(ε )+βσ2 (3) t t−1 t−1 t−1 t−1 t−1
where Zt ∼ N(0,1) and G(εt) is a probability transition function of εt satisfying
G(−εt) = 1 − G(εt).
Assume that ω,α,β,γ satisfy the necessary conditions for assuring that σt2 is positive and covariance-stationary.
a) Show that E[G(εt)|Ft−1] = E[G(−εt)|Ft−1] = 1/2. 3P Hint: Start writing E[G(εt)|Ft−1] by additionally conditioning on positive
and negative values of εt and account for the fact that εt |Ft−1 ∼ N (0, σt2 ).
b) Compare the Transition-GARCH(1,1) model with a simple GARCH(1,1) model. What are the similarities and what are the differences between the
two approaches? 2P
c) Show that, conditional on the information set Ft−1, ε2t and G(εt) are un- correlated.
d) Assume now that the transition function is given by:
G(εt) = 1{εt<0} (4)
where 1{.} denotes the indicator function.
1. Show that the transition function G(·) from Equation (4) satisfies the
condition: G(−εt ) = 1 − G(εt ). 1P
2. Which inequality relation between α and γ is reasonable under the
assumption of a leverage effect. 2P
3. Derive the News Impact Curve of the Transition-GARCH model with
the transition function given in Equation (4).
e) Now let the transition function be the logistic function
G(εt)= 1 , λ>0 (5)
1P |F ] of the conditional vari-
1 + exp(−λεt)
1. Show that the transition function G(·) from Equation (5) satisfies the
condition: G(−εt ) = 1 − G(εt ).
2. Calculate the s-step ahead forecast E[σ2
ance for s = 1, 2 for the transition function given in Equation (5). 2P
3. Derive the News Impact Curve of the Transition-GARCH model with
the transition function given in Equation (5). 1P
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