Financial Econometrics Open Book Final Examination WS 2020/21

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Financial Econometrics

Open Book Final Examination WS 2020/21

Problem 1:  Univariate Volatility Model

Consider the following conditional variance model for a daily log-return series s芒 of a financial asset:

s芒 = c + e芒

e芒 = 3芒尸7芒(2)

7芒(2) = u + ae芒(2)-〇 + ye芒(2)-〇 1(et − 1 <0} ,

where u > 0, a > 0, a + y > 0.  1(et − 1 <0}  denotes the indicator function, which is equal to 1, if e芒-〇  < 0 and 0 otherwise.

a) What is the idea behind the model specified in (3)? How is it different from         an ARCH(1) model?                                                                                           2 P

b)  Describe shortly the concept of the leverage effect in the context of condi-         tional volatility modeling.                                                                                   1 P

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 cou(7芒(2)′〇, e芒 N~芒-〇 ), where ~芒 is the information set avail- able up to time o. Under which parameter restrictions is cou(7芒(2)′〇, e芒 N~芒-〇 )

negative?                                                                                                             6 P

Hint:  Note  that   o 33 o(3)d3 = _尸   and   ＋o 33 o(3)d3 = 尸   with o(.)

be  the pdf of the standard normal distribution.  Moreover,  you  can use  the fact that cou[e芒(2) , 1(et<0}N~芒-〇 ] = 0 .

d)  Derive the conditional variance forecast over the next two periods E[7芒:(2)芒′2N~芒].6 P


Problem 2:  Best Volatility Model and Value-at-Risk

Assume you want to invest 10 000e  into a long position of a financial asset with a log-return s芒 . Table 2.1 summarises the outputs from estimating various GARCH models for s芒  such that:

s芒 = c + e芒  

e芒 = 3芒尸7芒(2)

3芒  r, (0, 1)

where 7芒(2) 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

cˆ

_0.02***

_0.03***

_0.03***

_0.03***

 

0.65***

0.07***

0.09***

0.03***

 

0.29***

0.12***

0.12***

0.24***

8ˆ

 

0.87***

0.86***

0.97***

 

 

 

0.01

_0.01

AIC

3.92

3.80

3.81

3.82

BIC

3.93

3.82

3.83

3.84

p-value ARCH LM

0.03

0.71

0.71

0.67

***  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 7芒(2)?         Explain your answer based on the results of the table.                                    3 P

b)  Please write down the equation for 7芒(2)  of the model you choose to be best         at point a). Please define correctly the parameters and their restrictions.     2 P

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 eˆ芒 = _0.015 and 芒 = 0.05.

Hint:  The  1% quantile of the standard normal distribution is -2.3263.             10 P


Problem 3:  Transition-GARCH Processes

Consider the following process:

y芒    =   c + e芒

e芒    =   尸7芒(2)3芒

7芒(2)    =   w + ae芒(2)-〇G(_e芒-〇 ) + ye芒(2)-〇G(e芒-〇 ) + 87芒(2)-〇

where Z芒  r(0, 1) and G(e芒 ) is a probability transition function of e芒 satisfying G(_e芒 ) = 1 _ G(e芒 ).

Assume that u, a, 8, y satisfy the necessary conditions for assuring that 7芒(2)  is positive and covariance-stationary.

a)  Show that E[G(e芒 )N~芒-〇 ] = E[G(_e芒 )N~芒-〇 ] = 1/2.                                          3 P Hint:  Start writing E[G(e芒 )N~芒-〇 ] by  additionally  conditioning  on positive          and negative values of e芒  and account for the fact that e芒 N~芒-〇  ～r(0, 7芒(2)) .

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?                                                                                                  2 P

c)  Show that, conditional on the information set ~芒-〇 , e芒(2)  and G(e芒 ) are un-         correlated.                                                                                                            2 P

d) Assume now that the transition function is given by:

G(e芒 ) = 1(et<0}                                                                                                                   (4)

where 1(.}  denotes the indicator function.

1.  Show that the transition function G(.) from Equation (4) satisfies the         condition: G(_e芒 ) = 1 _ G(e芒 ).                                                                  1 P

2. Which inequality relation between a and y is reasonable under the         assumption of a leverage effect.                                                                 2 P

3.  Derive the News Impact Curve of the Transition-GARCH model with         the transition function given in Equation (4).                                          1 P

e) Now let the transition function be the logistic function

G(e芒 ) = 1 + ezp(1)(_Ae芒) ,    A > 0                                                       (5)

1.  Show that the transition function G(.) from Equation (5) satisfies the         condition: G(_e芒 ) = 1 _ G(e芒 ).                                                                  1 P

2.  Calculate the s-step ahead forecast E[7芒(2)′5 N~芒] of the conditional vari-         ance for t = 1, 2 for the transition function given in Equation (5).         2 P

3.  Derive the News Impact Curve of the Transition-GARCH model with         the transition function given in Equation (5).                                          1 P















2022-03-10