这个作业是用Matlab完成非线性计量经济模型

Nonlinear econometrics for finance – Homework 3

1 Question 1 (50 points)

Consider the same C-CAPM model described in class, with CRRA utility function

u(ct) = c

1−γ

t

1 − γ

(1)

We have shown in class and in the second homework that the model provides the following conditions for all

time periods t = 1, 2, …, and all assets i = 1, 2, …, N

Et

”

β

ct+1

ct

−γ

1 + R

i

t+1

− 1

#

= 0. (2)

This provides a system of equations – one per asset – that we can use to estimate the parameters β and γ.

Indeed, the conditions in (2) are valid for any period t, and using the law of iterated expectations we can

write the unconditional expectation

E

”

β

ct+1

ct

−γ

1 + R

i

t+1

− 1

#

= 0 (3)

The GMM estimator solves the system of equations in (3) to find the parameter estimates for β and γ.

During the lectures, we also noticed that we can use equation (2) to create new moments conditions. We

have done this with the diffusion models in class, but the same logit applies here. Indeed, if we multiply the

conditional moment equation (2) by a function of variables contained in the information set of the investor,

we can obtain new moment conditions. To be concrete, remember that the notation Et means

E

”

β

ct+1

ct

−γ

1 + R

i

t+1

− 1

It

#

= 0 (4)

where It is all the information available for the inverstor at time t. This information includes past values of

consumption growth and returns,

It =

ct

ct−1

, R1

t

, …, RN

t

,

ct−1

ct−2

, R1

t−1

, …, RN

t−1

,

ct−2

ct−3

, R1

t−2

, …, RN

t−2

, …

(5)

This means that if we multiply any variable that is part of the information set It by one of the moment

conditions, the equation will still be equal to zero. The reason is that anything in the information set at time

t is a constant when computing the expected value at time t. Concretely, consider the moment condition

ct

ct−1

Et

”

β

ct+1

ct

−γ

1 + R

i

t+1

− 1

#

= 0. (6)

This is always true as we are multiplying ct

ct−1

by a conditional expected value that we know is equal to zero.

That means that if we take ct

ct−1

inside the expected value we still have an expected value equal to zero.

Et

“

ct

ct−1

β

ct+1

ct

−γ

1 + R

i

t+1

− 1

!# = 0. (7)

If this is the case, then the equation (7) is satisfied for any t and using the law of iterated expectations we

can use the unconditional moment condition

E

“

ct

ct−1

β

ct+1

ct

−γ

1 + R

i

t+1

− 1

!# = 0. (8)

as an additional moment to estimate our parameters.

This means that now for each asset i = 1, …, N we have two moment conditions to use for estimation

E

”

β

ct+1

ct

−γ

1 + R

i

t+1

− 1

#

= 0 (9)

E

“

ct

ct−1

β

ct+1

ct

−γ

1 + R

i

t+1

− 1

!# = 0. (10)

for a total of 2N moment conditions.

Notice how this is analogous to what we saw in the first homework with instrumental variables. Indeed,

we think of zt =

ct

ct−1

as an instrumental variable:

• zt =

ct

ct−1

is uncorrelated with the error in our nonlinear equation εt = β

ct+1

ct

−γ

1 + Ri

t+1

− 1,

because we have just shown

COV (zt, εt) = E (ztεt) = 0 (11)

because we know that E (εt) = 0.

• zt =

ct

ct−1

is correlated with the data, since it is certainly correlated with consumption growth in the

next period ct+1

ct

Data description The data in the file ccapmmonthlydata.xls are monthly data on consumption growth

and asset returns from February 1959 to November 1993 (not quarterly as in the first homework). The

first column contains the date; the second column contains the time series for the consumption growth ct+1

ct

.

Columns 3-13 are your asset returns for 11 asse

1. (10 points) Estimate the model using the usual moment conditions. This is equivalent to what you did

in Homework 2 with the quarterly data. Please use the HAC estimator for the standard errors.

(HINT: note that the data contains consumption growth ct+1

ct

for t = 1, …, T and not consumption levels

ct for t = 1, …, T. Modify the code from homework 2 accordingly, to compute the moments correctly.)

2. (5 points) Perform the Hansen test for overidentifying restrictions for this model. Explain your result.

3. (20 points) Use the lagged consumption growth ct

ct−1

as your instrumental variable. This means that

you need to add N moments to the usual moment conditions you used in the previous estimation. Reestimate the model using all the moment conditions (the original N moments and the new instrumental

variable moments). Please use the HAC estimator for the standard errors. Are your results different

from the estimates obtained in the first question? Why?

4. (5 points) Perform the Hansen test for overidentifying restrictions for this new estimates. Explain your

result.

5. (10 points) Is there any other variable that you could use as instrumental variable? Explain why

and write the moment conditions that you would use for estimation. (You don’t need to perform the

estimation in Matlab for this last question).

2 Question 2 (50 points)

1. (10 points) Estimate a GARCH(1,1)-M model using the Maximum Likelihood Estimator (see slides of

Lecture 6 for details on the model):

rt = βht + εt,

εt =

p

htut, with Et−1(ut) = 0 and Et−1(u

2

t

) = 1

ht = µ

∗ + δ

∗ht−1 + φ

∗

ε

2

t−1

.

2. (10 points) Compute standard errors (not using the gradient function in Matlab).

3. (4 points) Test whether the parameter β is significant. Test whether β = 5.

4. (8 points) Estimate a T-GARCH(1,1) model using the Maximum Likelihood Estimator (see slides of

Lecture 6 for details on the model):

rt =

p

htut, with Et−1(ut) = 0 and Et−1(u

2

t

) = 1

ht = µ

∗ + δ

∗ht−1 + φ

∗

ε

2

t−1 + ηε2

t−11(εt−1<0).

where 1(εt−1<0) is a variable that is equal to 1 when εt−1 < 0 and it is 0 otherwise

1(εt−1<0) =

(

1 if εt−1 < 0

0 otherwise

5. (10 points) Compute standard errors (not using the gradient function in Matlab).

6. (2 points) Test if the parameter η is significant.

7. (6 points) Plot the time series of conditional variances in both cases. Do you see anything interesting

in some time periods?

3