Financial Econometrics (ECOM40004/ECOM90011)

Assignment 2

Summary

• The assignment is marked out of 20. It is comprised of 2 parts.

• Each student has to hand in a SEPARATE WRITTEN report.

• Submission is electronic through LMS or via e-mail to: [email protected]

• Due date is 7pm Friday 13 September 2024. Late assignments will loose 2 marks per day.

Part 1 (4 marks)

• Consider the following VAR(1) for three stationary variables y , y , and y

1t 2t 3t

      

y ϕ ϕ ϕ y u

1t 11 12 13 1,t−1 1t

 y  =  ϕ ϕ ϕ  y + u  (1)

 2t   21 22 23  2,t−1   2t 

y ϕ ϕ ϕ y u

3t 31 32 33 3,t−1 3t

where

   

u σ σ σ

1t 11 12 13

 u  ∼ i.i.d.(0,Σ ), Σ =  σ σ σ 

 2t  u u  21 22 23 

u σ σ σ

3t 31 32 33

• Equation (1) can be written compactly as follows:

y = Φ y +u (2)

t 1 t−1 t

     

y ϕ ϕ ϕ u

1t 11 12 13 1t

where y =  y , Φ =  ϕ ϕ ϕ , u =  u 

t  2t  1  21 22 23  t  2t 

y ϕ ϕ ϕ u

3t 31 32 33 3t

• Equation (2) can be transformed using Wold representation theorem as follows:

∞

(cid:88)

y = u +Θ u +Θ u +... = Θ u (3)

t t 1 t−1 2 t−2 i t−i

i=0

1. Write the values of Θ and Θ in Equation (3) using the parameters of Equation (1) (i.e.

1 2

in ϕ and σ with i,j = 1,2,3). (1 mark)

ij ij

1

2. Write generalised impulse response of y to a shock of one-standard deviation to u at

1t 2t

time t+1 using the parameters of Equation (1) (i.e. in ϕ , σ with i,j = 1,2,3 and

ij ij

Σ ). (1 mark)

u

3. Write generalised impulse response of y to a shock of one-standard deviation to u at

3t 2t

time t+2 using the parameters of Equation (1) (i.e. in ϕ , σ with i,j = 1,2,3 and

ij ij

Σ ). (2 marks)

u

Part 2 (16 marks)

• The Excel file ‘Assignment1 S2 2023.xlsx’ contains daily changes (in basis points) of the sim-

ulated sovereign Credit Default Swap spreads (CDSs) for 18 European countries between

2006-2012.

1. Specify a VAR model to study the interactions among the 18 sovereign CDSs; report the

lag order selection test and indicate the optimal lag order of your specified VAR model

based on the Schwarz information criterion (SC). (1 mark)

2. Estimate a VAR(1) model for the provided daily changes of 18 sovereign CDSs; report

the stability condition of your estimated VAR(1) model.(1 mark)

3. Construct a network among the 18 sovereign CDSs, based on full-sample estimation of

the VAR(1) model, using Forecast Error Variance Decomposition (based on Cholesky

factorization of the variance-covariance matrix with the ordering provided, i.e. 1 to 18).

Report the network at 10-day-ahead forecast horizon and report the Dielbold-Yilmaz

(DY) Spillover Index of the network at this selected forecast horizon. (2 marks)

4. Re-constructtheFEVD-basednetworkat10-day-aheadforecasthorizon(basedonCholesky

factorization with the ordering provided) at two specified sample periods below:

– Before the Global Financial Crisis (GFC) network: based on observations from 17-

April 2006 to 30-March 2007,

– During the Global Financial Crisis network: based on observations from 3-March-

2008 to 13-February-2009,

a. Report the values of the DY Spillover Index at the above two sample periods, (2

marks)

b. Plot histograms of the network’s spillovers and its estimated kernel densities (based

on Gaussian kernel with optimal bandwidth) at the above two sample periods, (2

marks)

c. Using the values of the DY Spillover Index and the two estimated kernel densities of

the network, describe major changes in the shape of the network before and during

the GFC. (1 mark)

2

5. Estimate a VAR(1) on a rolling-window basis with a window length of 250 days, to

study the time-varying interactions among the 18 sovereign CDSs. Re-construct the

10-day-ahead network at each rolling sample using Generalised Forecast Error Variance

Decomposition.

a. Compute and plot the time-varying DY Spillover Index of the network. (2 marks)

b. Estimate the spillover density (using Gaussian kernel with optimal bandwidth, 1024

gridpoints)oftheconstructednetworkateachrollingsample;computetheKullback-

Leibler Information Criterion (KLIC) by comparing the spillover density at each

rolling sample, S , against the spillover density of the first rolling sample, denoted

r

S . Plot the time-varying KLIC series. (2 marks)

0

6. Construct a network among the 18 sovereign CDSs, based on full-sample estimation of

theVAR(1)model,usingaccumulatedGeneralisedImpulseResponseFunctionstoshocks

of 10 basis points to 18 sovereign CDSs. Report the network at 10-day-ahead horizon

and report the estimated spillover density for this constructed network. (3 marks)

3