编程代写 Microsoft Word - Tutorial 5 T2 2021.docx

Microsoft Word – Tutorial 5 T2 2021.docx

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University of Wales
School of Economics

Financial Econometrics
Tutorial 5

1. (Error correction and common trend)

Suppose that I(1) series 𝑦! and 𝑥! are cointegrated and 𝜀! = 𝑦! − 𝛽𝑥! is an independent white

noise process. Assume that Δ𝑥! = 𝛾Δ𝑥!”# + 𝜂! where 𝜂! is also an independent white noise

process. Here 𝛽 and 𝛾 are constant parameters. Show that the changes in 𝑦! and 𝑥! are

governed by the vector error correction model

Δ𝑥! = 𝛼#(𝑦!”# − 𝛽𝑥!”#) + 𝜙##Δ𝑥!”# + 𝑢#! ,

Δ𝑦! = 𝛼$(𝑦!”# − 𝛽𝑥!”#) + 𝜙$#Δ𝑥!”# + 𝑢$! .

Express the coefficients 𝛼#, 𝛼$, 𝜙##, 𝜙$#in terms of the original parameters 𝛽 and 𝛾. Express

the shocks 𝑢#! and 𝑢$! in terms of the white noise processes 𝜀! and 𝜂!. What is the common

trend in this example and why?

2. In light of stylized facts of financial returns, how likely is that an AR(p) or MA(q) or their

combination ARMA(p,q) model are suitable for modeling financial return series? Which

stylized facts are likely to be violated?

3. (Cointegration and error correction model)
This question is based on the data in the Excel file fisher_update.XLS. The file

contains 171 quarterly observations, from 1969Q4 to 2012Q2, on the Australian Consumer

price Index (P) and on the yield to maturity of 90-day bank accepted bills (R).

(a) Generate the inflation rate as: INF=400*(log(P(1))-log(P)). When we

construct the inflation rate this way, we lose the last observation, namely, 2012Q2. We

change the sample to 1984Q1 to 2012Q1, which is the post-float period of the exchange rate.

Plot R and INF. Comment on whether or not R and INF co-move.

(b) Throughout this and the following parts of the question, continue to use the sample

1984Q1-2012Q1. Assume that both R and INF are I(1) processes. Estimate the regression

R! = 𝛽% + 𝛽#INF! + 𝜀!

and perform an ADF test, without intercept and time trend, on the residuals from the

regression. What do you conclude?

(d) Regardless of your result in (c), assume that R! and INF! are cointegrated. If the

cointegration error 𝜀! = R! − 𝛽% − 𝛽#INF! is positive at 𝑡, what would you say about the

likely movements in R!&# and INF!&#?

(e) Estimate the following two error-correction equations separately using OLS

ΔR! = 𝑐# + 𝛼#(resid01)!”# +∑ (𝜙##,(ΔR!”( + 𝜙#$,(ΔINF!”())(*# + 𝑢#! ,

ΔINF! = 𝑐$ + 𝛼$(resid01)!”# + ∑ (𝜙$#,(ΔR!”( + 𝜙$$,(ΔINF!”())(*# + 𝑢$! .

Comment on your results. Do you observe error correction mechanism in the estimated

equations?

(f) Can you reduce the “size” of the model in (e) by dropping some lags? Re-estimate the

error-correction equations when insignificant lagged terms of ΔR! and ΔINF! are dropped

from the equations you estimated in part (e). Comment on the new results.

4. Simulation Exercise in Excel.

The Analysis ToolPak is a Microsoft Office Excel add-in program that is available when you install
Microsoft Office or Excel.

To use the Analysis ToolPak in Excel, however, you need to load it first.

1. Click the Microsoft Office Button , and then click Excel Options.
2. Alternatively you may get to Excel Options from open Excel file, File -> Options
3. Click Add-Ins, and then in the Manage box, select Excel Add-ins.
4. Click Go.
5. In the Add-Ins available box, select the Analysis ToolPak check box, and then click OK.
a. Tip If Analysis ToolPak is not listed in the Add-Ins available box, click Browse to locate it.
b. If you get prompted that the Analysis ToolPak is not currently installed on your computer,

click Yes to install it.
6. After you load the Analysis ToolPak, the Data Analysis command is available in

the Analysis group on the Data tab.

Generate 2 random walk series:

𝑦! = 𝑦!”# + 𝜀!, 𝜀! ∼ 𝑖𝑖𝑑 WN N(0,1)

𝑥! = 𝑥!”# + 𝑢!, 𝑢! ∼ 𝑖𝑖𝑑 WN N(0,1)

To do this in excel first generate two standard normal random variables. Data -> Data
Analysis -> Random number generation. We need:
Number of variables 2
Number of random numbers 1000

Distribution: Normal

This gives you two random normal variables. Set 𝑦# = 0, 𝑥# = 0. Generate 𝑦! , 𝑥! t>1 using

the equations above.

Regress y on x using Data -> Data Analysis -> Regression

Select range for y and x and press ok.

Analyse the output of the regression. Do you expect these results? What is going on?

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