# Advanced Time Series Analysis WS 0708

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```							Advanced Time Series Analysis WS 07/08
• 2 h per week lecture + 2 h PC lab (Kerstin Kehrle and Franziska Julia Peter) • PC lab uses GAUSS • Revise 4 h + x per week (Q4R)

• Exam: Either oral or written (open book) Material of lectures, script, reading list, chapters in textbooks
• Course plan • Prerequisites : Bachelor level exposure to Econometrics/Time Series Analysis

•Take notes ! • Script (download) • Textbooks: F. Hayashi (2000) Econometrics, Princeton J. Hamilton (1994) Time Series Analysis, Princeton W. Enders (1995) Applied Econometric Time Series, Wiley
1

Why follow the course? Time series techniques are essential in Economics & Finance
 Predictability of returns
Finance  Testing and estimating asset pricing models  Properties of price formation processes

 Properties of macroeconomic time series  Persistence of macro-shocks Economics

 Testing economic theories (PPT, Expectation Hypothesis of Term Structure)  Transmission of monetary policy
2

Agenda

 Basic concepts of time series analysis:

Stationarity, Ergodicity…

 Univariate time series (ARMA) Review  GARCH  Structural Vector Autoregressive Systems (SVAR)  Cointegration  What we have time for … Details: see course plan. Download from course page

3

for methods of analyzing economic time series with timevarying volatility (ARCH)

4

What is it? (1)
a) Daily close Dow Jones, from 08/23/1988 to 08/22/2000, daily frequency

xt (?)

b) Realisation of

xt  t  xt  t   t t  t xt

  0.08   0.2
t  1 / 250

 t  t ~ N (0,1)

5

What is it? (2)
a) Daily close Dow Jones, from 08/23/1988 to 08/22/2000, daily frequency

xt (?)

b) Realisation of

xt  t  xt  t   t t  t xt

  0.08   0.02
t  1 / 250

 t  t ~ N (0,1)

6

What is it? (3)
a) log of relative DAX change, from 01/02/1996 to 12/27/1996, daily frequency

xt (?)

b) Realisation of

xt  t   t t  t

  0.2
t  1 / 248

 t  t ~ N (0,1)

7

What is it? (4)
a) log of relative DAX change, from 01/02/1996 to 12/27/1996, daily frequency

xt (?)

b) Realisation of

xt  t   t t  t

  0.047
t  1 / 248

 t  t ~ N (0,1)

8

What is it? (5)
a) Realisation of

xt  t  xt    xt   t   xt t t  t

 3
xt (?)

  0.99   1.4
t  1 / 4

 t  t ~ N (0,1)
b) 3 month CHF LIBOR from 01/01/1974 to 01/01/2002, 3-month frequency

9

What is it? (6)
xt  t  xt    xt   t   xt t t  t
a) Realisation of

 3   0.99
xt (?)

  1.4
t  1 / 4

 t  t ~ N (0,1)
b) 3 month CHF LIBOR from 01/01/1974 to 01/01/2002, 3-month frequency

10

What is it? (7)
a) Price-dividend ratio S&P500 from 12/31/1947 to 12/31/1996, annual frequency

xt (?)

b) Realisation of

xt  t  xt    xt   t   t t  t

  23   0.5   0.9
t  1

 t  t ~ N (0,1)
11

What is it? (8)
a) Price-dividend ratio S&P500 from 12/31/1947 to 12/31/1996, annual frequency

xt (?)

b) Realisation of

xt  t  xt    xt   t   t t  t

  23   0.5   0.9
t  1

 t  t ~ N (0,1)

12

Assignments
Review statistical basics (e.g. Hamilton, 1994, p.739 ff.) Course dictionary: download from course page Random Variables and distributions

Expectation (mean, variance, higher moments) Joint distributions, covariance and correlation, Dependence and independence of random variables Conditional probability and conditional distribution
Conditional expectation and Independence

Hypothesis testing Estimation basics: OLS, Maximum Likelihood

13

It is important to distinguish the realisation from the process
stochastic process
Yt  Yt 1   t ,  t ~ N 0,1 Y0  0

Estimate by taking ensemble averages at each point
1 10000 s ˆ 1   Y1  -0.004 10000 s 1

Estimate by taking sample averages
ˆ 
2

1  Yt  6.377 T t 1

100

1 100 ˆ ˆ    Yt   2  25.130 T t 1

1 10000 s ˆ ˆ    Y1  1  0.991 10000 s 1 1 10000 s ˆ 100   Y100  0.023 10000 s 1
2 1





2

ˆ 

2 100

1 10000 s ˆ   Y100  100 10000 s 1





2

 99.028
14

It is important to distinguish the realisation from the process
stochastic process
Yt   t ,  t ~ N 0,1 Y0  0

Estimate by taking ensemble averages at each point
1 10000 s ˆ 1   Y1  -0.004 10000 s 1

Estimate by taking sample averages
1 100 ˆ    Yt  0.011 T t 1 1 100 ˆ ˆ    Yt   2  1.065 T t 1
2

1 10000 s 2 ˆ ˆ 1   Y1  1  1.001 10000 s 1 1 10000 s ˆ 100   Y100  0.000 10000 s 1 ˆ 
2 100





2

1 10000 s ˆ   Y100  100 10000 s 1





2

 0.996
15

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