time series analysis · the scope of our new journal re ects the diversity of themes that animate...

Time Series AnalysisSaarland University

PD Dr. Stefan Kloßner

Summer Term 2019U

N

IVE R S IT A

S

SA

RA V I E N

SI S

Time Series Analysis (SS 2019) Lecture 1 Slide 1

Topics of first lecture

Organizational issues

(Preliminary) course outline

Literature

Introduction


Organizational issues I

Lecture: Tue 08:30-10, Bldg. C3 1, Room 3.01

Tutorial: Mon 10-11:30, Bldg. C3 1, Room 3.01

Additional information & materials can be found at website

http://www.oekonometrie.uni-saarland.de

as well as moodle:

https://bit.ly/2UpSfsq

Contact: Bldg. C3 1, Room 2.19

Office hours: by appointment

Phone: +49 681 302 3179

E-Mail: [email protected]


http://www.oekonometrie.uni-saarland.de

https://bit.ly/2UpSfsq

Organizational issues II

Credit points: 6 ECTS (4 BP)

Exam: either written (2 hours) or oral (30 minutes)

Course is eligible forI Master BWL: Zusatzbereich BWLI Master Economics, Finance, and Philosophy: Pflichtbereich

EconometricsI Master Wirtschaftsinformatik: Okonometrie & Statistik


Aims and scope of Journal of Time Series Analysis (since 1980),according to journal’s homepage:During the last 25 years Time Series Analysis has become one ofthe most important and widely used branches of MathematicalStatistics. Its fields of application range from neurophysiology toastrophysics and it covers such well-known areas as economicforecasting, study of biological data, control systems, signalprocessing and communications and vibrations engineering.


This course: time series analysis with a view to financial econometrics(Finanzmarktokonometrie)

Journal of Financial Econometrics (since 2003), editor’sintroduction (vol. 1, no. 1, p. 1):Financial Econometrics has become one of the most active areasof research in econometrics. Twenty years ago, least-squaresmethods were the main econometric tool used to analyze issuessuch as efficient markets, tests of the capital asset pricing modelor arbitrage pricing theory, and stock returns forecasts. Theavailability of reliable financial data (often at very highfrequency) as well as increased computing power have spurredthe development of new and sophisticated econometrictechniques. These techniques are often unique to the field offinance.


They involve statistical modeling based on continuous time,focus on nonlinear features of time series, require a structuralapproach imposed by equilibrium or by the absence of arbitragefor pricing an increasingly complex array of financial products,and necessitate a delicate analysis of conditioning information.The Journal of Financial Econometrics intends to be thestatement of record for these developments.The scope of our new journal reflects the diversity of themesthat animate the field today. Estimation, testing, learning,prediction, and calibration in the framework of asset pricingmodels or risk management represent our core focus. Morespecifically, topics relating to volatility processes,continuous-time processes, dynamic conditional moments,extreme values, long memory, dynamic mixture models,endogenous sampling, transaction data, or microstructure offinancial markets will almost certainly appear in this journal. . . .


(Preliminary) outline

1 Introduction, notations, basics

2 Univariate linear time series models: ARIMA processes

3 Univariate non-linear time series models: ARCH and GARCHprocesses

4 Multivariate time series models


Literature I

Box & Jenkins: Time Series Analysis: Forecasting and Control,Holden-Day, San Francisco et al., 1970

Brockwell & Davis: Introduction to Time Series and Forecasting,2nd ed., Springer, New York et al., 2002

Brockwell & Davis: Time Series: Theory and Methods, 2nd ed.,Springer, New York et al., 2006

Brooks: Introductory econometrics for finance, Cambridge Univ.Press, Cambridge et al., 2002

Campbell, Lo & MacKinlay: The Econometrics of FinancialMarkets, 2nd ed., Princeton Univ. Press, Princeton, 1997

Franke, Hardle & Hafner: Einfuhrung in die Statistik derFinanzmarkte, 2. Aufl., Springer, Berlin et al., 2004


Literature II

Franses & van Dijk: Non-linear time series models in empiricalfinance, Springer, Cambridge Univ. Press, Cambridge et al., 2000

Gourieroux & Jasiak: Financial Econometrics, PrincetonUniversity Press, Princeton, 2001

Greene: Econometric Analysis, 7th ed., Prentice Hall Internat.,Upper Saddle River, 2012

Hamilton: Time Series Analysis, Princeton Univ. Press,Princeton, 1994

Mills: The econometric modeling of financial time series,Cambridge Univ. Press, Cambridge et al., 1993

Lutkepohl: New Introduction to Multiple Time Series Analysis,Springer, Berlin et al., 2005


Literature III

Rinne & Specht: Zeitreihen: Statistische Modellierung,Schatzung und Prognose, Vahlen, Munchen, 2002

Schlittgen, Streitberg: Zeitreihenanalyse, Oldenbourg, 9. Aufl.,Munchen, Wien, 2001

Schroder (Hrsg.): Finanzmarkt-Okonometrie, Schaffer-Poeschel,Stuttgart, 2002

Shiryaev: Essentials of Stochastic Finance: Facts, Models,Theory, Advanced Series on Statistical Science & AppliedProbability, Vol. 3, World Scientific, Singapur et al., 1999

Tsay: Analysis of Financial Time Series, Wiley, 3rd ed.,Hoboken, NJ, 2010

Remark: new editions exist of several of the books mentionedabove


Introduction

commodity prices in the future are uncertain stochastic models to describe this uncertainty

aim of these models:I assess the risk inherent in an investment

F Risikomessung und Value at RiskI decision on how to combine assets to form an ’optimal’

portfolio of assets (portfolio optimization)F Portfolio Selection

I pricing of derivative instruments (options, swaps, etc.) on thesecommodities (option pricing)

F Einfuhrung in die OptionsbewertungF Derivative Finanzinstrumente


Important tasks of time series analysis

finding an adequate stochastic model that properly describes thedata (model building)

testing the specified model

estimating model parameters

testing of hypotheses about the parameters

forecasting


RA Language and Environment for Statistical Computing

in practice: time series analysis essentially impossible withoutcomputer and suitable software

software used for this course: statistical software R (opensource, can be downloaded from www.r-project.org)

R is non-commercial, widely used in statistical and econometricresearch units, and gains importance in industry

many time series methods are available in R or some of itsuser-written packages

it is highly recommended to use R to get familiar with timeseries methods, however: knowing R not necessary for passingfinal exam

R is used in several other econometrics courses


Notations

Pt : price of some commodity at time t,1 Rt := Pt−Pt−1

Pt−1: returns, with −1 < Rt <∞ due to Pt > 0 ,

2 pt := lnPt : log-prices,3 rt := lnPt − lnPt−1: continuously compounded returns, with

rt = ln PtPt−1

= ln(1 + Rt), ert = 1 + Rt , and rt ≈ Rt for smallreturns.

If, for every t in some index set T , observations xt of someinteresting quantity are given, we think of xt as realisations of arandom process, a stochastic process (Xt)t∈T .


Example (prices)

2006 2008 2010 2012

2030

4050

6070

Date

Pt

closing prices of BMW (adjusted)


Example (log-prices)

2006 2008 2010 2012

3.0

3.5

4.0

Date

p t

logarithmic closing prices of BMW (adjusted)


Example (continuously compounded returns)

2006 2008 2010 2012

−0.

15−

0.10

−0.

050.

000.

050.

10

Date

r t

continuously compounded daily returns BMW (adjusted)


Definition (Stochastic process)

Given a probability space (Ω,F ,P), an index set T 6= ∅, and, forevery t ∈ T , a random variable Xt on (Ω,F ,P), we call

1 X := (Xt)t∈T stochastic process,

2 the mappingX•(ω) : T → R, t 7→ Xt(ω)

path of the stochastic process X (for every ω ∈ Ω),

3 (Xt)t∈T time series, if T is a subset of the integer numbers Z.


Remarks

The elements ω ∈ Ω are interpreted as scenarios, P then givesthe probability of the scenarios.

T is usually interpreted as a set of points in time.

Xt(ω) may then be interpreted as the process’ value at timet ∈ T , given that scenario ω ∈ Ω has occured.

If we have a path (xt)t∈T of some time series, we often call(xt)t∈T time series, too.

Usually we only observe some part of the path, i.e. (xt)t∈T for

some finite subset T ⊂ T .


Examples of time series: I

Air passengers on international flights

Month

Num

ber

(in th

ousa

nds)

1950 1952 1954 1956 1958 1960

100

200

300

400

500

600


Examples of time series: II

Mean annual temperature, New Haven, Conneticut

Year

Deg

rees

(Fa

hren

heit)

1910 1920 1930 1940 1950 1960 1970

4849

5051

5253

54


Examples of time series: III

Water depth of Lake Huron

Jahr

Dep

th (

feet

)

1880 1900 1920 1940 1960

576

577

578

579

580

581

582


Definition (Stationarity)

We call a stochastic process (Xt)t∈Z1 strongly stationary, if, for all m ∈ N, t1, . . . , tm ∈ Z, and h ∈ N,

the distributions of (Xt1 , . . . ,Xtm) and (Xt1+h, . . . ,Xtm+h) are

identical: (Xt1 , . . . ,Xtm)d= (Xt1+h, . . . ,Xtm+h),

2 (weakly) stationary or covariance stationary, if1 Xt has finite moments of second order for all t ∈ Z,2 EXt = EXt for all t, t ∈ Z,3 and Cov(Xt ,Xt) = Cov(Xt+h,Xt+h) for all h ∈ N0, t, t ∈ Z.


Remarks

It is possible to define (strict, weak) stationarity analogously forother index sets T as well.

In general, strict stationarity does not imply weak stationarity.

Strict stationarity together with existing second moments impliesweak stationarity.

In general, weak stationarity does not imply strict stationarity.

Weak stationarity implies strict stationarity, if all marginaldistributions of the time series are Gaussian.


TheoremIf (Xt)t∈Z is a strictly stationary time series and f : R→ R is ameasurable mapping, then the transformed process

f (X ) := (f (Xt))t∈Z

is also strictly stationary.

Remark:

An analogous assertion for weak stationarity does not hold, evenif the existence of second moments is guaranteed!


Examples

We take as given uncorrelated (independent) and identicallydistributed random variables εt with zero mean and variance σ2

ε

for t ∈ Z. Then, for every θ ∈ R, we can define a weakly(strictly) stationary process X θ by setting

X θt := εt + θεt−1.

Given uncorrelated random variables A and B with zero meanand unit variance, we can for every θ ∈ [−π, π] define a weaklystationary time series X θ by setting

X θt := A cos(θt) + B sin(θt).


Examples of paths of stationary time series I

−20 −10 0 10 20

−2

−1

01

2Paths of Xt

0.25 = εt + 0.25εt−1

t

x t


Examples of paths of stationary time series II

−20 −10 0 10 20

−1.

5−

1.0

−0.

50.

00.

51.

01.

5Paths of Xt

0.25 = Acos(0.25t) + Bsin(0.25t)

t

x t

ooo

A=−1, B=0.25A=0, B=0.5A=0.5, B=1


Introduction to complex numbers I(Mathematical) Interlude

In time series analysis, it is sometimes necessary to solveequations of the form x2 + 4 = 0.

We need to augment the real numbers R by moving on tocomplex numbers C.

We introduce the ’imaginary unit’ i with the property

i2 = −1 .

Complex numbers z ∈ C can be written as z = a + b i witha, b ∈ R. We call a = Re(a + b i) ’real part’ and b = Im(a + b i)’imaginary part’ of the complex number a + b i.

Most algebraic rules carry over to complex numbers.


Introduction to complex numbers IIAlgebraic rules

Addition: (a + b i) + (c + d i) = (a + c) + (b + d) i

Subtraction: (a + b i)− (c + d i) = (a − c) + (b − d) i

Multiplication: (a + b i) · (c + d i) = (ac − bd) + (ad + bc) i

Division:

a + b i

c + d i=

ac + bd

c2 + d2+

bc − ad

c2 + d2i (for c + d i 6= 0)

Absolute value: |a + b i | =√a2 + b2

For z ∈ C with z = a + b i, we define the conjugate of z byz = a − b i. We then have

z · z = |z |2 z + z = 2 · Re(z) z − z = 2 i · Im(z)


Introduction to complex numbers IIIFactorisation of polynomials

Huge advantage of complex numbers: for n ∈ N, a0, . . . , an ∈ R (C),every polynomial

p(x) =n∑

j=0

ajxj = anx

n + an−1xn−1 + · · ·+ a1x + a0

can be completely decomposed into linear factors, i.e. there existc , b1, . . . bn ∈ C with

p(x) = cn∏

j=1

(x − bj) = c · (x − b1) · (x − b2) . . . · (x − bn) .

Every polynomial of order n has n (possibly non-distinct) roots in C.


Introduction to complex numbers IVSolving equations in C (example)

Example: finding the roots of x2 − 2x + 5

Alternative 1:

x2 − 2x + 5 = 0⇔ (x − 1)2 = −4⇔ x − 1 = 2 i ∨ x − 1 = −2 i⇔ x = 1 + 2 i ∨ x = 1− 2 i

Alternative 2:

(’p–q–formula’: x = −p2±

√p2

4− q solves x2 + px + q = 0)

x = −−22

+√

44− 5 ∨ x = −−2

2−√

44− 5

⇔ x = 1 +√−4 ∨ x = 1−

√−4

⇔ x = 1 + 2 i ∨ x = 1− 2 i

(with√−a :=

√a i for a ∈ R+)


White noise, Gaussian processes

Definition (White noise, Gaussian process)1 A stochastic process is called Gaussian if all its marginal

distributions are Gaussian.

2 A stochastic process (εt)t∈T is called white noise, if:

E (εt) = 0 , Cov(εt , εs) =

σ2 , t = s

0 , t 6= s∀t, s ∈ T .

3 A white noise ε is called independent, if εt and εs arestochastically independent for all t 6= s.

4 A white noise that is also Gaussian is called Gaussian white noise.


Gaussian white noisePath of Gaussian white noise, with t = 1, . . . , 100, σ2 = 1

0 20 40 60 80 100

−2

−1

01

Gaussian white noise

t

ε t


Remarks

For a white noise ε, the random variables εt and εs are in generalonly uncorrelated, but not independent! Therefore, it is possiblethat Cov(ε2t , ε

2s ) 6= 0, although εt and εs are uncorrelated.

For an independent white noise ε, we have in particular: ε2t , ε2s

are stochastically independent, entailing Cov(ε2t , ε2s ) = 0 (given

that this covariance exists).

A Gaussian white noise is always independent, as normallydistributed random variables are uncorrelated if and only if theyare independent. Therefore, we have for Gaussian white noise:

εtiid∼ N(0, σ2).

(Gaussian) white noise is a central building block which is usedto construct more complicated stochastic processes.


Random Walk

Definition (Random Walk)1 Given a white noise (εt)t∈Z, we call a stochastic process (Xt)t∈Z

1 random walk (without drift), if we have for all t ∈ Z:Xt = Xt−1 + εt ,

2 random walk with drift α0 ∈ R, if we have for all t ∈ Z:Xt = α0 + Xt−1 + εt .

2 Given a white noise (εt)t∈N, we call a stochastic process (Xt)t∈N0

1 random walk (without drift) with initial value x ∈ R, if we have

for all t ∈ N0: Xt = x +t∑

n=1εn,

2 random walk with drift α0 ∈ R and initial value x ∈ R, if we

have for all t ∈ N0: Xt = x + α0t +t∑

n=1εn.


Random walk (without drift)Path of random walk, with t = 0, . . . , 100, εt

iid∼ N(0, 1), x = 0

0 20 40 60 80 100

−10

−8

−6

−4

−2

0

Random walk

t

Xt


Random walk with drift and initial valuePath of random walk with drift, with t = 0, . . . , 100, εt

iid∼ N(0, 1), x = −5,α0 = 0.3

0 20 40 60 80 100

−10

−5

05

1015

2025

Random walk with drift, initial value=−5

t

Xt


Remarks

Random walks without initial value and their distribution are notuniquely specified. It is possible to completely specify such aprocess by nailing down the process’ distribution at some timet0 ∈ Z.

For a random walk X , we call eX geometric or exponentialrandom walk. To emphasize the difference, X then is oftencalled arithmetic random walk.

A random walk is not stationary. For a random walk (withoutdrift) with initial value 0, we have Var(Xt) = σ2

εt and

Corr(Xt ,Xt+h) =√

tt+h

for all h ∈ N.


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