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Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe 1 and Stefan Mittnik 1 Ifo Institute for Economic Research, [email protected] SS 2016 1 / 442

Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, [email protected]

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Page 1: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Univariate Time Series Analysis

Klaus Wohlrabe1 and Stefan Mittnik

1 Ifo Institute for Economic Research, [email protected]

SS 2016

1 / 442

Page 2: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

1 Organizational Details and Outline

2 An (unconventional) introductionTime series CharacteristicsNecessity of (economic) forecastsComponents of time series dataSome simple filtersTrend extractionCyclical ComponentSeasonal ComponentIrregular ComponentSimple Linear Models

3 A more formal introduction

4 (Univariate) Linear ModelsNotation and TerminologyStationarity of ARMA ProcessesIdentification Tools

2 / 442

Page 3: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

5 Modeling ARIMA Processes: The Box-Jenkins ApproachEstimation of Identification FunctionsIdentificationEstimation

Yule-Walker EstimationLeast Squares EstimatorsMaximum Likelihood Estimator (MLE)

Model specificationDiagnostic Checking

6 PredictionSome TheoryExamplesForecasting in PracticeA second Case StudyForecasting with many predictors

7 Trends and Unit Roots3 / 442

Page 4: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Stationarity vs. NonstationarityTesting for Unit Roots: Dickey-Fuller-TestTesting for Unit Roots: KPSS

8 Models for financial time seriesARCHGARCHGARCH: Extensions

4 / 442

Page 5: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Table of content I1 Organizational Details and Outline

2 An (unconventional) introductionTime series CharacteristicsNecessity of (economic) forecastsComponents of time series dataSome simple filtersTrend extractionCyclical ComponentSeasonal ComponentIrregular ComponentSimple Linear Models

3 A more formal introduction

4 (Univariate) Linear ModelsNotation and Terminology

5 / 442

Page 6: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Table of content IIStationarity of ARMA ProcessesIdentification Tools

5 Modeling ARIMA Processes: The Box-Jenkins ApproachEstimation of Identification FunctionsIdentificationEstimation

Yule-Walker EstimationLeast Squares EstimatorsMaximum Likelihood Estimator (MLE)

Model specificationDiagnostic Checking

6 PredictionSome TheoryExamplesForecasting in Practice

6 / 442

Page 7: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Table of content IIIA second Case StudyForecasting with many predictors

7 Trends and Unit RootsStationarity vs. NonstationarityTesting for Unit Roots: Dickey-Fuller-TestTesting for Unit Roots: KPSS

8 Models for financial time seriesARCHGARCHGARCH: Extensions

7 / 442

Page 8: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Introduction

Time series analysis:Focus: Univariate Time Series and Multivariate TimeSeries Analysis.A lot of theory and many empirical applications with realdataOrganization:

12.04. - 24.05.: Univariate Time Series Analysis, sixlectures (Klaus Wohlrabe)31.05. - End of Semester: Multivariate Time SeriesAnalysis (Stefan Mittnik)15.04. - 27.05. mondays and fridays: Tutorials (Univariate):Malte Kurz, Elisabeth Heller

⇒ Lectures and Tutorials are complementary!

8 / 442

Page 9: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Introduction

Time series analysis:Focus: Univariate Time Series and Multivariate TimeSeries Analysis.A lot of theory and many empirical applications with realdataOrganization:

12.04. - 24.05.: Univariate Time Series Analysis, sixlectures (Klaus Wohlrabe)31.05. - End of Semester: Multivariate Time SeriesAnalysis (Stefan Mittnik)15.04. - 27.05. mondays and fridays: Tutorials (Univariate):Malte Kurz, Elisabeth Heller

⇒ Lectures and Tutorials are complementary!

8 / 442

Page 10: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Introduction

Time series analysis:Focus: Univariate Time Series and Multivariate TimeSeries Analysis.A lot of theory and many empirical applications with realdataOrganization:

12.04. - 24.05.: Univariate Time Series Analysis, sixlectures (Klaus Wohlrabe)31.05. - End of Semester: Multivariate Time SeriesAnalysis (Stefan Mittnik)15.04. - 27.05. mondays and fridays: Tutorials (Univariate):Malte Kurz, Elisabeth Heller

⇒ Lectures and Tutorials are complementary!

8 / 442

Page 11: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Introduction

Time series analysis:Focus: Univariate Time Series and Multivariate TimeSeries Analysis.A lot of theory and many empirical applications with realdataOrganization:

12.04. - 24.05.: Univariate Time Series Analysis, sixlectures (Klaus Wohlrabe)31.05. - End of Semester: Multivariate Time SeriesAnalysis (Stefan Mittnik)15.04. - 27.05. mondays and fridays: Tutorials (Univariate):Malte Kurz, Elisabeth Heller

⇒ Lectures and Tutorials are complementary!

8 / 442

Page 12: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Tutorials and Script

Script is available at: moodle website (see course website)Password: armaxgarchxScript is available at the day before the lecture (noon)All datasets and programme codesTutorial: Mixture between theory and R - Examples

9 / 442

Page 13: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Literature

Shumway and Stoffer (2010): Time Series Analysis andIts Applications: With R ExamplesBox, Jenkins, Reinsel (2008): Time Series Analysis:Forecasting and ControlLütkepohl (2005): Applied Time Series Econometrics.Hamilton (1994): Time Series Analysis.Lütkepohl (2006): New Introduction to Multiple Time SeriesAnalysisChatham (2003): The Analysis of Time Series: AnIntroductionNeusser (2010): Zeitreihenanalyse in denWirtschaftswissenschaften

10 / 442

Page 14: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Examination

Evidence of academic achievements: Two hour writtenexam both for the univariate and multivariate partSchedule for the Univariate Exam: 30/05 (to beconfirmed!!!)

11 / 442

Page 15: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Prerequisites

Basic Knowledge (ideas) of OLS, maximum likelihoodestimation, heteroscedasticity, autocorrelation.Some algebra

12 / 442

Page 16: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Software

Where you have to pay:STATAEviewsMatlab (Student version available, about 80 Euro)

Free software:R (www.r-project.org)Jmulti (www.jmulti) (Based on the book by Lütkepohl(2005))

13 / 442

Page 17: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Tools used in this lecture

usual standard lecture (as you might expected)derivations using the whiteboard (not available in thescript!)live demonstrations (examples) using Excel, Matlab,Eviews, Stata and JMultilive programming using Matlab

14 / 442

Page 18: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Outline

IntroductionLinear ModelsModeling ARIMA Processes: The Box-Jenkins ApproachPrediction (Forecasting)Nonstationarity (Unit Roots)Financial Time Series

15 / 442

Page 19: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Goals

After the lecture you should be able to ...... identify time series characteristics and dynamics... build a time series model... estimate a model... check a model... do forecasts... understand financial time series

16 / 442

Page 20: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

Organizational Details and Outline

Questions to keep in mind

General Question Follow-up QuestionsAll types of data

How are the variables de-fined?

What are the units of measurement? Do the data com-prise a sample? Ifo so, how was the sample drawn?

What is the relationship be-tween the data and the phe-nomenon of interest?

Are the variables direct measurements of the phe-nomenon of interest, proxies, correlates, etc.?

Who compiled the data? Is the data provider unbiased? Does the provider pos-sess the skills and resources to enure data quality andintegrity?

What processes generatedthe data?

What theory or theories can account for the relationshipsbetween the variables in the data?

Time Series dataWhat is the frequency ofmeasurement

Are the variables measured hourly, daily monthly, etc.?How are gaps in the data (for example, weekends andholidays) handled?

What is the type of mea-surement?

Are the data a snapshot at a point in time, an averageover time, a cumulative value over time, etc.?

Are the data seasonally ad-justed?

If so, what is the adjustment method? Does this methodintroduce artifacts in the reported series?

17 / 442

Page 21: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Table of content I1 Organizational Details and Outline

2 An (unconventional) introductionTime series CharacteristicsNecessity of (economic) forecastsComponents of time series dataSome simple filtersTrend extractionCyclical ComponentSeasonal ComponentIrregular ComponentSimple Linear Models

3 A more formal introduction

4 (Univariate) Linear ModelsNotation and Terminology

18 / 442

Page 22: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Table of content IIStationarity of ARMA ProcessesIdentification Tools

5 Modeling ARIMA Processes: The Box-Jenkins ApproachEstimation of Identification FunctionsIdentificationEstimation

Yule-Walker EstimationLeast Squares EstimatorsMaximum Likelihood Estimator (MLE)

Model specificationDiagnostic Checking

6 PredictionSome TheoryExamplesForecasting in Practice

19 / 442

Page 23: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Table of content IIIA second Case StudyForecasting with many predictors

7 Trends and Unit RootsStationarity vs. NonstationarityTesting for Unit Roots: Dickey-Fuller-TestTesting for Unit Roots: KPSS

8 Models for financial time seriesARCHGARCHGARCH: Extensions

20 / 442

Page 24: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Goals and methods of time series analysis

The following section partly draws upon Levine, Stephan,Krehbiel, and Berenson (2002), Statistics for Managers.

21 / 442

Page 25: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Goals and methods of time series analysis

understanding time series characteristics and dynamicsnecessity of (economic) forecasts (for policy)time series decomposition (trends vs. cycle)smoothing of time series (filtering out noise)

moving averagesexponential smoothing

22 / 442

Page 26: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Time Series

A time series is timely ordered sequence of observations.We denote yt as an observation of a specific variable atdate t .A time series is list of observations denoted as{y1, y2, . . . , yT} or in short {yt}Tt=1.What are typical characteristics of times series?

23 / 442

Page 27: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Economic Time Series: GDP I80

8590

9510

010

5G

DP

1990q1 1995q1 2000q1 2005q1 2010q1 2015q1time

Germany: GDP (seasonal and workday-adjusted, Chain index)

24 / 442

Page 28: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Economic Time Series: GDP II-4

-20

2Q

uart

erly

GD

P g

row

th

1990q1 1995q1 2000q1 2005q1 2010q1 2015q1time

Germany: Quarterly GDP growth

-10

-50

5Y

early

GD

P g

row

th

1990q1 1995q1 2000q1 2005q1 2010q1 2015q1time

Germany: Yearly GDP growth

25 / 442

Page 29: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Economic Time Series: Retail Sales0

5010

015

0R

etai

l Sal

es -

Cha

in In

dex

1950m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1time

Germany: Retail Sales - non-seasonal adjusted

26 / 442

Page 30: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Economic Time Series: Industrial Production20

4060

8010

012

0IP

1950m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1time

Germany: Industrial Production (non-seasonal adjusted, Chain index)

27 / 442

Page 31: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Economic Time Series: Industrial Production-2

0-1

00

1020

30M

onth

ly IP

gro

wth

1950m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1time

Germany: Monthly IP growth

-40

-20

020

40Y

early

IP g

row

th

1950m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1time

Germany: Yearly GIP growth

28 / 442

Page 32: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Economic Time Series: The German DAX0

5000

1000

015

000

DA

X

0 2000 4000 6000 8000Time

Level

-.06

-.04

-.02

0.0

2.0

4R

etur

n

0 2000 4000 6000 8000Time

Return

29 / 442

Page 33: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Economic Time Series: Gold Price0

500

1000

1500

2000

Gol

d

0 2000 4000 6000 8000 10000Time

Level

-.1

-.05

0.0

5R

etur

n

0 2000 4000 6000 8000 10000Time

Return

30 / 442

Page 34: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Further Time Series: Sunspots0

5010

015

020

0S

unsp

ots

1700 1800 1900 2000time

31 / 442

Page 35: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Further Time Series: ECG4

4.5

55.

56

EC

G

0 2000 4000 6000 8000 10000Time

32 / 442

Page 36: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Further Time Series: Simulated Series: AR(1)-2

-10

12

3A

R

0 20 40 60 80 100Time 33 / 442

Page 37: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Further Time Series: Chaos or a real time series?-1

0-5

05

1990m1 1995m1 2000m1 2005m1 2010m1 2015m1time 34 / 442

Page 38: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Further Time Series: Chaos?-4

-20

24

0 20 40 60 80 100Time 35 / 442

Page 39: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Time series Characteristics

Characteristics of Time series

TrendsPeriodicity (cyclicality)SeasonalityVolatility ClusteringNonlinearitiesChaos

36 / 442

Page 40: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Necessity of (economic) forecasts

Necessity of (economic) Forecasts

For political actions and budget control governments needforecasts for macroeconomic variablesGDP, interest rates, unemployment rate, tax revenues etc.marketing need forecasts for sales related variables

future salesproduct demand (price dependent)changes in preferences of consumers

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Page 41: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Necessity of (economic) forecasts

Necessity of (economic) Forecasts

retail sales company need forecasts to optimizewarehousing and employment of stafffirms need to forecasts cash-flows in order to account ofilliquidity phases or insolvencyuniversity administrations needs forecasts of the number ofstudents for calculation of student fees, staff planning,space organizationmigration flowshouse prices

38 / 442

Page 42: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Components of time series data

Time series decomposition

Time Series

Trend Cyclical

Seasonal Irregular

39 / 442

Page 43: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Components of time series data

Time series decomposition

Classical additive decomposition:

yt = dt + ct + st + εt (1)

dt trend component (deterministic, almost constant overtime)ct cyclical component (deterministic, periodic, mediumterm horizons)st seasonal component (deterministic, periodic; more thanone possible)εt irregular component (stochastic, stationary)

40 / 442

Page 44: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Components of time series data

Time series decomposition

Goal:Extraction of components dt , ct and st

The irregular component

εt = yt − dt − ct − st

should be stationary and ideally white noise.Main task is then to model the components appropriately.Data transformation maybe necessary to account forheteroscedasticity (e.g. log-transformation to stabilizeseasonal fluctuations)

41 / 442

Page 45: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Components of time series data

Time series decomposition

The multiplicative model:

yt = dt · ct · st · εt (2)

will be treated in the tutorial.

42 / 442

Page 46: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

Simple Filters

series = signal + noise (3)

The statistician would says

series = fit + residual (4)

At a later stage:

series = model + errors (5)

⇒ mathematical function plus a probability distribution ofthe error term

43 / 442

Page 47: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

Linear Filters

A linear filter converts one times series (xT ) into another (yt ) bythe linear operation

yt =+s∑

r=−q

ar xt+r

where ar is a set of weights. In order to smooth local fluctuationone should chose the weight such that∑

ar = 1

44 / 442

Page 48: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

The idea

yt = f (t) + εt (6)

We assume that f (t) and εt are well-behaved.Consider N observations at time tj which are reasonably closein time to ti . One possible smoother ist

y∗ti = 1/N∑

ytj = 1/N∑

f (tj) + 1/N∑

εtj ≈ f (ti) + 1/N∑

εtj(7)

if εt ∼ N(0, σ2), the variance of the sum of the residuals isσ2/N2.The smoother is characterized by

span, the number of adjacent points included in thecalculationtype of estimator (median, mean, weighted mean etc.)

45 / 442

Page 49: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

Moving Average

Used for time series smoothing.Consists of a series of arithmetic means.Result depends on the window size L (number of includedperiods to calculate the mean).In order to smooth the cyclical component, L shouldexceed the cycle lengthL should be uneven (avoids another cyclical component)

46 / 442

Page 50: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

Moving Average

MA(yt ) =1

2q + 1

+q∑r=−q

yt+r

L = 2q + 1

where the weights are given by

ar =1

2q + 1

47 / 442

Page 51: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

Moving Average

Example: Moving Average (MA) over 3 PeriodsFirst MA term: MA2(3) = y1+y2+y3

3

Second MA term: MA3(3) = y2+y3+y43

48 / 442

Page 52: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

Moving Average

49 / 442

Page 53: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

Moving Average

0

20

40

60

80

100

120

140

50 55 60 65 70 75 80 85 90 95 00 05 10

IP

0

20

40

60

80

100

120

140

50 55 60 65 70 75 80 85 90 95 00 05 10

M A(3)

0

20

40

60

80

100

120

50 55 60 65 70 75 80 85 90 95 00 05 10

M A(5)

0

20

40

60

80

100

120

50 55 60 65 70 75 80 85 90 95 00 05 10

M A(7)

0

20

40

60

80

100

120

50 55 60 65 70 75 80 85 90 95 00 05 10

M A(9)

0

20

40

60

80

100

120

50 55 60 65 70 75 80 85 90 95 00 05 10

M A(11)

0

20

40

60

80

100

120

50 55 60 65 70 75 80 85 90 95 00 05 10

M A(25)

0

20

40

60

80

100

120

50 55 60 65 70 75 80 85 90 95 00 05 10

M A(51)

20

30

40

50

60

70

80

90

100

110

50 55 60 65 70 75 80 85 90 95 00 05 10

M A(101)

⇒ the larger L the smoother and shorter the filtered series50 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

EXAMPLE

Generate a random time series (normally distributed) withT = 20

Quick and dirty: Moving Average with ExcelNice and Slow: Write a simple Matlab program forcalculating a moving average of order LAdditional Task: Increase the number of observations toT = 100, include a linear time trend and calculate differentMAsVariation: Include some outliers and see how thecalculations change.

51 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

Exponential Smoothing

weighted moving averageslatest observation has the highest weight compared to theprevious periods

yt = wyt + (1− w)yt−1

Repeated substitution gives:

yt = wt−1∑s=0

(1− w)syt−s

⇒ that’s why it is called exponential smoothing, forecasts arethe weighted average of past observations where the weightsdecline exponentially with time.

52 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

Exponential Smoothing

Is used for smoothing and short–term forecastingChoice of w :

subjective or through calibrationnumbers between 0 and 1Close to 0 for smoothing out unpleasant cyclical or irregularcomponentsClose to 1 for forecasting

53 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

Exponential Smoothing

yt = wyt + (1− w)yt−1 w = 0.2

54 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Some simple filters

Exponential Smoothing

0

20

40

60

80

100

120

140

50 55 60 65 70 75 80 85 90 95 00 05 10

IP

20

40

60

80

100

120

50 55 60 65 70 75 80 85 90 95 00 05 10

w=0.05

0

20

40

60

80

100

120

50 55 60 65 70 75 80 85 90 95 00 05 10

w=0.2

0

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40

60

80

100

120

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50 55 60 65 70 75 80 85 90 95 00 05 10

w=0.95

55 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Trend Component

positive or negative trendobserved over a longer time horizonlinear vs. non–linear trendsmooth vs. non–smooth trends⇒ trend is ’unobserved’ in reality

56 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Trend Component: Example

0

4

8

12

16

20

24

25 50 75 100 125 150 175 200

Linear trend with a cyclical component

0.0

0.5

1.0

1.5

2.0

2.5

3.0

25 50 75 100 125 150 175 200

Nonlinear trend with cyclical component

57 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Why is trend extraction so important?

The case of detrending GDPtrend GDP is denoted as potential outputThe difference between trend and actual GDP is called theoutput gapIs an economy below or above the current trend? (Or is theoutput gap positive or negative?)⇒ consequences for economic policy (wages, prices etc.)Trend extraction can be highly controversial!

58 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Linear Trend Model

Year Time (xt ) Turnover (yt )05 1 206 2 507 3 208 4 209 5 710 6 6

yt = α + βxt

59 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Linear Trend Model

Estimation with OLS

yt = α + βxt = 1.4 + 0.743xt

Forecast for 2011:

y2011 = 1.4 + 0.743 · 7 = 6.6

60 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Quadratic Trend Model

Year Time (xt ) Time2 (x2t ) Turnover (yt )

05 1 1 206 2 4 507 3 9 208 4 16 209 5 25 710 6 36 6

yt = α + β1xt + β2x2t

61 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Quadratic Trend Model

yt = α + βxt + β2x2t = 3.4− 0.757143xt + 0.214286x2

t

Forecast for 2011:

y2011 = 3.4− 0.757143 · 7 + 0.214286 · 72 = 8.6

62 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Exponential Trend ModelYear Time (xt ) Turnover (yt )05 1 206 2 507 3 208 4 209 5 710 6 6

yt = αβxt1

⇒ Non-linear Least Squares (NLS) orLinearize the model and use OLS:

log yt = logα + log(β1)xt

⇒ ’relog’ the model63 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Exponential Trend Model

Estimation via NLS:

yt = α + β1xt

= 0.08 · 1.93xt

Forecast for 2011:

y2011 = 0.08 · 1.937 = 15.4

64 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Logarithmic Trend Model

Year Time (xt ) log(Time) Turnover (yt )05 1 log(1) 206 2 log(2) 507 3 log(3) 208 4 log(4) 209 5 log(5) 710 6 log(6) 6

Logarithmic Trend:yt = α + β1 log xt

65 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Logarithmic Trend Model

Estimation via OLS:

yt = α + β1 log xt = 1.934675 + 1.883489 · log yt

Forecast for 2011:

Y2011 = 1.934675 + 1.883489 · log(7) = 5.6

66 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Comparison of different trend models

1

2

3

4

5

6

7

8

2005 2006 2007 2008 2009 2010

TURNOVER Linear TrendLogarithmic Trend Quadratic TrendExponential Trend

67 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Detrending GDP

60

70

80

90

100

110

120

92 94 96 98 00 02 04 06 08 10 12

GDP Linear TrendLogarithmic Trend Quadratic Trend

68 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Which trend model to choose?Linear Trend model, if the first differences

yt − yt−1

are stationaryQuadratic trend model, if the second differences

(yt − yt−1)− (yt−1 − yt−2)

are stationaryLogarithmic trend model, if the relative differences

yt − yt−1

yt

are stationary69 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

The Hodrick-Prescott-Filter (HP)

The HP extracts a flexible trend. The filter is given by

minµt

T∑t=1

[(yt − µt )2 + λ

T−1∑t=2

{(µt+1 − µt )− (µt − µt−1)}2] (8)

where µt is the flexible trend and λ a smoothness parameterchosen by the researcher.

As λ approaches infinity we obtain a linear trend.Currently the most popular filter in economics.

70 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

The Hodrick-Prescott-Filter (HP)

How to choose λ?Hodrick-Prescot (1997) recommend:

λ =

100 for annual data1600 for quarterly data14400 for monthly data

(9)

Alternative: Ravn and Uhlig (2002)

71 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

The Hodrick-Prescott-Filter (HP)

76

80

84

88

92

96

100

104

108

112

92 94 96 98 00 02 04 06 08 10 12

IP Lambda = 14,400Lambda a.t. Ravn and Uhlig (2002) Lambda = 50,000Lambda = 10,000,000

72 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

The Hodrick-Prescott-Filter (HP)

-16

-12

-8

-4

0

4

8

12

92 94 96 98 00 02 04 06 08 10 12

Lambda = 14,400 Lambda a.t. Ravn and Uhlig (2002)Lambda = 50,000 Lambda = 10,000,000

73 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Problems with the HP-Filter

λ is a ’tuning’ parameterend of sample instability⇒ AR-forecasts

74 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Case study for German GDP: Where are we now?80

8590

9510

010

5G

DP

1990q1 1995q1 2000q1 2005q1 2010q1 2015q1time

Germany: GDP (seasonal and workday-adjusted, Chain index)

75 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

HP-Filter

-6

-4

-2

0

2

4

75

80

85

90

95

100

105

110

92 94 96 98 00 02 04 06 08 10 12 14

GDP Trend Cycle

Hodrick-Prescott Filter (lambda=1600)

76 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

Can we test for a trend?

Yes and noIf the trend component significant?several trends can be significantTrend might be spuriousIs it plausible that there is a trend?A priori information by the researcherunit roots

77 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Trend extraction

EXAMPLE

Time series: Industrial Production in Germany(1991:01-2014:02)

Plot the time series and state which trend adjustmentmight be appropriatePrepare your data set in Excel and estimate various trendsin EviewsWhich trend would you choose?

78 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Cyclical Component

Cyclical Component

is not always present in time seriesIs the difference between the observed time series and theestimated trend

In economicscharacterizes the Business cycledifferent length of cycles (3-5 or 10-15 years)

79 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Cyclical Component

Cyclical Component: Example

-15

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-5

0

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10

70

80

90

100

110

120

92 94 96 98 00 02 04 06 08 10 12 14

IP Trend Cycle

Hodrick-Prescott Filter (lambda=14400)

80 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Cyclical Component

Cyclical Component: Example II0

5010

015

020

0S

unsp

ots

1700 1800 1900 2000time

81 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Cyclical Component

Cyclical Component: Example III4

4.5

55.

56

EC

G

0 2000 4000 6000 8000 10000Time

82 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Cyclical Component

Can we test for a cyclical component?

Yes and nosee the trend sectionDoes a cycle make sense?

83 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Seasonal Component

Seasonal Componentsimilar upswings and downswings in a fixed time intervalregular pattern, i.e. over a year

050

100

150

Ret

ail S

ales

- C

hain

Inde

x

1950m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1time

Germany: Retail Sales - non-seasonal adjusted

84 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Seasonal Component

Types of Seasonality

A: yt = mt + St + εt

B: yt = mtSt + εt

C: yt = mtStεt

Model A is additive seasonal, Models B and C containsmultiplicative seasonal variation

85 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Seasonal Component

Types of Seasonality

if the seasonal effect is constant over the seasonal periods⇒ additive seasonality (Model A)if the seasonal effect is proportional to the mean⇒ multiplicative seasonality (Model A and B)in case of multiplicative seasonal models use thelogarithmic transformation to make the effect additive

86 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Seasonal Component

Seasonal AdjustmentSimplest Approach to seasonal adjustment:

Run the time series on a set of dummies without a constant(Assumes that the seasonal pattern is constant over time)the residuals of this regression are seasonal adjustedExample: Monthly data

yt =12∑

i=1

βiDi + εt

εt = yt −12∑

i=1

βDi

yt ,sa = εt + mean(yt )

The most well known seasonal adjustment procedure:CENSUS X12 ARIMA

87 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Seasonal Component

Seasonal Adjustment: Dummy Regression Example

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-5

0

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10

15

80

90

100

110

120

130

140

1992 1994 1996 1998 2000 2002 2004 2006 2008

Residual Actual Fitted

88 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Seasonal Component

Seasonal Adjustment: Example

90

95

100

105

110

115

92 94 96 98 00 02 04 06 08 10

Dummy Approach

92

96

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104

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112

92 94 96 98 00 02 04 06 08 10

Arima X12

80

90

100

110

120

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140

92 94 96 98 00 02 04 06 08 10

Original Series

Seasonal Adjustement Retail Sales

89 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Seasonal Component

Seasonal Moving Averages

For monthly data one can employ the filter

SMA(yt ) =12yt−6 + yt−5 + yt−4 + . . .+ yt+6 + 1

2yt+6

12

or for quarterly data

SMA(yt ) =12yt−2 + yt−1 + yt + yt+1 + 1

2yt+2

4

Note: The weights add up to one!Standard moving average not applicable

90 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Seasonal Component

Seasonal Moving Averages: Retail Sales Example

0

20

40

60

80

100

120

50 55 60 65 70 75 80 85 90 95 00 05 10

Moving Seasonal Average Dummy Approach91 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Seasonal Component

Seasonal Differencing

seasonal effect can be eliminated using the a simple linearfilterin case of a monthly time series: ∆12yt = yt − yt−12

in case of a quarterly time series: ∆4yt = yt − yt−4

92 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Seasonal Component

Seasonal Differencing: Retail Sales Example

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-8

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16

50 55 60 65 70 75 80 85 90 95 00 05 10

Year Differenced

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-10

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20

30

50 55 60 65 70 75 80 85 90 95 00 05 10

Differenced Month

93 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Seasonal Component

Can we test for seasonality?

Yes and noDoes seasonality makes sense?Compare the seasonal adjusted and unadjusted serieslook into the ARIMA X12 outputBe aware of changing seasonal patterns

94 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Seasonal Component

EXAMPLE

Time series: seasonally unadjusted Industrial Production inGermany (1950:01-2011:02)

Remove the seasonality by a moving seasonal filterTry the dummy approachFinally, use the ARIMAX12-ApproachStart the sample in 1991:01 and compare all filters with thefull sample

95 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Irregular Component

Irregular Component

erratic, non-systematic, random "residual" fluctuations dueto random shocks

in naturedue to human behavior

no observable iterations

96 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Irregular Component

Can we test for an irregular component?

YESseveral tests available whether the irregular component isa white noise or not

97 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Simple Linear Models

White Noise

A process {yt} is called white noise if

E(yt ) = 0γ(0) = σ2

γ(h) = 0 for |h| > 0

⇒ all yt ’s are uncorrelated. We write: {yt} ∼WN(0, σ2)

98 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Simple Linear Models

White Noise

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-2

0

2

4

10 20 30 40 50 60 70 80 90 100-4

-2

0

2

4

10 20 30 40 50 60 70 80 90 100-3

-2

-1

0

1

2

3

10 20 30 40 50 60 70 80 90 100

-3

-2

-1

0

1

2

3

10 20 30 40 50 60 70 80 90 100-3

-2

-1

0

1

2

3

10 20 30 40 50 60 70 80 90 100-3

-2

-1

0

1

2

3

10 20 30 40 50 60 70 80 90 100

-3

-2

-1

0

1

2

3

10 20 30 40 50 60 70 80 90 100-4

-2

0

2

4

10 20 30 40 50 60 70 80 90 100-3

-2

-1

0

1

2

3

10 20 30 40 50 60 70 80 90 100

White Noise with Variance = 1

99 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Simple Linear Models

White Noise

-3

-2

-1

0

1

2

10 20 30 40 50 60 70 80 90 100

WN with Variance = 1

-6

-4

-2

0

2

4

6

10 20 30 40 50 60 70 80 90 100

White Noise with Variance = 2

-30

-20

-10

0

10

20

30

10 20 30 40 50 60 70 80 90 100

White Noise with Variance = 10

-400

-300

-200

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0

100

200

300

10 20 30 40 50 60 70 80 90 100

White Noise with Variance = 100

100 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Simple Linear Models

Random Walk (with drift)

A simple random walk is given by

yt = yt−1 + εt

By adding a constant term

yt = c + yt−1 + εt

we get a random walk with drift. It follows that

yt = ct +t∑

j=1

εj

101 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Simple Linear Models

Random Walk: Examples

-15

-10

-5

0

5

10

15

10 20 30 40 50 60 70 80 90 100

102 / 442

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Univariate Time Series Analysis

An (unconventional) introduction

Simple Linear Models

Random Walk with Drift: Examples

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Page 107: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Simple Linear Models

EXAMPLE

Fun with Random WalksGenerate 50 different random walksPlot all random walksTry different variances and distributions

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Page 108: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Simple Linear Models

Autoregressive processes

especially suitable for (short-run) forecastsutilizes autocorrelations of lower order

1st order: correlations of successive observations2nd order: correlations of observations with two periods inbetween

Autoregressive model of order p

yt = α + β1yt−1 + β2yt−2 + . . .+ βpyt−p + εt

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Page 109: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Simple Linear Models

Autoregressive processesNumber of machines produced by a firm

Year Units2003 42004 32005 22006 32007 22008 22009 42010 6

⇒ Estimation of an AR model of order 2

yt = α + β1yt−1 + β2yt−2 + εt

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Page 110: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Simple Linear Models

Autoregressive processes

Estimation Table:Year Constant yt yt−1 yt−22003 1 42004 1 3 42005 1 2 3 42006 1 3 2 32007 1 2 3 22008 1 2 2 32009 1 4 2 22010 1 6 4 2

⇒ OLSyt = 3.5 + 0.8125yt−1 − 0.9375yt−2

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Page 111: Univariate Time Series Analysis · Univariate Time Series Analysis Univariate Time Series Analysis Klaus Wohlrabe1 and Stefan Mittnik 1 Ifo Institute for Economic Research, wohlrabe@ifo.de

Univariate Time Series Analysis

An (unconventional) introduction

Simple Linear Models

Autoregressive processes

Forecasting with an AR(2) model:

yt = 3.5 + 0.8125yt−1 − 0.9375yt−2

y2011 = 3.5 + 0.8125y2010 − 0.9375y2009

= 3.5 + 0.8125 · 6− 0.9375 · 4= 4.625

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