Univariate Time Series Analysis - uni-muenchen.de · Univariate Time Series Analysis Organizational Details and Outline Introduction Time series analysis: Focus: Univariate Time Series

Univariate Time Series Analysis


Klaus Wohlrabe1 and Stefan Mittnik

1 Ifo Institute for Economic Research, [email protected]

SS 2016

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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

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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


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

8 Models for financial time seriesARCHGARCHGARCH: Extensions

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Organizational Details and Outline

Table of content I1 Organizational Details and Outline



4 (Univariate) Linear ModelsNotation and Terminology

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Table of content IIStationarity of ARMA ProcessesIdentification Tools




6 PredictionSome TheoryExamplesForecasting in Practice

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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


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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!

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Introduction




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Introduction




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Introduction




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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

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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

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Examination

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

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Prerequisites

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

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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))

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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

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Outline

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

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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

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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?

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An (unconventional) introduction

Table of content I1 Organizational Details and Outline



4 (Univariate) Linear ModelsNotation and Terminology

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Table of content IIStationarity of ARMA ProcessesIdentification Tools




6 PredictionSome TheoryExamplesForecasting in Practice

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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


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Goals and methods of time series analysis

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

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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

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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?

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Economic Time Series: GDP I80

8590

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DP

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

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

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Economic Time Series: GDP II-4

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Germany: Yearly GDP growth

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Economic Time Series: Retail Sales0

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Germany: Retail Sales - non-seasonal adjusted

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Economic Time Series: Industrial Production20

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0IP

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

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

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Economic Time Series: Industrial Production-2

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Germany: Yearly GIP growth

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Economic Time Series: The German DAX0

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Economic Time Series: Gold Price0

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Further Time Series: Sunspots0

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Further Time Series: ECG4

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

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

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Further Time Series: Chaos?-4

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Characteristics of Time series

TrendsPeriodicity (cyclicality)SeasonalityVolatility ClusteringNonlinearitiesChaos

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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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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

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Components of time series data

Time series decomposition

Time Series

Trend Cyclical

Seasonal Irregular

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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)

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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)

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The multiplicative model:

yt = dt · ct · st · εt (2)

will be treated in the tutorial.

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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

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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

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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.)

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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)

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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

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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

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Some simple filters

Moving Average

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Some simple filters

Moving Average

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M A(101)

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



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.

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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.

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Some simple filters


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

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Some simple filters


yt = wyt + (1− w)yt−1 w = 0.2

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Some simple filters


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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

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Trend extraction

Trend Component: Example

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Linear trend with a cyclical component

0.0

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Nonlinear trend with cyclical component

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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!

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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

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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

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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

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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

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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

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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

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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

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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

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Trend extraction

Comparison of different trend models

1

2

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5

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2005 2006 2007 2008 2009 2010

TURNOVER Linear TrendLogarithmic Trend Quadratic TrendExponential Trend

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Trend extraction

Detrending GDP

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92 94 96 98 00 02 04 06 08 10 12

GDP Linear TrendLogarithmic Trend Quadratic Trend

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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



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.

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Trend extraction


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)

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Trend extraction


76

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

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Trend extraction


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

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Trend extraction

Problems with the HP-Filter

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

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Trend extraction

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

8590

9510

010

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DP

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

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

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Trend extraction

HP-Filter

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Hodrick-Prescott Filter (lambda=1600)

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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

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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?

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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)

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Cyclical Component

Cyclical Component: Example

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IP Trend Cycle

Hodrick-Prescott Filter (lambda=14400)

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Cyclical Component

Cyclical Component: Example II0

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Cyclical Component

Cyclical Component: Example III4

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Cyclical Component

Can we test for a cyclical component?

Yes and nosee the trend sectionDoes a cycle make sense?

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Seasonal Component

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

050

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ail S

ales

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Germany: Retail Sales - non-seasonal adjusted

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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

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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

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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

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Seasonal Component

Seasonal Adjustment: Dummy Regression Example

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Residual Actual Fitted

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Seasonal Component

Seasonal Adjustment: Example

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Dummy Approach

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Arima X12

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Original Series

Seasonal Adjustement Retail Sales

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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

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Seasonal Component

Seasonal Moving Averages: Retail Sales Example

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Moving Seasonal Average Dummy Approach91 / 442



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

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Seasonal Component

Seasonal Differencing: Retail Sales Example

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Differenced Month

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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

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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

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Irregular Component

Irregular Component

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

in naturedue to human behavior

no observable iterations

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Irregular Component

Can we test for an irregular component?

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

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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)

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White Noise

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White Noise with Variance = 1

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White Noise

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WN with Variance = 1

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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

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Random Walk: Examples

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Random Walk with Drift: Examples

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EXAMPLE

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

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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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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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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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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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Univariate Time Series Analysis - uni-muenchen.de · Univariate Time Series Analysis Organizational Details and Outline Introduction Time series analysis: Focus: Univariate Time Series