Forecasting Stock Market

Time Series

Regina Fuchs, 0103290

Richard Sellner, 0150085

Outline

• Description of the Data

• Model free procedures (Double

Exponential Smoothing, Holt-Winters

Method)

• Model based procedures (ARIMA,

GARCH)

• Forecasting and comparing the results

The Data

• Daily stock market data from 1st of

January 1998 to 31st December 2007

• Data Source: Datastream,

www.datastream.com

• We investigate Austrian, German and US

stock market data (ATX, DAX, Dow Jones)

• 2608 observation of each time series.

• 5 day week.

Graphical Representation

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98 99 00 01 02 03 04 05 06 07

ATXINDX

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DAXINDX

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DJINDUS

Augmented Dickey-Fuller

Test Stat. – Diff.Test Stat. – Level Index

-48.78-2.36DJ

-48.58-1.27 DAX

-17.832.57 ATX

Critical values: 1% (-3.43), 5% (-2,86). Time series seem to have unit roots.

Augmented Dickey Fuller test confirms this Hypothesis. For this reason, we

decided to compute 1st differences of the time series.

Time series in first differences

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98 99 00 01 02 03 04 05 06 07

ATXD

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DAXD

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DJD

We used 01-01-98 to 31-12-06 to fit the

model and 01-01-07 to 31-12-07 for dynamic

forecast evaluation.

Double Exponential Smoothing

1

1

)1(

)1(

−

−

−+=−+=

ttt

ttt

TLT

LxL

αααα

• Time series are integrated of order one -> no single exponential smoothing.

• Literature suggests values of alpha between 0.1 and 0.3.

• We fixed value of alpha to 0.1, 0.3 and 0.9 respectively and additionally fitted an alpha to the data by minimizing MSE.

Holt-Winters Method

• We fixed an alpha of 0.1 and compared it with

the alpha E-Views suggests.

11

11

)1()(

))(1(

−−

−−

−+−=+−+=

tttt

tttt

TLLT

TLxL

ββαα

DES,HW (ATX)

• DES: E-Views suggests an alpha of 0.54

• HW: alpha = 1, beta = 0.01 -> weak dependency

on past observation, strong trend.

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ATXATX DES 0.1ATX DES 0.3

ATX DES 0.5ATX DES 0.9

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ATXATX Holt-Winters 1ATX Holt-Winters 0.1

DES, HW (DAX)

• DES: EViews suggests an alpha of 0.49.

• HW: Yields similar results as for ATX, alpha =1,

beta = 0.

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DAXDAX Holt-Winters 1DAX Holt-Winters 0.1

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DAXDAX DES 0.1DAX DES 0.3

DAX DES 0.5DAX DES 0.9

DES, HW (DJ)

• E-Views suggests an alpha of 0.49

• HW: alpha = 0.99, beta = 0.

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DJDJ Holt-Winters 1DJ Holt-Winters 0.1

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DJDJ DES 0.1DJ DES 0.3

DJ DES 0.5DJ DES 0.9

ACF/PACF

ATX (Levels, 1st Diff.) DAX (Levels, 1st Diff.) DJ (Levels, 1st Diff.)

ARIMA

• In order to fit a model to the time series we start visually inspecting the time series (ACF, PACF).

• The autocorrelations do not show a noticeable pattern.

• For this reason, we tried any ARIMA process from ARIMA (1,1,1) to ARIMA(8,1,8).

• We forecasted the model with the smallest AIC.

Forecast for ATX using

ARIMA(7,1,7)

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ATX ATX ARIMA(7,1,7)

Forecast DAX using ARIMA (8,1,8)

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DAX ARIMA(8,1,8) DAX

Forecast for Dow Jones using

ARIMA (8,1,4)

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DJ DJ ARIMA(8,1,4)

Forecast using ARIMA-GARCH

models

1

2

1101

2)|( −−− ++==

+=

ttttt

tt

hhE

X

βεααεε

εµ

• Since Engel (1982) it has become very popular

in Finance to model volatility explicitly.

• We tried several ARIMA specifications of the

GARCH(1,1) model and performed a forecast for

the specification with the smallest AIC.

Forecast for ATX using

ARIMA(6,1,3) - GARCH

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ATX ATX ARIMA(6,1,3)-GARCH

Forecast for DAX using

ARIMA(3,1,3)-GARCH

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DAX DAX ARIMA(3,1,3)-GARCH

Forecast Dow Jones using

ARIMA(2,1,1)-GARCH

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DJ DJ ARIMA(2,1,1)-GARCH

Evaluation: Root Mean squared

prediction error

∑+−=

−− −=

N

mNt

tt xxmRMSE

1

21

1 ))1(ˆ( 261=m 2608=N

ATX DAX DJ

DES 1268.63 1139.35 1018.56

DES 0.1 1648.60 432.92 732.51

DES 0.3 1136.65 1139.35 1018.56

DES 0.9 288.26 2325.69 5314.55

HW 701.65 995.74 753.30

ARIMA 373.66 788.96 550.21

GARCH 633.17 330.86 397.25

Thanks for your Attention!!