**Time Series Analysis**

Conclusion

In summary, time series analysis represents an

important field of statistics

and is

used

in

various fields

of application.

"The best qualification of a prophet is to have a good memory."

Marquis of Halifax

Advantages and Disadvantages

of Time Series Analysis in General

Classical Component Model

of Time Series Analysis

Specific Models

of Time Series Analysis

Typical Examples of a Time Series

Classical Component Model

of Time Series Analysis

Daily

closing prices

of market-listed derivatives

Monthly measured

unemployment rates

Annual

production rates

of a steel plant

Quarterly

revenue

of a company

Time Series Analysis in General

Classical Component Model

of Time Series Analysis

A time series is a sequence of observed values

of a certain characteristic, which are chronologically in succession and mostly periodical

among the same carrier.

Definition of Time Series

Advantages

Valuable to identify

seasonal variations

Often practiced method

, probably more then theoretical methods

Disadvantages

There are

no completely true components

Validity of hypotheses

are only partially verifiable

Components have to be

functions of time

Moving average

Exponential smoothing

Trend extrapolation

Box-Jenkins

Moving Average

Aim:

calculate an average

for the trend of historic data and

smooth out a time series

Differentiation into:

simple, exponential and weighted

moving average

Arithmetic mean of defined number of

successive values

within a specific time frame

Categorization into moving average of

even and uneven

order

Application of the Moving Average

Moving average of

uneven

order:

Moving average of

even

order:

Application of the Moving Average

Database for

Example in Lecture

Advantages and Disadvantages

of the Moving Average

Advantages

Simple, practicable, and effective

in use

Basic mathematical tools

are sufficient

Easy in terms of

comprehensibility

Disadvantages

Difficult to decide about

number of considered values

Simple moving average states an

equal weighting

for all values

It is suggested that the time series has an

regular cycle

Exponential Smoothing

Mathematical-statistical method for

short-term

to maximum

medium-term

horizons

Differentiation into:

single, double

and

triple

exponential smoothing

Recent

observed values are

more meaningful

for the prediction of the future than

earlier observed

Values are exponentially

increasing

from

past

to

present

Application of Exponential Smoothing

Application of Exponential Smoothing

Advantages and Disadvantages

of Exponential Smoothing

Trend Extrapolation

Simplest way to

define forecasts

Extrapolation relates to a

single time series

Historic data with

regularities

is continued into future

--> Trends in data must be

valid for future

as well

Based on the regularities in the historic data

trends are extrapolated into the future

Application of Trend Extrapolation

Possibility to use the

free-hand method

by sense of proportion

Mathematical way is

more appropriate

Different mathematical approaches

--> linear, parabolic, exponential and logistic trend, as well as gompertz curve

Application of Trend Extrapolation

Box-Jenkins methodology

Box-Jenkins methodology

Origin:

George Box and Gwilym Jenkins

in 1970

Main condition for the application:

stationarity

(mean variance and autocorrelation function that are essentially constant through time)

Principle of parsimony

: The more parameters to estimate, the more errors.

Aim: finding a good model

that describes how observations in a time series are related to each other

Comparison to classical model

Consideration of stochastic processes

instead of deterministic processes in order to model a time series.

Classical model: Cyclical movements are modeled as stationary processes around the deterministic trend.

Box-Jenkins:

random shocks are considered

and can have a

permanent effect on subsequent time series

data.

Basic Box-Jenkins Processes

Autoregressive

Process AR (p)

Basic Box-Jenkins Processes

Autoregressive Moving Average Process

ARMA

(p, q)

The Box-Jenkins model building process

Single

Exponential Smoothing:

Advantages

Simple

, robust, easy to use

Possible to utilize

very short time

series

Not an art

to

update

if new data is available

Disadvantages

Overly simplistic

and inflexible

Not optimal

for capturing any

linear dependence

in data

Hard

to forecast in the

long run

as values are

highly influenced

by

recent

happenings in the history of the time series

Database for

Example in Lecture

Application of Exponential Smoothing

Moving Average

Process MA (q)

Treatment of non-stationary processes:

ARIMA

(p, d, q)

Application of the trend extrapolation based on the linear trend states a

mathematical function

Additive

Multiplicative

Hybrid form

Advantages and Disadvantages

of Trend Extrapolation

Advantages

Simple

to use

Time

as

only element

to change in function

Can be used to define

short-, medium- and long-term forecasts

Disadvantages

Fluctuations in business cycle are

not considered

In a practical view it is

no longer sufficient

to extrapolate trends

Regularities of the past are

suggested to be valid

in future as well

Smooth Component

Advantages

Wide spectrum of software programs

enables simulation of various types of models with correspondent results

Consideration of random shocks

which have a permanent effect on subsequent time series data

Reduction of errors

due to the principle of parsimony

Disadvantages

Difficult interpretation

of the model

Need of time series with a

minimum of 50 observations

Modeling process requires

high investment of time and resources

to build a satisfactory model

Database for

Example in Lecture

Application of Trend Extrapolation

Advantages and Disadvantages

of the Box-Jenkins