Time-Series Analysis and Forecasting – Part III To read at home

Time-Series Analysis and Forecasting – Part III

To read at home

Time-Series Data

Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 19-3

Time-Series Plot

the vertical axis measures the variable of interest

the horizontal axis corresponds to the time periods

U.S. Inflation Rate

0.002.004.006.008.00

10.0012.0014.0016.00

197

5

197

7

197

9

198

1

198

3

198

5

198

7

198

9

199

1

199

3

199

5

199

7

199

9

200

1

Year

Infl

ati

on

Rat

e (

%)

A time-series plot is a two-dimensional plot of time series data

The problem of comparability of levels of time series

Jointing (смыкание) of time series

Since time series is formed during the long period of time, its levels are

frequently incomparable

Reasons of the incomparability1. Change of prices.2. Different methods of calculation of

the same indicator. 3. Change of «borders» (organizational,

administrative)

The method of jointing time series is often used to ensure the comparability of data. It is

necessary to have a transitional link (переходное звено) for jointing time series. Transitional link – is the period of time, for

which the investigated indicator was calculated using the old method (in old borders) and the new method (in new borders). A transitional

coefficient for this transitional is calculated, the transitional coefficient spreads over all the

previous period of time

Production of oil, mln t 2005 2006 2007 2008 2009

Before merger 6600 6700 6900 - -

After merger - - 7500 7800 7900

1,0876900

7500К

Transitional coefficient

06y 6700 1,087 7283

05y 6600 1,087 7174

Production of oil, mln t 2005 2006 2007 2008 2009

Before merger 6600 6700 6900 - -

After merger - - 7500 7800 7900

Comparable series 7174 7283 7500 7800 7900

Analysis of the main tendency of time series

The levels of time series are formed under the influence of lots of

factors. They can be divided into 5 groups

Time-Series Components

Time Series

Cyclical Compo-

nent

Irregular Compo-

nent

Trend Compo-

nent

Seasonality Component

Secular Compo-

nent

1. Determining (Определяющие) factors have a constant and strong

influence on the examined indicator. They determine the main tendency (the

trend) of time series


Upward trend

Trend Component

Long-run increase or decrease over time (overall upward or downward movement)

Data taken over a long period of time

Sales

Time


Downward linear trend

Trend Component

Trend can be upward or downward Trend can be linear or non-linear

Sales

Time Upward nonlinear trend

Sales

Time

(continued)

There are 4 kinds of charge or variations involved in time series analysis. They are:2.Secular trend Ut. In the secular trends the value of variable

tends to increase or decrease over a long period of time. The steady increase of the cost of living recorded by the Сonsumer price index is an example of secular trend. From year to year, the cost of living varies a great deal, but if we examine long-period, we see that the trend is toward a steady increase.

3. Cyclical fluctuation Vt. The most common example of

cyclical fluctuation is the business cycle. Over time, there are years when the business cycle hits a peak above the trend line. At other times, business activity is likely to slump, hitting a low point below the trend line. The time between hitting peaks and falling to low points is at least one year and it can be as many as 15 or 20 years.


Cyclical Component

Long-term wave-like patterns Regularly occur but may vary in length Often measured peak to peak or trough to

trough

Sales1 Cycle

Year

4. Seasonal variation St involves pattern of change

within year that tend to be repeated from year to year. For example the consumption of drinks, juices, ice cream and other. Seasonal factors give rise to oscillations relative to the main tendency


Seasonal Component

Short-term regular wave-like patterns Observed within 1 year Often monthly or quarterly

Sales

Time (Quarterly)

Winter

Spring

Summer

Fall

Winter

Spring

Summer

Fall

5. Irregular variation εt. The value of variable may be

completely unpredictable changing in random manner. For example, the Iraqi situation in 1990, the ruble devaluation in 1998 and the others. Random factors cause the random fluctuations of levels of series (for example, weather factor)


5. Irregular variation εt. The value of variable may be completely

unpredictable changing in random manner. For example, the Iraqi situation in 1990, the ruble devaluation in 1998 and the others.

Thus each value of time series could be presented as follows:


Ut Vt St t.ty


Irregular Component

Unpredictable, random, “residual” fluctuations Due to random variations of

Nature Accidents or unusual events

“Noise” in the time series


Time-Series Component Analysis

Used primarily for forecasting Observed value in time series is the sum or product of

components Additive Model

Multiplicative model (linear in log form)

where Tt = Trend value at period t

St = Seasonality value for period t

Ct = Cyclical value at time t

It = Irregular (random) value for period t

ttttt ICSTX

ttttt ICSTX


Smoothing the Time Series

Calculate moving averages to get an overall impression of the pattern of movement over time

This smooths out the irregular component

Moving Average: averages of a designatednumber of consecutivetime series values

Method of interval enlargement

Method of interval enlargement consists in replacement of initial

levels of series by the average values, which are calculated for

the enlarged intervals

Month yt Quarterly sumsAverage

monthly value (per quarter)

1 5.1

2 5.4 15.7 5.23

3 5.2

4 5.3

5 5.6 16.7 5.57

6 5.8

7 5.6

8 5.9 17.6 5.87

9 6.1

10 6.0

11 5.9 18.1 6.03

12 6.2

The End of Part III

To be continued


Documents

Time-Series Analysis and Forecasting – Part III To read at home