Datta Meghe Institute of Management Studies Quantitative Techniques Unit No.:04 Unit Name: Time...

Datta Meghe Institute of

Management Studies

Quantitative Techniques

Unit No.:04

Unit Name: Time Series Analysis and Forecasting

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Management Studies

Time Series Analysis and Forecasting - Components of Time Series, Trend - Moving averages, semi-averages and least-squares, seasonal variation, cyclic variation and irregular variation, Index numbers, calculation of seasonal indices, Additive and multiplicative models, Forecasting, Non linear trend – second degree parabolic trends

SYLLABUS


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Learning ObjectivesLearning Objectives

• Describe what forecasting is• Explain time series & its components• Smooth a data series• Moving average• Exponential smoothing• Forecast using trend models• Measuring forecast error• The multiplicative time series model• Naïve extrapolation• The mean forecast model• Weighted moving average models

What Is Forecasting?What Is Forecasting?

• Process of predicting a future event

• Underlying basis of all business decisions– Production– Inventory– Personnel– Facilities

• Used when situation is vague & little data exist– New products– New technology

• Involve intuition, experience

• e.g., forecasting sales on Internet

Qualitative Methods

Forecasting ApproachesForecasting Approaches

Quantitative Methods

• Used when situation is ‘stable’ & historical data exist– Existing products– Current technology

• Involve mathematical techniques

• e.g., forecasting sales of color televisions

Forecasting ApproachesForecasting Approaches

• Used when situation is vague & little data exist– New products– New technology

• Involve intuition, experience

• e.g., forecasting sales on Internet

Qualitative Methods Quantitative Methods


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Quantitative Forecasting Quantitative Forecasting

• Select several forecasting methods• ‘Forecast’ the past• Evaluate forecasts • Select best method• Forecast the future• Monitor continuously forecast accuracy


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Quantitative Forecasting MethodsQuantitative Forecasting Methods


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Quantitative

Forecasting


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Quantitative

Forecasting

Time Series

Models


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Causal

Models

Quantitative Forecasting Methods


Quantitative

Forecasting

Time Series

Models


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Causal

Models



Quantitative

Forecasting

Time Series

Models

Exponential

Smoothing

Trend

Models

Moving

Average


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Causal

Models



Quantitative

Forecasting

Time Series

Models

RegressionExponential

Smoothing

Trend

Models

Moving

Average


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Causal

Models



Quantitative

Forecasting

Time Series

Models

RegressionExponential

Smoothing

Trend

Models

Moving

Average

CasualModels


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What is a Time Series?What is a Time Series?

• Set of evenly spaced numerical data– Obtained by observing response variable at

regular time periods• Forecast based only on past values

– Assumes that factors influencing past, present, & future will continue

• Example– Year: 1995 1996 1997 1998 1999– Sales: 78.7 63.5 89.7 93.2 92.1


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Time Series vs. Cross Sectional Data


Time series data is a sequence of observations

– collected from a process – with equally spaced periods of time.


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Contrary to restrictions placed on cross-sectional data, the major purpose of forecasting with time series is to extrapolate beyond the range of the explanatory variables.



Time series is dynamic, it does change over time.


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When working with time series data, it is paramount that the data is plotted so the researcher can view the data.


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Time Series ComponentsTime Series Components


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Trend


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Trend Cyclical


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Trend

Seasonal

Cyclical


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Trend

Seasonal

Cyclical

Irregular


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Trend ComponentTrend Component

• Persistent, overall upward or downward pattern• Due to population, technology etc.• Several years duration

Mo., Qtr., Yr.

Response

© 1984-1994 T/Maker Co.


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Trend ComponentTrend Component

• Overall Upward or Downward Movement• Data Taken Over a Period of Years

Sales

Time

Upward trend


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Cyclical ComponentCyclical Component

• Repeating up & down movements• Due to interactions of factors influencing economy• Usually 2-10 years duration

Mo., Qtr., Yr.

Response Cycle


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Cyclical ComponentCyclical Component

• Upward or Downward Swings• May Vary in Length• Usually Lasts 2 - 10 Years

Sales

Time

Cycle


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• Regular pattern of up & down fluctuations• Due to weather, customs etc.• Occurs within one year

Seasonal ComponentSeasonal Component

Mo., Qtr.

Response

Summer

© 1984-1994 T/Maker Co.


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• Upward or Downward Swings• Regular Patterns• Observed Within One Year

Seasonal ComponentSeasonal Component

Sales

Time (Monthly or Quarterly)

Winter


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Irregular ComponentIrregular Component

• Erratic, unsystematic, ‘residual’ fluctuations• Due to random variation or unforeseen events

– Union strike– War

• Short duration & nonrepeating

© 1984-1994 T/Maker Co.


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Random or Irregular ComponentRandom or Irregular Component

• Erratic, Nonsystematic, Random, ‘Residual’

Fluctuations

• Due to Random Variations of

– Nature

– Accidents

• Short Duration and Non-repeating


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Time Series ForecastingTime Series Forecasting


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TimeSeries


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TimeSeries

Trend?


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TimeSeries

Trend?Smoothing

Methods

NoNo


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TimeSeries

Trend?Smoothing

MethodsTrend

Models

YesYesNoNo


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TimeSeries

Trend?Smoothing

MethodsTrend

Models

YesYesNoNo

ExponentialSmoothing

MovingAverage


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Linear

TimeSeries

Trend?SmoothingMethods

TrendModels

YesNo


Quadratic Exponential Auto-Regressive

MovingAverage


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Time Series Analysis


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Plotting Time Series Data

09/83 07/86 05/89 03/92 01/95

Month/Year

0

2

4

6

8

10

12

Number of Passengers

(X 1000)Intra-Campus Bus Passengers

Data collected by Coop Student (10/6/95)


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Moving Average MethodMoving Average Method


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Linear

TimeSeries


TrendModels

YesNo



MovingAverage


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• Series of arithmetic means • Used only for smoothing

– Provides overall impression of data over time


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• Series of arithmetic means • Used only for smoothing

– Provides overall impression of data over time

Used for elementary forecasting


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Moving Average GraphMoving Average Graph

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2

4

6

8

93 94 95 96 97 98

Year

Sales Actual


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Moving Average [An Example]

Moving Average [An Example]

You work for Firestone Tire. You want to smooth random fluctuations using a 3-period moving average.

1995 20,0001996 24,0001997 22,0001998 26,0001999 25,000


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Moving Average [Solution]

Moving Average [Solution]

Year Sales MA(3) in 1,0001995 20,000 NA1996 24,000 (20+24+22)/3 = 221997 22,000 (24+22+26)/3 = 241998 26,000 (22+26+25)/3 = 241999 25,000 NA


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Moving Average

Year Response Moving Ave

1994 2 NA

1995 5 3

1996 2 3

1997 2 3.67

1998 7 5

1999 6 NA

94 95 96 97 98 99

8

6

4

2

0

Sales


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Exponential Smoothing MethodExponential Smoothing Method


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Linear

TimeSeries


TrendModels

YesNo



MovingAverage


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Exponential Smoothing MethodExponential Smoothing Method

• Form of weighted moving average– Weights decline exponentially– Most recent data weighted most

• Requires smoothing constant (W)– Ranges from 0 to 1– Subjectively chosen

• Involves little record keeping of past data


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Exponential Smoothing [An Example]

Exponential Smoothing [An Example]

You’re organizing a Kwanza meeting. You want to forecast attendance for 1998 using exponential smoothing ( = .20). Past attendance (00) is:

1995 41996 61997 51998 31999 7

© 1995 Corel Corp.


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Exponential SmoothingExponential Smoothing

Time YiSmoothed Value, Ei

(W = .2)Forecast

Yi + 1

1995 4 4.0 NA

1996 6 (.2)(6) + (1-.2)(4.0) = 4.4 4.0

1997 5 (.2)(5) + (1-.2)(4.4) = 4.5 4.4

1998 3 (.2)(3) + (1-.2)(4.5) = 4.2 4.5

1999 7 (.2)(7) + (1-.2)(4.2) = 4.8 4.2

2000 NA NA 4.8

Ei = W·Yi + (1 - W)·Ei-1Ei = W·Yi + (1 - W)·Ei-1

^


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Exponential Smoothing [Graph]Exponential Smoothing [Graph]

0

2

4

6

8

93 96 97 98 99Year

Attendance

Actual


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Forecast Effect of Smoothing Coefficient (W)

Forecast Effect of Smoothing Coefficient (W)

Weight

W is... Prior Period2 Periods

Ago3 Periods

Ago

W W(1-W) W(1-W)2

0.10 10% 9% 8.1%

0.90 90% 9% 0.9%

Yi+1 = W·Yi + W·(1-W)·Yi-1 + W·(1-W)2·Yi-2 +...^


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Linear Time-Series Forecasting Model



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Linear

TimeSeries


TrendModels

YesNo



MovingAverage


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• Used for forecasting trend• Relationship between response variable Y &

time X is a linear function• Coded X values used often

– Year X: 1995 1996 1997 19981999

– Coded year: 0 1 2 34

– Sales Y: 78.7 63.5 89.7 93.292.1


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Linear Time-Series ModelLinear Time-Series Model

Y

Time, X 1

Y b b Xi i 0 1 1

b1 > 0

b1 < 0


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Linear Time-Series Model [An Example]

Linear Time-Series Model [An Example]

You’re a marketing analyst for Hasbro Toys. Using coded years, you find Yi = .6 + .7Xi.

1995 11996 11997 21998 21999 4

Forecast 2000 sales.

^


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Linear Time-Series [Example]Linear Time-Series [Example]

Year Coded Year Sales (Units)1995 0 11996 1 11997 2 21998 3 21999 4 42000 5 ?

2000 forecast sales: Yi = .6 + .7·(5) = 4.1

The equation would be different if ‘Year’ used.

^


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The Linear Trend Model

iii X..XbbY 743143210 Year Coded Sales

94 0 2

95 1 5

96 2 2

97 3 2

98 4 7

99 5 6

0

1

2

3

4

5

6

7

8

1993 1994 1995 1996 1997 1998 1999 2000

Projected to year 2000

CoefficientsIntercept 2.14285714X Variable 1 0.74285714

Excel Output


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Time Series Plot

01/93 01/94 01/95 01/96 01/97

Month/Year

16

17

18

19

20

Number of Surgeries

(X 1000)

Surgery Data(Time Sequence Plot)

Source: General Hospital, Metropolis


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Time Series Plot [Revised]

01/93 01/94 01/95 01/96 01/97

Month/Year

183

185

187

189

191

193

Number of Surgeries

(X 100)

Revised Surgery Data(Time Sequence Plot)



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Seasonality Plot

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Month

99.7

99.9

100.1

100.3

100.5

Monthly Index

Revised Surgery Data

(Seasonal Decomposition)



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Trend Analysis

12/92 10/93 8/94 6/95 9/96 2/97 12/97

Month/Year

18

18.3

18.6

18.9

19.2

19.5

Number of Surgeries

(X 1000)

Revised Surgery Data(Trend Analysis)



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Quadratic Time-Series Forecasting Model



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Linear

TimeSeries


TrendModels

YesNo



MovingAverage


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• Used for forecasting trend• Relationship between response variable Y & time X

is a quadratic function• Coded years used


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• Used for forecasting trend

• Relationship between response variable Y & time X is a quadratic function

• Coded years used

• Quadratic model

Y b b X b Xi i i 0 1 1 11

21


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Y

Year, X 1

Y

Year, X 1

Y

Year, X 1

Quadratic Time-Series Model Relationships

Quadratic Time-Series Model Relationships

Y

Year, X 1

b11 > 0b11 > 0

b11 < 0b11 < 0


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Quadratic Trend Model

2210 iii XbXbbY

22143308572 iii X.X..Y

Excel Output

Year Coded Sales

94 0 2

95 1 5

96 2 2

97 3 2

98 4 7

99 5 6

CoefficientsIntercept 2.85714286X Variable 1 -0.3285714X Variable 2 0.21428571


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Exponential Time-Series ModelExponential Time-Series Model


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Linear

TimeSeries


TrendModels

YesNo



MovingAverage


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Exponential Time-Series Forecasting Model

Exponential Time-Series Forecasting Model

• Used for forecasting trend• Relationship is an exponential function• Series increases (decreases) at increasing

(decreasing) rate


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Exponential Time-Series Model Relationships

Exponential Time-Series Model Relationships

Y

Year, X 1

b1 > 1

0 < b1 < 1


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Exponential Weight [Example Graph]

94 95 96 97 98 99

8

6

4

2

0

Sales

Year

Data

Smoothed


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CoefficientsIntercept 0.33583795X Variable 10.08068544

Exponential Trend Model

iXi bbY 10 or 110 blogXblogYlog i

Excel Output of Values in logs

iXi ).)(.(Y 21172

Year Coded Sales

94 0 2

95 1 5

96 2 2

97 3 2

98 4 7

99 5 6

antilog(.33583795) = 2.17antilog(.08068544) = 1.2

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• Described what forecasting is

• Explained time series & its components

• Smoothed a data series– Moving average– Exponential smoothing

• Forecasted using trend models

SUMMARY

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LONG & SHORT QUESTION

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• Statistics : Theory and Practice

- R.S.N. Pillai & Bhagwati

Fundamentals of Statistics

- S.C. Gupta

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