KVANLI
Sponsored Links
This presentation is the property of its rightful owner.
1 / 35

Chapter 4 Time Series Analysis and Forecasting PowerPoint PPT Presentation


  • 101 Views
  • Uploaded on
  • Presentation posted in: General

KVANLI PAVUR KEELING. Chapter 4 Time Series Analysis and Forecasting. Chapter Objectives. At the completion of this chapter, you should be able to: ∙ Deseasonalize a time series by first calculating the seasonal indexes

Download Presentation

Chapter 4 Time Series Analysis and Forecasting

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


KVANLI

PAVUR

KEELING

Chapter 4Time Series Analysis and Forecasting


Chapter Objectives

  • At the completion of this chapter, you should be able to:

    ∙ Deseasonalize a time series by first calculating the

    seasonal indexes

    ∙ Discuss the nature of additive and multiplicative

    seasonality

    ∙ Estimate the trend, cyclical and noise components

    ∙ Forecast a time series containing trend

    ∙ Calculate price indexes, including a Laspeyres and

    Paasche index


What is a Time Series?

  • A time series consists of a variable (such as Sales) recorded across time

  • Example: tYearSales (millions of $)

    1 1985 1.7

    2 1986 2.4

    3 1987 2.8

    4 1988 3.4

    .

    .

    22 2006 9.6

    23 2007 10.7

This is y1

This is an example of annual data

This is y23


Time Series Data

  • Time series data can be:

    ∙ annual (one value for each year)

    ∙ quarterly (4 values for each year)

    ∙ monthly (12 values for each year)

  • Each time series value is made up of 3 or 4 components. These are:

    ∙ Trend (TR) ∙ Seasonality (S)

    ∙ Cyclic (C) ∙ Irregular or noise (I)

Monthly or quarterly data only


Trend

  • Trend is the long-term growth or decline in the time series

  • Trend usually follows a straight line

  • Examples of linear trend are illustrated in the next two slides


Yt

Yt

t

t

(a) Increasing trend

(b) Decreasing trend

Linear Trends


11.0 –

10.0 –

9.0 –

8.0 –

7.0 –

6.0 –

5.0 –

4.0 –

3.0 –

2.0 –

1.0 –

Trend

Number of employees (thousands)

|

2000

|

2001

|

2002

|

2003

|

2004

|

2005

|

2006

|

2007

t

Employees Example

We’ll take a closer look at this example in the slides to follow


Curvilinear Trend

  • Trend can also be curvilinear

  • Curvilinear trend is also called quadratic trend

  • Curvilinear trend is demonstrated in the following three slides

  • In this chapter, we will pay little attention to curvilinear trend

  • The macros will assume that trend is linear


Yt

Yt

t

t

(b)

(a)

Examples of Curvilinear Trend


Yt

Yt

t

t

(d)

(c)

Examples of Curvilinear Trend


300 –

200 –

100 –

Power consumption (million kwh)

|

1998

|

1999

|

2000

|

2001

|

2002

|

2003

|

2004

|

2005

|

2006

|

2007

t (time)

An Illustration of Curvilinear Trend

The increase in power consumption slows down over time and the trend is not linear

Figure 4.1


Seasonality

  • Seasonality is predictable variation within a year

  • For example, sales are always high in December

  • Seasonality only exists for monthly or quarterly data

  • Two types of seasonal variation

    ∙ Additive seasonality

    ∙ Multiplicative seasonality

  • Seasonal variation will be discussed later in this chapter

∙ Usually the case

∙ Assumed in the

Excel macros


Cyclical Variation

  • Cyclical variation is nonseasonal movement about the trend

  • The next slide illustrates a time series containing cyclical movement (corporate taxes paid by a textile company over a 25-year period)

  • This time series does not exhibit a trend (long-term upward or downward growth)


4.0 –

3.5 –

3.0 –

2.5 –

2.0 –

1.5 –

1.0 –

Corporate taxes

(millions of dollars)

1

2

3

|

1980

|

1990

|

2000

|

2005

Textile Example


Irregular Activity

  • Engineers refer to this as “noise”

  • This is what is left over after measuring the seasonal, trend, and cyclic activity


Combining the Components

  • If the seasonality is assumed to be additive, each yt is the sum of its four components

    yt = St+TRt + Ct+ It

  • If the seasonality is assumed to be multiplicative, each yt is the product of its four components

    yt = St ∙ TRt ∙ Ct ∙ It

The seasonal component (St) is omitted for annual data

Multiplicative seasonality is the usual situation and assumed in the Excel macros


Capturing the Trend

  • We will illustrate this using annual data which has no seasonality

This is the trend line


Finding the Trend Line

  • The equation of the trend line is

    yt= b0+ b1t

  • b0 is called the intercept (and is fairly boring)

  • b1 is called the slope (and is pretty interesting)

  • The calculations necessary to find the slope and intercept are shown on the next slide


Example in Section 2

  • ytis the number of employees (in thousands) for eight years

  • t ytt∙yt

    1 1.1 1.1

    2 2.4 4.8

    3 4.6 13.8

    .

    .

    8 11.289.6

    48.3 276.3

Let A = the sum of the time series values

So, A = 48.3

Let B = the sum of the right-hand column

So, B = 276.3

Let T = the number of time periods. So, T = 8


Equation of the Trend Line

  • First, find the slope:

  • Next, find the intercept:


The Example in Section 2

Carry a lot of decimal places

The trend line is = -.279 + 1.404t

OK to round now


Interpreting the Slope

  • The slope of the trend line is a very interesting value

  • Here, b1 is 1.404

  • Since the number of employees each year (yt) is measured in thousands, then the number of employees in this company is increasing 1,404 (on the average) each year


Forecasting – Extending the Trend

  • If you assume the linear growth or decline as described by the trend line continues for another year, a simple forecast can be obtained from this trend line

  • For example, what would be your forecast for the year 2008?

  • This is time period t = 9

  • Use this value for t in the trend line equation


Forecasting – Extending the Trend

  • This would be -.279 + 1.404(9) = 12.357

  • The forecast for 2008 is 12,357 employees

The forecast for 2008

The forecast period

sample data


Measuring Cyclic Activity – Annual Data

  • We’ll assume the multiplicative model, where each time series value is the product of its components

  • Since this is annual data, there is no seasonal component and

    yt = TRt ∙ Ct ∙ It

  • The trend line values ( values) contain trend only


The Estimated Number of Employees

  • The estimated number of employees in each time period using the trend line:

    = -.279 + 1.404(1) = 1.125

    = -.279 + 1.404(2) = 2.529

    = -.279 + 1.404(3) = 3.933

    = -.279 + 1.404(8) = 10.953


Measuring Cyclic Activity – Annual Data

  • By dividing the yt values by the values, you can eliminate the trend components

  • We’ll call these ratios the cyclic components, even though they contain noise (It)

  • There is no way to separate out the noise component but it can be reduced when using monthly or quarterly data (illustrated later)


tytytyt/yt

^

^

11.11.125.978

22.42.529.949

34.63.9331.169

45.45.3371.012

55.96.741.875

68.08.145.982

79.79.5491.016

811.210.9531.022

Trend and Cyclical Activity

Trend activity

Cyclical activity


Ct

1.15 –

1.10 –

1.05 –

1.00 –

.95 –

.90 –

Start

End

|

1

|

2

|

3

|

4

|

5

|

6

|

7

|

8

t

2000

2002

2004

2007

Plot of Cyclical Components


Yt

11.0 –

10.0 –

9.0 –

8.0 –

7.0 –

6.0 –

5.0 –

4.0 –

3.0 –

2.0 –

1.0 –

Actual yt

^

yt =−.279 + 1.404t

(trend line)

Number of employees (thousands)

|

2000

|

2001

|

2002

|

2003

|

2004

|

2005

|

2006

|

2007

t

Cyclical Activity


Yt

Trend

2000 –

1500 –

1000 –

500 –

100 units

100 units

Actual time series

Units sold

100 units

|

Winter

2005

|

Winter

2006

|

Winter

2007

t

Additive Seasonal Variation

Figure 4.16


Yt

700 –

600 –

500 –

400 –

300 –

200 –

100 –

TRt = 100 + 20t

Sales (tens of thousands of dollars)

Estimated sales using trend and seasonality

|

1

|

2

|

3

|

4

|

5

|

6

|

7

|

8

|

9

|

10

|

11

|

12

|

13

|

14

|

15

|

16

|

17

|

18

|

19

|

20

t

JetskiSales – Additive Seasonality

Figure 4.17


Yt

2000 –

1500 –

1000 –

500 –

250 units

Trend

180 units

Units sold

Actual time series

100 units

|

Winter

2005

|

Winter

2006

|

Winter

2007

t

Heat Pump Sales – Multiplicative Seasonality

Figure 4.18


Yt

700 –

600 –

500 –

400 –

300 –

200 –

100 –

TRt = 100 + 20t

Sales (tens of thousands of dollars)

Estimated sales using trend and seasonality

|

1

|

2

|

3

|

4

|

5

|

6

|

7

|

8

|

9

|

10

|

11

|

12

|

13

|

14

|

15

|

16

|

17

|

18

|

19

|

20

t

Jetski Sales Multiplicative Season Variation

Figure 4.19


Procedure with Monthly or Quarterly Data

  • The Excel macros assume multiplicative seasonality, where yt = St ∙ TRt ∙ Ct ∙ It

  • Determine the seasonal components (St values)

  • Deseasonalize the data

  • Determine the trend components (TRtvalues) using the deseasonalized data

  • Determine the cyclic components (Ct values)

  • Determine the irregular (noise) components (It values)


  • Login