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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

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Chapter 4 Time Series Analysis and Forecasting

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Chapter 4 time series analysis and forecasting

KVANLI

PAVUR

KEELING

Chapter 4Time Series Analysis and Forecasting


Chapter objectives

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

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

  • 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

  • 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


Linear trends

Yt

Yt

t

t

(a) Increasing trend

(b) Decreasing trend

Linear Trends


Employees example

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

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


Examples of curvilinear trend

Yt

Yt

t

t

(b)

(a)

Examples of Curvilinear Trend


Examples of curvilinear trend1

Yt

Yt

t

t

(d)

(c)

Examples of Curvilinear Trend


An illustration 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

  • 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

  • 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)


Textile example

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

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

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

Capturing the Trend

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

This is the trend line


Finding 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

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

Equation of the Trend Line

  • First, find the slope:

  • Next, find the intercept:


The example in section 2

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

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

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 trend1

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

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

  • 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 data1

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)


Trend and cyclical activity

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


Plot of cyclical components

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


Cyclical activity

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


Additive seasonal variation

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


Jetski sales additive seasonality

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


Heat pump sales multiplicative seasonality

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


Jetski sales multiplicative season variation

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

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)


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