Chapter 4 Time Series Analysis and Forecasting

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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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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

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2002

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2003

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2004

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2005

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2006

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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
300 –

200 –

100 –

Power consumption (million kwh)

|

1998

|

1999

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2000

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2001

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2002

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2003

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2004

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2005

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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

∙ 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

^

^

1 1.1 1.125 .978

2 2.4 2.529 .949

3 4.6 3.933 1.169

4 5.4 5.337 1.012

5 5.9 6.741 .875

6 8.0 8.145 .982

7 9.7 9.549 1.016

8 11.2 10.953 1.022

Trend and Cyclical Activity

Trend activity

Cyclical activity

Ct

1.15 –

1.10 –

1.05 –

1.00 –

.95 –

.90 –

Start

End

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1

|

2

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3

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4

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5

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6

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7

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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

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2002

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2003

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2004

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2005

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2006

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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

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Winter

2007

t

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

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1

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2

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3

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4

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5

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6

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7

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8

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9

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10

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11

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12

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13

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14

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15

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16

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17

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18

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19

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20

t

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

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1

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2

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3

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4

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5

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6

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7

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8

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9

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10

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11

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12

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13

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14

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15

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16

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17

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18

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19

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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)