Chapter 15

Time Series: Descriptive

Analyses, Models, and

Forecasting

Descriptive Analysis: Index Numbers

Index Number – a number that measures the change in a

variable over time relative to the value of the variable

during a specific base period

Simple Index Number – index based on the relative

changes (over time) in the price or quantity of a single

commodity

0

1 0 0

1 0 0i

t

T im e s e r ie s v a lu e a t t im e tIn d e x n u m b e r a t t im e t

T im e s e r ie s v a lu e a t b a s e p e r io d

YI

Y

Descriptive Analysis: Index Numbers

Laspeyres and Paasche Indexes compared•The Laspeyres Index weights by the purchase quantities

of the baseline period

•The Paasche Index weights by the purchase quantities of

the period the index value represents.

•Laspeyres Index is most appropriate when baseline

purchase quantities are reasonable approximations of

purchases in subsequent periods.

•Paasche Index is most appropriate when you want to

compare current to baseline prices at current purchase

levels

Descriptive Analysis: Index Numbers

Calculating a Laspeyres Index

Collect price info for the k price series to be used, denote

as P1t, P2t…Pkt

Select a base period t0

Collect purchase quantity info for base period, denote as

Q1t0, Q2t0…..Qkt0

Calculate weighted totals for each time period using the

formula

Calculate the index using the formula

0

1

k

i t i t

i

Q P

0

1

0 0

1

1 0 0

k

i t i t

i

t k

i t i t

i

Q P

I

Q P

Descriptive Analysis: Index Numbers

Calculating a Paasche Index

Collect price info for the k price series to be used, denote

as P1t, P2t…Pkt

Select a base period t0

Collect purchase quantity info for every period, denote as

Q1t, Q2t…..Qkt

Calculate the index for time t using the formula

1

0

1

1 0 0

k

i t i t

i

t k

i t i t

i

Q P

I

Q P

Descriptive Analysis: Exponential

Smoothing

Exponential smoothing is a type of weighted

average, that applies a weight w to past and

current values of the time series.

Exponential smoothing constant (w) lies between 0

and 1, and smoothed series Et is calculated as:

1 1

2 2 1

3 3 2

1

(1 )

(1 )

(1 )t t t

E Y

E w Y w E

E w Y w E

E w Y w E

Descriptive Analysis: Exponential

Smoothing

Selection of smoothing constant w is made by

researcher.

Small values of w give less weight to current value,

yield a smoother series

Large values of w give more weight to current

value, yield a more variable series

Time Series Components

4 components of time series models:

Tt – secular or long-term trend

Ct – cyclical trend

St – seasonal effect

Rt – residual effect

These 4 components are from a widely used model called the additive model:

Yt = Tt + Ct + St + Rt

Forecasting: Exponential

Smoothing

Calculation of Exponentially Smoothed Forecasts

Calculate exponentially smoothed values E1, E2,…Et for

observed time series Y1, Y2,…Yt.

Used last smoothed value to forecast the next time series

value

Assuming that Yt is relatively free of trend and seasonal

components, use the same forecast for all future values of

Yt:

1t t t t tF E F w Y F

2 1

3 1

t t

t t

F F

F F

Forecasting Trends: The Holt-

Winters Model

The Holt-Winters model adds a trend component to the forecast.

Calculating Components of the Holt-Winters Model

1. Select exponential smoothing constant w

2. Select trend smoothing constant v

3. Calculate the two components Et and Tt from time series Ytbeginning at time t=2

E2=Y2

T2=Y2-Y1

E3=wY3+(1-w)(E2+T2)

T3=v(E3-E2)+(1-v)T2

…

Et=wYt+(1-w)(Et-1+Tt-1)

Tt=v(Et-Et-1)+(1-v)Tt-1

Forecasting Trends: The Holt-

Winters Model

Holt-Winters Forecasting

1. Calculate the exponentially smoothed and

trend components Et and Tt for each observed

value of Yt (t >2)

2. Calculate the one-step-ahead forecast using

Ft+1=Et+Tt

3. Calculate the k-step-ahead forecast using

Ft+k=Et+kTt

Measuring Forecast Accuracy:

MAD and RMSE

Measures of Forecast Accuracy for m Forecasts

Mean absolute Deviation (MAD)

Mean absolute percentage error (MAPE)

Root mean squared error (RMSE)

1

m

t t

t

Y F

M A Dm

1

1 0 0

m

t t

t t

Y F

YM A P E

m

2

1

m

t t

t

Y F

R M S Em

Forecasting Trends: Simple Linear

Regression

Simple Linear Regression is the simplest inferential forecasting model

After fitting the regression line to existing data, the least squares model can be used to forecast future values of the dependent variable

Two problems are associated with using a LSM to forecast time series:

1. Forecasting falls outside of the experimental region, increases width of prediction intervals

2. Cyclical effects are not built in to the model, and introduce the problem of correlated error

Seasonal Regression Models

Use of multiple regression model with

dummy variables to describe seasonal

differences is common

In the following example, dummy variables

are set up for Quarters 1, 2 and 3

Model that reflects seasonal component and

expected growth in usage is:

0 1 2 1 3 2 4 3( )

tE Y t Q Q Q

Seasonal Regression Models

Data to be used is in the following table:Quarterly Power Loads (megawatts) for a Southern Utility Company, 1992-2003

Year Quarter Power Load Year Quarter Power Load

1992 1 68.8 1998 1 130.6

2 65.0 2 116.8

3 88.4 3 144.2

4 69.0 4 123.3

1993 1 83.6 1999 1 142.3

2 69.7 2 124.0

3 90.2 3 146.1

4 72.5 4 135.5

1994 1 106.8 2000 1 147.1

2 89.2 2 119.3

3 110.7 3 138.2

4 91.7 4 127.6

1995 1 108.6 2001 1 143.4

2 98.9 2 134.0

3 120.1 3 159.6

4 102.1 4 135.1

1996 1 113.1 2002 1 149.5

2 94.2 2 123.3

3 120.5 3 154.4

4 107.4 4 139.4

1997 1 116.2 2003 1 151.6

2 104.4 2 133.7

3 131.7 3 154.5

4 117.9 4 135.1

Seasonal Regression Models

Result of regression analysis is:

Seasonal Regression Models

Forecast results and actual values for 2004

are:

Seasonal Regression Models

Use of multiplicative models often provides a better

forecasting model when the time series is changing at an

increasing rate over time.

Multiplicative model for Power Load problem would be:

Taking antilogarithm of both sides shows multiplicative

nature:

0 1 2 1 3 2 4 3ln ( )

tE Y t Q Q Q

0 1 2 1 3 2 4 3e x p e x p e x p e x p

tY t Q Q Q

Constant Secular

trend

Seasonal Component Residual

Component

Autocorrelation and the Durbin-

Watson Test

A residual pattern as illustrated here suggests that autocorrelationmay be an issue.

Autocorrelation is the correlation between time series residuals at different points in time. Correlation between neighboring residuals is called first-order autocorrelation

Autocorrelation and the Durbin-

Watson Test

Durbin-Watson d statistic is calculated to test for the presence of first-order autocorrelation

If residuals are uncorrelated, then d ≈ 2.

If residuals are positively autocorrelated, then d<2, and if autocorrelation is very strong, d ≈ 0.

If residuals are negatively autocorrelated, then d>2, and if autocorrelation is very strong, d ≈ 4.

2

1

2

2

1

: 0 4

n

t t

t

n

t

t

R R

d R a n g e o f d d

R

Autocorrelation and the Durbin-

Watson Test

One-Tailed Test Two-Tailed Test

H0:No first-order autocorrelation H0:No first-order autocorrelation Ha:Positive first-order autocorrelation (or Ha:Negative first-order autocorrelation)

Ha:Positive or negative first-order autocorrelation

Test Statistic

2

1

2

2

1

n

t t

t

n

t

t

R R

d

R

Rejection region: d< dL, Rejection region: d< dL,/2 or (4-d)< dL,/2

[or (4-d)< dL, if Ha:Negative first-order autocorrelation]

Where dL, is the lower tabled value corresponding to k independent variables and n observations.

The corresponding upper value dU, defines a

“possibly significant” region between dL, and dU,

Where dL,/2 is the lower tabled value corresponding to k independent variables and n observations. The corresponding upper value

dU,/2 defines a “possibly significant” region

between dL,/2 and dU,/2