Statistics for business decision making and analysis robert stine and foster chapter 27

27.1 Decomposing a Time SeriesSmoothing: Monthly Shipments Example Red: 13 month moving average Green: seasonally adjusted... 27.2 Regression ModelsPolynomial Trends Monthly shipments:

Time Series

Chapter 27

27.1 Decomposing a Time Series

Based on monthly shipments of computers and electronics in the US from 1992 through

2007, what would you forecast for the future?

 Use methods for modeling time series,

including regression.

 Remember that forecasts are always

extrapolations in time.

27.1 Decomposing a Time Series

timeplot, such as that of monthly shipments

of computers and electronics shown below

27.1 Decomposing a Time Series

 Forecast: a prediction of a future value of a

time series that extrapolates historical

patterns.

 Components of a time series are:

 Trend: smooth, slow meandering pattern.

 Seasonal: cyclical oscillations related to seasons.

 Irregular: random variation.

27.1 Decomposing a Time Series

Smoothing

 Smoothing: removing irregular and seasonal

components of a time series to enhance the

visibility of the trend.

 Moving average: a weighted average of adjacent values of a time series; the more terms that are

averaged, the smoother the estimate of the trend.

27.1 Decomposing a Time Series

Smoothing

seasonal component of a time series

seasonally adjusted, for example,

unemployment rates

27.1 Decomposing a Time Series

Smoothing: Monthly Shipments Example

Red: 13 month moving average

Green: seasonally adjusted.

27.1 Decomposing a Time Series

Smoothing: Monthly Shipments Example

Strong seasonal component (three-month

cycle).

27.1 Decomposing a Time Series

Exponential Smoothing

 Exponentially weighted moving average (EWMA):

a weighted average of past observations with

geometrically declining weights.

 EWMA can be written as

Hence, the current smoothed value is the

weighted average of the current observation and the prior smoothed value.

1

) 1

s

27.1 Decomposing a Time Series

Exponential Smoothing

becomes

values trail behind the observations

27.1 Decomposing a Time Series

Exponential Smoothing

Monthly Shipments Example (w = 0.5)

27.1 Decomposing a Time Series

Exponential Smoothing

Monthly Shipments Example (w = 0.8)

27.2 Regression Models

 Leading indicator: an explanatory variable

that anticipates coming changes in a time

series.

 Leading indicators are hard to find.

 Predictor: an ad hoc explanatory variable in

a regression model used to forecast a time

series (e.g., time index, t)

27.2 Regression Models

Polynomial Trends

time series that uses powers of t as

0

27.2 Regression Models

Polynomial Trends

Monthly shipments: Six-degree polynomial

The high R 2 indicates a great fit to historical

data.

27.2 Regression Models

Polynomial Trends

Monthly shipments: Six-degree polynomial

The model has serious problems forecasting

27.2 Regression Models

Polynomial Trends

have high powers of the time index

4M Example 27.1:

PREDICTING SALES OF NEW CARS Motivation

The U.S auto industry neared collapse in

2008-2009 Could it have been anticipated from historical trends?

4M Example 27.1:

PREDICTING SALES OF NEW CARS Motivation –

Timeplot of quarterly sales (in thousands)

Cars in blue; light trucks in red

4M Example 27.1:

PREDICTING SALES OF NEW CARSMethod

Use regression to model the trend and seasonal

components apparent in the timeplot Use a

polynomial for trend and three dummy variables for the four quarters

Let Q1 = 1 if quarter 1, 0 otherwise;

Q2 = 1 if quarter 2, 0 otherwise;

Q3 = 1 if quarter 3, 0 otherwise

The fourth quarter is the baseline category

Consider the possibility of lurking variables (e.g., gasoline prices).

4M Example 27.1:

PREDICTING SALES OF NEW CARS Mechanics

Linear and quadratic trend fit to the data

Linear appears more appropriate

4M Example 27.1:

PREDICTING SALES OF NEW CARS Mechanics

Estimate the model

Check conditions before proceeding with

4M Example 27.1:

PREDICTING SALES OF NEW CARSMechanics

Examine residual plot.

This plot, along with the Durbin-Watson statistic

D = 0.84, indicates dependence in the residuals.

Cannot form confidence or prediction intervals.

4M Example 27.1:

PREDICTING SALES OF NEW CARS Message

A regression model with linear time trend and seasonal factors closely predicts sales of

new cars in the first two quarters of 2008,

but substantially overpredicts sales in the

last two quarters

27.2 Regression Models

Autoregression

prior values of the response as predictors

response in a time series

27.2 Regression Models

Autoregression

one lag:

autoregression, denoted as AR(1)

t t

y = β0 + β1 −1 + ε

27.2 Regression Models

Autoregression

Example: AR(1) for Monthly Shipments

27.2 Regression Models

Autoregression

Scatterplot of Shipments on the Lag

Indicates a strong positive linear association.

27.2 Regression Models

Forecasting an Autoregression

Example: Use AR(1) to forecast shipments

For Jan 2008, use observed shipment for

Dec 2007:

1

9706

0 9000

0

ˆt = + yt−y

billion

yˆJan.2008 ≈ $31.7385

27.2 Regression Models

Forecasting an Autoregression

For Feb 2008, there is no observed shipment for Jan 2008 Use forecast for Jan 2008:

Once forecasts are used in place of

observations, the uncertainty compounds and

is hard to quantify.

billion

yˆFeb.2008 = 0 9000 + 0 9706 ( 31 785 ) ≈ $ 31 751

27.3 Checking the Model

Autoregression and the Durbin-Watson

Statistic

Example: Residuals from sixth-degree

polynomial trend fit to monthly shipments

plotted over time

27.3 Checking the Model

Autoregression and the Durbin-Watson

Statistic

Example: Residuals from sixth-degree

polynomial trend fit to monthly shipments

plotted over their lag

27.3 Checking the Model

Autoregression and the Durbin-Watson Statistic

 Residual plots show that the sixth-degree

polynomial leaves substantial dependence in the residuals.

 This dependence or correlation between

adjacent residuals is known as autocorrelation

(this first order autocorrelation is denoted as r 1 ).

27.3 Checking the Model

Autoregression and the Durbin-Watson

Statistic

the autocorrelation of the residuals in a

1

1

2 2

2 1

r e

27.3 Checking the Model

Timeplot of Residuals

Useful for identifying outliers (e.g., April 2001).

27.3 Checking the Model

Summary

Examine these plots of residuals when fitting a

time series regression:

 Timeplot of residuals;

 Scatterplot of residuals versus fitted values; and

 Scatterplot of residuals versus lags of the

residuals.

4M Example 27.2:

FORECASTING UNEMPLOYMENT Motivation

Using seasonally adjusted unemployment

data from 1980 through 2008, can a time

series regression predict the rapid increase

in unemployment that came with the

recession of 2009?

4M Example 27.2:

FORECASTING UNEMPLOYMENT Motivation

4M Example 27.2:

FORECASTING UNEMPLOYMENTMethod

Use a multiple regression of the percentage

unemployed on lags of unemployment and a

time trend In other words, use a combination

of an autoregression with a polynomial trend.

The scatterplot matrix shows linear association

and possible collinearity; hopefully the lags will capture the effects of important omitted

variables.

4M Example 27.2:

FORECASTING UNEMPLOYMENT Mechanics

Estimate the model

4M Example 27.2:

FORECASTING UNEMPLOYMENT Mechanics

All conditions for the model are satisfied;

proceed with inference

model explains statistically significant

variation The fitted equation is

) (

164

0 192

0 794

0 086

0

y

4M Example 27.2:

FORECASTING UNEMPLOYMENT Message

A multiple regression fit to monthly

unemployment data from 1980 through

2008 predicts that unemployment in

January 2009 will be between 7.02% and

7.66%, with 95% probability Forecasts for February and March call for unemployment

to rise further to 7.48% and 7.64%,

respectively

4M Example 27.3:

FORECASTING PROFITS Motivation

Forecast Best Buy’s gross profits for 2008

Use their quarterly gross profits from 1995

to 2007

4M Example 27.3:

FORECASTING PROFITS Method

Best Buy’s profits have not only grown

nonlinearly (faster and faster), but the

growth is seasonal In addition, the

variation in profits appears to be increasing with level Consequently, transform the

data by calculating the percentage change

year-over-year percentage changes

4M Example 27.3:

FORECASTING PROFITS Method

Timeplot of year-over-year percentage

change

4M Example 27.3:

FORECASTING PROFITS Method

Scatterplot of the year-over-year percentage

change on its lag.

4M Example 27.3:

FORECASTING PROFITS Mechanics

Estimate the model

4M Example 27.3:

FORECASTING PROFITS Mechanics

All conditions for the model are satisfied;

proceed with inference.

The fitted equation has R 2 = 71.0% with s e =

7.37.

The F-statistic shows that the model is

statistically significant Individual t-statistics

show that each slope is statistically significant.

4M Example 27.3:

FORECASTING PROFITS Mechanics

Forecast for the first quarter of 2008:

prediction interval includes zero It is

[-6.5% to 25%]

% 345

9 )

282

11 ( 383

0 )

318

0 ( 443

0 )

285

2 ( 911

0 971

.

2

y

4M Example 27.3:

FORECASTING PROFITS Message

The time series regression that describes

year-over-year percentage changes in

gross profits at Best Buy is significant and explains 70% of the historical variation It

predicts profits in the first quarter of 2008 to grow about 9.3% over the previous year;

however, the model can’t rule out a much

larger increase (25%) or a drop (about

6.5%)

Best Practices

forecast

autocorrelation

Best Practices (Continued)

of a model

 Don’t summarize a time series with a histogram

unless you’re confident that the data don’t have a pattern

 Avoid polynomials with high powers.

 Do not let the high R 2 of a time series regression

convince you that predictions from the regression will be accurate.

Pitfalls (Continued)

also have to be forecast