ECON 332

Business Forecasting Methods

Chapter 2
Prof. Kirti K. Katkar
2-2

Six Considerations Basic to Successful

Forecasting
• Decision environment and loss function
• Forecast object
• Forecast statement
• Forecast horizon
• Information set
• Methods and complexity, the parsimony principle, the
shrinkage principle
2-3

Forecast Process

• Define objectives
• Determine forecast entity
• Identify time horizon
• Data considerations
• Model Selection
• Model evaluation
• Forecast preparation
• Forecast discussion and closure
• Tracking results
• Adjusting model/ forecast
2-4

Model Selection Guide

2-5

Data Patterns: Power of Graphics

• Trend
• Seasonal
• Cyclical
• Irregular
• Structural shifts
2-6
U.S. Real Gross Domestic Product (GDP)

Potential Forecasting Methods

•Holt’s exponential smoothing
•Linear regression trend
•Causal regression
•Time series decomposition
2-7
Leo Burnett Billings (LBB) in U.S.

Potential Forecasting Methods

•Non-linear regression trend
•Causal regression
2-8
Private Housing Starts (PHS)

Potential Forecasting Methods

•Winter’s exponential smoothing
•Linear regression trend w/ seasonal adjustment
•Causal regression
•Time series decomposition
2-9

Review of Statistics
• Distributions
– Normal distribution
– Student’s t distribution
• Descriptive Statistics: Using numbers to describe phenomena
– Central tendency
– Dispersion
• Measures of central dispersion
– Mean
– Median
– Mode
• Measures of Dispersion
– Range
– Variance
– Standard deviation
2-10
Distributions
• Standard Normal distribution
− x2

φ(x) = e 2

2π
We always use
Tables for these
Where μ = 0 and σ = 1 distributions
• Student’s t distribution
 n +1
Γ  n +1
2 − 2
sn(x) = 1  2  1 + x 
nπ  n   n 
Γ 
2

where n is the degrees of freedom for a sample size of (n+1)

2-11

Central Tendency Illustration

2-12

Central Tendency Illustration: Mode

Sales Number of
Mode: Response that Occurrences
occurs most frequently 1 2
2 3
Mode = 4 3 3
4 6
5 4
6 2
7 3
8 1`
13 1
TOTAL 25
2-13

Central Tendency Illustration: Median

Median: Value that splits responses into two
equal parts when they are ordered from low to
high
1 1 2 2 2 3 3 3 4 4 4 4 4 4 5 5 5 5 6 6 7 7 7 8 13

12 Values Median 12 Values

If the series has even numbers of responses, the

median will be a value that is not observed.
In that case,

Median = ((n/2)th Value + ((n/2)+1)th Value)/2

2-14

Central Tendency Illustration: Mean

Mean: Arithmetic average of all numbers in a data

series (entire population)

µ ==∑ Xi / N
N

i =1

= 115/25 = 4.6

If the data series represents a sample, mean is

designated as
_ n

X = ∑ Xi / n = 115/25 = 4.6
i =1
2-15

Stationary Series
2-16

Dispersion Illustrations

• Range = Greatest value – Smallest value

= 13 – 1
= 12
• Variance for population: σ =
2 ∑ ( Xi − µ ) 2

• Standard Deviation for population = Variance

2-17

Dispersion Illustrations (Contd.)

∑ ( Xi − X ) 2

• Variance for Sample: s2 = (n − 1)

• Standard Deviation for sample s = Variance

2-18
Dispersion Illustration: Variance &
Standard Deviation
2-19
The Normal Distribution

.Fully defined by mean and

Standard deviation
.Bell shape and symmetrical
Around mean
. μ±1σ include 68% area
. μ±2σ include 95% area
. μ±3σ include 99% area
. Standard Normal
Distribution
is called Z distribution. The
Transformation( Xto−Zµis
) simple:
Z=
σ
2-20

Illustration of Z Transformation
2-21
The Standard Normal Distribution (Z)
Table
2-22

The Standard Normal Distribution (Z)

Table (Contd.)
2-23

Distribution of Sample Statistics

• When the sample size is large, sample means

are distributed normally
• The distribution of sample means (X’s) is
centered on the population mean μ
• The standard deviation of the sample means,
called standard error of the mean, is
σ
σx =
n
2-24 Illustrative Example of using Sample Means:
Chance of selecting a good sample that represents
population
Suppose sample size n=100, X=300 and σ=60.
Would this be a good sample if the true population mean μ=288?
Since sample means are normally distributed we can answer this
question using Z transformation and the table:
σx = σ
Z = (X – μ)/σx , and ,
n

Therefore, Z=(300-288)/(60/√100)
= 12/6 = 2
From Z distribution table, area between 0 and 2 is 0.4772
As a result, area beyond 2 is 0.5-0.4772=0.0228

Thus there is only 2.28% chance that sample selected will have
mean > 300.
2-25

Student’s t Distribution

• This is used when the population standard

deviation is not known or when the sample
size is too small
• The probability of a large deviation from the
mean is greater in the t as against the normal
distribution
• Just like the Z transformation for normal
distribution, t transformation is used for
Student’s t distribution.
2-26

Student’s t Distribution (Contd.)

• t = X −µ
s/ n
• t distribution depends on degrees of freedom
df which is equal to (n-1)

• For df > 29, use the inf row in the t distribution

table
2-27

Student’s t Distribution Table

2-28

Student’s t Distribution Table (Contd.)

2-29

Statistical Inference: From Sample to

Population
• Focus always on population
• Sample useful in making some inference about population: e.g.
per capita income in Pennsylvania to ensure a new service will
succeed
• Sample statistic is best point estimate of the corresponding
population parameter but could be wrong
• Interval estimate of parameter more useful and accurate
μ = X ± t(s/√n), where
t = value from t table for (n-1) df,
s/√n = standard error of the sample mean, and
X = sample mean
2-30

Example of Statistical Inference: From

Sample to Population
• Suppose a sample of 100 responses gives a mean of
$15,000 per capita income and a standard deviation of
$5,000
• The best point estimate of sample mean would be
$15,000
• A 95% confidence interval estimate would be:
μ = 15,000 ± 1.96(5,000/√100)
= 15,000 ± 980
Thus, $14,020 ≤ μ ≤ $15,980
where t = 1.96, value from t table for 99 df
with each tail being 0.025
2-31

Hypothesis Testing

• Set up two mutually exclusive and exhaustive

hypothesis to test
• Null hypothesis Ho : μ = μ0
• Alternative hypothesis H1 : μ ≠ μ0
• Use t test after deciding the confidence level:
usually 95% for business applications
• Significance level α = 1 – Confidence level:
usually 5% for business applications
2-32

Hypothesis Testing (Contd.)

•Type I Error: Occurs if we reject Ho when in

fact it is true
•Type II Error: Occurs if we fail to reject Ho that
is in fact incorrect
2-33
Hypothesis Testing (Contd.)
• Type I and II errors are related: by reducing the
chance of Type I error we increase the chance of
Type II error and vice versa
• If the cost of Type I error is large we use lower α –
say 1% or less
• Hypothesis tests may be one or two-tailed
Case I. A two-tailed test. H0 : μ=μ0 H1 : μ≠μ0
Case II. A lower-tailed test. H0 : μ≥μ0 H1 : μ<μ0
Case III. A upper-tailed test. H0 : μ≤μ0 H1 : μ>μ0

• For each hypothesis test, t-value is calculated (tcalc )

and compared with the tcalccritical value from the t
distribution (tT ). If > tT , reject the null
hypothesis.
2-34

Example of Two-tailed Test

• A sample of 49 people in a city resulted in a mean X of $200

entertainment expenses per month and a standard deviation s
of 84. The national average μ is $220. The hypothesis are:
H0 : μ = 220 and H1 : μ ≠ 220
• In this case, tcalc = (200-220)/(84/√49)
= -20/12 = -1.67
• If the confidence level is 95%, i.e. α = 0.05, the critical value
of t is ±1.96.
• Since tcalc < 1.96, we fail to reject H0 . That is, the
entertainment expenses per month in this city are not different
from the national average.
2-35

Example of Lowered-tailed Test

• A sample of 25 people resulted in a mean X of 1.2 six-packs of

soft-drink consumption per week and a standard deviation s of
0.6. The national average μ is 1.5. The hypothesis are:
H0 : μ ≥ 1.5 and H1 : μ < 1.5
• In this case, tcalc = (1.2-1.5)/(0.6/√25)
= -0.3/0.12 = -2.5
• If the confidence level is 95%, i.e. α = 0.05, the critical value
of t for 24 df is ±1.711.
• Since tcalc > 1.711, we reject H0 . tcalc falls in the rejection
region. And the soft-drink consumption per week is different
(lower) from the national average
2-36

Example of Upper-tailed Test

• A sample of 144 women in a city resulted in a mean X of 45

hours worked per week and a standard deviation s of 29. The
national average μ is 40. The hypothesis are:
H0 : μ ≤ 40 and H1 : μ > 40
• In this case, tcalc = (45-40)/(29/√144)
= 5/2.42 = 2.07
• If the confidence level is 95%, i.e. α = 0.05, the critical value
of t is ±1.96.
• Since tcalc < 1.645, we fail to reject H0 . That is, the hours
worked in this city are different (higher) from the national
average.
2-37

Correlation
• It is a measure of association between two variables. e.g. sales
and advertising, income and taxes etc.
• One measure is Pearson product-moment correlation
coefficient – ρ for population and r for sample. We will simply
call it correlation coefficient.
• If X and Y are two variables of interest, the degree of linear
association between them is given by the correlation
coefficient as
∑ ( X − x) (Y − y)
2 2

r = ∑ ( X − x) ∑ (Y − y)
2 2

• ρ is calculated similarly except the means are population

means rather than the sample means
• R ranges in value from -1 to +1 with -1 being a perfect
negative correlation and +1 being the perfect positive
correlation
2-38
Scatterplots with corresponding
Correlation Coefficients
2-39

Scatterplots with corresponding

Correlation Coefficients (Contd.)
2-40

Testing for linear association between two

variables
• H0 : ρ = 0 H1 : ρ ≠ 0

r −0
• tcalc = (1 − r 2 ) /( n − 2)

• Note we lose two degrees of freedom for estimating two

statistics : mean and correlation coefficient
• For α =.05, and df =3, the critical value of t is ±3.182
• For r = -0.85 ( Plot D), tcalc = -2.795. Since this is within the
±3.182 range, we fail to reject the null hypothesis even though
r = -0.85 indicates a strong negative correlation. This is result
of very small sample size.
2-41

Autocorrelations
• For time series evaluation, the measure of correlation
used is called autocorrelation
• Let rk = Autocorrelation for k period lag
Yt = Value of time series at time t
Yt-k = Value of time series at time t-k
y = Mean of the time series
Then
∑
n−k
t=
(Yt − k − y )(Yt − y )

∑
n
t − y)
2
rk = t =1
(Y
2-42

Characteristics of Autocorrelations

• If the time series is stationary, rk diminishes

rapidly to zero
• If there is a trend, rk declines toward zero slowly
• If seasonal pattern exists, the value of rk will be
significantly from zero at k = n*4 for quarterly data,
k = n*12 for monthly data, where n = 1,2,…..,n
• A k period plot of autocorrelation is called An
Autocorrelation Function (ACF) or a correlogram
2-43

U.S. Real Gross Domestic Product (GDP)

2-44

ACF Values for U.S. Real GDP

2-45

Figure 2-9
2-46
ACF for GDP Change
Figure 2-10