Linear Trend Lines

Yt = b0 + b1 Xt




Where Yt is the dependent variable being
forecasted
Xt is the independent variable being used
to explain Y. In Linear Trend Lines, Xt is
assumed to be t.
b1 is the slope of the line, determined by
Excel
b0 is the y intercept of the line, determined
by Excel
Tools Data Analysis
Regression
Coefficient of Determination:
R-square



Proportion of variation in Y around its
mean that is accounted for by the
regression model
0 <= R2 <= 1
Will always increase as add more
independent variables into regression
model. Use adjusted R2 to compare
when more than one independent
variable is used
Standard Error of the line: Se



The standard deviation of estimation
errors
The measure of amount of scatter
around the regression line
Can be used as a rough rule of thumb
for predicting level of accuracy.
Excel’s Trend Function

=trend(known y-range, known x-range,
new x)



Where known y-range are the cells that
hold known values for the y variable
Where known x-range are the cells that
hold known values for the x variable
Where new x is the cell or value for which
the y variable is to be forecasted
Stationary Seasonal Effects
A d d itiv e S e a s o n a l E ffe c ts
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
18
19
20
21
22
23
24
25
T im e P e r io d
M u ltip lic a tiv e S e a s o n a l E ffe c ts
1
2
3
4
5
6
7
8
9
10
11
12
13
14
T im e P e r io d
15
16
17
Text Use of Multiplicative
Seasonal Indices (pg. 532)
1.
2.
3.
4.
Create a trend model and calculate the
estimated value for each observation
Calculate the ratio of the actual value to the
predicted value for each observation
Use the average of the values for each
seasonal period to compute the seasonal
index
Multiply any forecast produced by the trend
model by the appropriate seasonal index
Use Solver to Identify Seasonal
Indices and Trendline
1.
Program linear trendline formula for trend
forecast, referring to input data cells for b0 and
b1
2.
3.
4.
Program seasonal adjustment formula,
referring to input data cells for seasonal indices
Program MAPE or MSE calculations
Program Solver to Min MAPE/MSE
By Changing seasonal indices, b0 and b1
Subject to average seasonal index = 100%
and seasonal indices>=0
Forecasting periods 37 and 38
for the Vintage Case


Y37 = 185.8 + .372*37 = 199.63
Seasonal forecast for 37 = seasonal
index for 37 * Y37
=1.44* 199.63 = 288.4


Y38 = 185.8 + .372*38 = 200
Seasonal forecast for 38 = 1.29*200
= 259
Simple Linear Regression:
Example
You want to examine
the linear dependency
of the annual sales of
produce stores on their
size in square footage.
Sample data for seven
stores were obtained.
Find the equation of
the straight line that
fits the data best.
Store
Square
Feet
Annual
Sales
($1000)
1
2
3
4
5
6
7
1,726
1,542
2,816
5,555
1,292
2,208
1,313
3,681
3,395
6,653
9,543
3,318
5,563
3,760
Scatter Diagram: Example
Annua l Sa le s ($000)
12000
10000
8000
6000
4000
2000
0
0
Excel Output
1000
2000
3000
4000
S q u a re F e e t
5000
6000
Equation for the Sample
Regression Line: Example
Yˆi  b0  b1 X i
 1636.415  1.487 X i
From Excel Printout:
C o e ffi c i e n ts
I n te r c e p t
1 6 3 6 .4 1 4 7 2 6
X V a ria b le 1 1 .4 8 6 6 3 3 6 5 7
Graph of the Sample
Regression Line: Example
Annua l Sa le s ($000)
12000
10000
8000
6000
4000
2000
0
0
1000
2000
3000
4000
S q u a re F e e t
5000
6000
Interpretation of Results:
Example
Yˆi  1636.415 1.487 Xi
The slope of 1.487 means that for each increase of
one unit in X, we predict the average of Y to
increase by an estimated 1.487 units.
The model estimates that for each increase of one
square foot in the size of the store, the expected
annual sales are predicted to increase by $1487.