BBA315 2016 Week07 Simple Linear Regression

28/04/2016
BBA315 Business Forecasting
Week #7
Simple Linear Regression
Week #7
Agenda
Reading Game contributions Review of the Unit so far Next week’s Quiz Refresher on Simple Regression
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1
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Standard Error of the Estimate
The standard error of the estimate measures the amount by which the actual Y values differ from the estimated, or Ŷ, values.
• Simple regression analysis gives us the ability to estimate the mathematical relationship between a dependent variable (usually called Y) and an independent variable (usually called X)
Coefficient of Determination: R2 is the % of variance in Y explained by X
OLS estimates have good forecast properties
Week
1 2 3 4 5 6 7 8 9 10
Sum Average
Advertising Sales ($100s) (1,000s)
Y2
X2 X*Y
Ŷ
Error (Y-Ŷ)
9
10
100 81 90 9.01 0.99
SEE is the standard deviation of the errors
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Standard Error of the Estimate
To estimate standard deviation (σ) we start with the variance and take the square root to obtain the standard deviation. For simple linear regression the estimate of σ2 is the average squared residual.
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regression.
Simple Linear Regression
General form: b0 is the point at which the straight line intersects the Y axis (value of Y when x=0) b1 indicates how much of value of Y changes when the value of X changes one unit
10
100 225 150
15
225 144 180
5
25 36 30
12
144 100 120
17
289 225 255
20
400 441 420
112 1488 1522 1480
11.2
b1

n XY X Y n X 2 ( X )2
b0 Y b1 X
b1 = (10*1480 ‐ 114*112)/(10*1522‐114*114) = 0.914
Standard Coefficients Error t Stat
Lower P-value 95%
Upper 95%
Lower 95.0%
Upper 95.0%
0.784 2.025 0.387 0.709 -3.886 5.454 -3.886 5.454
Advertising ($100s)
• The dependent variable (Y) is the variable for which we want to make a prediction, while the independent variable (X) is as a predictor
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Determining the parameters of a straight line
Week
1 2 3 4 5 6 7 8 9 10
Sum Average
Advertising ($100s) 9 7 5 14 15 12 6 10 15 21
114 11.4
Sales (1,000s)
Y2
X2
X*Y
10
100 81 90
6
36 49 42
5
25 25 25
12
144 196 168
25
1
9
10
2
7
6
20
3
5
5
Sales (1,000s)
4
14
12
15
5
15
10
6
12
15
10
7
6
5
5
8
10
12
9
15
17
0
10
21
20
0
Advertising Vs Sales over time
y = 0.9137x + 0.7842
5
10
15
20
25
Advertising ($100s)
Y=b0+b1X1
Ordinary Least Squares
The method of least squares is used to calculate the equation of a straight line that minimises the sum of squared distances (SSE) between the X‐Y data points and the line, as measured in the vertical (Y) direction
0.914 0.164 5.566 0.001 0.535 1.292 0.535 117.292
Testing the goodness of fit
Test for overall model significance: F test
Test for individual variable significance: t tests
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To estimate σ, use s
s estimates the standard deviation σ of the error
term ε in the statistical model for simple linear
b0 = 11.2 – 0.914*11.4 = 0.784
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2
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Regression Equation
Sample regression line:
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Y= 0.784+0.914X+ε
7
6
36 49 42 7.18 -1.18
5
5
25 25 25 5.35 -0.35
14
12
1S4t4anda1r9d6Erro1r68and S13u.5m8 -1.58
15
10
100of Sq2u2a5red E1r5r0ors a14re.49 -4.49
12
15
225 1m44inim1is8e0d 11.75 3.25
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.891
R Square
0.795
Adjusted R Square
0.769
Standard Error
2.448
Observations
10
ANOVA
Regression Residual Total
df
6
5
25 36 30 6.27 -1.27
10
12
144 100 120 9.92 2.08
15
17
289 225 255 14.49 2.51
21
20
400 441 420 19.97 0.03
114
112 1488 1522 1Байду номын сангаас80 104.94 0.00
11.4
11.2
(Y-Ŷ)^2
0.986 1.392 0.124 2.482 20.153 10.574 1.603 4.323 6.304 0.001 47.94
13
14
Example of Simple Regression with Excel
Week
1 2 3 4 5 6 7 8 9 10