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Forecasting Using the Simple Linear Regression
Model and Correlation
What is a forecast?
Using a statistical method on past data to predict the future.
Using experience, judgment and surveys to predict the future.
Why forecast?
to enhance planning.to force thinking about the future.to fit corporate strategy to future conditions. to coordinate departments to the same future.to reduce corporate costs.
Kinds of Forecasts
Causal forecasts are when changes in a variable (Y) you wish to predict are caused by changes in other variables (X's).
Time series forecasts are when changes in a variable (Y) are predicted based on prior values of itself (Y).
Regression can provide both kinds of forecasts.
Types of Relationships
Positive Linear Relationship Negative Linear Relationship
Types of Relationships
Relationship NOT Linear No Relationship
(continued)
Relationships
If the relationship is not linear, the forecaster often has to use math transformations to make the relationship linear.
Correlation Analysis
•Correlation measures the strength of the linear relationship between variables.
•It can be used to find the best predictor variables.
• It does not assure that there is a causal relationship between the variables.
The Correlation Coefficient
• Ranges between -1 and 1.
• The Closer to -1, The Stronger Is The Negative Linear Relationship.
• The Closer to 1, The Stronger Is The Positive Linear Relationship.
• The Closer to 0, The Weaker Is Any Linear Relationship.
Y
X
Y
X
Y
X
Y
X
Y
X
Graphs of Various Correlation (r) Values
r = -1 r = -.6 r = 0
r = .6 r = 1
The Scatter Diagram
0
20
40
60
80
0 20 40 60
X
Y
Plot of all (Xi , Yi) pairs
The Scatter Diagram
Is used to visualize the relationship and to assess its linearity. The scatter diagram can also be used to identify outliers.
Regression Analysis
Regression Analysis can be used to model causality and make predictions.
Terminology: The variable to be predicted is called the dependent or response variable. The variables used in the prediction model are called independent, explanatory or predictor variables.
Simple Linear Regression Model
•The relationship between variables is described by a linear function.•A change of one variable causes the other variable to change.
• Population Regression Line Is A Straight Line that Describes The Dependence of One Variable on The Other
Population Y intercept
Population SlopeCoefficient
Random Error
Dependent (Response) Variable
Independent (Explanatory) Variable
Population Linear Regression
ii iY X PopulationRegressionLine
= Random Error
Y
X
How is the best line found?
Observed Value
Observed Value
YX iX
i
ii iY X
Sample Linear Regression
Sample Y Intercept
SampleSlopeCoefficient
Sample Regression Line Provides an Estimate of The Population Regression Line
0 1i iib bY X e
provides an estimate of
provides an estimate of
0b
1b
Sample Regression Line
Residual
Y
Simple Linear Regression: An Example
You wish to examine the relationship between the square footage of produce stores and their annual sales. Sample data for 7 stores were obtained. Find the equation of the straight line that fits the data best
Annual Store Square Sales
Feet ($1000)
1 1,726 3,681
2 1,542 3,395
3 2,816 6,653
4 5,555 9,543
5 1,292 3,318
6 2,208 5,563
7 1,313 3,760
The Scatter Diagram
0
2000
4000
6000
8000
10000
12000
0 1000 2000 3000 4000 5000 6000
Square Feet
An
nu
al
Sa
les
($00
0)
Excel Output
The Equation for the Regression Line
i
ii
X..
XbbY
4871415163610
From Excel Printout:
CoefficientsIntercept 1636.414726X Variable 1 1.486633657
Graph of the Regression Line
0
2000
4000
6000
8000
10000
12000
0 1000 2000 3000 4000 5000 6000
Square Feet
An
nu
al
Sa
les
($00
0)
Y i = 1636.415 +1.487X i
Interpreting the Results
Yi = 1636.415 +1.487Xi
The slope of 1.487 means that 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 1 square foot in the size of the store, the expected annual sales are predicted to increase by $1487.
The Coefficient of Determination
SSR regression sum of squares
SST total sum of squaresr2 = =
The Coefficient of Determination (r2 ) measures the proportion of variation in Y explained by the independent variable X.
Coefficients of Determination (R2) and Correlation (R)
r2 = 1,
Y
Yi = b0 + b1Xi
X
^
r = +1
Coefficients of Determination (R2) and Correlation (R)
r2 = .81, r = +0.9
Y
Yi = b0 + b1Xi
X
^
(continued)
Coefficients of Determination (R2) and Correlation (R)
r2 = 0, r = 0
Y
Yi = b0 + b1Xi
X
^
(continued)
Coefficients of Determination (R2) and Correlation (R)
r2 = 1, r = -1
Y
Yi = b0 + b1Xi
X
^
(continued)
Correlation: The Symbols
• Population correlation coefficient (‘rho’) measures the strength between two variables.
• Sample correlation coefficient r estimates based on a set of sample observations.
Regression StatisticsMultiple R 0.9705572R Square 0.94198129Adjusted R Square 0.93037754Standard Error 611.751517Observations 7
Example: Produce Stores
From Excel Printout
Inferences About the Slope
• t Test for a Population SlopeIs There A Linear Relationship between X and Y ?
1
11
bS
bt
•Test Statistic:
n
ii
YXb
)XX(
SS
1
21
and df = n - 2
• Null and Alternative Hypotheses
H0: 1 = 0 (No Linear Relationship) H1: 1 0 (Linear Relationship)
Where
Example: Produce Stores
Data for 7 Stores: Estimated Regression Equation:
The slope of this model is 1.487.
Is Square Footage of the store affecting its Annual Sales?
Annual Store Square Sales
Feet ($000)
1 1,726 3,681
2 1,542 3,395
3 2,816 6,653
4 5,555 9,543
5 1,292 3,318
6 2,208 5,563
7 1,313 3,760
Yi = 1636.415 +1.487Xi
t Stat P-valueIntercept 3.6244333 0.0151488X Variable 1 9.009944 0.0002812
H0: 1 = 0
H1: 1 0
.05
df 7 - 2 = 5
Critical value(s):
Test Statistic:
Decision:
Conclusion:
There is evidence of a linear relationship.t0 2.5706-2.5706
.025
Reject Reject
.025
From Excel Printout
Reject H0
Inferences About the Slope: t Test Example
Inferences About the Slope Using A Confidence Interval
Confidence Interval Estimate of the Slopeb1 tn-2 1bS
Excel Printout for Produce Stores
At 95% level of Confidence The confidence Interval for the slope is (1.062, 1.911). Does not include 0.
Conclusion: There is a significant linear relationship between annual sales and the size of the store.
Lower 95% Upper 95%Intercept 475.810926 2797.01853X Variable 11.06249037 1.91077694
Residual Analysis
Is used to evaluate validity of assumptions. Residual analysis uses numerical measures and plots to assure the validity of the assumptions.
Linear Regression Assumptions
1. X is linearly related to Y.
2. The variance is constant for each value of Y (Homoscedasticity).
3. The Residual Error is Normally Distributed.
4. If the data is over time, then the errors must be independent.
Residual Analysis for Linearity
Not Linear Linear
X
e eX
Y
X
Y
X
Residual Analysis for Homoscedasticity
Heteroscedasticity Homoscedasticity
e
X
e
X
Y
X X
Y
Residual Analysis for Independence: The Durbin-Watson Statistic
It is used when data is collected over time.
It detects autocorrelation; that is, the residuals in one time period are related to residuals in another time period.
It measures violation of independence assumption.
n
ii
n
iii
e
)ee(D
1
2
2
21 Calculate D and
compare it to the value in Table E.8.
Preparing Confidence Intervals for Forecasts
Interval Estimates for Different Values of X
X
Y
X
Confidence Interval for a individual Yi
A Given X
Confidence Interval for the mean of Y
Y i = b0 + b1X i
_
Estimation of Predicted Values
Confidence Interval Estimate for YX
The Mean of Y given a particular Xi
n
ii
iyxni
)XX(
)XX(
nStY
1
2
2
21
t value from table with df=n-2
Standard error of the estimate
Size of interval vary according to distance away from mean, X.
Estimation of Predicted Values
Confidence Interval Estimate for Individual Response Yi at a Particular Xi
n
ii
iyxni
)XX(
)XX(
nStY
1
2
2
21
1
Addition of 1 increases width of interval from that for the mean of Y
Example: Produce Stores
Yi = 1636.415 +1.487Xi
Data for 7 Stores:
Regression Model Obtained:
Predict the annual sales for a store with
2000 square feet.
Annual Store Square Sales
Feet ($000)
1 1,726 3,681
2 1,542 3,395
3 2,816 6,653
4 5,555 9,543
5 1,292 3,318
6 2,208 5,563
7 1,313 3,760
Estimation of Predicted Values: Example
Find the 95% confidence interval for the average annual sales for stores of 2,000 square feet
n
ii
iyxni
)XX(
)XX(
nStY
1
2
2
21
Predicted Sales Yi = 1636.415 +1.487Xi = 4610.45 ($000)
X = 2350.29 SYX = 611.75 tn-2 = t5 = 2.5706
= 4610.45 612.66
Confidence interval for mean Y
Confidence Interval Estimate for YX
Estimation of Predicted Values: Example
Find the 95% confidence interval for annual sales of one particular store of 2,000 square feet
Predicted Sales Yi = 1636.415 +1.487Xi = 4610.45 ($000)
X = 2350.29 SYX = 611.75 tn-2 = t5 = 2.5706
= 4610.45 1687.68
Confidence interval for individual Y
n
ii
iyxni
)XX(
)XX(
nStY
1
2
2
21
1
Confidence Interval Estimate for Individual Y