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Regression Analysis

Gordon Stringer

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Regression Analysis

Regression Analysis: the study of the relationship between variables

Regression Analysis: one of the most commonly used tools for business analysis

Easy to use and applies to many situations

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Regression Analysis

Simple Regression: single explanatory variable

Multiple Regression: includes any number of explanatory variables.

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Regression Analysis

Dependant variable: the single variable being explained/ predicted by the regression model (response variable)

Independent variable: The explanatory variable(s) used to predict the dependant variable. (predictor variable)

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Regression Analysis

Linear Regression: straight-line relationship Form: y=mx+b

Non-linear: implies curved relationships, for example logarithmic relationships

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Data Types

Cross Sectional: data gathered from the same time period

Time Series: Involves data observed over equally spaced points in time.

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Graphing Relationships

Highlight your data, use chart wizard, choose XY (Scatter) to make a scatter plot

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Scatter Plot and Trend line

Click on a data point and add a trend line

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Scatter Plot and Trend line Now you can see if there is a relationship

between the variables. TREND uses the least squares method.

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Correlation

CORREL will calculate the correlation between the variables

=CORREL(array x, array y)

or… Tools>Data Analysis>Correlation

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Correlation

Correlation describes the strength of a linear relationship

It is described as between –1 and +1 -1 strongest negative +1 strongest positive 0= no apparent relationship exists

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Simple Regression Model

Best fit using least squares method Can use to explain or forecast

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Simple Regression Model

y = a + bx + e (Note: y = mx + b) Coefficients: a and b Variable a is the y intercept Variable b is the slope of the line

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Simple Regression Model

Precision: accepted measure of accuracy is mean squared error

Average squared difference of actual and forecast

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Simple Regression Model

Average squared difference of actual and forecast

Squaring makes difference positive, and severity of large errors is emphasized

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Simple Regression Model

Error (residual) is difference of actual data point and the forecasted value of dependant variable y given the explanatory variable x.

Error

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Simple Regression Model

Run the regression tool. Tools>Data Analysis>Regression

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Simple Regression Model Enter the variable data

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Simple Regression Model Enter the variable data y is dependent, x is independent

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Simple Regression Model Check labels, if including column labels Check Residuals, Confidence levels to

displayed them in the output

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Simple Regression Model The SUMMARY OUTPUT is displayed

below

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Simple Regression Model Multiple R is the correlation coefficient =CORREL

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Simple Regression Model R Square: Coefficient of Determination =RSQ Goodness of fit, or percentage of variation

explained by the model

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Simple Regression Model Adjusted R Square =

1- (Standard Error of Estimate)2 /(Standard Dev Y)2

Adjusts “R Square” downward to account for the number of independent variables used in the model.

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Simple Regression Model Standard Error of the Estimate Defines the uncertainty in estimating y with

the regression model =STEYX

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Simple Regression Model Coefficients:

– Slope– Standard Error– t-Stat, P-value

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Simple Regression Model Coefficients:

– Slope = 63.11– Standard Error = 15.94– t-Stat = 63.11/15.94 = 3.96; P-value = .0005

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Simple Regression Model y = mx + b

Y= a + bX + e Ŷ = 56,104 + 63.11(Sq ft) + e

If X = 2,500 Square feet, then

$213,879 = 56,104 + 63.11(2,500)

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Simple Regression Model Linearity Independence Homoscedasity Normality

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Simple Regression Model Linearity

Square Feet Line Fit Plot

0

50,000

100,000

150,000

200,000

250,000

300,000

350,000

1,500 2,000 2,500 3,000 3,500 4,000

Square Feet

Co

st

Cost Predicted Cost

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Simple Regression Model Linearity

Square Feet Residual Plot

-100000

-50000

0

50000

100000

1,500 2,000 2,500 3,000 3,500 4,000

Square Feet

Re

sid

ua

ls

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Simple Regression Model Independence:

– Errors must not correlate– Trials must be independent

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Simple Regression Model Homoscedasticity:

– Constant variance– Scatter of errors does not change from trial to

trial– Leads to misspecification of the uncertainty in

the model, specifically with a forecast– Possible to underestimate the uncertainty– Try square root, logarithm, or reciprocal of y

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Simple Regression Model Normality:

• Errors should be normally distributed

• Plot histogram of residuals

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Multiple Regression Model Y = α + β1X1 + … + βkXk + ε

Bendrix Case

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Regression Modeling Philosophy Nature of the relationships Model Building Procedure

– Determine dependent variable (y)– Determine potential independent variable (x)– Collect relevant data– Hypothesize the model form– Fitting the model– Diagnostic check: test for significance