Regression Analysis

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# Regression Analysis - PowerPoint PPT Presentation

Regression Analysis. Gordon Stringer . 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. Regression Analysis.

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### Regression Analysis

Gordon Stringer

Gordon Stringer, UCCS

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

Gordon Stringer, UCCS

Regression Analysis
• Simple Regression: single explanatory variable
• Multiple Regression: includes any number of explanatory variables.

Gordon Stringer, UCCS

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)

Gordon Stringer, UCCS

Regression Analysis
• Linear Regression: straight-line relationship

Form: y=mx+b

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

Gordon Stringer, UCCS

Data Types
• Cross Sectional: data gathered from the same time period
• Time Series: Involves data observed over equally spaced points in time.

Gordon Stringer, UCCS

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

Gordon Stringer, UCCS

Scatter Plot and Trend line
• Click on a data point and add a trend line

Gordon Stringer, UCCS

Scatter Plot and Trend line
• Now you can see if there is a relationship between the variables. TREND uses the least squares method.

Gordon Stringer, UCCS

Correlation
• CORREL will calculate the correlation between the variables
• =CORREL(array x, array y)

or…

• Tools>Data Analysis>Correlation

Gordon Stringer, UCCS

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

Gordon Stringer, UCCS

Simple Regression Model
• Best fit using least squares method
• Can use to explain or forecast

Gordon Stringer, UCCS

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

Gordon Stringer, UCCS

Simple Regression Model
• Precision: accepted measure of accuracy is mean squared error
• Average squared difference of actual and forecast

Gordon Stringer, UCCS

Simple Regression Model
• Average squared difference of actual and forecast
• Squaring makes difference positive, and severity of large errors is emphasized

Gordon Stringer, UCCS

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

Gordon Stringer, UCCS

Simple Regression Model
• Run the regression tool.
• Tools>Data Analysis>Regression

Gordon Stringer, UCCS

Simple Regression Model
• Enter the variable data

Gordon Stringer, UCCS

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

Gordon Stringer, UCCS

Simple Regression Model
• Check labels, if including column labels
• Check Residuals, Confidence levels to displayed them in the output

Gordon Stringer, UCCS

Simple Regression Model
• The SUMMARY OUTPUT is displayed below

Gordon Stringer, UCCS

Simple Regression Model
• Multiple R is the correlation coefficient
• =CORREL

Gordon Stringer, UCCS

Simple Regression Model
• R Square: Coefficient of Determination
• =RSQ
• Goodness of fit, or percentage of variation explained by the model

Gordon Stringer, UCCS

Simple Regression Model

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.

Gordon Stringer, UCCS

Simple Regression Model
• Standard Error of the Estimate
• Defines the uncertainty in estimating y with the regression model
• =STEYX

Gordon Stringer, UCCS

Simple Regression Model
• Coefficients:
• Slope
• Standard Error
• t-Stat, P-value

Gordon Stringer, UCCS

Simple Regression Model
• Coefficients:
• Slope = 63.11
• Standard Error = 15.94
• t-Stat = 63.11/15.94 = 3.96; P-value = .0005

Gordon Stringer, UCCS

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)

Gordon Stringer, UCCS

Simple Regression Model
• Linearity
• Independence
• Homoscedasity
• Normality

Gordon Stringer, UCCS

Simple Regression Model
• Linearity

Gordon Stringer, UCCS

Simple Regression Model
• Linearity

Gordon Stringer, UCCS

Simple Regression Model
• Independence:
• Errors must not correlate
• Trials must be independent

Gordon Stringer, UCCS

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

Gordon Stringer, UCCS

Simple Regression Model
• Normality:
• Errors should be normally distributed
• Plot histogram of residuals

Gordon Stringer, UCCS

Multiple Regression Model
• Y = α + β1X1 + … + βkXk + ε
• Bendrix Case

Gordon Stringer, UCCS

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

Gordon Stringer, UCCS