Issues Regarding Regression Models

(Lesson - 06/C)

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Collinearity

• A perfect linear relationship between two (or more) independent variables is called collinearity (multi-collinearity) • Under this condition, the least-square regression coefficients cannot be uniquely defined.

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Collinearity

• A strong but less than perfect linear relationship between the independent variables can cause: 1. Regression coefficients to be unstable, 2. Standard errors to the coefficients become large, hence, confidence intervals for coefficients become large and coefficients become imprecise, Dr. C. Ertuna 3

Collinearity Mesurement

• One of the measures to determine the impact of Collinearity on the precision of the estimates is called the “Variance Inflation Factor (VIF).” Dr. C. Ertuna 4

Collinearity Detection

• Wrong signs for the coefficients • Drastic changes in the coefficients in terms of size and/or sign as a new variable is added to the equation.

• High VIF Dr. C. Ertuna 5

Collinearity: Remedies

• • There is no Quick Fix for collinearity, Some strategies: 1. Variable selection for the model: Based on correlation matrix, some of the highly correlated variables could be excluded from the model, 2. Ridge Regression instead Ordinary Least Squared Regression (OLR).

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Unusual Data

A single observation that is substantially different from all other observations can make a large difference in the results of your regression analysis.

If a single observation (or small group of observations) substantially changes your results, you would want to know about this and investigate further.

There are three ways that an observation can be unusual.

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Unusual Data

•

Outliers

: In linear regression, an outlier is an observation with large residual. In other words, it is an observation whose dependent-variable value is unusual given its values on the predictor variables. An outlier may indicate a sample peculiarity or may indicate a data entry error or other problem. Dr. C. Ertuna 8

Unusual Data

•

Leverage

: An observation with an extreme value on a predictor variable is called a point with high leverage. Leverage is a measure of how far an independent variable deviates from its mean. These leverage points can have an unusually large effect on the estimate of regression coefficients. Dr. C. Ertuna 9

Unusual Data

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Influence

: An observation is said to be influential if removing the observation substantially changes the estimate of coefficients. Influence can be thought of as the product of leverage and outlierness. Dr. C. Ertuna 10

Outliers and Influential Data

• An outlier is an observation whose dependent variable value is unusual given the value of the independent variable • Not all outliers has an important effect on the intercept and/or slope of the regression.

• For an outlier to be influential it should be away from the mean of the independent variable.

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Influential Data: Diagnosis

Cook’s D • If Cook’s distance for a particular observation is greater than a cutoff point than that observation could be considered as influential data.

• One such cutoff point is – D i > 4 / (n-k-1) – Where, k = number of independent variables Dr. C. Ertuna 12

Influential Data Diagnostics on SPSS

Standardized DfBETA(s):

• Change in the regression coefficient that results from the deletion of the i th case. A standardized DfBETA value is computed for each case for each regression coefficient generated by a model.

• Cut-off Points > 0 means case i increases the slope < 0 means case i decreases the slope |DfBETA(s)| > 2 strong indication of influence |DfBETA(s)| > 2/sqrt(n) might be problem Dr. C. Ertuna 13

Influential Data: Remedies

• The unusual data need to be investigated – For example, it may stem from an error in data entry • The model could be re-specified, robust estimation methods could be used, • An influential data could only be discarded if it is a truly bad data and cannot be corrected.

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Checking the Assumptions

• • • • • There are assumptions that need to be met to accept the results of Regression analysis and use the model for future decision making:

Linearity Independence Normality

of errors (No autocorrelation), of errors,

Constant Variance

of errors (Homoscadasticity ).

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Tests for Linearity

Linearity:

• Plot dependent variable against each of the independent variables separately.

• Decide whether linear regression is a “Reasonable” description of the tendency in the data.

– Consider curvilinear pattern, – Consider undue influence of one data point on the regression line, etc.

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Nonlinear Relationships

Diminishing Returns Relationship of Advertising versus Sales Advertising

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Nonlinear Relationships

Diminishing Returns Relationship of Advertising versus Sales Advertising

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3 2 1 0 -1 -2 -3 3 2 1 0 -1 -2 -3

Analysis of Residuals

(a) Nonlinear Pattern

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Tests for Independence

Independence of Errors:

• Plot residuals against time (Residual-Time Plot) – Residuals form y-axis, time form x-axis – If the residuals group alternately into positive and negative clusters then that indicates auto-correlation • Ljung-Box Test (Note that only one lag version is applied here) Dr. C. Ertuna 22

Residuals-Time Plot

• Notice the tendency of the residuals to group alternately into positive and negative clusters.

• That is an indication that the residuals are not independent but auto-correlated.

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Residual-Time Plot

10 20

Time

30 23 40

1 0 -1 -2 -3 3 2 1 0 -1 -2 -3 3 2

Analysis of Residuals

(a) Independent Residuals (b) Residuals Not Independent

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Time

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Time

Ljung-Box Test

• Compute LB Test Statistics for one lag (Q (1) ) –

Q(1) = (n(n-2)/ (n-1) ) * Correl(Data_Range_ 1 , Data_Range_ 2 )^2

• Compare LB against Chi-square_alpha-value –

Chiinv ( alpha / tails, 1)

• Ho: Q(1)

<

Chi-square_alpha Dr. C. Ertuna 25

Non-Independence: Remedies

• EGLS (Estimated Generalized Least Squares) Methods – Prais-Winsten – Cochrane-Orcutt (Note that these are effective only for first-order autocorrelation.) Dr. C. Ertuna 26

Tests for Normality

Normality of Errors:

• Normal-Quantile Plot of Residuals (Errors) • Compute Skewness • Compute Kurtosis • Jarque-Bera Test Dr. C. Ertuna 27

Normal-Quantile Plot of Residuals

• Sort Residuals (min => max) • Create a Rank column • Compute z-scores =NORMINV((rank-0.5)/N,0,1) • Plot z-scores (x) and residuals (y) • For normality the plot should be reasonably linear.

Normal Quantile Plot of Errors

3 2 1 0 -1 -15 -2 -3 -10 -5 0 5 10

Normal Quantiles

15 20 Dr. C. Ertuna 28

Jarque-Bera Test

(in Excel)

• Compute JB-Test Statistics –

JB = (n/6)*Skew(Data_Range)^2 + + (n/24) * ( Kurt(Data_Range)^2

• Compute p-value by using the formula –

Chdist(JB,2)

• Ho: Data is normally distributed –

Note that JB is very sensitive to sample size, and p_values are not uniformly distributed, hence danger in committing Type I error

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Non-Normality: Remedies

To stabilize error variance, one of the most frequently used technique is data transformation.

• X and/or Y values could be transformed by employing power to those variables, • y (or x) => y p (or x p ) where p = -2, -1, -½, ½, 2, 3 Dr. C. Ertuna 30

Tests for Constant Variance

Constant Variance of Errors:

• Plot residuals against y-estimates: – Residuals form y-axis and estimated y-values form x axis.

– When errors get larger (or smaller) as y-values increase that would indicate non-constant variance.

• Plot residuals against each x: – Residuals form y-axis and x-values form x-axis.

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Analysis of Residuals

3 2 1 0 -1 -2 -3

x 1

(a) Variance Decreases as x Increases

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Analysis of Residuals

3 2 1 0 -1 -2 -3

x 1

(b) Variance Increases as x Increases

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3 2 1 0 -1 -2 -3

Analysis of Residuals

x 1

(c) Constant Variance

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Non-Constant Variance: Remedies

• Transform dependent variable (y) – y => y p where p = -2, -1, -½, ½, 2, 3 • Weighted Least Square Regression Method Dr. C. Ertuna 35

Next Lesson

(Lesson - 07/A)

Qualitative & Judgmental Forecasting Methods

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