Posted: January 28th, 2022

Quantative Analysis essay

EFIM10014

Quantitative Analysis in Management

Regression Analysis: Model Validation II

Sophie Lythreatis

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Content:

• There are 4 assumptions of the regression model

• 1st assumption of the regression model

• 2nd assumption of the regression model

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Assumptions of the Regression Model

• The mathematics underlying the regression procedure is based upon a number of assumptions

• If these are not valid, then even though the regression procedure will produce a regression line, it could be totally meaningless as a predictive tool

• We need to ensure the assumptions are valid

• The four main assumptions are:

• Constant error variance (homoscedasticity)

• Normality of residuals

• Independent residuals (no autocorrelation)

• Independence of explanatory/independent variables (no multicollinearity)

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1. Constant error variance (homoscedasticity)

When we don’t have homoscedasticity, we have

HETEROSCEDASTICITY

What does this mean?

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• The errors terms are assumed to be:

Homoscedastic (The same variance at every X)

• This assumption means that the variance of the

residuals is constant for all values of a given

explanatory/independent variable.

• The case of unequal error variances is called

heteroscedasticity (this is a problem!)

Homoscedasticity

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How do we check for heteroscedasticity?

• The easiest way to check for this is a scatterplot of the residuals against each explanatory/independent variable

• A residual plot is a graph that shows the residuals/errors on the vertical axis and the independent variable on the horizontal axis.

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Residual plot against x

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Residual Plot Against x

Residual Plot Against x

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Nonconstant Variance

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Heteroskedasticity Heteroskedasticity No Heteroskedasticity

Residual Plots

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Residuals seem slightly more widely scattered in the middle, so it seems

there is a possibility of mild heteroscedasticity.

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Extreme Heteroscedasticity

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Consequences and Cure for Heteroscedasticity

• This “fan shaped” pattern is the classic indication of heteroscedasticity

• Heteroscedasticity leads to the standard error of the regression coefficient being inaccurate. This means C.I and H.T. for this coefficient could be misleading

• There are two ways to deal with heteroscedasticity: • Use Weighted Least Squares as the regression technique. Use a

logarithmic transformation of the response variable

• Curing heteroscedasticity is beyond the scope of this unit.

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2. Normality of residuals

The error terms/residuals are assumed to be:

1. Homoscedastic (constant error variance)

2. Normally distributed

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Normality of Residuals • The residuals should form a normal distribution

• There are many formal tests available for this (Chi Squared, Shapiro- Wilks, Anderson-Darling, Lilliefors, Q-Q Plot etc).

• This assumption is usually satisfied for most data sets, unless the residuals are severely non-normal.

• As a quick practical step it is not unusual just to plot a histogram of the residuals and qualitatively observe whether or not there is marked deviation from a normal distribution.

• If the residuals appear non-normally distributed, there are transformation of variable techniques available but these are beyond the scope of this unit.

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Normality of residuals

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Histogram of Residuals

The residuals closely resemble a normal distribution indicating no significant

issue with this assumption.

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In this video, we looked at 2 assumptions of the

regression model that need to be satisfied to

validate our model.

In the next video, we look at the remaining 2

assumptions of the regression model.

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