CHAPTER-14

INTRODUCTION TO CORRELATION & REGRESSION ANALYSIS

ByDR. PRASANT SARANGI

Key concepts:

Introduction to Correlation Analysis Rank Correlation Linear Regression Analysis Multiple Regression Analysis

CORRELATION ANALYSIS

• Positive Correlation• Negative Correlation• Linear Correlation and • Non-linear Correlation

Positive Correlation

• Two variables are said to be positively correlated when the movement of the one variable leads the movement of the other variable in the same direction.

• There exists direct relationship between the two variables.

Negative Correlation

• Correlation between two variables is said to be negative when the movement of one variable leads to the movement in the other variable in the opposite direction.

• Here there exists inverse relationship between the two variables.

Linear Correlation

• The correlation between two variables is said to be linear where the points when drawn is a graph represents a straight line.

• Non-linear Correlation A relationship between two variables is said to be non-linear if a unit change

in one variable causes the other variable to change in fluctuations. If X is changed then corresponding values of Y will not change in the same

proportion.

Methods of Measuring Correlation

• The Graphical MethodThe correlation can be graphically shown by using scatter diagrams. Scatter diagram reveals two important useful information. Firstly, through this diagram, one can observe the patterns between two

variables which indicate whether there exists some association between the variables or not.

Secondly, if an association between the variables is found, then it can be easily identified regarding the nature of relationship between the two (whether two variables are linearly related or non-linearly related).

• Karl Pearson’s Coefficient of Correlation Karl Pearson’s coefficient of correlation (developed in 1986) measures

linear relationship between two variables under study. Since, the relationship is expressed is linear, hence, two variables change in a fixed proportion. This measure provides the answer of the degree of relationship in real number, independent of the units in which the variables have been expressed, and also indicates the direction of the correlation.

• Direct method

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Assumed Mean Method

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Assumptions of Coefficient of Correlation

1. The Value of the Coefficient of Correlation Lies between -1 (minus one) to +1 (plus one).

2. The Value of the Coefficient of Correlation is Independent of the Change of Origin and Change of Scale of Measurement

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Rank Correlation Coefficient

There are three different situations of applying the Spearman’s rank correlation coefficient.

• When ranks of both the variables are given• When ranks of both the variables are not given and • When ranks between two or more observations in a series are

equal

• When Ranks of Both the Variables are Given

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When Ranks of both the Variables are not Given

•In such cases, each observation in the series is to be ranked first.

•The selection of highest value depends on the researcher.

• In other words, either the highest value or the lowest value will be ranked 1 (one) depends upon the decision of the researcher.

• When Ranks between Two or More Observations in a Series are Equal• The ranks to be assigned to each observation are an average of the ranks

which these observations would have got, if they differed from each other.

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Simple Linear Regression Model

What do we use regression models for:

1. Estimate a relationship among economic variables, such as y = f(x).

2. Test hypotheses

3. Forecast or predict the value of one variable, y, based on the value of another variable, x.

Dependent and Independent Variables

Dependent variable - the variable we are trying to explain

Independent (or explanatory) variables - variables that we think cause movements in the dependent variable

Simple Regression Model

Y = dependent variableX = independent variable

Model is: Y = α + Xα is the intercept or constant is the slope coefficient

Linearity

Models that are linear in the variables and in the coefficients:Y = α + X

Models that are nonlinear in the variables but linear in the coefficients: Y = α + X2

Models that are nonlinear in the variables and in the coefficients:Y = α + X

Some models that are nonlinear can be made linear in the coefficients:

Y = e α X

take logs:ln Y = α + ln X

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E(Y|X)

E(Y|X)

AverageExpenditure

X (income)

E(Y|X)= α + X

=E(Y|X)

X

An Example showing income and average expenditure

Error Term

Y is a random variable composed of two parts:

I. Systematic component: E(Y) = α + X This is the mean of Y.

II. Random component: u = Y - E(Y | X) = Y - α - X

u is called the stochastic or random error.

Together E(Y) and u form the model: Y = α + X + u

Sources of error term

• Dependent variable measured with error• Model left out relevant variables• Wrong functional form• Inherent randomness of behaviour

True Relationship

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The Estimated Model

We use the data on Y and X to come up with guesses for α and . These estimated parameters or coefficients are

α and cap

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Our estimated, or “fitted”, model gives the predicted value for Y for any given X:

Yi = α + Xi

The residual is the difference between the actual or observed value of Y and the predicted value:

ui = Yi - Yi = Yi - α - Xi

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