# Regression Analysis | Business Statistics Notes | B.Com Notes Hons & Non Hons | CBCS Pattern

## Meaning of Regression Analysis

Regression is the measure of the average relationship between two or more variable in terms of the original units of the data. It is a statistical tool with the help of which the unknown values of one variable can be estimated from known values of another variable.

In the words of Ya Lum Chou, “Regression analysis attempts to establish the nature of the relationship between variables – that is, to study the functional relationship between the variable and thereby provide a mechanism for prediction, or forecasting.”

It is clear from above definitions that regression analysis is a statistical device with the help of which we are in a position to estimate the unknown values of one variable from known values of another variable.

 Table of Contents 1. Meaning and Properties of Regression Analysis 2. Significance and Limitations of Correlation Analysis 3. Various Types of Regression analysis 4. Difference between Correlation and Regression 5. Regression lines. Why there are two regression lines? 6. Standard error of estimates

### Characteristics of regression coefficients:

1. Both regression co-efficients will have the same sign.

2. If one regression co-efficient is above unity, then the other regression co-efficient should be below unity.

3. If both the regression co-efficient are negative, correlation co-efficient should be negative

4. Regression co-efficients are independent of change of origin but not of scale.

## Uses and Significance of Regression Analysis

The following are main Advantages of regression analysis:

(1) Helpful to statisticians: The study of regression helps the statisticians to estimate the most probable value of one variable of a series for the given values of the other related variables of the series.

(2) Nature of relationship: Regression is useful in describing the nature of the relationship between two variables.

(3) Estimation of relationship: Regression analysis is widely used for the measurement and estimation of relationship among economic variables.

(4) Predictions: Regression analysis is helpful in making quantitative predictions on the basis of estimated relationship among variables.

(5) Policy formulation: The predictions made on the basis of estimated relationship are used in policy making.

## Limitations of Regression analysis

The following are the main limitation of regression:

1) No change in relationship: Regression analysis is based on the assumption that while computing regression equation; the relationship between variables will not change.

(2) Conditions: The application of regression analysis is based on certain conditions like, for existence of linear relationship between the variables; exact values are needed for the independent variable.

(3) Spurious relationships: There may be nonsense and spurious regression relationships. In such case, the regression analysis is of no use.

## Kinds of Regression Analysis

Kinds of regression may be studied on the basis of:

I. Change in proportions.

II. Number of variation.

(I) Basis of change in proportion: There are two important regressions on the basis of change in proportion. They are:

(a) Linear regression: Regression is said to be linear when one variable move with the other variable in fixed proportion

(b) Non-linear regression: Regression is said to be non-linear when one variable move with the other variable in changing proportion.

(II) On the basis of number of variables: On the basis of number of variables, regression may be:

(a) Simple regression: When only two variables are studied it is a simple regression.

(b) Partial regression: When more than two variables are studied keeping other variables constant, it is called partial regression.

(c) Multiple regressions: When at least three variables are studied and their relationships are simultaneously worked out, it is a case of multiple regressions.

## Distinguish between correlation and regression.

There are some basis difference between correlation and regression:

(1) Nature of relationship: Correlation explains the degree of relationship, whereas regression explains the nature of the relationship.

(2) Causal relationship: Correlation does not explain the cause behind the relationship whereas regression studies the cause and effect relationship.

(3) Prediction: Correlation does not help in making prediction whereas regression enables us to make prediction.

(4) Origin and scale: Correlation coefficient is independent of the change of origin and scale, whereas regression coefficient is independent of change of origin but not of scale.

(5) Nature of variables: Correlation analysis does make any difference between dependent and independent variable. On the other hand, regression analysis makes difference between dependent and independent variable.

## Regression lines:

A line of regression by the method of “least square” shows an average relationship between variables under study. This regression line can be drawn graphically or derived algebraically. A line fitted by method of least square is known as the line of best fit. There are two regression lines:-

Regression line of x on y: Regression line of x on y is used to predict x for a given value of y. The regression equation of x on y is x=a+by.

Regression line of y on x: Regression line of y on x is used to predict y for a given value of x. The regression equation of y on x is y=a+bx

Why do we generally have two regression equations?

Two regression lines: We know that there are two lines of regression: - x on y and y on x. For these lines, the sum of the square of the deviations between the given values and their corresponding estimated values obtained from the line is least as compared to other line. One regression line cannot minimise the sum of squares for both the variables that is why we are getting two regression lines. (We get one regression line when r = +1 and Two regression lines will be at right angles when r = 0.)

## Standard error of estimate

With the help of regression equations, perfect prediction of values is not possible. In order to measure the accuracy of estimated figures, a statistical tool is used which is known as standard error of estimate. Calculation of standard error of estimate, symbolized as Sxy similar to standard deviation. Standard deviation measures the dispersion about an average, such as mean. The standard measure of estimate measures the dispersion about an average line, called the regression line. The formula for calculating the standard error of estimate is:

The standard error of estimate measures the accuracy of the estimated figures. The smaller the values of standard error of estimate, the closer will the actual value and estimated value. If standard error of estimate is zero, then there is no variation.