Answer to Question #298387 in Microeconomics for Jordan

Question

A detailed lecture note on analysis of variance (ANOVA) along with regression

Answer 1

ANOVA

Analysis of variance (ANOVA) is a statistical analysis method that divides observed aggregate variability within a data set into two parts: systematic factors and random factors. Random factors have no statistical impact on the provided data set, but systematic factors do. In a regression analysis, analysts utilize the ANOVA test to examine the impact of independent variables on the predictor variables.

The Formula for ANOVA is:

F = $\frac{MST}{MSE}$

where:

F=ANOVA coefficient

MST = Mean sum of squares due to treatment

MSE=Mean sum of squares due to error

​

The ANOVA test is the first stage in determining which factors influence a particular data set. Following the completion of the test, an analyst does additional testing on the methodical variables that demonstrably contribute to the data set's inconsistency. The ANOVA test findings are used in an f-test by the analyst to create extra data that is consistent with the proposed regression models.

The ANOVA test compares more than two groups at the same time to see whether there is an association between them. The F statistic (also known as the F-ratio) produced by the ANOVA formula allows for the study of many sets of data to assess the variability between and within samples.

If there is no significant difference between the tested groups, which is known as the null hypothesis, the F-ratio statistic of the ANOVA will be near to 1. The F-distribution is the distribution of all possible values of the F statistic. This is actually a collection of distribution functions having two distinct numbers known as the numerator and denominator degrees of freedom.

ANOVA calculations give information regarding degrees of variability inside a regression model and serve as the foundation for tests of significance.

DATA = FIT + RESIDUAL, the basic regression line principle, is reformulated as follows:

(y_i - $\bar{y}$) = ($\^{y}$_i - $\bar{y}$) + (y_i - $\^{y}$_i).

REGRESSION

Regression is a statistical approach used in finance, investment, and other fields to identify the strength and type of a relationship between one predictor variable (typically represented by Y) and a set of other variables (known as independent variables).

Simple linear regression and multiple linear regression are the two most common forms of regression, while non-linear regression methods exist for more complex data and analysis. Multiple linear regression utilizes two or more independent variables to describe or forecast the outcome of the dependent variable Y, whereas simple linear regression utilizes one independent variable to explain or forecast the outcome of the dependent variable Y.

The general form of each type of regression is:

• Simple linear regression: Y = a + bX + u
• Multiple linear regression: Y = a + b₁X₁ ⁺ b₂X₂ + b₃X₃ + ... + b_tX_t + u

Where:

• Y = the variable you are attempting to forecast (dependent variable).
• X = the variable you're utilizing to forecast Y. (independent variable).
• a = the intercept.
• b = the slope.
• u = the regression residual.