Regression analysis formula can help business owners and leaders alike to make wise decisions. But what exactly is the regression analysis formula, what is the purpose of it and what are some examples for reference?

Let's dive in.

Regression analysis, which is used across many disciplines like finance, refers to "a set of statistical methods used for the estimation of relationships between a dependent variable and one or more independent variables," according to the Corporate Finance Institute. "It can be utilized to assess the strength of the relationship between variables and for modeling the future relationship between them."

In short, regression analysis is a prediction tool that's used to determine the dependent variable with the facilitation of one or more independent variables.

Regression analysis includes three variations:

- Linear
- Multiple Linear
- Nonlinear

The most common regression analysis models are simple linear and multiple linear. Meanwhile, nonlinear regression analysis is commonly used for far more nuanced data sets in which the dependent and independent variables have a nonlinear relationship with one another.

Linear regression analysis is based on six key assumptions, according to the Corporate Finance Institute:

- "The dependent and independent variables show a linear relationship between the slope and the intercept."
- "The independent variable is not random."
- "The value of the residual (error) is zero."
- "The value of the residual (error) is constant across all observations."
- "The value of the residual (error) is not correlated across all observations."
- "The residual (error) values follow the normal distribution."

Simple linear regression is the simplest of models. It explores the relationship between just one dependent variable and just one independent variable. It uses the following equation: Y (the dependent variable) = a (intercept) + bX (slope/independent explanatory) variable) + ϵ (residual *error*).

Then there is multiple linear regression analysis, which is very similar to the simple linear model, except that there are multiple independent variables used in this model. The conditions are otherwise the same. The multiple linear regression analysis uses the following equation: Y (the dependent variable) = a (intercept) + bX (slope/independent (explanatory variable) + cX (slope/independent explanatory variable) + dX (slope/independent explanatory variable) + ϵ (residual *error*).

Why use regression analysis? The purpose of regression analysis is, in short, to determine which of the independent variables are related to the dependent variable — and what those relationships are. With that information, one can then predict an outcome based on historical data.

"While running a regression analysis, the main purpose of the researcher is to find out the relationship between the dependent variable and the independent variable," according to Wall Street Mojo. "In order to predict the dependent variable one or multiple independent variables are chosen which can help in predicting the dependent variable. Regression analysis helps in the process of validating whether the predictor variables are good enough to help in predicting the dependent variable."

Regression analysis can be used in a variety of ways in finance and beyond.

"Regression analysis has several applications in finance," according to the Corporate Finance Institute. "For example, the statistical method is fundamental to the Capital Asset Pricing Model (CAPM). Essentially, the CAPM equation is a model that determines the relationship between the expected return of an asset and the market risk premium. The analysis is also used to forecast the returns of securities based on different factors, or forecast the performance of a business."

It's clear, then, that regression analysis can be significantly helpful in making wise business and finance decisions.

"Regression is a very useful statistical method," according to Wall Street Mojo. "For any business decision in order to validate a hypothesis that a particular action will lead to the increase in the profitability of a division can be validated based on the result of the regression between the dependant and independent variables. Regression analysis equation plays a very important role in the world of finance. A lot of forecasting is done using regression analysis. For example, the sales of a particular segment can be predicted in advance with the help of macroeconomic indicators that has a very good correlation with that segment. Both linear and multiple regressions are useful for practitioners in order to make predictions of the dependent variables and also validate the independent variables as a predictor of the dependent variables."

Here are two examples of regression analysis in finance.

- Beta and CAPM: Finance uses regression analysis to calculate the Beta (the volatility of returns that are relative to the market) for a stock, which can be done in Excel.
- Forecasting Revenues and Expenses: Finance also uses regression analysis (usually multiple regression analysis( in order to forecast financial statements for companies. This is used in order to determine the ways in which changes in assumptions of drivers of a business will impact future expenses and, ultimately, revenue.

Here are some examples of when you might put regression analysis into practice outside of finance:

- You want to try to find out the relation between the number of hours driven by an Uber driver and the age of the Uber driver. A company like Uber might actually use regression analysis in order to determine whether the relationship between two variables (the hours driven vs. the age) is validated by the regression equation.
- You want to try to find out the relation between the number of baked goods produced and the number of bakers in the kitchen. A bakery might use regression analysis in order to determine whether the relationship between two variables (the number of baked goods vs. the number of bakers) is validated by the regression equation.
- You want to try to find out the relation between the percentage of passing grades in a classroom and the number of years of experience a teacher has. A school might use regression analysis in order to determine whether the relationship between two variables (the percentage of passing grades in a classroom and the number of years of experience a teacher has) is validated by the regression equation.

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