Regression analysis is a type of data analysis that gives small business owners detailed insights that improve their products and services. Small business owners use regression analysis to examine the influence of one or more independent variables on a dependent variable.

Businesses that often use regression analysis include insurance companies, pharmaceutical companies, credit card companies, and finance companies.

Here’s a look at how regression analysis works and how it applies to financing.

## What is Regression Analysis?

Regression analysis helps business owners identify which variables have an impact on a specific topic of interest. This helps them plan more strategically for the company’s financial future. Regression analysis is built around two variables:

- Dependent Variable: This is the main factor that you’re either trying to understand or trying to predict. It is the variable being tested and measured, and is dependent on the independent variable (you may also see it referred to as a “response variable”).
- Independent Variables: This variable has a direct effect on the dependent variable, and is the one that will be changed,.

There are three types of regression that are relevant in business: Simple linear, multiple linear, and nonlinear regression. Most of the time, businesses are using a linear regression model because it fits predictor variables.

## Regression Analysis in Finance

In real world examples of financial modeling, entrepreneurs use regression analysis to estimate the strength of the relationship between variables and subsequently forecast this relationship’s future behavior. It fits in any setting where we hypothesize there is (or not) a correlation between two or more variables.

In finance, this goes hand-in-hand with the Capital Asset Pricing Model (CAPM). The CAPM determines the relationship between an asset’s expected return and the associated market risk premium. A financial analyst would use this to forecast returns and the operational performance of your business.

## Regression Analysis – Linear Model Assumptions

Linear regression is a type of analysis that assesses whether one or more predictor variables explain the dependent variable. These variables are represented as ‘x’ and ‘y’. (x)- representing independent and (y)- representing dependent.

There are four assumptions associated with a linear regression model which includes:

- Linearity: This means that the relationship between the x-axis and the mean of y-axis is linear. The easiest method of detecting if an assumption is met is by creating a scatter plot for (x) versus (y). Scatter plots provide a visual to see if there is a linear relationship between the two variables. When the points in the plot appear as though they could fall along a straight line, then there is some type of linear relationship between the two variables and this assumption is met.

- Homoscedasticity: This indicates that variance of residual is the same for any value of (x). Heteroscedasticity is usually shown by a cluster of points that is wider as the values for the predicted dependent variable get larger. For homoscedasticity entrepreneurs look at a scatterplot between each independent variable and the dependent variable.

- Independence: This shows that observations are independent of each other and there is no correlation between consecutive residuals in the time series data. An easier way to test if this assumption is met is to look at a residual time series plot: plot of residuals vs. time. Business owners use the Durbin-Watson test to measure if the assumption is met.

- Normality: For any fixed value of (x), (y) is normally distributed. You can use quantile-quantile plots (Q-Q plots). Q-Q plots tend to be easier, especially for testing a smaller sample size. A Q-Q plot is a type of plot used to determine whether or not the residuals of a model follow a normal distribution. If the points on the plot form a relatively straight diagonal line, then the normality assumption is met.

## Regression Analysis – Simple Linear Regression

The objective, when using simple linear regression, is to get the predicted values of an output variable (a response) based on the value of an input (a predictor) variable. Simple linear regression is used to model the relationship between two continuous variables. It is a tool commonly used in financial analysis and has also been referred to as “ordinary least squares” (OLS regression).

Using scatter plots or scatterplot matrices, you can determine correlation which supplies a measure of the linear association between pairs of variables.

“Covariance” is the formula used to calculate the relationship between the two variables. This calculation shows you the direction of the relationship. So, if one variable increases and the other variable also increases, then the covariance would be positive. If one variable goes up and the other goes down, then the covariance would be negative.

To better interpret and use the covariance in forecasting, it has to be standardized. The result of this is the correlation calculation. The correlation calculation takes the covariance and divides it by the product of the standard deviation of the two variables, making the correlation between a value of -1 and +1. A correlation of +1 can suggest that both variables move positively with each other and a -1 proposes they are negatively correlated.

Here is an example of a simple linear regression equation:

y = bx + a

(y) Is the value we are trying to forecast – the dependent variable

(b) Is the slope of the regression line

(x) Is the value of our independent value – the dependent variable

(a) Represents the y-intercept

## Regression Analysis – Multiple Linear Regression

Multiple linear regression or multiple regression analysis is a statistical technique that is used to predict the outcome of a variable based on the value of two or more variables. The dependent variable is the variable that you want to predict. The variables used to predict the value of the dependent variable are referred to as independent or explanatory variables.

Here’s an example of multiple regression as a formula:

*yi *=*β*0 + *β*1 *x i*1 +*β*2 *x i*2 +…+*βp x ip *+*ϵ*

where, for i=n observations:

yi = dependent variable

xi = explanatory variables

β0 =y-intercept (constant term)

βp = slope coefficients for each explanatory variable

ϵ = the model’s error term (also known as the residuals)

Multiple linear regression is based on five assumptions:

1. A linear relationship between the dependent and a number of independent variables:

The best way to check the linear relationships is to create scatter plots and then visually inspect the scatterplots for linearity. If it’s not linear, the data is transformed using statistical software, such as SPSS.

2. The independent variables are not highly correlated with each other:

The data should not show multicollinearity, which occurs when the independent variables (explanatory variables) are highly correlated. The best method to test for the assumption is the Variance Inflation Factor method.

3. The variance of the residuals is constant:

Multiple linear regression assumes that the amount of error in the residuals is similar at each point of the linear model – also known as homoscedasticity.

4. Independence of observation:

This assumes that the observations should be completely independent of one another or that the values of residuals are independent. The Durbin Watson statistic is used to test it.

5. Multivariate normality

Multivariate normality occurs when residuals are normally distributed. It can be tested using two methods, including a histogram with a superimposed normal curve or the Normal Probability Plot method.

## Regression Analysis Tools

To find more efficient ways to implement regression analysis, here are some tools you can use:

- Microsoft Excel: It remains one of the most popular statistical tools used to calculate regression models for finances. It is a more basic tool than it is complex.
- Python: Serves as a more complex regression analysis tool for businesses. Due to its high-grade coding language, businesses opt to use Python in order to detect relationships within their data.
- R: This statistical software is touted as one of the best. It is free for computing and runs on various platforms including Windows and MacOS. They also offer an FAQ-section on how to get started and get it installed.

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