## What Is Multiple Linear Regression?

Multiple linear regression is a statistical technique that is used to predict the value of a dependent variable (also known as an outcome variable) based on the values of one or more independent variables (also known as predictor variables).

The goal of multiple linear regression is to find the best-fitting line or curve that represents the relationship between the dependent variable and the independent variables.

In order to understand multiple linear regression, it is important to first understand linear regression.

### Linear regression

Linear regression is a statistical technique that is used to predict the value of a dependent variable based on the value of one independent variable.

For example, let’s say that you are interested in predicting someone’s weight based on their height.

In this case, weight would be the dependent variable and height would be the independent variable.

You could use linear regression to find the best-fitting line or curve that represents the relationship between weight and height.

Once you have this line or curve, you could then use it to predict someone’s weight based on their height.

Multiple linear regression is similar to linear regression, except that it uses multiple independent variables instead of just one.

In this case, if you wanted to predict someone’s weight, you would use more than just their height and consider other lifestyle factors and so on.

#### Simple and Multiple Linear Regression

## How are multiple regression models used in finance?

Multiple regression models are often used in finance to predict the future value of a dependent variable (such as a stock’s price) based on the values of one or more independent variables (such as other stock prices).

For example, let’s say that you are interested in predicting the future price of Company A’s stock based on the current prices of Company B’s stock and Company C’s stock.

In this case, the future price of Company A’s stock would be the dependent variable and the current prices of Company B’s stock and Company C’s stock would be the independent variables.

You could use multiple linear regression to find the best-fitting line or curve that represents the relationship between the future price of Company A’s stock and the current prices of Company B’s stock and Company C’s stock.

Once you have this line or curve, you could then use it to predict the future price of Company A’s stock based on the current prices of Company B’s stock and Company C’s stock.

It should be noted that models like this are only reliable when all variables impacting the output are included and when the past is going to be a good predictor of the future.

When this is not true, these models can do poorly.

It can also help ascertain the relative strength or weakness of a relationship.

For example, if you wanted to determine what factors into the stock price of a gold miner, you could use the spot price of gold and the inflation rate. (The inflation rate can be a useful proxy for determining the relative easiness or tightness of monetary policy that, in turn, directs money and credit flows.)

**Related**: Can You Use Linear Regression to Predict Stock Prices?

## Example: Fama-French Model

The Fama-French three-factor model is a multi regression model that describes stock returns in terms of three factors:

- market risk
- size risk, and
- value risk

The model was developed by Eugene Fama and Kenneth French in 1990.

The market risk factor is the return of the market portfolio. The size risk factor is the return of a portfolio consisting of small stocks minus the return of a portfolio consisting of large stocks.

The value risk factor is the return of a portfolio consisting of high book-to-market ratio stocks minus the return of a portfolio consisting of low book-to-market ratio stocks.

The Fama-French three-factor model has been shown to explain a significant amount of the variation in stock returns. It is a popular model for use in academic research and has been used in a variety of applications.

The model has come under criticism in recent years, with some critics arguing that it does not accurately reflect the true nature of stock returns.

However, the model continues to be a useful tool for understanding the drivers of stock returns.

#### Fama French Three Factor Model

### Fama-French Derivative Models

Since the original 3-factor model was published, a number of derivative models have been developed.

The idea is that the 3-factor model is not inclusive of all variables and can be improved upon.

The most notable of these is the Carhart four-factor model, which adds a momentum factor to the Fama-French three-factor model.

Other derivative models include the five-factor model, the six-factor model, and the seven-factor model.

The Carhart four-factor model has been shown to provide a better explanation of stock returns than the original Fama-French three-factor model.

However, there is debate as to whether the additional factors truly improve the explanatory power of the model or if they are simply capturing known risk factors in a different way.

The Fama-French three-factor model continues to be a useful tool for understanding stock returns and is likely to remain an important part of financial research in the years to come.

## What are the benefits of using multiple linear regression?

There are many benefits of using multiple linear regression, including:

**1.** Multiple linear regression can be used to predict the value of a dependent variable based on the values of one or more independent variables.

**2.** Multiple linear regression can be used to find the best-fitting line or curve that represents the relationship between the dependent variable and the independent variables.

**3.** Multiple linear regression can be used to determine the relative strength or weakness of the relationship between the dependent variable and the independent variables.

**4.** Multiple linear regression can be used to assess the impact of multiple independent variables on a dependent variable.

**5.** Multiple linear regression can be used to identify which independent variables are most important in predicting the value of a dependent variable.

## What are the limitations of using multiple linear regression?

There are several limitations of using multiple linear regression, including:

**1.** Multiple linear regression requires that all variables impacting the output are included in the model. If any important variables are omitted, the results of the multiple linear regression will be inaccurate.

**2.** Multiple linear regression only works if the past is a good predictor of the future. If the variables included in the model are not good predictors of the dependent variable, then the results of the multiple linear regression will not be accurate.

**3.** Multiple linear regression can be affected by outliers, or data points that are far from the rest of the data. Outliers can have a significant impact on the results of a multiple linear regression and should be removed from the data set if possible.

**4.** Multiple linear regression can be affected by multicollinearity, which is when two or more independent variables are highly correlated with each other. This can cause problems because it can be difficult to determine which independent variable is having the biggest impact on the dependent variable.

**5.** Multiple linear regression can be affected by heteroscedasticity, which is when the variance of the dependent variable is not constant across all values of the independent variables. This can cause problems because it can make it difficult to accurately interpret the results of the multiple linear regression.

## Difference Between Linear and Multiple Regression

The main difference between linear and multiple regression is that linear regression only uses one independent variable to predict the value of a dependent variable, while multiple regression uses two or more independent variables to predict the value of a dependent variable.

Both linear and multiple regression are useful tools for predictive analysis.

However, multiple regression is generally more accurate than linear regression because it can take into account the relationships between multiple independent variables and the dependent variable.

Naturally, there tends to be more than one input that affects an output.

For example, if one wants to predict the inflation rate through multiple regression, the inputs might be:

- The unemployment rate
- Various interest rates (e.g., short-term rate, 10-year rate, 30-year mortgage rate)
- The growth rate of the economy
- The price of oil, commodities, and other industrial inputs

Each of these inputs will have some effect on inflation, but they will not have an equal effect. Therefore, it is important to use multiple regression in order to determine the relative importance of each input.

Additionally, multiple regression can also account for nonlinear relationships between the independent variables and the dependent variable.

## FAQs – Multiple Linear Regression

### Why use a multiple regression over a simple OLS regression?

A multiple regression can be used when there are two or more independent variables that are thought to impact the dependent variable.

A simple OLS regression can only be used when there is one independent variable.

### What is multicollinearity and how can it affect a multiple linear regression?

Multicollinearity is when two or more independent variables are highly correlated with each other.

This can cause problems because it can make it difficult to determine which independent variable is having the biggest impact on the dependent variable.

### How can outliers affect a multiple linear regression?

Outliers can have a significant impact on the results of a multiple linear regression and should be removed from the data set if possible.

### What is the best software for multiple linear regression?

There is no definitive answer to this question as different software packages have different features and capabilities.

Some popular software packages for performing multiple linear regression include Excel, R, SAS, and SPSS.

### How do I interpret the results of a multiple linear regression?

The results of a multiple linear regression can be interpreted in terms of the coefficients, p-values, and R-squared value.

The coefficients represent the estimated effect of each independent variable on the dependent variable.

The p-values represent the probability that each coefficient is not statistically significant.

The R-squared value represents the percentage of variation in the dependent variable that is explained by the independent variables.

**Related**

### What is the difference between linear and multiple regression?

The main difference between linear and multiple regression is that linear regression only uses one independent variable to predict the value of a dependent variable, while multiple regression uses two or more independent variables to predict the value of a dependent variable.

Both linear and multiple regression are useful tools for predictive analysis.

However, multiple regression is generally more accurate than linear regression because it can take into account the relationships between multiple independent variables and the dependent variable.

Single-variable regression models tend to be overly simplistic.

For example, if there was a regression between ice cream and temperature, a single-variable model might suggest that ice cream causes hot weather.

### What are the assumptions of multiple linear regression?

The assumptions of multiple linear regression include linearity, normality, homoscedasticity, and independence.

These assumptions must be met in order for the results of the multiple linear regression to be reliable.

### What is heteroscedasticity and how can it affect a multiple linear regression?

Heteroscedasticity is when the variance of the dependent variable is not constant across all values of the independent variables.

This can cause problems because it can make it difficult to accurately interpret the results of the multiple linear regression.

### What is the difference between a dependent and an independent variable?

A dependent variable is the variable that is being predicted by the multiple linear regression.

An independent variable is a variable that is used to predict the value of the dependent variable.

Independent variables can be either categorical or continuous.

### Do traders use multiple regression analysis?

Multiple regression analysis can be a useful form of analysis when used appropriately.

Some traders use multiple regression analysis to help identify relationships between different markets and to predict future market movements.

However, it is important to remember that multiple regression analysis is only one tool that can be used for trading and that it should not be relied upon exclusively.

There are sophisticated entities that use a variety of statistical techniques to trade the markets.

### What are some of the limitations of multiple linear regression?

Multiple linear regression is a powerful predictive tool, but it does have some limitations.

One limitation is that it can only be used to predict variables that are linearly related to the independent variables.

Another limitation is that multicollinearity can make it difficult to accurately interpret the results of the multiple linear regression.

Outliers can also cause problems with multiple linear regression.

Finally, multiple linear regression assumes that the data is free of error.

Despite these limitations, multiple linear regression can be a useful tool for predictive analysis.

## Conclusion – Multiple Linear Regression

Multiple linear regression is a statistical technique that is used to estimate the relationships between a response variable and one or more predictor variables.

The response variable is also known as the dependent variable, and the predictor variable is also known as the independent variable.

Multiple linear regression can be used to estimate the relationships between a response variable and a set of predictor variables, which can be continuous or categorical.

The coefficient estimates produced by multiple linear regression are used to determine the strength of the relationships between the response variable and the predictor variables, and whether these relationships are statistically significant.

Multiple linear regression can be useful in finance because it can be used to estimate the relationships between outputs (e.g., stock returns) and a set of predictor variables, such as market return, size, value, momentum, and volatility.

The coefficient estimates can be used to make investment decisions, such as deciding which stocks to buy or sell, and when to buy or sell them.