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# Regression Coefficients: How To Calculate Them

## Sarah Thomas

Subject Matter Expert

Learn what regression coefficients are, how to calculate the coefficients in a linear regression formula, and much more

Regression coefficients are one of the main features of a regression model. They are the values you use to study the relationship between variables.

In this article, we'll explore what regression coefficients are and how they're calculated. Plus why they are crucial for understanding the results of a regression analysis.

## Regression Coefficients

Regression is a method we use in statistical inference to study the relationship between a dependent variable and one or more independent variables.

In a regression equation, the coefficients are the numbers that sit directly in front of the independent variables.

$Y= a + bX$

In this simple regression, b represents a regression coefficient.

It sits directly in front of the independent variable x.

The intercept of the regression, a, is also a coefficient, but we ‌simply refer to it as the intercept, constant, or the $\beta_0$ of the equation. The intercept tells you the expected value of Y, when all of the independent variables in the model are equal to zero.

## What Are Regression Coefficients?

In math, coefficients refer to the numbers by which a variable in an equation is multiplied. Regression coefficients, therefore, are the numbers multiplying the independent variables in the regression equation.

Each regression coefficient represents the estimated size and direction of the relationship between the dependent variable—also called the response variable—and a particular independent variable—also called a predictor variable.

## 3 Examples

Let’s take a look at a few examples.

### Example 1: Simple Linear Regression

In the simple linear regression below, 2 is the regression coefficient on X. In this model, Y is the dependent variable and X is the independent variable.

$Y = 3 + 2X$

In the case of a simple linear regression, the relationship between X and Y is approximated by a straight line, and the regression coefficient gives you the slope of that line. A positive coefficient tells you there is a positive relationship between the independent variable X and the dependent variable Y. The actual value of the coefficient tells you that for every 1-unit increase in X, the model estimates that, on average, Y will increase by 2 units.

### Example 2: Multiple Linear Regression

Regressions can have more than one independent variable, and therefore, more than one regression coefficient.

In the regression below, $X_1$ and $X_2$, are both independent variables, so the values 1.5 and -0.8 are both regression coefficients. The value 1.5 indicates that a one-unit increase in $X_1$ is associated with a 1.5 increase in Y. The value -0.8 indicates that a one-unit increase in $X_2$ is associated with a 0.8 decrease in Y.

$Y= 30 + 1.5X_1 -0.8X_2$

### Example 3: Regression Coefficient Notation

Prior to estimating the parameters of your model, you may use a regression like the one below.

$Y= a +b_1X_1+b_2X_2$

Here, we use lowercase letters and subscripts as placeholders for the coefficients. The letter a represents the intercept, and $b_1$ and $b_2$ represent the coefficients on the predictor variables. We use subscripts to indicate which coefficients correspond to which variables.

An alternative notation for writing a regression equation is to use the Greek letter $\beta$ to represent the coefficients. In the equation below, $\beta_0$ is the intercept, and $\beta_1$ to $\beta_n$ represent regression coefficients for the predictor variables.

#### Population Regression Equation

$Y=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}+...+\beta_{n}X_{n}$

You can also distinguish between the population regression equation and the estimated sample regression equation by using lowercase letters for x and y, and ^ symbol over coefficients in the sample regression.

#### Sample Regression Equation

$\hat{y}=\hat{\beta_{0}}+\hat{\beta_{1}}x_{1}+\hat{\beta_{2}}x_{2}+...+\hat{\beta_{n}}x_{n}$

## Is The Regression Coefficient The Same As The Correlation Coefficient?

The regression coefficient and the correlation coefficient are two statistical measures we commonly use to evaluate the relationship between two variables, but they are not the same thing.

A regression coefficient is an estimate of the change in the dependent variable, Y, that results from a change in the independent variable, X. In other words, it tells us how much Y changes when X changes. In a standard linear regression, the coefficient measures the change in Y that results from a one-unit increase in X.

A correlation coefficient, on the other hand, measures the strength of the relationship between two variables. One of the most common correlation coefficients is the Pearson correlation coefficient (r), which measures the strength and direction of the linear relationship between two variables. Unlike a regression coefficient, the correlation coefficient ranges from -1 to 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation.

## How Do You Calculate The Regression Coefficient in a Linear Regression Equation?

In a simple linear regression—a linear regression with one dependent and ‌one independent variable—the regression coefficient is the slope of the regression line.

You can find the regression coefficient by dividing the covariance of your independent and dependent variables by the variance of the dependent variable. You can also calculate the coefficient using software like Excel, R, or Stata.

$\beta_0=\dfrac{Cov(X_i,Y_i)}{Var(X_i)}$

Calculating the slope of a simple linear regression involves finding the value that minimizes the sum of squared errors between the predicted values given by the model and the actual values in your data.

## How To Interpret The Sign of Coefficients in a Regression Analysis?

The sign of a regression coefficient tells you the direction of the relationship between a particular predictor variable and the response variable.

### Positive Regression Coefficents

A positive regression coefficient indicates a positive relationship between the predictor variable and the response variable.

This means that as the predictor variable increases, the response variable also tends to increase. As the predictor variable decreases, the response variable also tends to decrease.

### Negative Regression Coefficents

A negative regression coefficient indicates a negative relationship between the predictor variable and the response variable. This means that as the predictor variable increases, the response variable tends to decrease, and as the predictor variable decreases, the response variable tends to increase. In other words, the independent variable and the dependent variable tend to move in opposite directions.

## How To Interpret the Coefficient Values in a Regression Analysis?

In a standard linear regression, the value of a regression coefficient gives you the estimated average change in the dependent variable in response to a one-unit change in the independent variable.

For example, if you have the linear regression equation Y= 10 + 2X, the coefficient of 2 tells you that for every 1 unit increase in X, the model predicts that Y will increase by 2.

In a modified linear regression model or a nonlinear regression, the interpretation of the coefficients can start to change. For example, in a linear-log regression, the coefficient divided by 100 gives us the estimated change in Y associated with a 1% increase in X.

## FAQs

### How many regression coefficients should I have in my regression?

This depends on the number of independent variables in your model. In a bivariate linear regression, there are two regression coefficients: the intercept and one coefficient on the single explanatory variable in the model. For every additional independent variable you add to your model, you will have one additional regression coefficient.

### Can a regression coefficient be greater than 1?

Yes. Careful not to confuse the regression correlation with the regression coefficient. The correlation coefficient always falls within the range of -1 and 1, but the regression coefficient can be greater than 1. Remember, in a linear regression, the regression coefficient tells you the estimated change in Y that results from a one-unit increase in X.

### If I get a negative value for my regression coefficient, did I do something wrong?

No. Regression coefficients can be positive or negative. A negative regression coefficient indicates a negative relationship between your predictor variable and your response variable. When the coefficient is negative, your model estimates that Y (the response variable) tends to decrease as X (the predictor variable) increases.

### Why are regression coefficients sometimes called parameter estimates?

Like all other statistical inference methods, regression is a tool we use to make inferences about a population using sample data. We use sample data points to calculate a sample regression equation.

This sample regression equation gives us an estimate of what the population regression equation is. In other words, it gives us an estimate of what the true relationship is between the variables in our model. In this way, the regression coefficients we calculate are estimates of population parameters.

### What is the relationship between R-squared and regression coefficients?

R-squared $(R^2)$ is a statistical measurement, which is also called the coefficient of determination. It’s a measure of how much of the variation in your independent variable, y, can be explained by variation in the independent variables in your regression. We use R-squared as an indicator of how strong a model is.

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