When conducting regression analysis, we obtain the intercept and coefficient estimates for each independent variable. These values, both intercept and coefficients, can be positive or negative.

Many of our colleagues often inquire about the interpretation of these values when they are negative. The question is, are negative intercepts or coefficient estimates permissible?

In a previous article, I discussed negative coefficient estimates for independent variables. In this article, I will delve into whether a negative intercept in a linear regression equation is acceptable.

**Understanding Intercept in Regression Equation**

For those utilizing linear regression analysis, whether simple linear regression or multiple linear regression, specifying the equation is essential.

In this equation specification, we denote “b_{0}” or “a” as the constant or intercept. Here are examples of equation specifications:

Y = b_{0} + b_{1}X + e ………… (1)

Y = b_{0} + b_{1}X_{1} + b_{2}X_{2} + … + b_{n}X_{n} + e ……….. (2)

The first equation represents simple linear regression as it has only one independent variable (X). In this equation, if the value of variable X = 0, the predicted value of Y equals the intercept.

The same principle applies to the second equation, which represents multiple linear regression. It’s called multiple linear regression because it has more than one independent variable. In the second equation, if all independent variables are zero, the predicted value of Y will be equal to the intercept or constant.

From these equations, we understand that when independent variables are zero, the predicted value of the dependent variable equals the intercept. Furthermore, we know that the intercept value is obtained from the estimation of linear regression.

**Interpreting Negative Intercept in Regression**

In linear regression analysis using the least squares method, the intercept value is not always positive. In some cases, we may encounter a negative intercept.

Of course, this often raises questions among many people. Is it acceptable for the intercept to be negative? How do we interpret this negative intercept value?

To facilitate understanding, let’s create a simple example. Suppose there is a study examining the impact of the number of products sold on a company’s profit.

In this context, we can specify the equation, where the variable being influenced is called the dependent variable (Y), and the influencing variable is called the independent variable (X).

In this equation, we consider the number of products sold as the independent variable (X) and the company’s profit as the dependent variable (Y).

After performing simple linear regression analysis, we obtain the intercept and regression coefficient estimates as follows:

Y = -250 + 14X

From this equation, we can see that the intercept value is -250 and the regression coefficient estimate for the variable X is 14. Based on this example, if the variable X is zero, then:

Y = -250 + 14X

Y = -250 + 14(0)

Y = -250 + 0

Y = -250

Thus, we can interpret that if the variable X is zero, then the prediction for variable Y is -250. In the context of the variables used, this means that if there are no products sold (X=0), the company will incur a loss of 250 units. Does this make sense?

This negative intercept value can make sense because it can be interpreted that even if there are no sales, the company still has to bear fixed costs and operational expenses. So, a loss of 250 units reflects the costs that the company must incur even if no products are sold.

**Interpretation of Negative Intercept, Yet Not Logical**

In the first example, the intercept value can be interpreted logically. However, what if the resulting intercept value doesn’t make sense?

Let’s consider another example. Suppose a researcher wants to examine the effect of study hours on exam scores. In this context, study hours refer to the time spent studying outside of class in preparation for exams.

Based on this example, we can designate study hours as the independent variable (X) and exam scores as the dependent variable (Y). After conducting simple linear regression analysis, the obtained intercept and coefficient estimates are as follows:

Y = -10 + 1.5X

From these estimates, if study hours are zero (no study hours outside of class), then the predicted exam score is:

Y = -10 + 1,5X

Y = -10 + 1.5(0)

Y = -10 + 0

Y = -10

According to this calculation, if there are no study hours (X=0), the predicted exam score becomes -10. Does this make sense?

We know that exam scores usually range from 0 to 100. Surely, a predicted exam score of -10 doesn’t make sense, does it?

So, are negative intercept values permissible? In the first example, the interpretation of the intercept still makes sense, but not in the second example. What should we do then?

To address illogical intercept values, we can consider several approaches. One of them is to add context or constraints to our model to make the interpretation of the intercept more realistic. Alternatively, we can choose another model that better fits our data.

Therefore, it’s important to always check and consider the logic behind the intercept values obtained from linear regression analysis, especially when their interpretation doesn’t align with reality or common sense.

However, despite this phenomenon, I conclude that negative intercept values can be disregarded, provided that all assumptions of linear regression are met. Conducting classical assumption tests is necessary to ensure the attainment of the Best Linear Unbiased Estimator (BLUE). This conclusion reflects my opinion, and if it’s inaccurate, I welcome corrections in the comments section.

Additionally, in practical research, it’s rare to find instances where all independent variables have a value of zero, so even with a negative intercept, the predicted value of the dependent variable still approximates reality.

Furthermore, the interpretation of analysis results mainly focuses not on the intercept but on the slope or coefficient estimates to determine predictions of the dependent variable. For example, if product sales increase by 10 units, what happens to the company’s profit?

That’s all for this article. I hope my insights provide valuable insights to those in need. Thank you for visiting Kanda Data’s website and reading this article. See you in the next article.