Econometrics: how to interpret a regression model?

In the subject of Data Analysis o Econometricsa fundamental aspect that we study with our students from IE University is to build, analyze and interpret regression models applied to economics and business.

In this article we explain some keys to better understand it.

Table of contents

1. What is a regression model?

2. Mathematical form of the regression

3. Demand as a regression function

4. A numerical example of a regression model

5. What is R² and how is it interpreted?

What is a regression model?

An econometric regression model is a statistical tool used to study the relationship between a dependent variable (Y) and a dependent variable (Y). dependent variable (Y) and one or more independent variables (X)which are presumed to influence the dependent variable.

It is commonly used in economics to model the behavior of economic variables, such as the demand for a product, the relationship between supply and price, or the impact of government policies. impact of government policies on the economy. on the economy.

The basic idea behind a regression model is to estimate a mathematical equation representing the relationship between the relationship between the dependent variable and the independent variables.

Mathematical form of a regression

The equation takes the following form: 

Y = β0 + β1X1 + β2X2 + ... + βkXk + ε 

Where Y is the dependent variable, X1, X2, ..., Xk are the independent variables, β0 is the intercept, β1, β2, ..., βk are the coefficients, and ε is the error or residual term. 

The coefficients represent the marginal effect of each independent variable on the dependent variable, holding all other variables constant. To estimate the coefficients, the regression model uses a set of observations, which includes the values of the dependent variable and the independent variables for a sample of individuals or entities.

Once the model is estimated, it can be used to make predictions about predictions about the dependent variable based on the values of the independent variables. It can also be used to test hypotheses about the relationship between the variables, such as whether a particular independent variable has a significant effect on the dependent variable or whether the coefficients are statistically different from zero.

In general, econometric regression models provide a powerful tool for analyzing the economic relationships between economic relationships and making predictions about the behavior of economic variables.

Demand as a regression function

An example of an econometric regression econometric regression model is the study of the relationship between the price of a product and the demand for the same product. Suppose we want to estimate how the demand for a product varies as a function of price and other factors such as consumer income and the price of substitutes.

In this case, we could define demand as the dependent variable (Y), and price (X1), income (X2) and the price of substitute products (X3) as independent variables. 

The general regression equation would be: 

Y = β0 + β1X1 + β2X2 + β3X3 + ε 

where β0 is the intercept, β1, β2, and β3 are the coefficients that measure the relationship between demand and the price, income, and price of substitute products, respectively, and ε is the error or residual term that captures other factors influencing demand. 

Once the model is estimated, a number of predictions can be made. predictions can be made about the demand for the product. For example, one could use the model to estimate how demand would change if the price of the product is reduced by a certain percentage, holding other factors constant. One could also use the model to analyze how changes in income would affect demand. changes in income or in the price of substitute products affect the demand for the product.

A numerical example of a regression model

Suppose that research was conducted in a city to analyze the relationship between the price of a product (in euros), the average income of consumers (in thousands of euros) and the price of substitute products (in euros) with the demand for the product (in units). 

Data were collected from 100 observations and the following multiple linear regression model was estimated multiple linear regression model (i.e., with more than one variable):

Demand = 200 - 2.5 Price + 0.6 Rent + 0.8 * Substitute_price

Where Price is the independent variable that measures the price of the product, Income is the independent variable that measures the average income of consumers and Price_substitutes is the independent variable that measures the price of substitute products. 

The estimated coefficients indicate that a decrease of one dollar in the price of the product is associated with an increase of 2.5 units in the demand for the product, a decrease in the average consumer income by one thousand euros is associated with a decrease of 0.6 units in the demand for the product, and a one euro increase in the price of substitute products is associated with a 0.8 unit increase in the demand for the product. 0.8 units in the demand for the product, holding all other factors constant.

What is R² and how is it interpreted?

To interpret the coefficient of determination (R²)we must remember that this coefficient measures the total variability of Y that is explained by the model.

Their values always range between 0 and 1The closer to zero, the poorer the quality of the model, and the closer to 1, the higher the quality of the model.

In this case, we are going to assume that the coefficient of determination is 0.75, which means that the model explains 75% of the total variability in the demand for the product, while the remaining 25% is explained by other factors that are not included in the model.

In short, this model indicates that a lower price of the product itself and a higher price of substitute products will increase demand, while a decrease in consumer income will decrease demand. The coefficient of determination indicates that the model is relatively good at explaining variability in product demand.

Summary table

Definition	Mathematical form	Utility	R²
Model that studies the relationship between a dependent variable (Y) and one or more independent variables (X).	Y = β0 + β1X1 + β2X2 + ... + βkXk + ε	Regression models are used to make predictions about Y and to test hypotheses about variables.	Measures the total variability of Y that is explained by the model.  Their values always range between 0 and 1.

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