Non-linear regression can be used to predict market demand by modeling the relationship between the predictor variables (such as price, income, advertising, etc.) and the response variable (demand). The model can then be used to predict demand at different values of the predictor variables.

Non-linear regression can also be used to estimate market elasticities, which are a measure of how demand responds to changes in the predictor variables. For example, a market elasticity of -1 means that a 1% increase in price will lead to a 1% decrease in demand.

An analogy for a non-linear regression model would be a person's weight as a function of their height. A linear regression model would suggest that a person's weight is directly proportional to their height, meaning that if a person's height increases by one unit (say, one inch), their weight will also increase by a certain amount (say, one pound).

However, we know that this is not the case - a person's weight does not increase linearly as their height increases. Instead, there is a curvilinear relationship between height and weight, such that a person of average height will weigh more than a person who is very short, but a person who is very tall will not necessarily weigh more than a person of average height. This curvilinear relationship can be modeled using a non-linear regression model.

Curvilinear Relationship

A curvilinear relationship is a type of relationship between two or more variables in which the change in one variable is not proportional to the change in the other variable. Curvilinear relationships are often represented by a curve, and they can be either positive or negative.

Positive curvilinear relationships occur when the variables increase or decrease together, and the relationship between the variables is nonlinear. Negative curvilinear relationships occur when the variables move in opposite directions, and the relationship between the variables is also nonlinear.

Polynomial Regression

The most common type of non-linear regression is polynomial regression, which models the relationship between the response variable and the predictor variable(s) as a polynomial. Polynomial regression can be used to model non-linear relationships between the predictor variable(s) and the response variable, even if the relationship is not linear in the parameters.

Other types of non-linear regression include logistic regression, which is used to model binary data, and spline regression, which is used to model data that is not evenly spaced.