Describe the following functional forms of a regression model clearly explaining under
what circumstances they can be suitably used. Give examples where possible.
c) A linlogmodel
d) A log-linmodel.
e) A polynomialregressionmodels.
c) A linlog model: In a linlog model, one variable is linearly related to the logarithm of another variable. This form of regression can be used when there is a clear expectation or theoretical basis that the dependent variable has a logarithmic relationship with the independent variable.
For example, suppose we want to analyze the impact of advertising expenditure on sales. In some cases, it is believed that the effect of advertising on sales is not linear but rather has a diminishing return as advertising expenditure increases. In this situation, we can use a linlog model to capture this relationship, where the dependent variable (sales) is linearly related to the logarithm of the independent variable (advertising expenditure).
d) A log-lin model: In a log-lin model, the dependent variable is modeled as a linear function of the logarithm of one or more independent variables. This form of regression is typically used when there is an expectation or evidence that the dependent variable has an exponential growth or decay pattern with respect to the independent variable(s).
For example, if we want to analyze the effect of time on population growth, it is common to assume that population growth follows an exponential pattern. In this case, we can use a log-lin model where the dependent variable (population) is modeled as a linear function of the logarithm of the independent variable (time).
e) Polynomial regression models: Polynomial regression allows us to model the relationship between the dependent variable and the independent variable(s) using higher-order polynomial functions. This form of regression is suitable when the underlying relationship between the variables is nonlinear and cannot be adequately captured by a simple linear regression.
For example, let's say we want to analyze the relationship between age and income. It is likely that the relationship is not linear, as income might increase rapidly in the early years of a person's career and then level off later. In such cases, we can use a polynomial regression model to capture the nonlinearity by including higher-order polynomial terms along with the linear term.
It's important to note that the choice of the appropriate functional form depends on the specific data and the underlying theory or knowledge about the relationship. Experimentation, plotting the data, and examining the residuals are often helpful in determining the most suitable functional form for a regression model.