1) Describe the following functional forms of a regression model clearly explaining under
what circumstances they can be suitably used. Give examples where possible.
a.A model without intercept.
b.A log-linearmodel.
c.A linlogmodel
d.A log-linmodel.
e.A polynomial regression models.
a. A model without intercept: A regression model without an intercept term assumes that the relationship between the independent variables and the dependent variable passes through the origin (0,0). This form can be suitably used when we have prior knowledge or strong theoretical reasons to believe that the intercept term should be zero. For example, in some physical or biological systems, it is plausible to assume that a zero value of the independent variable will result in a zero value of the dependent variable.
Example: In a study analyzing the relationship between the weight of an object and its volume, we can expect that when the weight is zero, the volume will also be zero. Hence, a model without an intercept can be used in this case.
b. A log-linear model: A log-linear model assumes that the relationship between the independent variables and the dependent variable is linear when the dependent variable is logged. This form is suitable when we expect a constant percentage change in the dependent variable for a one-unit change in the independent variable. Log transformations are often used when dealing with exponential growth or decay.
Example: In economics, the relationship between income and expenditure is often described using a log-linear model. An increase in income will usually lead to a proportional increase in expenditure rather than an absolute increase.
c. A lin-log model: A lin-log model assumes that the relationship between the independent variables and the dependent variable is linear when the independent variable is logged. This form is suitable when we expect a constant change in the dependent variable for a one-unit change in the logarithm of the independent variable. Lin-log models are useful when dealing with variables that have exponential effects on the dependent variable.
Example: In environmental science, the relationship between pollution and population can be described using a lin-log model. As the population increases exponentially, the pollution level increases linearly.
d. A log-lin model: A log-lin model assumes that the relationship between the independent variables and the dependent variable is linear when the dependent variable is logged. This form is suitable when we expect a constant change in the dependent variable for a one-unit change in the independent variable. Log transformations are often used when dealing with variables that have multiplicative effects on the dependent variable.
Example: In marketing, the relationship between advertising expenditure and sales can be described using a log-lin model. As advertising expenditure increases, the sales increase by a constant percentage.
e. Polynomial regression models: Polynomial regression models involve fitting a polynomial equation to the data. It allows for non-linear relationships between the independent and dependent variables. Polynomial models can be suitable when the relationship between variables is not well-explained by a linear model alone. The degree of the polynomial determines the complexity of the relationship that can be captured.
Example: In physics, the relationship between distance and time during free-fall of an object can be better described using a polynomial regression model. The equation could include terms for distance^2, distance^3, etc., to account for the non-linear acceleration due to gravity.