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.
a) A model without intercept is a regression model that does not include a constant term. This type of model is typically used when the dependent variable is expected to be zero when all of the independent variables are zero. For example, a model without intercept could be used to predict the number of sales of a product based on the price of the product and the number of advertisements. In this case, the number of sales would be expected to be zero when the price and number of advertisements are both zero.
b) A log-linear model is a regression model in which the dependent variable is transformed using the natural logarithm. This type of model is typically used when the dependent variable is expected to have an exponential relationship with the independent variables. For example, a log-linear model could be used to predict the number of sales of a product based on the price of the product and the number of advertisements. In this case, the number of sales would be expected to increase exponentially as the price and number of advertisements increase.
a) A model without intercept: A regression model without an intercept (or constant term) is a model where the regression line passes through the origin (0,0) rather than having a separate intercept term. This means that the predicted value of the dependent variable will be zero when all the independent variables are zero.
Circumstances where this form is suitably used:
1. When it is theoretically justifiable: In some cases, it may make sense to exclude the intercept term if there is a clear rationale to assume that the dependent variable is inherently zero when all the independent variables are zero. For example, in physics, an object's acceleration is typically assumed to be zero when there is no force acting upon it.
2. When the intercept term is not meaningful: In certain situations, the intercept term may not have any practical interpretation or may lead to incorrect inferences. For instance, in economic models, the intercept term sometimes represents a base level of consumption or production, and excluding it can provide more accurate estimates.
Example: Consider a model that predicts the weight of an object (dependent variable) based on its volume (independent variable). If it is assumed that an object with zero volume also has zero weight, a model without an intercept can be used.
b) A log-linear model: A log-linear regression model is a form of regression model where the dependent variable is transformed by taking the natural logarithm (log) of it, while the independent variables remain in their original form. This transformation helps to model the relationship between the variables in a way that the dependent variable changes proportionally to the independent variables.
Circumstances where this form is suitably used:
1. When the relationship is multiplicative: Log-linear models are suitable when the relationship between variables is multiplicative rather than additive. This means that a change in the independent variable leads to a proportional change in the dependent variable. For example, in economics, the relationship between income and consumption is often assumed to be multiplicative.
2. When the data has a wide range or heteroscedasticity: Log-linear models can also be useful when the data exhibits heteroscedasticity, meaning that the variability of the dependent variable changes across different levels of the independent variables. Taking the logarithm of the dependent variable can help normalize its distribution and reduce heteroscedasticity.
Example: Suppose we want to predict sales (dependent variable) based on advertising expenditure (independent variable). If we assume that a doubling of the advertising budget leads to a proportional doubling of sales, a log-linear model can be used to capture this relationship.
a) A model without intercept refers to a regression model where the y-intercept term is excluded. In other words, it assumes that the regression line passes through the origin (0,0) rather than having a constant term. This model is suitable when there is strong theoretical or empirical evidence that the relationship between the independent variables and the dependent variable starts at zero. It is commonly used when working with ratios or percentages, where a value of zero on the independent variable(s) would result in a zero value in the dependent variable.
Example: Suppose you are analyzing the relationship between the number of hours studied (X) and the test score (Y) for a group of students. If it is believed that without any studying, a student should expect a test score of zero, then a model without intercept can be used. The regression equation would be Y = βX, where β represents the coefficient of the independent variable.
b) A log-linear model refers to a regression model where the dependent variable is transformed using a natural logarithm, resulting in a linear relationship between the predictors and the logged outcome variable. This model is appropriate when the relationship between the predictors and the outcome is expected to be multiplicative rather than additive. It is commonly used when analyzing variables measured on a multiplicative scale, such as income or population growth.
Example: Suppose you are studying the relationship between advertising expenditure (X) and sales revenue (Y) for a company. It is believed that a 10% increase in advertising expenditure leads to a certain percentage increase in sales revenue, rather than a fixed amount. In this case, a log-linear model can be used. The regression equation would be ln(Y) = βX, where ln represents the natural logarithm and β represents the coefficient of the independent variable.