Functional Forms for Simple Linear Regression

Most Financial and economic data exhibit non-linear relationships between the dependent and independent variables. Estimating such data using a simple linear regression model would lead to the dependent variable being understated for some ranges of the independent variable. Thus, we need to transform the data and make it suitable for use in linear regression. The functional forms Include:

1. The log-lin model
2. The lin-log model
3. The log-log model

Log-Lin Model.

In this model, the dependent variable is changed into a logarithmic form, and the independent variable is linear:

$$lnY_i= b_0+b_1 X_1$$

The slope coefficient for this model will be the relative change in the dependent variable for an absolute change in the independent variable. One must be careful when using the log-lin model to make a forecast. The predicted value will be the antilog of the forecasted value. We can only compare models with the same form of the dependent variable.

Lin-Log Model.

In the lin-log model, the independent variable is logarithmic, but the dependent variable is linear.

$$Y_i=b_0+b_1 lnX_i$$

The slope coefficient for this model gives the absolute change in the dependent variable for a relative change in the independent variable.

Log-Log Model.

In this model, both the dependent and independent variables are linear in their logarithmic forms, also known as the double-log model.

$$lnY_i=b_0+b_1 lnX_i$$

The log-log model is useful when we want to compute the elasticities. This is because the slope coefficient is the relative change in the independent variable for a relative change in the dependent variable.

Examining the goodness of a fit measures is the key to fitting the best functional form of simple linear regression.

Question

An analyst was using the model below to investigate the relationship of Revenues (Rev) and adverts (Ad):

$$Rev_i=b_0+b_1 ln⁡(Ad)_i$$

The functional form that the analyst is using is most likely to be:

1. Log-Lin Model.
2. Lin-Log Model
3. A simple linear model with no transformation

Solution

The correct answer is B:  The model above is a Lin-Log model because the independent variable Adverts is logarithmic, and the dependent variable is linear.

A is incorrect: In the Log-Lin Model, the dependent variable is logarithmic, and the independent variable is linear.

C is incorrect: The independent variable is logarithmic, so the equation has been transformed.

Reading 0: Introduction to Linear Regression

LOS 0 (h) Describe different functional forms of simple linear regressions