Functional Forms for Simple Linear Regression

To address non-linear relationships, we employ various functional forms to potentially convert the data for linear regression. Here are three commonly used log transformation functional forms:

1. Log-lin model: In this log transformation, the dependent variable is logarithmic, while the independent variable is linear. It is represented as shown below.$$ lnY=b_0+b_1X_i. $$

   The slope coefficient in the log-lin model is the relative change in the dependent variable for an absolute change in the independent variable.

   When utilizing a log-lin model, caution must be exercised when making forecasts. For example, in the predicted regression equation like \(Y=-3+5X\), if X is equal to 1, the \(ln{Y}=-3\), then,

   $$ Y=e^{-3}=0.0498 $$

   Moreover, the lin-lin model cannot be compared with the log-lin model without the transformation. As such, we need to transform \(R^2\) and F-statistic.

2. Lin-log model: In this case, the dependent variable is linear, while the independent variable is logarithmic. It is represented as follows:
   \(Y_i=b_0+b_1lnX_i\).

   The slope coefficient in the lin-log model is responsible for the absolute change in the dependent variable for a relative change in the independent variable.

3. Log-log model: In this log transformation, both the dependent and independent variables are logarithmic. It is represented as \(lnY_i=b_0+b_1lnX_i\). The slope coefficient in the log-log model is the relative change in the dependent variable for a relative change in the independent variable. In other words, if X increases by 1%, Y will change by \(b_1\).

Selecting the Correct Functional Form

To settle on the correct functional form, consider the following goodness of fit measures:

1. Coefficient of determination \((R^2)\). A high value is better.
2. F-statistic. The high value of the F-statistic is better.
3. Standard error of the estimate \((S_e)\). A low value of \(S_e\) is better.

In addition to the factors cited above, the patterns in residuals can also be analyzed when evaluating a model. In a good model, residuals are random and uncorrelated.

Question 1

Which of the following statements about the log-lin model is most likely correct:

1. The dependent variable is linear, while the independent variable is logarithmic.
2. Both the dependent and independent variables are logarithmic
3. The dependent variable is logarithmic, while the independent variable is linear.

The correct answer is c.

In the log-lin model, the dependent variable (\(Y\)) is logarithmic, as represented by $$lnY = b_{0} + b_{1}X_{i}$$ While the independent variable (\(X\)) is linear.

A is incorrect. It describes the lin-log model, where the dependent variable is linear and the independent variable is logarithmic.

B is incorrect. It describes the log-log model, where both the dependent and independent variables are logarithmic.