# 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,$$Y=e^{-3}=0.9102$$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.

Aside from the factors cited above, the patterns in residuals can also be analyzed when evaluating a model. Residuals are random and uncorrelated in a good model.

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