###### Net Present Value of an Investment Pro ...

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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:

**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.**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.

**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 \(b_1\).

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

- Coefficient of determination \((R^2)\). A high value is better.
- F-statistic. The high value of the F-statistic is better.
- 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.