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Functional Forms for Simple Linear Regression

Functional Forms for Simple Linear Regression

For non-linear relationships, we can use several functional forms to potentially transform the data to be used in linear regression. The following are the three often-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 \(lnY_i = b_0 + b_1 X_i\). The slope coefficient in the log-lin model is the relative change in the dependent variable for an absolute change in the independent variable. That is, a one-unit increment in \(X\) triggers a \(100b_1\) percent change in \(Y\).
  2. Lin-log model: In this case, the dependent variable is linear while the independent variable is logarithmic. It is represented as \(Y_i = b_0 + b_1 lnX_i\). 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. Put another way, if X increases by 1%, \(Y\) will change by \(b_1\) per 100 units.
  3. Log-log model: In this log transformation, both the dependent and independent variables are logarithmic. It is represented as \(lnY_i = b_0 + b_1 lnX_i\). 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 (R2). High value of is better.
  2. F-statistic. High value of F-statistic is better.
  3. Standard error of the estimate (Se). A low value of Se 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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