Partial or Marginal Effects in Other Functional Forms (transformed variables)

Essentially, we are interested in the “marginal effect” of an independent variable (x) on a dependent variable (y). Mathematically, this is expressed as (frac{partial y}{partial x}). In the familiar form of the linear regression, the marginal effect of the indepedent varaibles on the dependent varaibles is just the coefficient:

$$
y = beta_0 + beta_1 x_1 + beta_2 x_2 + u
$$

$$
frac{partial y}{partial x_1} = 0 + beta_1 + 0 +0
$$

Log-log (log-linear)

For a **log-log**, also called the **log-linear**functional form however, the marginal effect (frac{partial y}{partial x}) has (x) within its expression, which can be interpreted as: “the estiamted effect of (x) is dependent on the value of (x)”:

$$
ln y = beta_1 + beta_2 ln x +u
$$

$$
y = e^{beta_1 + beta_2 ln x +u}
$$

$$
frac{partial y}{partial x} = e^{beta_1 + beta_2 ln x + u} times (beta_2 frac{1}{x})
$$

(recall the chain rule for taking the partial derivative with respect of (x)). The marginal effect (frac{partial y}{partial x}) is not the most interpretable thing. So instead, recognizing that the first factor is equal to (y), we simplify by dividing both sides of the equation by (y) and multiplying both sides by (x):

$$
frac{frac{partial y}{y}}{frac{partial x}{x}} = beta_2
$$

This expression can be interpreted as “a 1 percent increase in x is associated with a (beta_2) percent increase in y, (controlling for all other factors if other covariates are included in the regression)”b>. (beta_2) is therefore also called the elasticity of (y) with respect to (x).

Level-log and log-level (semi-log)

There are two types of semi-log functional forms: the **level-log** ((y= beta_1 + beta_2 ln x + u)) and the **log-level** (ln y = beta_1 + beta_2 x + u).

Here are the derivations of the interpretations of (beta) in each:

level-log

$$
$y= beta_1 + beta_2 ln x +u $
$$

$$
frac{partial y}{partial x} = frac{beta_2}{x}
$$

$$
frac{partial y}{partial x /x} = beta_2
$$

The interpretation of (beta_2) therefore is: “A 1 percent increase in (x) is associated with a (beta_2/100) change in (y) (controlling for all other factors if other covariates are included in the regression)”

log-level
$$
ln y = beta_1 + beta_2 x + u
$$

$$
y = e^{beta_1 + beta_2 x + u }
$$

$$
frac{partial y}{partial x} = e^{beta_1 + beta_2 x + u } times beta_2
$$

$$
frac{partial y / y}{partial x} = beta_2
$$

The interpretation of (beta_2) therefore is: “A unit increase in (x) is associated with a (100(e^{beta_2}-1)) percent change in (y) (controlling for all other factors if other covariates are included in the regression)”

Leave a Reply

Your email address will not be published. Required fields are marked *