- #1
fny
Say you have a log-level regression as follows:
$$\log Y = \beta_0 + \beta_1 X_1 + \beta_1 X_2 + \ldots + \beta_n X_n$$
We're trying come up with a meaningful interpretation for changes Y due to a change in some Xk.
If we take the partial derivative with respect to Xk. we end up with
$$\frac{dY}{Y} = \beta_k \cdot dX_k$$
which implies that if Xk. increases by 1, you expect Y to increase by 100βk percent.
Can someone walk through the calculus to get from this
$$\frac{\partial}{\partial X_k} \log{y}= \frac{\partial}{\partial X_k} (\beta_0 + \beta_1 X_1 + \beta_1 X_2 + \ldots + \beta_n X_n)$$
to this
$$\frac{dY}{Y} = \beta_k dX_k$$?
I'm particularly confused about how one transitions from a partial derivate to a total derivative.
$$\log Y = \beta_0 + \beta_1 X_1 + \beta_1 X_2 + \ldots + \beta_n X_n$$
We're trying come up with a meaningful interpretation for changes Y due to a change in some Xk.
If we take the partial derivative with respect to Xk. we end up with
$$\frac{dY}{Y} = \beta_k \cdot dX_k$$
which implies that if Xk. increases by 1, you expect Y to increase by 100βk percent.
Can someone walk through the calculus to get from this
$$\frac{\partial}{\partial X_k} \log{y}= \frac{\partial}{\partial X_k} (\beta_0 + \beta_1 X_1 + \beta_1 X_2 + \ldots + \beta_n X_n)$$
to this
$$\frac{dY}{Y} = \beta_k dX_k$$?
I'm particularly confused about how one transitions from a partial derivate to a total derivative.