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A Differential of Multiple Linear Regression

  1. Oct 4, 2017 #1

    fny

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    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.
     
  2. jcsd
  3. Oct 4, 2017 #2

    andrewkirk

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    In general one cannot make that transition. The second last formula is correct but the last is not, unless there is no dependence between ##X_k## and any of the other ##X_j##s. A corrected version of the last formula is:
    $$
    \frac{dY}{Y} = \sum_{j=1}^n \beta_j dX_j
    $$

    To get the total derivative wrt ##X_k## we use the total derivative formula, for the case where ##Y## is a function of ##X_1,...,X_n##:

    $$\frac{dY}{dX_k}=\sum_{j=1}^n \frac{\partial Y}{\partial X_j} \frac{dX_j}{dX_k}$$

    In this case we have ##Y = \exp\left(\beta_0 + \sum_{k=1}^j \beta_j X_j\right)## so that ##\frac{\partial Y}{\partial X_j} = \beta_jY##, and we also have ##\frac{d X_k}{d X_k}=1##, so that the total derivative becomes:
    $$\frac{dY}{dX_k}=Y\left(\beta_k + \sum_{\substack{j=1\\j\neq k}}^n \beta_j \frac{dX_j}{dX_k}\right)$$

    This reduces to the formula you wrote above for the total derivative if all the ##\frac{dX_j}{dX_k}## are zero, ie if there are no dependences between ##X_k## and any of the other ##X_j##.

    What we can say is that ##Y## increases by ##Y\beta_k\delta X_k## if ##X_k## increases by ##\delta X_k## and all other variables do not change.
     
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