Changing the unit of measurement of any independent variable, where log of the independent variable appears in the regression:

Question

BlackTom AI · Accepted Answer

The question asks about what happens to a regression model when the unit of measurement of an independent variable changes, given that the log of the independent variable appears in the regression.
First, consider a model of the form Y = β0 + β1 log(X) + ε. If you change the unit of X by a constant scaling factor k (i.e., X becomes kX), then log(kX) = log(k) + log(X).
Thus the model with the transformed predictor is Y = β0 + β1 log(kX) = β0 + β1[log(k) + log(X)] + ε = [β0 + β1 log(k)] + β1 log(X) + ε. In this rewritten form, the intercept is shifted to β0 + β1 log(k), while the slope on log(X) remains β1.
Consequently, changing the unit of X (through a multiplicative scaling that affects the log of X) affects the intercept coefficient but leaves the slope associated with log(X) unchanged.
Option analysis:
- 'affects only the intercept coefficient.': This aligns with the algebra above, since the intercept absorbs the additive log(k) term while the slope on log(X) stays the same. This reasoning is consistent with the effect of re-scaling the predictor inside a logarithm.
- If there were an option claiming the slope changes, that would be incorrect because the β1 parameter multiplies log(X), which, after the unit change, only picks up a constant shift inside the log, not a change in the coefficient multiplying log(X).
- If there were an option saying nothing changes, that would be incorrect for the same reason: the intercept must adjust to account for the additive log(k) term introduced by the unit change.
- If there were an option suggesting both intercept and slope change, that would misstate the effect of a simple multiplicative unit change inside a log-transformed predictor, since the multiplicative change translates to an additive shift inside the log function, isolating the effect to the intercept.
Overall, the correct interpretation is that changing the unit of measurement of X, with log(X) in the model, affects only the intercept coefficient and not the slope on log(X).

AS.440.606.50.SP25 M06: Assignment

Changing the unit of measurement of any independent variable, where log of the independent variable appears in the regression:

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