Consider the following estimated regression model [math: ln⁡(yi)^=0.034−0.023xi]\widehat {\ln (y_i)}=0.034-0.023x_i where [math: ln⁡(.)]\ln (.) denotes the natural log. Which of the following is a correct interpretation?[Fill in the blank]

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BlackTom AI · Accepted Answer

Let’s parse the given regression model carefully and translate it into an interpretable statement. The model is ln(y_i) = 0.034 − 0.023 x_i, where ln(.) is the natural log. This is a log-linear model, meaning the dependent variable is in logs while the independent variable x is in levels.

Option a: A 1 unit decrease in x is associated with a 2.3 percent increase in the expected value of y. This matches the standard interpretation: the coefficient −0.023 on x implies that for a one-unit increase in x, ln(y) decreases by 0.023, which approximates a 2.3% decrease in y when you exponentiate back to the level of y. Conversely, a one-unit decrease in x would increase y by about 2.3%. Since the exact percent change is approximately 100*(exp(-0.023) − 1) ≈ −2.27% for an increase, the corresponding decrease yields about +2.27%. So the direction and magnitude are consistent with the log-linear interpretation.

Option b: A 1 percent decrease in x is associated with a 0.023 unit increase in the expected value of y. This misinterprets both the unit of x (it is not measured in percent here) and the scale of y (the model uses ln(y), not y directly, and the coefficient does not translate to a 0.023 unit increase in y). Therefore this is incorrect.

Option c: A 1 percent increase in x is associated with a 2.3 unit increase in the expected value of y. Again, the percent change in x is not the same as the unit change used to estimate ln(y). Also, the 2.3 unit increase in y is not the proper interpretation since the model does not directly estimate y in level units, but ln(y). This statement confuses the logarithmic transformation with equal-magnitude level changes.

Option d: A 1 unit increase in x is associated with a 0.00023 percent decrease in the expected value of y. The magnitude here is far too small and the direction is opposite of what the coefficient would imply for a unit increase in x. The correct expansion would yield a roughly 2.3% change (in the opposite direction), not a 0.00023% change, so this is incorrect.

Consider the following estimated regression model [math: ln⁡(yi)^=0.034−0.023xi]\widehat {\ln (y_i)}=0.034-0.023x_i where [math: ln⁡(.)]\ln (.) denotes the natural log. Which of the following is a correct interpretation?[Fill in the blank]

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