Misspecified model: a variable grows at a constant rate over time

sales_t = b₀ + b₁t + e_t

sales_t: quarterly sales
b₀: intercept term
b₁: slope
t: time variable (quarter number)
e_t: random error

The equation above might be misspecified; especially, actual sales have been increasing at a fairly constant rate over time.

Which of the following data transformations should be applied to the dependent variable in the equation above to best address the concern on the misspecification?

A. Lagged transformation.
B. Logarithmic transformation.
C. First difference transformation.

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Answer: B

A logarithmic transformation of the dependent variable is the most appropriate transformation to apply when the variable grows at a constant rate over time:

ln(sales_t) = a^* + b^*t + e_t

The slope of this equation (b^*) equals the nominal constant rate. The effective rate equals e^b*-1.

ln(sales_t) = a^* + b^*t + e_t
ln(sales_t-1) = a^* + b^*(t-1) + e_t

ln(sales_t) - ln(sales_t-1) = b^*
ln((sales_t)/(sales_t-1)) = ln(e^b*)

(sales_t)/(sales_t-1) = e^b*
(sales_t)/(sales_t-1) - 1 = e^b* - 1: effective rate