Step 5:

Having estimated the linear model specification, we will next estimate the multiplicative power function specification. This specification assumes that, instead of Cases being a function of the sum of the predictor variables multiplied by a coefficient (that is estimated), Cases is the product of the predictor variables, each raised to the power of an estimated coefficient. This is easy to do, because if we take the natural logarithm of each side of the multiplicative power function model, it becomes a linear model of logs of the variables, and we can use standard linear regression to estimate the coefficients. To estimate the power function specification we must, therefore, apply a natural logarithm transformation to the dependent variable (Cases) and all the continuous (ratio or interval scaled) predictor variables. However, we do not apply the natural logarithm transformation to factor variables.

Example –

A power function model of egg sales as a function of egg prices, cereal prices, and chicken prices looks like Sale = beta0 x (EggPr)^beta1 x (CerealPr)^beta2 x (ChickenPr)^beta3 x First.Week, where beta0, beta1, beta2, and beta3 are the parameters we wish to estimate (x are multiplication). We transform this model into a form we can apply linear regression to by taking the natural logarithm of both sides, resulting in the equation log(Sales) = log(beta0) + beta1 x [log(eggPr)] + beta2 x [log(CerealPr)] + beta3 x [log(ChickenPr)] + beta4 x First.Week. This is now a linear model with target variable being log(Sales) rather than Sales, and the predictor variables being the terms inside the square brackets. When we run a multiple linear regression with these logged variables, the resulting estimated intercept is log(beta0). To find beta0, we simply take the inverse, exp(intercept) = exp(log(beta0)) = beta0. Similarly, the exponents in the power function formulation, beta1, beta2, and beta3, are the estimated coefficients.

Fit a OLS MLR to all the explanatory variables in the data set minus Week (no interaction terms). Using these same variables apply a multiplicative power function to the same variables. Compare the results of the two estimated models.

Question 17: Compare the results of the two estimated models. Question to consider when comparing the two models: Which fits the data better?, How do the estimated coefficients differ across these two models, particularly their signs and significance?, Although the magnitudes are very different between the two models, are the larger magnitudes in one model the same as the larger magnitudes in the other?, Which variables appear to have effects that are “robust” to the selection of a particular functional form (that is, remain fairly similar in sign and relative magnitude and significance in both models)?