Comparing coefficients in a regression

[frederick lim]

Hi, I have the following linear regression model specified. These are at the consumer order level, in which the DV measures the amount of discount coupon ($) consumers desire, which is a function of product price ($), market price ($) and durability (measure in months) of the product.

DISCOUNT_COUPON = PRODUCT_PRICE + MARKET_PRICE + PRODUCT_DURABILITY + CONTROLS

The estimated coefficients of the regression model is:

DISCOUNT_COUPON = 0.10*PRODUCT_PRICE + 0.02*MARKET_Price + 0.01*PRODUCT_DURABILITY + CONTROLS

Is it valid to make the following observations:

1. Consumers are 5 times (=0.10/0.02) more sensitive to product price than to market price.
2. Consumers are 10 times (=0.1/0.01) more sensitive to product price than to durability of the product.

Overall, I would like to know if I can simply compare the magnitude of the coefficients in the same regression model for (a) both coefficients have the same unit of measure (i.e. $), and (b) both coefficients have different unit of measure (i.e. one in $ and the other in months).

Thank you.

[Clyde Schechter]

So first, there is the question of whether this model is useful. Have you checked its fit to the data?

Assuming the model is OK, I would say that the first inference is fine but the second is not, and for precisely the reason you allude to: the unit of measure.

The problem with the second inference is easily seen by considering what would happen if instead of using the prices in dollars you used them in, say, Euros, and if you measured durability in years instead of months. You would get a radically different ratio of price sensitivity to durability sensitivity.

When you are dealing with attributes that cannot be measured on a common scale, there is no way to make the kind of comparison you are trying to make in 2.

[frederick lim]

Hi Clyde, thanks for the reply and great comments. I appreciate it. The model fit is okay, so glad to know the first inference can be made. BTW, can the same inference be made for (1) if the coefficient for MARKET_PRICE is now -0.02 instead of 0.02? So we simply consider the absolute ratio.

Then, extending the first inference based on -0.02 and given 1:5 ratio, I would like to make the statement that "the company can compensate a decrease of $1 in market price with a decrease of 20 cents in its own product price". In your opinion, is this a fair statement to arrive to based on the results above?

Thanks.

[Clyde Schechter]

Well, you have to be a bit more careful in your language. You can still say that there is five times greater product price sensitivity than there is market price sensitivity--but you need to be very clear that the sensitivity (marginal effect on the discount coupon) runs in opposite directions here.

I would like to make the statement that "the company can compensate a decrease of $1 in market price with a decrease of 20 cents in its own product price". In your opinion, is this a fair statement to arrive to based on the results above?

I can't be certain, but probably not. The problem is that your language implies causality. If the data were obtained as a result of an experiment in which product price and market price were manipulated (and preferrably people were randomly assigned to different manipulations) then this statement would be justified. But if this is observational data, you really can't make this kind of causal inference. On top of that, it is conceivable that if the company changes its product price in response to a shift in market price, that might in turn set off another change in market price which then throws off everything. So for all these reasons, I think that this one is probably not supported by this data and analysis.