price elasticity: Linear regression low r square

I faced an interview question for a job where interviewer asked me suppose your r square is very low (between 5 to 10%) for a price elasticity model.

How would you solve this question? Anything that a typical econometrics guy would do?

Edit: I have posted this question in stats.stackexchange.com forum for understanding this problem from the perspective of model diagnostics and feature engineering

• Cross-posted question, see stats.stackexchange.com/questions/277292/… – keepAlive May 3 '17 at 10:23
• @Tnerual, i have followed the instructions in this meta discussion and made it suitable for the other forum instead of copy-pasting question word by word as suggested by Scortchi in the accepted answer. Thanks – Enthusiast May 3 '17 at 10:27

I would raise two issues about the dataset to which the regression line was fitted:

The range of prices in the dataset. If the datapoints all lie within a very narrow range of prices, then even small variation (whether real or due to measurement error) in the associated quantities can lead to a low coefficient of determination $R^2$. In terms of the formula:

$$R^2 = 1 - \frac{\text{Residual sum of squares}}{\text{Total sum of squares}}$$

a narrow range of prices tends to result in little of the total sum of squares being explained and most of it being residual, resulting in a low $R^2$. Note that it is assumed here that the regression takes price as the independent variable and quantity as the dependent variable.

The number of datapoints. If the number is small, then, even within its range of prices, the dataset may happen to be unrepresentative of the distribution of points within the population of interest. This can result in the squared residuals calculated from the dataset being on average either much smaller or much larger than is representative of the population. Thus it is possible that $R^2$ has been correctly calculated from data which is not representative of the population.

• Thanks for your answer. In line with your answer, firstly, How much data is considered good enough in such scenario? And secondly how do we decide if range of prices is small or not? i mean is there any test that needs to be done or we can judge that by visually plotting that or by checking percentiles of prices? – Enthusiast May 3 '17 at 10:36
• This answer was prepared and posted in response to the question as originally posted, before I had seen the edit. – Adam Bailey May 3 '17 at 10:39
• @Enthusiast Re number of datapoints, with 3 or 4 points, the risk that they are unrepresentative is quite high, with 30-40, if randomly sampled, it's very low. But there is no clear cut-off in between. – Adam Bailey May 3 '17 at 10:52
• @Enthusiast Re range of prices, one test would be to divide the dataset in two according to whether price is in the upper of lower half of the range of prices. Then consider the means and standard deviations of the quantities within each half of the dataset. If the standard deviations are much larger than the difference in the means, then one would expect a low $R^2$. Please note both this and my comment on number of datapoints are brief suggestions and no doubt much more could be said. – Adam Bailey May 3 '17 at 11:09