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I have a log-log model as follows:

ln quality = ln price + predictor_2 + predictor_3

I ran a regression and using the coefficient values obtained, I predicted log quality values and then I plotted a curve of predicted log quality (y-axis) vs. log price (x-axis). I feel it is better to have a plot of predicted quality vs. price (instead of the log of these variables). Can I use the same coefficients obtained from the log-log model regression to predict quality using price (note: not the log of price), predictor_2 and predictor_3 values? Or should I apply some conversion formula to obtain different coefficient values for the independent variables?

Rnovice
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  • So you have a regression. $log q= \alpha \log p + \beta x_1+\gamma x_2$, and you want to use $\alpha,\beta, \gamma$ to predict $q$ (not $\log q$) as a function of $p$ (not $\log p$). You can just exponentiate both sides to obtain $q= p^\alpha e^{x_1+x_2} $. This is just a transformation, I am not sure it does anything useful. In fact, it is not useful to interpret the regression as you would normally do because of the inclusion of $e^{x_1+x_2}$ as a term – tintinthong Aug 21 '17 at 22:34
  • @tintinthong, Thanks for this. This helps. It's just that a plot of q vs. p is more intuitive to interpret than log q vs. log p. Also, shouldn't the transformed equation be >q=p^α e^βx1 e^γx2? – Rnovice Aug 22 '17 at 01:59
  • No probs. Yes you are spot on. I made a mistake there – tintinthong Aug 22 '17 at 02:08

1 Answers1

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Your true model being $$q=p^ \alpha\,e^{\beta x_1+\gamma x_2} \tag 1$$ for conveniency you made it linear using a logarithmic transform $$\log (q)= \alpha \log( p) + \beta x_1+\gamma x_2\tag 2$$ But you must take care that these two models are not the same since you minimize with respect to the parameters $\alpha,\beta,\gamma$ two different objective functions $$SSQ_1=\sum_{i=1}^n \left(p_i^ \alpha\,e^{\beta x_{1i}+\gamma x_{2i}} -q_i\right)^2$$ $$SSQ_2=\sum_{i=1}^n \left(\alpha \log (p) + \beta x_{1i}+\gamma x_{2 i}-\log(q_i)\right)^2$$ and this is not the same story.

If you want to be rigorous, since using $(2)$ you have reasonable and consistent estimates, use them as initial values for a nonlinear regression.

In fact, if the errors are small, you could demonstrate that using $(2)$ is almost equivalent to a minimization of the sum of the squares of the relative errors on $q$ while $(1)$ corresponds to the minimization of the sum of the squares of the absolute errors on $q$ (which, I suppose, is your goal).

If you do not want to use nonlinear regression, you could use what is proposed here which is easy to extend to your case (three linear equations for your three parameters). It would be a good compromise between $(1)$ and $(2)$.

Claude Leibovici
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