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Consider the two regression equations: \begin{align} Y &= \alpha+\beta X+\varepsilon \\ \log Y &= \gamma+\delta X+\eta \end{align} Is there a "simple" approximated relation between the coefficient estimated using the first equation ($\hat{\beta}$) and the one got by the second ($\hat{\delta}$) that can be used to get a sense of how approximately $\hat{\delta}$ looks like without running the regression? e.g. something like $\hat{\beta} \sim \hat{\delta}/\bar{Y}$, at least for some range of values of the variables or some hypothesis on the true DGP.

It was suggested to me by a fellow student, I cannot see why this should hold but I was not able to disprove it neither.

EDIT: I found this other related question Using results of regression on raw observation values to approximate results of regression on relative change between observations (Simple, Linear), where the asker in the similar case in which the second model is a log-log: $$ \log Y=\gamma+\delta \log X+\eta $$ proposes the formula: $$ \delta=\beta\times \frac{x_T}{\bar{y}} $$ where I guess that $T$ is the last observation (it is a time series example). There is a similar answer to the one of Ami Tavory, but not much discussion too.

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at least for some range of values of the variables

Say you start with

$$ Y=\alpha+\beta X+\varepsilon $$

and

$$ \tag{1} \log Y =\gamma+\delta X+\eta. $$

Taking the exponent of both sides of (1), we get

$$ \tag{2} Y = e^{\log Y} = e^{\gamma+\delta X+\eta}. $$

For small values of $y$, we can use the Taylor approximation of $e^x$ on the RHS of (2):

$$ \tag{3} Y \simeq 1 + \gamma + \delta X + \eta. $$

So, if $Y$ takes small enough values in all of your dataset, you can apprximate $\gamma \sim 1 - \alpha$ and $\delta \sim \beta$.

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  • $\begingroup$ Last sentence $1+\gamma \approx \alpha$, isn't it? And a capital $Y$ in eq. (3)? $\endgroup$ – Richard Hardy Nov 1 '16 at 11:59
  • $\begingroup$ Also, you needn't restrict yourself to $x \approx 0$: you could take the Taylor series around any arbitrary value of x and truncate to only the 0th and 1st order terms in order to obtain a linear approximation. $\endgroup$ – stuart10 Nov 1 '16 at 15:42
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    $\begingroup$ I don't think (3) is correct - it should be $ Y \approx e^{\gamma +\eta} + \delta e^{\gamma +\eta} X + \ldots$ . This reveals a problem with this approximation - the shift from assumed additive noise to assumed multiplicative noise. This can be dealt with e.g. if we assume that the noise is Gaussian and $\eta$ is small wrt $\gamma$. $\endgroup$ – stuart10 Nov 1 '16 at 15:48
  • $\begingroup$ @Ami Tavory thanks for the answer, you mean for values of y near 1 not 0, right? the exponential approx works when the exponent is small, so when the log is small, that is y near 1. $\endgroup$ – Matteo Nov 3 '16 at 11:28
  • $\begingroup$ @stuart10 could you elaborate a bit on your comment? why the approximation does not work when there is a stochastic term and how Gaussian noise can fix it? $\endgroup$ – Matteo Nov 3 '16 at 11:29

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