Assume you have an equation where the dependent variable is LN(Yt/Yt-1) and the independent variable is measured as LN(Xt/Xt-1). How do I interpret a coefficient value of 0.15?

Both variables are measured as logarithms of their change between time points i.e. growth rates.

Context: I am studying the relationship between cryptocurrency growth and some explanatory variables including: stock exchange composite indices, hash rate of the cryptocurrency network, number of transactions and exchange trading volume.

  • $\begingroup$ Are you estimating the model with or without a constant term? Can you describe what it looks like? $\endgroup$
    – Emil
    Commented Apr 24, 2018 at 16:34
  • $\begingroup$ Hi @Emil, I'm estimating an ARDL model with constant term. I'm also estimating a normal OLS model with constant term. Equation: LN(Yt/Yt-1) = a + B1(LN(Xt/Xt-1) + B2(LN(Xt/Xt-1) + B3(LN(Xt/Xt-1) etc. $\endgroup$
    – Henk
    Commented Apr 25, 2018 at 9:29

1 Answer 1


If we define $y_t := \ln{\frac{Y_t}{Y_{t-1}}}$ and similarly $x_t := \ln{\frac{X_t}{X_{t-1}}}$, the interpretation of the coefficients of the OLS regression $y_t = \alpha + \beta x_t + \varepsilon_t$ is pretty much the very same in every other linear regression, since that's exactly what this is: a (simple) linear regression (if you have other variables in the model in the same fashion, it's no longer simple but it's still linear). In other words, $\beta$ is the expected change in $y_t$ for a one-unit change in $x_t$.

The difference in your case is that $y_t$ and $x_t$ are themselves a transformation (log-returns) of the original prices, so the interpretation that would address the $X_t$ and $Y_t$ is "a 1% change in $\frac{X_t}{X_{t-1}}$ translates to a $\beta$% change (increase or decrease, depending on the sign of $\beta$) in $\frac{Y_t}{Y_{t-1}}$. In other words, the coefficients become what economists call elasticities.

Two great resources to read that might help you more are this and this. My answer is in fact a watered down version of one of the answers in the first link.

A last note is that since your data consist of time series and cryptocurrencies are notoriously volatile, I suspect (emphasis on that verb) that this will create a significant presence of heteroscedasticity. Therefore, perhaps a model like the above is not the way to go, and a multivariate ARMA-GARCH approach might be more suited. Hope this helps.

  • 1
    $\begingroup$ shouldn't the interpretation be in terms of percentage points and not percents since we are talking about unit changes here? If the regression was log-log, then I agree that we would be talking about percents. $\endgroup$ Commented Nov 10, 2022 at 8:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.