# How to interpret the relationship between two growth rate variables (i.e. changes between time points)?

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.

• Are you estimating the model with or without a constant term? Can you describe what it looks like?
– Emil
Commented Apr 24, 2018 at 16:34
• 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.
– Henk
Commented Apr 25, 2018 at 9:29

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.