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Emil
Emil
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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 $y_t = \alpha + \beta x_t + \varepsilon_t$, then the interpretation here 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 "if we observe a"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.

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 $y_t = \alpha + \beta x_t + \varepsilon_t$, then the interpretation here 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 "if we observe 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.

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.

Source Link
Emil
Emil
  • 1.1k
  • 1
  • 8
  • 12

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 $y_t = \alpha + \beta x_t + \varepsilon_t$, then the interpretation here 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 "if we observe 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.