Structural shocks in economic models are unobservable and country-specific? Let's consider an exogenous oil price shock ($e_t$) as an example in the following equation $oil_t = \rho \; oil_{t-1} + e_t$, where $oil_t$ is one variable in a VAR system. We normally want to recover $e_t$ based on some structural model we have for some given economy. So I guess that makes $e_t$ also country-specific, right?
So, for example, we cannot just look visually at oil-price time series data and say that oil price shocks are more volatile or less volatile. Like it depends on the economy of interest?
So although oil prices may be more or less volatile in general for a given period, it does not say anything about oil price shocks for a given economy, I guess. We need a structural model to recover the oil-price shocks to see how it behaves, right?
If $oil_t$ is the worldwide price of oil, neither $\rho$ nor $e_t$ are determined in a country specific model (unless the country's economy is so large that it affects global oil demand, which is not the case here).
Because there is no feedback from domestic variables to the worldwide oil price $e_t$ is the structural shock if AR(1) is an appropriate model of the oil price.The residual of the regression of $oil_t$ on $oil_{t-1}$  $\widehat{e_{AR(1),t}}$ is already an estimate for the structural shock $e_t$. So you can assess the volatility of the structural oil price shock $e_t$ based on $\widehat{e_{AR(1),t}}$. For an AR(1)-process it is possible to do this by looking at a plot of the original series.