Why Conditional Variance and Observed Log Returns Differ in Scale

Summary

The question concerns a chart that overlays exchange-rate log differences with a conditional volatility series and observes that the latter is much smaller, despite similar-looking peaks in other examples. The response emphasizes that log returns and conditional volatility are different quantities, so their plotted values are not directly comparable as if they were the same measure.

The apparent relationship can depend on the time frame being considered. The short answer offers no specific model, diagnostic, data transformation, or plotted example, so it does not establish whether the chart is correctly constructed. In particular, it does not clarify whether the displayed volatility is a standard deviation or a variance, or whether the two series use compatible units and scaling. A careful interpretation therefore requires checking the volatility definition, return interval, and plot scale before comparing peak magnitudes.

Key ideas

• Log returns and conditional volatility represent different quantities and need not have similar plotted magnitudes.
• The time frame can affect how the relationship between returns and conditional volatility appears.
• Check whether the volatility series is variance or standard deviation before interpreting its scale.
• A chart alone cannot confirm correctness without knowing the model, units, and scaling.

Tags

# Interpretation conditional volatility plot


# Interpretation conditional volatility plot












I have plotten the log differences of exchange rates and in the same plot, I show the conditional volatility $\sigma_t^2$. The conditional volatility follows approximately the same path, but is much smaller. Is this correct? Because most figures that I have seen, is that the peaks are approximately the same size.

Thanks in advance!

## Answer by Vitomir (score 0)

https://quant.stackexchange.com/a/46014

In my understanding that truly depends on how large the time frame you consider is. Indeed, on one side you have log-returns on the other conditional volatility, which are two different concepts.

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