Interpreting Regressions of Squared Stock and Index Returns

Summary

The document explains a regression in which a stock’s squared return is related to the current squared index return, the stock’s own lagged squared return, and the index’s lagged squared return. Squared returns discard direction and serve as a simple proxy for the magnitude of daily price moves, so the regression is interpreted as examining how stock volatility relates to market volatility and its own recent history.

The response expects positive lag coefficients when volatility is persistent: unusually large moves tend to be followed by large moves, while quieter periods often remain quiet. This connects the specification to volatility research and models such as GARCH, which represent time-varying volatility. The interpretation is descriptive and does not establish causation; coefficients also depend on the chosen model and data. Squared returns are a crude volatility measure, and the document does not discuss inference, alternative volatility estimators, or how to distinguish contemporaneous co-movement from predictive effects of lagged variables.

Key ideas

• Squared returns measure return magnitude without retaining whether the move was positive or negative.
• The regression relates stock return magnitude to current market volatility and lagged stock and market volatility.
• Positive lag coefficients are consistent with volatility persistence across periods.
• The specification is related to volatility studies and models such as GARCH.
• Regression associations alone do not establish causal effects.

Tags

# What does a regression of squared returns of stock on squared index returns and lags show?


# What does a regression of squared returns of stock on squared index returns and lags show?












We have a squared stock return at t regressed on 3 variables: squared index return, squared stock return at t-1, and squared index return at t-1.

My two questions would be: 1. What does this test for 2. What would positive/negative coefficients of each variable show?

Thanks!

## Answer by Alex C (score 0, accepted)

https://quant.stackexchange.com/a/50157

Squared return is basically a measure (the simplest measure) of volatility, it shows how big the day's return is without looking at whether it is positive or negative. Periods of time with big squared returns are volatile periods (The Great Recession of 2008-2009 for example).

So the regression you propose basically shows how volatility of a stock today is affected by today's index volatility as well as yesterday's volatility in the index and in the stock. The lag coefficients will presumably be positive, showing that volatility movements are persistent from one day to the next. Big vol yesterday predicts big vol today and vice versa low vol yesterday is usually followed by low vol today (with exceptions of course, surprises do happen).

These kinds of regressions are often performed in volatility studies. They are also related to volatility models like GARCH and so on that try to track volatility over time. This kind of research has shown that market volatility is not constant but changes in predictable ways.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.