# 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!

• You should set up your regression after you figure out what you want to test for, not before. Dec 7, 2019 at 19:24
• @Mild_Thornberry, this is for an assignment, so if I was 100% of what I am testing for, I wouldn’t be questioning it, no? Dec 7, 2019 at 19:52
• @AlexC that is the right answer - I would convert to real from commengt
– Attack68
Dec 7, 2019 at 20:50

## 1 Answer

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.