Assume we decompose the daily (log) returns of a stock as beta times market movement plus an idiosyncratic part. If this is done ex-ante, what proportion of the variance is explained by the market component vs. the idiosyncratic part? I am looking more for rule of thumb/experience of others, based on, say, U.S. equities in the S&P 1500. Another way of putting this is "what is the (average/rule of thumb/expected/garden-variety) Pearson correlation coefficient between the return of a stock and the return of the market?"
An Axioma research paper from August 2011, Using Multiple Risk Models for Superior Portfolio Management… A Practice Not Just For Quants, answers exactly your question, I believe. Note the graphs at the top of page 8. They compare their medium-horizon fundamental and statistical factor models from January 2008 to January 2009. At the start of the period, they find about 30% of active risk comes from common fundamental factors (as a % of variance), with the rest accounted for by stock-specific factors. The corresponding figure for statistical factors is 40%. During the 2008-2009 crisis, this figure rose to as high as 50% for fundamental factors and close to 70% for statistical factors. Other research included in this paper suggests these numbers would have been even higher for their short-horizon risk models. I would guess that common risk proportion decreased to levels similar to early 2008 more recently, and may even be lower than that over longer horizons.
I believe that over the last 3 years or so the $R^2$ of a regression of actual returns on model returns was in the low 30s % using the medium-horizon fundamental factor model, which generally agrees with their data and implies that common risk proportion continued to decline into 2009 and 2010.
Addendum: The previous research concerned the $R^2$ for daily returns, which was the essence of your question. However, some additional research shows that the monthly $R^2$ is much higher, currently at around 60%, and has been trending higher for some time now. In 1995 the $R^2$ was consistently under 20%. In 2000 it was 20-30%, occasionally increasing to over 40% during crisis periods. Even during the relatively calm mid-2000s $R^2$ was consistently between 30% and 40%.
Many people today talk about there being a "correlation bubble" in equities, but considering how long these trends have gone on, I'd be surprised to find the $R^2$ drop to anywhere near the 1990s levels any time soon. Henceforth I think we can rely on at least 40% (30%) of monthly (daily) returns being attributable to various broad factors, possibly much higher.
BTW, besides the overall market itself, the most significant factors are industry, size, momentum, and volatility.
I just want to give a qualitative assessment to your question:
Volatility of a market is different than the volatility of a stock. Similarly like Copeland and Antikarov (2001) say that "...the volatility of a gold mine is different than the volatility of a gold..."
If you want to quantitatively compute the percentage of a stock's volatility affected by market's volatility, then I would do back calculation using options formula.
The term 'rule of thumb' is ambiguous here. Because I don't think there are any rule of thumb, you just need to do the number crunching.
However there are some stable characteristic through time linked to correlation. For instance it is a common fact that the hierarchy of correlation within different market is relatively stable. US equities are less correlated than Europe. pre-crisis it was around 40 for the us, and 60 for Europe I think. Obviously it shooted up after.
So my first answer was off base. For some reason I was thinking first moment (idiosyncratic returns), but he's looking for second moment (idiosyncratic volatility). There is a line of research on the returns to portfolios sorted on idiosyncratic volatility and I was hoping that there were descriptive statistics that said "fraction $\rho$ of stock/portfolio vol is market and $1 - \rho$ of stock/portfolio vol is idiosyncratic". But I couldn't find any stats. So I guess you have to run your own models and find out.
If you're thinking about implementing this, I would check out the literature. It may save you some time. Ang, Hodrick, Xing, and Zhang (Journal of Finance 2006) say that high ivol has low returns. But Bali and Cakici (Journal of Finance and Quantitative Analysis 2008) show that their results are do to sorting techniques and that there's really no return (pos or neg) to high ivol.
I may be off base here. But in terms of econometrics, when you calculate beta, you've assumed that the idiosyncratic portion is white noise. So the idiosyncratic portion is mean zero and normally distributed so it accounts for none of the return. So the ratio of market:idiosyncratic is 1:0.
By definition the idiosyncratic return should be white noise. Maybe you want to add another risk factor to your model and find the projection of stock returns on that factor to find the split between the market and that other factor?