Which CEOs drive a Rolls to work? That could be an important question for government regulators looking to crack down on corporate fraud.

It's not that executives who spend money on lavish luxury goods are more likely to commit fraud. But they may be more likely to sit at the head of a company where fraud is occurring, according to a new paper from the National Bureau of Economic Research.

The paper, authored by researchers Robert Davidson, Aiyesha Dey and Abbie Smith, argues that CEOs with a habit of conspicuous consumption -- buying expensive houses, boats or cars, say -- tend to work at companies where fraud takes place. More fraud, anyway, than you'd find at a company where the chief executive doesn't flash so much cash.

The paper's authors say that these companies tend to have a "relatively loose control environment," which stems from a certain managerial style. And while the authors stress their sample size is small, they nevertheless note that where you find big-spender CEOs, you're also likely to find reporting errors, inefficiency, bankruptcy and low investment in long-term growth.

The findings are of particular interest, since corporate fraud cases are on the rise -- and executive pay is also creeping up at many companies. The boost in executive pay is likely due to the widespread practice of peer benchmarking, wherein companies race to pay their C-suite employees more and more simply so the executives don't leave for a job elsewhere.

The NBER paper isn't the only piece of research in recent months to suggest that what CEOs do with their free time can provide insight into how they run their companies.

In March, another NBER paper, by a separate group of authors, suggested that unmarried CEOs are more likely to spend money in relatively risky areas like R&D and acquisitions, while married CEOs tend to take a safer, more conservative approach.

And in May, researchers Matthew Cain and Stephen McKeon found that CEOs with a license to pilot small aircraft -- "a risky activity," the authors note -- are also likely to make risky moves in business, suggesting that an individual executive's appetite for danger can sometimes end up jeopardizing a whole company.