Yawn. I want to write "Correlation does not indicate causation" across the top of this news article and give if back to the journalist with an F. I'm not even a finance guy and there are four five reasons off the top of my head that this effect could appear in the data even ignoring the hypothesized cause.
1. CEOs are largely paid in warrants. When they're getting paid a lot its because the stock price is high, largely due to ordinary variation. Reversion towards the mean/the markets' search for an equilibrium price naturally puts the brakes on performance in such situations. The authors say that their effect is stronger the more incentive pay CEOs receive, which fits neatly into this notion.
2. Ceteris paribus, a higher-paid CEO is likely to be a newer CEO since market wages are increasing faster than wages at any one post. I downloaded the paper - they don't control for this.
3. Ceteris paribus, a higher-paid CEO is likely to be at a firm that needs a CEO more desperately. The CEO post is often the last job a motivated, successful person will ever hold, even if they do well and the company runs into headwinds that aren't their fault (or the company does well but the price is weak, the firm gets bought out, and John or Jane Q. CEO is suddenly redundant). So if you want to lure real leadership to a firm that's in the pits, pony up, and if you see a firm paying higher than its peers, it may indicate distress for the firm.
4. Agency theory when applied to this situation allows us to see what happens when there are conflicts of interest among parties and there are incomplete contracts such that parties can try and extract rents from one another. An easily cowed board of directors, or one that's too sympathetic to the opinions of management, will be much less willing to question the CEO's appointee's estimate of how much the CEO should be paid. But it may be the board that's the reason the firm is underperforming.
5. CEO pay is procyclical, with salaries soaring during booms and dropping back during busts. But forward stock price performance is higher the lower P/E is, and so we could naturally expect that stock prices do their best just as CEO pay reaches its lowest point (or is increasing the slowest, in the case of a mild bust). The authors' analysis won't pick that up either, since it's a cross-section.