“Mancini really is magic,” proclaims The Sun newspaper, and others agree. Since Roberto Mancini replaced Mark Hughes as manager of Manchester City, the world’s richest football club, City have won four games straight. There are whispers that the flaxen-haired Italian might even win the Premier League.

Stefan Szymanski and I argue in our book Soccernomics that few club managers have any effect on their teams’ performances. Yet the cult of the manager – reminiscent of the cult of miracle-working chief executives in the business world circa 2000 – thrives. And studies show that after clubs sack managers, their results tend to perk up. So is Mancini magic or irrelevant?

Admittedly, over a single season the relationship between wages and league position is weaker: only about 70 per cent. That’s because in the short run, injuries, luck, referees’ mistakes and other chance factors intervene. But over a long period, these chance factors cancel each other out. Then the best-paying club wins. That’s why I bet on wealthy City to finish in the top four. City’s Emmanuel Adebayor earns about twice as much as the best-paid player at his old club, Arsenal.

Some people have objected that clubs with the best-paid players also usually have the best-paid managers. However, whereas players are paid almost solely for their contribution to results, managers are paid more as marketing men. The manager is the club’s spokesman and talisman. He must look good, speak appropriately, and boast a glittering playing career. Mancini does. Yet a manager’s looks and performance in press conferences does not help his team win. Nor does his playing record. When Stefan analysed more than 100 managers in England from 1974 to 1994, he found no correlation between their playing and managerial success. In short, managers are not primarily paid for their contribution to results.

Nonetheless, when a club changes managers, results usually improve. Sue Bridgewater of Warwick Business School analysed sackings in the premier league from 1992 to 2008 and found: “There is a boost for a short honeymoon period.”

It’s easy to explain why. A typical club earns on average 1.3 points a match. Usually, a club sacks its manager when it’s averaging only one point a match – that is, at a low point in the cycle. Any statistician can predict what should happen after a low point: whether or not the club sacks its manager, or changes its brand of teacakes, its performance will probably regress to the mean. Simply put, from a low point you are always likely to improve. The club may have hit that low point owing to bad luck, or injuries, or a tough run of fixtures or – as perhaps in City’s case – the time it takes for a largely new team to gel. Whatever the reason for hitting a low, things will almost inevitably improve afterwards. The new manager rarely causes the pendulum to swing. He’s just the beneficiary of the swing.

Eventually results regress to the mean. Prof Bridgewater found that three months after a sacking, the typical club was averaging the standard 1.3 points a game. Sheikh Mansour, City’s billionaire owner, should have just stuck with Hughes and waited for results to rebound, but in business doing nothing is often the hardest thing.

Historians used to believe in the “Great Man Theory of History”. The idea was that great men – Genghis Khan or Napoleon – caused historical change. Historians binned the notion long ago, and now even business magazines have, but it’s sweet to see that the theory has an afterlife in football.