Start with the actual unemployment rate in (say) Canada over (say) the last 20 years (the years of targeting 2% inflation). Calculate the average unemployment rate over that 20 years. Call it "Ua".
Now imagine a policy counterfactual. Suppose that the Bank of Canada had done exactly what it did do, plus or minus a lot of random stupid things. But all those random stupid things had no effect on average inflation, or on trend inflation, over that same 20 years. Because they were mean zero random stupid things, and there were enough of them that their effects on inflation all cancelled out on average. Call the average unemployment rate in that counterfactual "Uc".
Would you expect Uc to equal Ua?
I wouldn't. I would expect Uc to be bigger than Ua. Doing random stupid things makes things like unemployment worse on average, for a given average inflation rate.
Or imagine a second policy counterfactual. Suppose the Bank of Canada had had a crystal ball. So it did a lot fewer things that seemed sensible at the time but look random and stupid from the benefit of 20/20 hindsight. Again, assume possession of the crystal ball has no effect on average inflation and on trend inflation. Call the average unemployment rate in that second counterfactual "Ub".
Would you expect Ub to equal Ua?
I wouldn't. I would expect Ub to be less than Ua. Not doing things that were random and stupid from 20/20 hindsight would make things like unemployment better on average, for a given average inflation rate.
Only in a linear model, where the actual unemployment and inflation are a linear function of deviations of monetary policy from trend, or deviations from optimal monetary policy, would Ua=Ub=Uc.
Even if you have a natural rate model, where the average monetary policy (the inflation target) has zero effect on unemployment, the average level of unemployment will depend on the variance of monetary policy around that average, and on the covariance of monetary policy with respect to other shocks hitting the economy. Unless the model is linear, and the short run Phillips Curves are not curves, but are straight lines.
That is one of the most important reasons why it matters to get monetary policy right. The booms and busts do not all cancel out on average. You can maybe define the natural rate independently of mean monetary policy. But you cannot define the natural rate independently of the variance and covariances of monetary policy. You cannot empirically estimate some policy-invariant natural rate simply by taking some sort of smoothed average trend of the actual rates. If you do that over a period when monetary policy made bigger than normal mistakes (from hindsight), like recently, and if I am right about the direction of non-linearity, you will overestimate the natural rate of unemployment.
Brad DeLong says (correctly, I think) that the New Keynesian research program included:
"3. Business cycle fluctuations in production are best analyzed from a starting point that sees them as fluctuations around the sustainable long-run trend (rather than as declines below some level of potential output)."
Funnily enough, both Old Keynesians, and Milton Friedman's "plucking model", see fluctuations more as declines below some level of potential output than as fluctuations around the sustainable long run trend. They are (mostly) fluctuations below the sustainable long run trend. The plucking model is very non-linear. Recessions can be as big as monetary policy is too tight. But there is a limit to how big booms can be, however loose monetary policy is. Recessions and booms don't average out. With an extreme version of the plucking model, where there are no booms at all, the average measures the average recession.
The data seem to support the plucking model. [Update: see Robert Waldmann's closely related post, which provides links to more evidence.] My eyeballs say it looks like that too. Unemployment meanders along, not doing much, then it suddenly jumps up, then slowly declines back towards its previous trend. The bigger the previous jump up, the bigger the eventual decline.
Even if there were no trend in inflation between now and before the recession, it is not correct to take the average unemployment rate over the last few years as an empirical estimate of the natural rate. Your sample reflects a period when the variances and covariances of monetary policy were worse than normal, and worse than we hope they will be in future.