Here is a simple way to think about the correlations between nominal interest rates and inflation rates.
We need to distinguish between three "Fisher curves": the long run; the medium run; and the short run Fisher curve.
We need to distinguish between: the actual inflation rate; the central bank's inflation target; and what people expect the central bank's inflation target to be (the "expected inflation target").
Start with the (green) Long Run Fisher curve. It shows a positive correlation between nominal interest rate and inflation. This is what you observe when you vary the inflation target, and vary the expected inflation target, so that the two are always equal. It has a slope of one (assuming super-neutrality of money). If the inflation target were 0%, the nominal interest rate would have to equal the (real) natural rate of interest. Fluctuations in the natural rate would cause the LRFC to shift horizontally back and forth (not shown). If we look at long run or cross-country data, where inflation targets differ a lot, we should observe the LRFC.
The (blue) Medium Run Fisher curve shows what happens when we hold the inflation target fixed, and assume the expected inflation target equals the actual inflation target. We observe zero correlation between nominal interest rate and inflation. This is an immediate consequence of rational expectations on the part of the central bank, and the implied orthogonality of the central bank's inflation forecast errors with its information set (that information set presumably includes the nominal interest rate). Canada, over the last years of targeting 2% inflation, should look like the MRFC.
The (red) Short Run Fisher Curve shows what happens when we hold the expected inflation target fixed, and vary the actual inflation target. We observe a negative correlation between nominal interest rates and inflation. For example, if the central bank has a lower inflation target than people expect, it would need to set a higher nominal interest rate, to make real interest rates exceed the natural rate, to ensure that actual inflation is below what people expect. When we have a transition from one inflation target to the next, and if the expected inflation target does not immediately adjust, we would observe the SRFC.
For example: if the Bank of Canada cut the inflation target from 2% to 1% the blue MRFC would shift vertically down. If people immediately adjusted their expectations of the inflation target, the red SRFC would shift left, to intersect the blue MRFC on the green LRFC. Nominal interest rates would fall by 1%. But if the expected inflation target stayed constant in the short run, the red SRFC would not shift, and nominal interest rates would rise, to the point where the MRFC and SRFC intersect.
Merry Christmas everyone!