WASHINGTON (Reuters) - Chicago Federal Reserve President Austan Goolsbee said on Friday continued high consumer price index readings were concerning, but he remains focused on how the Fed's targeted personal consumption price expenditures index behaves.

"We've had multiple inflation readings that were higher than we wanted" for the CPI, Goolsbee said on Fox Business, but PCE "is the better measure...If we start getting better readings that show us that arc of inflation coming down...that will make us feel a lot better about where we are...If PCE is reinflating - we will stabilize prices."

"One month is no months," Goolsbee said, reflecting Fed policymakers' reluctance to put too much weight on a single bad data point.

But the higher-than-expected CPI readings seen in January, February and March amount to "real months" of bad data that the Fed will now have to parse in deciding whether progress towards lower inflation has stalled or will continue.

The central bank sets its 2% inflation target using the separate PCE price index, which is based on the same consumption data used to calculate gross domestic product, and therefore weights things like housing and healthcare costs differently than the CPI, which is based on a survey of consumer spending on a defined basket of goods and services.

It tends to be lower than the CPI, and even after the jump in March consumer prices, analysts this week said it was still possible PCE inflation might show a small drop for the month when the next round of data is released on April 26.

Goolsbee did not detail his policy views or predict what the Fed might do in coming meetings after a week in which investors pounced on evidence of persistent inflation to push back their own expectations for rate cuts.

But his comments do show the influence coming data releases will have on Fed policy, even among those who tended to be more optimistic that inflation will continue falling.

Goolsbee repeated for example that he is closely watching housing costs, which have defied policymaker expectations that an easing in shelter inflation was imminent.

Shelter and rising fuel prices accounted for much of the higher-than-expected consumer inflation in March, and Goolsbee said the Fed's job of restoring price stability would be tough unless housing costs behave close to how they did before the pandemic.

"The most important number to be watching on the inflation front here in the immediate term is what is happening with housing," Goolsbee said. "If that doesn't go down to something like it was pre-COVID we will have a hard time getting the overall back to target."

Shelter costs account for about a third of the CPI and were rising around 3.2% annually in the years before the pandemic; the figure in March was 5.7%.

The Fed next meets on April 30-May 1 and is now all but certain to keep the policy interest rate steady in the current 5.25% to 5.5% range.