Six Month Lag

Inflation, finance, economics.

Core CPI report for Jan 2025, released Feb 12, 2025

CPI data reweighted by Jason Furman

Core CPI increased 0.4% in the month of January, with headline CPI increasing 0.5%. Over the past year all items CPI increased by 3.3%. Bloomberg reports that Federal Reserve Chairman Powell state, “I would say we’re close, but not there on inflation.”

Over at Bluesky and X, Jason Furman presents the CPI report, with prices reweighted to track the PCE: see the image above. He notes that his weights have been overestimating the PCE figures lately, so perhaps the PCE-equivalent adjustment should be moderated. The one month figure is too noisy to pay attention to. The one year figure is dated. My appendix shows that the 6 month figure is the sweet spot for Core CPI as well. With 6 month inflation at 3.4% and rising, we are a ways from the Fed’s long run 2.0% target.

If you must forecast, forecast often. That said, based on this report I would expect a shift to more restrictive language in the Fed’s statement during the March 18-19 meeting barring a very dramatic shift in the data between now and then.


Core PCE inflation report for December 2024, released on January 31, 2025

Core PCE prices rose 2.8% over the preceding year, the same as last month’s report. The six month annualized figure was somewhat more reassuring, dropping from 2.4% to 2.3%. Given this data, we would expect centered inflation to be lower than it was 6 months ago with 85% confidence, based on historical experience and ignoring what we know about future tariff increases.

The six month annualized lagged percentage change is a better predictor of centered one year inflation than other lags. The six month perspective suggests that as of December 2024, the Fed was making steady progress towards its goal of 2% inflation. The higher 2.8% figure should be interpreted as the level of inflation in the economy as of 6 months ago. In the graph the 2 indicators show some contrast: the green 6 month lag shows a modest decline while the blue bars showing the 12 month lagged percentage change are relatively flat.

Bear in mind though that we’d expect an average divergence of 0.7% between the six month figure and the expected centered 12 month figure. Prospects of higher tariffs moving forward increase estimated uncertainty and suggest less complacency regarding future inflation. But as of December 2024 at least, inflation was declining towards its long run target.


Review of Core PCE report for November, Released on Dec 20, 2024, Part II

Here is a more proper update of the graph in my paper, reflecting the inflation data as it was initially reported at the time, relative to the revised centered one year percentage change in prices represented by the thick blue line.

The data starts in March 2019, because there were delays in reporting core PCE prices due to a US government shutdown in Dec 22, 2018 to Jan 25, 2019. The updated graph reflects similar data on the right of the chart, with 6 month annualized inflation being slightly lower than inflation measured over the previous year, and inflation generally easing over the previous six months.


Review of Core PCE report for November, Released on Dec 20, 2024

With the next PCE report coming on Jan 31, 2025, I thought I would review the report released last month. The above graph is a little different than the last post as it only uses the latest revised data and doesn’t bother to dig out inflation rates as they were originally reported. Comparison of the red, green, and inchworm lines, as well as the purple probability line, with the last post shows that the effects of the approximation are more than trivial.

One year core PCE inflation came in at 2.8%: inflation over the past year gives us an idea of general price pressures as of 6 months ago, halfway through the time span. The annualized six month lag provides the best estimate of centered inflation as of November 2024: that was 2.4%, a little lower.

Price pressures eased: given historical experience we’d expect November’s centered inflation to come in lower than centered inflation in May (6 months prior) about 80% of the time.

The Federal Reserve has a long term target of 2.0% inflation: the latest report indicates steady progress towards that goal as of November 2024.

ETA Jan 26, 2025: See the next post for a much better graph.


One month centered inflation vs annualized 3, 6, and 12 month lagged inflation

Centered 12 month inflation and 3, 6, and 12 month lagged inflation. Also the probability that inflation rose over the past 6 month. 2019-2024.

I showed a chart plotting errors for various annualized lagged percentage changes in a previous post. Here we can get an idea of what they look like. This chart from my paper shows annualized lagged percentage changes relative to underlying contemporaneous inflation as measure by the one year centered percentage change shown in the thick blue line. The red dashed line shows three month lagged percentage changes: they move around too much to provide insight into the underlying trend. The one year lagged percentage change, which is generally reported as the contemporaneous inflation rate, provides a somewhat outdated perspective (six months outdated) to existing underlying inflation. The sweet spot is in the six or seven month range: the thin green line shows the annualized six month percentage change. It fits the thick blue line visibly better than the 2 alternatives shown.

The bottom half of the chart indicates inflation’s direction, whether the centered thick blue line is rising or falling over the previous 6 months, something which is only known retrospectively. Formally, it is the probability that inflation is rising: when it is above say 80% it predicts rising inflation and when it’s below say 20% it predicts falling inflation. The probabilities are based on the price data alone. Such lack of sophistication is helpful to the extent that at least the forecast process (or rather the nowcast process) can be readily understood, unlike models where there are many contributing variables.

The data goes through July 2024 and reflects the late August report. I hope to post something more up to date soon.


Annualization

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There are a number of ways of taking a one month or six month percentage change and stating it in annual terms, for apples-to-apples comparisons. The simplest method is to just multiply by 12 in the case of one month or 2 in the case of the six month percentage change. I call that simple annualization, though it could also be called multiplicative annualization. It does not reflect compounding: twelve consecutive monthly percentage changes of 1% will result in a percentage change of 12.68% over an entire year, somewhat over 12.00%.


To take compounding into account, use a formula like this for monthly percentage changes:

\( Geometrically \, Annualized \, Inflation = \left [ \left ( \frac{P_m}{P_{m-1}} \right )^{12}-1 \right ] *100 \)


A similar formula could be used to annualized 6 month percentage changes:

\( Geometrically \, Annualized \, Inflation = \left [ \left ( \frac{P_m}{P_{m-6}} \right )^{2}-1 \right ] *100 \)

Above, the current month’s prices are divided by prices 6 months ago. To inflate them, they are taken to the power of 2 (since there are 2 six-month periods in a year). Further discussion and a more general formula are presented in the paper. Simple annualization isn’t an awful approximation for geometric annualization, but there’s no reason not to insert the proper formula into your spreadsheet.

 


Estimating one year inflation centered on the current month rather than six months prior

Media discussions of inflation releases like this one or this one highlight one year and one month percentage changes in prices. The one year figure is stale: half of it reflects economic developments that are over six months old. The one month figure is up to date but too volatile to be useful.

What we would like is a one year percentage change centered over the current period, but naturally that won’t be available for another six months. However, we can estimate that figure with annualized percentage changes taken over various lags. The best estimate for one year centered inflation turns out to be six or seven months long. Here’s the first chart measuring the errors for various lags: we want errors to be low:

For both 1959-2023 and a subset of that era when inflation fluctuated a lot, annualized six month lagged percentage changes approximated one year centered inflation best. One month inflation annualized inflation produces errors about 2 and a half times as high. The one year lagged percentage change, emphasized in most inflation reports, has higher errors than any lag between 4 and 11 months.

Memo to reporters: calculate the annualized 6 month lagged percentage change! And download my paper! The next post will discuss annualization in greater detail and provide a formula.

 


First post: six month lag

Glasses on Calendar

Greetings and welcome to Six Month Lag, a blog focusing on inflation, finance, and economics. I started the blog to promote my paper, Forecasting the Present: Optimal Lags for Contemporaneous Core PCE Inflation. The optimal lag for nowcasting core PCE inflation turned out to be 6 or 7 months at the time of this post. I make no promises regarding what future research will show for other time series. That said, the internet appendix to my paper shows that the six month annualized percentage change has lowest errors relative to centered one year inflation for headline CPI, core CPI, and unadjusted PCE inflation as well. So I am hopeful that I won’t have to change my website name.

Here’s a summary of the paper:

Percentage changes bracketing a given date gauge inflation best but can only be measured retrospectively. Geometrically annualized percentage changes over the previous 6 or 7 months provide the best timely approximation of underlying core PCE inflation according to a range of criteria. Decimal lags permit the optima to be stated with greater precision. Probabilistic modeling of inflation’s level and direction can be conducted using estimates of the error’s variance and kurtosis. A review of Federal Reserve statements during the COVID and post-COVID era suggests that greater focus on the geometrically annualized 6 month percentage change could have provided earlier warning of incipient inflation.

Future posts will march through some of the points of the paper in a conversational style. I also plan occasional commentary on monthly inflation data releases.