The Bureau of Economic Analysis reported today that U.S. real GDP grew at an annual rate of 1.3% in the first quarter of 2007, moving our recession probability index up to 16.9%. This post provides some background on how that index is constructed and what the latest move up might signify.

What sort of GDP growth do we typically see during a recession? It is easy enough to answer this question just by selecting those postwar quarters that the National Bureau of Economic Research (NBER) has determined were characterized by economic recession and summarizing the probability distribution of those quarters. A plot of this density, estimated using nonparametric kernel methods, is provided in the following figure- (figures here are similar to those in a paper I wrote with UC Riverside Professor Marcelle Chauvet, which appeared last year in Nonlinear Time Series Analysis of Business Cycles). The horizontal axis on this figure corresponds to a possible rate of GDP growth (quoted at an annual rate) for a given quarter, while the height of the curve on the vertical axis corresponds to the probability of observing GDP growth of that magnitude when the economy is in a recession. You can see from the graph that the quarters in which the NBER says that the U.S. was in a recession are often, though far from always, characterized by negative real GDP growth. Of the 45 quarters in which the NBER says the U.S. was in recession, 19 were actually characterized by at least some growth of real GDP.

One can also calculate, as in the blue curve below, the corresponding characterization of expansion quarters. Again, these usually show positive GDP growth, though 10 of the postwar quarters that are characterized by NBER as part of an expansion exhibited negative real GDP growth.

The observed data on GDP growth can be thought of as a mixture of these two distributions. Historically, about 20% of the postwar U.S. quarters are characterized as recession and 80% as expansion. If one multiplies the recession density in the first figure by 0.2, one arrives at the red curve in the figure below. Multiplying the expansion density (second figure above) by 0.8, one arrives at the blue curve in the figure below. If the two products (red and blue curves) are added together, the result is the overall density for GDP growth coming from the combined contribution of expansion and recession observations. This mixture is represented by the yellow curve in the figure below.

It is clear that if in a particular quarter one observes a very low value of GDP growth such as -6%, that suggests very strongly that the economy was in recession that quarter, because for such a value of GDP growth, the recession distribution (red curve)is the most important part of the mixture distribution (yellow curve). Likewise, a very high value such as +6% almost surely came from the contribution of expansions to the distribution. Intuitively, one would think that the ratio of the height of the recession contribution (the red curve) to the height of the mixture distribution (the yellow curve) corresponds to the probability that a quarter with that value of GDP growth would have been characterized by the NBER as being in a recession. Actually, this is not just intuitively sensible, it in fact turns out to be an exact application of Bayes’- Law. The height of the red curve measures the joint probability of observing GDP growth of a certain magnitude and the occurrence of a recession, whereas the height of the yellow curve measures the unconditional probability of observing the indicated level of GDP growth. The ratio between the two is therefore the conditional probability of a recession given an observed value of GDP growth. This ratio is plotted as the red curve in the figure below.

Such an inference strategy seems quite reasonable and robust, but unfortunately it is not particularly useful–- for most of the values one would be interested in, the implication from Bayes’- Law is that it’-s hard to say from just one quarter’-s value for GDP growth what is going on. However, there is a second feature of recessions that is extremely useful to exploit–- if the economy was in an expansion last quarter, there is a 95% chance it will continue to be in expansion this quarter, whereas if it was in a recession last quarter, there is a 75% chance the recession will persist this quarter. Thus suppose for example that we had observed -10% GDP growth last quarter, which would have convinced us that the economy was almost surely in a recession last quarter. Before we saw this quarter’-s GDP number, we would have thought in that case that there’-s a 0.75 probability of the recession continuing into the current quarter. In this situation, to use Bayes’- Law to form an inference about the current quarter given both the current and previous quarters’- GDP, we would weight the mixtures not by 0.2 and 0.8 (the unconditional probabilities of this quarter being in recession and expansion, respectively), but rather by magnitudes closer to 0.75 and 0.25 (the probabilities of being in recession this period conditional on being in recession the previous period). The ratio of the height of the resulting new red curve to the resulting new yellow curve could then be used to calculate the conditional probability of a recession in quarter t based on observations of the values of GDP for both quarters t and t – 1. Starting from a position of complete ignorance at the start of the sample, we could apply this method sequentially to each observation to form a guess about whether the economy was in a recession at each date given not just that quarter’-s GDP growth, but all the data observed up to that point.

Once can also use the same principle, which again is nothing more than Bayes’- Law, working backwards in time–- if this quarter we see GDP growth of -6%, that means we’-re very likely in a recession this quarter, and given the persistence of recessions, that raises the likelihood that a recession actually began the period before. The farther back one looks in time, the better inference one can arrive at. Seeing this quarter’-s GDP numbers helps me make a much better guess about whether the economy might have been in recession the previous quarter. We then work through the data iteratively in both directions–- start with a state of complete ignorance about the sample, work through each date to form an inference about the current quarter given all the data up to that date, and then use the final value to work backwards to form an inference about each quarter based on GDP for the entire sample.

All this has been described here as if we took the properties of recessions and expansions as determined by the NBER as given. However, another thing one can do with this approach is to calculate the probability law for observed GDP growth itself, not conditioning at all on the NBER dates. Once we’-ve done that calculation, we could infer the parameters such as how long recessions usually last and how severe they are in terms of GDP growth directly from GDP data alone, using the principle of maximum likelihood estimation. It is interesting that when we do this, we arrive at estimates of the parameters that are in fact very similar to the ones obtained using the NBER dates directly.

What’-s the point of this, if all we do is use GDP to deduce what the NBER is eventually going to tell us anyway? The issue is that the NBER typically does not make its announcements until long after the fact. For example, the most recent release from the NBER Business Cycle Dating Committee was announced to the public in July 2003. Unfortunately, what the NBER announced in July 2003 was that the recession had actually ended in November 2001–- they are telling us the situation 1-1/2 years after it has happened.

Waiting so long to make an announcement certainly has some benefits, allowing time for data to be revised and accumulating enough ex-post data to make the inference sufficiently accurate. However, my research with the algorithm sketched above suggests that it really performs quite satisfactorily if we just wait for one quarter’-s worth of additional data. Thus, for example, with the advance 2007:Q1 GDP data just released, we form an inference about whether a recession might have started in 2006:Q4. The graph below shows how well this one-quarter-delayed inference would have performed historically. Shaded areas denote the dates of NBER recessions, which were not used in any way in constructing the index. Note moreover that this series is entirely real-time in construction–- the value for any date is always based solely on information as it was reported in the advance GDP estimates available one quarter after the indicated date.

Although the sluggish GDP growth rates of the past year have produced quite an obvious move up the recession probability index, it is still far from the point at which we would conclude that a recession has likely started. At Econbrowser we will be following the procedure recommended in the research paper mentioned above–- we will not declare that a recession has begun until the probability rises above 2/3. Once it begins, we will not declare it over until the probability falls back below 1/3.

So yes, the ongoing sluggish GDP growth has come to a point where we would worry about it, but no, it’-s not at the point yet where we would say that a recession has likely begun.

[James Hamilton is professor of economics at the University of California, San Diego. The above is cross-posted from Econbrowser].

Very interesting post.

Is there any way you can make the x-axes of the density graphs scaled by logarithmic growth, instead of percentage growth?

Alex, actually I left out this detail– the axes and all the statistical analysis in fact are based on logarithmic growth. The measure is always 100 times the difference of the natural logarithms of real GDP between consecutive quarters.

ahhh… sweet. Thanks.