As far back as last December's post-FOMC press conference, Yellen was asked just this question, to which she replied, "I think it’s a myth that expansions die of old age." But nine months later, with the policy rate no greater now than it was then, it's worth revisiting this issue, particularly ahead of this week's FOMC meeting.

Yellen's views are likely influenced by the work of San Francisco Federal Reserve Research Director Glenn Rudebusch, who recently published an

*Economic Letter*on this topic, updating his research with Francis Diebold from 1990. In the update, Rudebusch concludes, "a long recovery appears no more likely to end than a short one." In more technical terms, he explains, "a statistical test cannot reject the hypothesis that ... the probability of a recession in any month is independent of the age of the recovery." But is the situation really that straightforward?
In this post, I'll consider Rudebusch's conclusion in light of the small sample size available. And as a check, I'll apply a related approach to see whether it produces the same results. I'll conclude by considering the implications for the upcoming FOMC meeting and for the financial markets more generally.

**Rudebusch's approach**

The essence of Rudebusch's approach is to compare the histogram of the durations of observed economic recoveries to two hypothetical probability densities:

- the probability density that one would expect to observe for recovery durations if recession risk were
*independent*of the age of the recovery - the probability density that one would expect to observe for recovery durations if recession risk were
*dependent*on the age of the recovery.

*Which of these two densities more closely matches the actual data?*It turns out that with only eleven observations, the answer depends on the way we construct our histogram. The next graph shows a histogram with four bins, giving the impression that recession risk does not depend on a recovery's age.

The next graph is produced using the same data but with five bins rather than four. In this case, we get the impression that the data more closely matches the red probability density -- ie,, it appears as if recession risk

*does*increase with the age of the recovery.
Of course, econometricians don't reach conclusions by eyeballing histograms, and Rudebusch bases his analysis on a specific nonparametric test statistic used to differentiate between these probability distributions. Based on this statistic, Rudebusch finds that he is unable to reject his default assumption that the probability of entering a recession is independent of the age of the expansion.

Rudebusch's test statistic has two quite useful properties. First, it doesn't depend on any particular assumption about the nature of the dependence between the age of the recovery and the probability of entering a recession. For example, the intensity of recession risk could increase linearly with the expansion's duration, or it could increase logarithimically with the expansion's duration. Or it could follow some other pattern. Second, this test statistic doesn't require us to have a good estimate of the expected duration of the recovery.

However, these two benefits come with a cost -- namely, that this test statistic isn't such a reliable discriminator between our null and alternative hypotheses (ie, between the red and blue lines above). In statistical terms, we say the

*power*of the test is low.
For example, consider this next histogram, showing durations of simulated expansions under the explicit assumption that recession risk increases with the age of the expansion.

Because it's humped rather than monotonically decreasing, I'd be inclined to conclude that this histogram more closely matches the red probability density in the first graph above. And of course that's to be expected, because we simulated this data from a model in which recession risk increases with the age of the recovery.

But the value of Rudebusch's test statistic in this case, at 0.153, is only in the 83rd percentile of the test statistic density for a sample size equal to eleven. (I simulated the test statistic from our simulated expansion data.). In other words, this test statistic is telling us that the probability that this simulated data displays duration-dependence is only 0.83. That probability sounds meaningful -- and it is, in the general sense of the word 'meaningful'. But statisticians have been trained to use cut-off levels of 0.90 or even 0.95 for rejecting null hypotheses. And someone using a critical rejection probability that high would conclude that the data fails to reject the null hypothesis -- ie, he would conclude that the data does not display age-dependent recession risk.

In his recent Economic Letter, Rudebusch doesn't mention the critical threshold he uses to draw his inference regarding recession risk. But the value of the test statistic I calculate using post-war expansion data is 0.175, putting it in the 90.6 percentile of the simulated test statistic distribution. In other words, if recession risk really were independent of the age of an expansion, the probability of observing this test statistic with a sample of eleven observed expansions, would be only 0.094. As a result, it appears considerably more likely to me that the data came from a probability density like the red one above -- ie, that the risk of entering a recession does depend on the age of the expansion.

**A parametric test of duration-dependence**
Rudebusch's nonparametric approach avoids making a specific assumption about the functional form that recession risk intensity might take. But if we're willing to make an assumption about this dependence, we can use more powerful test statistics to differentiate between our two hypotheses of age-dependent and age-independent recession risks.

In our case, we'll follow the example of other research done in this area and assume that the intensity of recession risk increases linearly over time. (See, for example,

In our case, we'll follow the example of other research done in this area and assume that the intensity of recession risk increases linearly over time. (See, for example,

*The Duration of Business Cycle Expansions and Contractions: Are There Change-Points in Duration Dependence?*__by Vitor Castro)__. In that case, expansion durations come from the Weibull distribution, which has a probability density with the following functional form (from wikipedia):
Here, x is the duration of the recovery; lambda is a scale parameter determining the length of the typical duration; and kappa is the parameter determining the rate at which the intensity of recession risk increases over time. If kappa has a value of 1, then there is no change in the intensity of recession risk with the age of the recovery. If kappa has a value greater than 1, then the risk intensity of a recession increases with the age of the recovery --ie, recoveries do die of old age.

As a starting point, we can choose lambda and kappa so that the expected value and standard deviation of this probability density match the sample mean (58.4 months) and sample standard deviation (35 months) of the eleven post-war recoveries in our sample.

In this case, we obtain a value of lambda of 65.5 and a value of kappa of 1.72. So already, in our effort to match the mean and standard deviation of the theoretical and empirical distributions, we get an estimate of lambda that is well above 1 -- consistent with recession risk being dependent on the age of the recovery.

A more precise way to determine these parameter values is to use maximum likelihood estimation. In particular, given our sample of eleven recoveries, the maximum likelihood estimate of lambda is 65.9, and the maximum likelihood estimate of kappa is 1.84.

An estimate of 1.84 suggests the actual value of kappa is probably well above the level of 1 -- consistent with the conclusion that recession risk does depend on the age of a recovery. But it also could be the case that our estimate of kappa is simply too imprecise to reach a conclusion, particularly given our limited amount of data.

To address this issue, I simulate a sample of estimates from a sample of recovery durations, all produced under the assumption that the value of kappa is 1 -- ie, that recessions don't die of old age. (In these simulations, the value of lambda is set equal to the maximum likelihood estimate of lambda given the actual data, under the assumption that the value of kappa is equal to 1.) This allows us to ask the question:

As it happens, under the maintained assumption that kappa is equal to 1, the probability of obtaining an estimate of 1.84 from a sample size of 11 is 0.046. In other words the probability that we'd obtain an estimate for kappa of 1.84 in this case is just under 5%. We can reject the assumption that recession risk is age-independent in favor of the hypothesis that recession risk is age-dependent, with more than a 95% level of confidence that we're reaching the correct conclusion.

Applying Rudebusch's nonparametric approach to the eleven post-war economic recoveries, we concluded that recession risk does depend on the age of the recovery, with a 90.6% degree of confidence. By making an assumption that the intensity of recession risk increases linearly with the age of the recovery, we were able to reach the same conclusions with a 95.4% degree of confidence.

But the point here is that even with a limited amount of data, it seems more likely that recoveries die of old age than that they never grow old. And this result seems fairly robust to whether one takes a parametric approach or a nonparametric approach.

First, it's possible he's applying a relatively strict cut-off value for rejecting his null hypothesis that recessions don't die of old age. For example, if he insists on being 95% confident in rejecting the null hypothesis, then even a test statistic with a probability value of 0.094 will cause him to accept the null hypothesis that recessions don't die of old age. But in this case, we might ask whether age-independence is the more appropriate hypothesis to designate as the default assumption. And we might ask whether such a strict cut-off value is really warranted in this situation.

Second, the approach taken here differs in two other respects from the approach reported by Rudebusch. First, he uses a truncated probability density to reflect the view that the NBER business cycle dating committee will not identify a business cycle peak only a few months after a trough in the cycle. This appears to be standard practice in the literature, though I don't see an especially strong motivation for doing this when I read the

At any rate, it's clear there is room for further analysis into this issue, and I'll be happy to post further results as and when I consider the effects of truncation and the inclusion of the current recovery in the data set.

If recession risk is independent of the age of the recovery, the next recession would be expected in just under five years. But if recession risk does depend on the age of the recovery, as suggested by our analysis of post-war recoveries, we'd expect the next recession in just under two years. (The expected value of the duration of a recovery conditional on the recovery already lasing 87 months, is 110 months.)

Various FOMC members have discussed the importance of being able to lower the policy rate sufficiently in response to the next recession. But in order to lower the policy rate sufficiently, FOMC members first need to raise it. For example, if the FOMC raised the IOER by 25 bp at every meeting with a scheduled press conference (ie, four meetings per year), starting with this week's meeting, the IOER would be 2.5% by the time the next recession was expected -- assuming recession risk depends on the age of the recovery. And given the current low likelihood of rate hikes at both the September and December meetings this year, a more reasonable expectation for the policy rate would be 2.25% -- a level that doesn't give the FOMC much room to maneuver with the onset of the next recession.

On the other hand, FOMC members clearly don't want to increase the IOER so quickly that they trigger a recession. Their task then is to navigate between the Scylla and Charybdis -- neither hiking so quickly that they catalyze the next recession nor hiking so slowly that the get caught with insufficient policy ammunition when the next recession arrives.

Of course, Yellen has told us that she believes recession risk is independent of the age of a recovery. And to the extent that view is more generally shared among her colleagues on the FOMC, they may be inclined to move more slowly than four 25 bp hikes per year -- though the most recent set of interest rate projections offered by the FOMC is consistent with four such hikes per year.

But on the other hand, given the desire for the IOER to be at least 3% by the time of the next recession, as per David Reifschneider's paper,

As a starting point, we can choose lambda and kappa so that the expected value and standard deviation of this probability density match the sample mean (58.4 months) and sample standard deviation (35 months) of the eleven post-war recoveries in our sample.

In this case, we obtain a value of lambda of 65.5 and a value of kappa of 1.72. So already, in our effort to match the mean and standard deviation of the theoretical and empirical distributions, we get an estimate of lambda that is well above 1 -- consistent with recession risk being dependent on the age of the recovery.

A more precise way to determine these parameter values is to use maximum likelihood estimation. In particular, given our sample of eleven recoveries, the maximum likelihood estimate of lambda is 65.9, and the maximum likelihood estimate of kappa is 1.84.

An estimate of 1.84 suggests the actual value of kappa is probably well above the level of 1 -- consistent with the conclusion that recession risk does depend on the age of a recovery. But it also could be the case that our estimate of kappa is simply too imprecise to reach a conclusion, particularly given our limited amount of data.

To address this issue, I simulate a sample of estimates from a sample of recovery durations, all produced under the assumption that the value of kappa is 1 -- ie, that recessions don't die of old age. (In these simulations, the value of lambda is set equal to the maximum likelihood estimate of lambda given the actual data, under the assumption that the value of kappa is equal to 1.) This allows us to ask the question:

*Assuming recessions don't die of old age, how likely is that we'd observe an estimate for kappa of 1.84?*As it happens, under the maintained assumption that kappa is equal to 1, the probability of obtaining an estimate of 1.84 from a sample size of 11 is 0.046. In other words the probability that we'd obtain an estimate for kappa of 1.84 in this case is just under 5%. We can reject the assumption that recession risk is age-independent in favor of the hypothesis that recession risk is age-dependent, with more than a 95% level of confidence that we're reaching the correct conclusion.

**Comparison of the two sets of results**Applying Rudebusch's nonparametric approach to the eleven post-war economic recoveries, we concluded that recession risk does depend on the age of the recovery, with a 90.6% degree of confidence. By making an assumption that the intensity of recession risk increases linearly with the age of the recovery, we were able to reach the same conclusions with a 95.4% degree of confidence.

But the point here is that even with a limited amount of data, it seems more likely that recoveries die of old age than that they never grow old. And this result seems fairly robust to whether one takes a parametric approach or a nonparametric approach.

*Why might Rudebusch conclude otherwise?*I can think of two reasons.First, it's possible he's applying a relatively strict cut-off value for rejecting his null hypothesis that recessions don't die of old age. For example, if he insists on being 95% confident in rejecting the null hypothesis, then even a test statistic with a probability value of 0.094 will cause him to accept the null hypothesis that recessions don't die of old age. But in this case, we might ask whether age-independence is the more appropriate hypothesis to designate as the default assumption. And we might ask whether such a strict cut-off value is really warranted in this situation.

Second, the approach taken here differs in two other respects from the approach reported by Rudebusch. First, he uses a truncated probability density to reflect the view that the NBER business cycle dating committee will not identify a business cycle peak only a few months after a trough in the cycle. This appears to be standard practice in the literature, though I don't see an especially strong motivation for doing this when I read the

*Business-Cycle Dating Procedure*from the NBER. Second, he includes the ongoing recovery in his sample, whereas I've excluded it in this analysis, since we haven't yet observed the duration of this recovery. On the other hand, we know that the duration is at least 87 months, and perhaps this data should be included. But if we're going to reach a different conclusion on the basis of a single observation, perhaps we should be less confident than was Yellen when she labelled age-dependence a "myth".At any rate, it's clear there is room for further analysis into this issue, and I'll be happy to post further results as and when I consider the effects of truncation and the inclusion of the current recovery in the data set.

**Implications for the upcoming FOMC decision and for the market generally***When should we expect the next recession, and what value should we expect for the policy rate when the next recession arrives?*If recession risk is independent of the age of the recovery, the next recession would be expected in just under five years. But if recession risk does depend on the age of the recovery, as suggested by our analysis of post-war recoveries, we'd expect the next recession in just under two years. (The expected value of the duration of a recovery conditional on the recovery already lasing 87 months, is 110 months.)

Various FOMC members have discussed the importance of being able to lower the policy rate sufficiently in response to the next recession. But in order to lower the policy rate sufficiently, FOMC members first need to raise it. For example, if the FOMC raised the IOER by 25 bp at every meeting with a scheduled press conference (ie, four meetings per year), starting with this week's meeting, the IOER would be 2.5% by the time the next recession was expected -- assuming recession risk depends on the age of the recovery. And given the current low likelihood of rate hikes at both the September and December meetings this year, a more reasonable expectation for the policy rate would be 2.25% -- a level that doesn't give the FOMC much room to maneuver with the onset of the next recession.

On the other hand, FOMC members clearly don't want to increase the IOER so quickly that they trigger a recession. Their task then is to navigate between the Scylla and Charybdis -- neither hiking so quickly that they catalyze the next recession nor hiking so slowly that the get caught with insufficient policy ammunition when the next recession arrives.

Of course, Yellen has told us that she believes recession risk is independent of the age of a recovery. And to the extent that view is more generally shared among her colleagues on the FOMC, they may be inclined to move more slowly than four 25 bp hikes per year -- though the most recent set of interest rate projections offered by the FOMC is consistent with four such hikes per year.

But on the other hand, given the desire for the IOER to be at least 3% by the time of the next recession, as per David Reifschneider's paper,

*Gauging the Ability of the FOMC to Respond to Future Recessions*(cited by Yellen at Jackson Hole), it strikes me as quite likely that FOMC members will hike at a faster pace than is currently reflected in the graph below, showing current Fed Funds futures pricing along with the most recent set of interest rate projections from the FOMC.
For example, the Dec18 Fed Funds futures contract is currently priced at a rate of 0.835, whereas the median FOMC projection corresponding to that time is roughly 140 bp higher. (The reported Fed Funds rates have been about 10 bp below the IOER in recent months.)

Given market pricing, and given the interest of many FOMC members in raising the IOER suitably before the onset of the next recession, I believe it makes sense to sell Fed Funds futures contracts expiring in 2017 and 2018, though liquidity in that complex is modest further along the curve. As a more liquid alternative, I would suggest paying fixed in the OIS swap market in the <2Y segment. For example, the 2Y OIS rate is currently 0.65% -- only 25 bp above the current Fed Funds rate.

Given the paucity of data, we can't be certain that recoveries die of old age. But our analysis of post-war recoveries suggests a considerable likelihood that they do. In particular, I don't believe the data supports Yellen's claim that age-dependence is a "myth". More important, I don't believe the majority of Yellen's colleagues on the FOMC can be confident about this conclusion either.

Given the proximity of the upcoming Presidential election, I suspect most FOMC members will opt for a 'lame duck hike' at the December meeting rather than risk triggering a bout of market volatility with potential to affect the outcome of the election.

Looking beyond 2016, I suspect the actual path of the IOER will be lower than the path most recently forecast by FOMC members but greater than the path currently consistent with pricing in the market.

Given the proximity of the upcoming Presidential election, I suspect most FOMC members will opt for a 'lame duck hike' at the December meeting rather than risk triggering a bout of market volatility with potential to affect the outcome of the election.

Looking beyond 2016, I suspect the actual path of the IOER will be lower than the path most recently forecast by FOMC members but greater than the path currently consistent with pricing in the market.