Given that economic targets are now interdependent with the Federal Open Market Committee (FOMC) projections of rate firming times, we can use statistical analysis to get a stronger sense on the likelihood of future FOMC activities. For example, macroeconomists generally expect rate firming in 2015 or later. And markets have recently responded, perhaps over-responded, to Federal Reserve Chairman Bernanke's recent communications. Nonetheless not all market participants- notably equity investors- could state that there is some chance of rate firming prior to 2015, and that this chance can come from multiple quantitative analyses directly from the FOMC material. With a conclusion that the rate firming probability prior to 2015 has edged down, to near 20%.
We begin with market participants, who have seen the typical FOMC future frequency data (e.g., Figure 2 here) that has been available for quite some time. The distribution is somewhat unchanged from the previous projection summaries (e.g., see March data here), while the markets have recently priced in a substantially greater probability of rate firming sooner than 2015. The Figure 2 data shows that FOMC participants expect a full 3/4 percent rise in rates, within the first year it happens. Yet in recent months the long-term government bonds, which Chairman Bernanke stated would be among the first assets to be tapered, have seen their yields rise by nearly this amount.
So let's go back into the FOMC economic data and use a more traditional probabilistic approach of calculating the relevant portions from the FOMC projection data. In the June projections, there are 4 of 19 participants who judge rate firming by 2015, or nearly a 20% chance. The important binomial variance equation is appropriate to determine the confidence interval surrounding this 20% chance. Keep in mind that the variance associated with these discrete categorical probabilities is equal to n*p*q, when n is 19 participants, p is 20%, and q is 100%-p.
This gives a symmetrical confidence interval, which is better served for estimates closer to half, and not for estimates near either end (e.g., what we have here at 20%). In this case, since the estimate is closer to 0% than it is to 50%, an asymmetrical confidence interval would better fit on the lower end of this probability distribution. We'd also be suffering from having an overall insignificant small sample of 19 (and similarly trained) economists. Statistically relevant minimal sample sizes for these types of probability problems are generally magnitudes greater than this. Nonetheless, geometrically extrapolating our confidence interval would get a range of 20%+5%, for the standard range about the 20% FOMC projection estimate. In order to get this geometric process for the confidence interval surrounding the 20% probability, we take the interval from p^(1/U), to p^U. Where U is the exponential rate of the arithmetic (symmetrical) standard error we showed above. Our objective is an upward error that is clearly slightly larger than our downward error, for cases where the central estimate is well below 50%. In this case the upward and downward errors are different by such a fractional amount that it would be absorbed into the rounding of +5%.
Another approach to this probability problem is through the more recent economic thresholds being communicated by the FOMC. Now what do we learn about these "barriers", which provide the chance for the FOMC to discuss rate firming? In this case one main threshold is a 6.5% unemployment rate, and the approach to this problem is similar to a discrete approximation of an options pricing model. So let's see how the 2014 FOMC's unemployment rate forecast has evolved over time. We are using the 2014 unemployment rate forecast, since it is the reference just "prior to 2015". See the chart below.
The 2014 the FOMC unemployment rate forecast has improved over time (see purple projection data in the chart above), since late 20122 when they began releasing this forecast. However it has not been a smooth reduction in the forecasted unemployment rate, reflecting a similar 0.2% standard error in each FOMC report's forecast error among the 19 FOMC participants. Bear in mind that the reduction in the central unemployment rate forecast reflects only a temporary optimistic bias in the past FOMC forecast of this 2014 value. Additionally, another recent pattern is that as the FOMC forecast evolves over time, the downward standard error is about 0.05% less than the upward standard error. This is parallel to the above topic on asymmetry in the frequency distribution. And it is particularly unfortunate in the case of economic thresholds, since financial market participants are trying to understand the likelihood of being on the other side of an economic threshold, at various future times. In our case, we see through the red marker in the chart above, that the 6.5% unemployment rate threshold would be demanding to achieve prior to 2015.
The most recent May unemployment rate was 7.6%, or 1.1% above the 6.5% threshold. Of course this 7.6% level on the recent unemployment rate was known at the time of the FOMC participants were surveyed in June. See the blue data in the chart above to see the known actual unemployment rate during recent years.
The standard errors associated with this rate are more symmetrical. And they reflect a mixture of the historical U.S. Department of Labor's survey sampling error, with the contemporaneous standard error in the unemployment rate trend in-between the FOMC meetings. This standard error is roughly half that of the FOMC standard error, though closer to two-thirds the FOMC standard error recently. This information shows the reduced accuracy of the FOMC in projecting future unemployment rates, relative to the survey errors themselves. Meanwhile the underlying unemployment rate had rapidly fallen during our recent recovery from the global financial crisis, but this unemployment rate would need to courageously maintain something close to this fast rate of descent still, in order to achieve the 6.5% unemployment rate threshold prior to 2105.
With the FOMC central tendency of a 2014 unemployment rate at 6.6%, and a 0.1% downward standard error, this implies a one standard deviation exception from forecast for the FOMC to cross the 6.5% unemployment rate threshold prior to 2015. Despite the small downward-trending bias we discussed in the FOMC 2014 unemployment rate forecast, there is still a nearly 65% of this typical sampling distribution residing within one standard deviation. Therefore about half of the remainder (i.e., about 20%) would be the FOMC probability of crossing the economic threshold prior to 2015. To be clear, the Federal Reserve Chairman Bernanke did state that this target would only start the conversation about rate firming. So notwithstanding recent market reaction, just looking at the rate firming frequency distribution from the participants (see Figure 2 from the link above), we see that we are able to better narrow the statistical confidence intervals about the near 20% probability of rate firming prior to 2015. And at least a three-fourths probability of rate firming, in 2015 or beyond.
We begin with market participants, who have seen the typical FOMC future frequency data (e.g., Figure 2 here) that has been available for quite some time. The distribution is somewhat unchanged from the previous projection summaries (e.g., see March data here), while the markets have recently priced in a substantially greater probability of rate firming sooner than 2015. The Figure 2 data shows that FOMC participants expect a full 3/4 percent rise in rates, within the first year it happens. Yet in recent months the long-term government bonds, which Chairman Bernanke stated would be among the first assets to be tapered, have seen their yields rise by nearly this amount.
So let's go back into the FOMC economic data and use a more traditional probabilistic approach of calculating the relevant portions from the FOMC projection data. In the June projections, there are 4 of 19 participants who judge rate firming by 2015, or nearly a 20% chance. The important binomial variance equation is appropriate to determine the confidence interval surrounding this 20% chance. Keep in mind that the variance associated with these discrete categorical probabilities is equal to n*p*q, when n is 19 participants, p is 20%, and q is 100%-p.
This gives a symmetrical confidence interval, which is better served for estimates closer to half, and not for estimates near either end (e.g., what we have here at 20%). In this case, since the estimate is closer to 0% than it is to 50%, an asymmetrical confidence interval would better fit on the lower end of this probability distribution. We'd also be suffering from having an overall insignificant small sample of 19 (and similarly trained) economists. Statistically relevant minimal sample sizes for these types of probability problems are generally magnitudes greater than this. Nonetheless, geometrically extrapolating our confidence interval would get a range of 20%+5%, for the standard range about the 20% FOMC projection estimate. In order to get this geometric process for the confidence interval surrounding the 20% probability, we take the interval from p^(1/U), to p^U. Where U is the exponential rate of the arithmetic (symmetrical) standard error we showed above. Our objective is an upward error that is clearly slightly larger than our downward error, for cases where the central estimate is well below 50%. In this case the upward and downward errors are different by such a fractional amount that it would be absorbed into the rounding of +5%.
Another approach to this probability problem is through the more recent economic thresholds being communicated by the FOMC. Now what do we learn about these "barriers", which provide the chance for the FOMC to discuss rate firming? In this case one main threshold is a 6.5% unemployment rate, and the approach to this problem is similar to a discrete approximation of an options pricing model. So let's see how the 2014 FOMC's unemployment rate forecast has evolved over time. We are using the 2014 unemployment rate forecast, since it is the reference just "prior to 2015". See the chart below.
The most recent May unemployment rate was 7.6%, or 1.1% above the 6.5% threshold. Of course this 7.6% level on the recent unemployment rate was known at the time of the FOMC participants were surveyed in June. See the blue data in the chart above to see the known actual unemployment rate during recent years.
The standard errors associated with this rate are more symmetrical. And they reflect a mixture of the historical U.S. Department of Labor's survey sampling error, with the contemporaneous standard error in the unemployment rate trend in-between the FOMC meetings. This standard error is roughly half that of the FOMC standard error, though closer to two-thirds the FOMC standard error recently. This information shows the reduced accuracy of the FOMC in projecting future unemployment rates, relative to the survey errors themselves. Meanwhile the underlying unemployment rate had rapidly fallen during our recent recovery from the global financial crisis, but this unemployment rate would need to courageously maintain something close to this fast rate of descent still, in order to achieve the 6.5% unemployment rate threshold prior to 2105.
With the FOMC central tendency of a 2014 unemployment rate at 6.6%, and a 0.1% downward standard error, this implies a one standard deviation exception from forecast for the FOMC to cross the 6.5% unemployment rate threshold prior to 2015. Despite the small downward-trending bias we discussed in the FOMC 2014 unemployment rate forecast, there is still a nearly 65% of this typical sampling distribution residing within one standard deviation. Therefore about half of the remainder (i.e., about 20%) would be the FOMC probability of crossing the economic threshold prior to 2015. To be clear, the Federal Reserve Chairman Bernanke did state that this target would only start the conversation about rate firming. So notwithstanding recent market reaction, just looking at the rate firming frequency distribution from the participants (see Figure 2 from the link above), we see that we are able to better narrow the statistical confidence intervals about the near 20% probability of rate firming prior to 2015. And at least a three-fourths probability of rate firming, in 2015 or beyond.
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