### Likelihood Ratios and their use in Recession Indicators

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In medicine, likelihood ratios improve patient outcomes and refine drug regimens by assessing the reliability of common diagnostic tests. In finance, likelihood ratios can quantify the reliability of an economic indicator such as one designed to identify recessions.

Specifically, using likelihood ratios, I will show that the ECRI coincident index is a poor tool for recognizing recessions, while their weekly leading index is a better indicator.

In medicine, likelihood ratios are used to estimate how much the probability that a patient has a particular disease changes from before a diagnostic test is given to after its result is known. (For those interested, precise definitions of the various terms, calculation methods and application to medicine are described here and here.) One can use the same concept to calculate the probability of the economy being in recession when an indicator is positive or negative.

At any particular time a recession indicator can be sending one of four messages, depending on the actual state of the economy and where it is to the recession “trigger” value that causes its prediction to change. The possibilities are a correct recession call (true positive), a false recession call (false positive), a valid all-is-well (true negative), or that the indicator is missing the existence of a recession (false negative). How often any one of the conditions occurs, together with the length of the observation period and the length of the recessions, are the raw data for my analysis.

**Likelihood ratios for the ECRI coincident index**

Let’s assess the value of testing for a recession using the year-on-year growth rate of the ECRI coincident index (ECRI-COg) and applying the concept of likelihood ratios. Specifically, I will calculate the probability that this indicator has correctly identified recessions when it declines to or below a level of 2%. This indicator and criterion was used by ECRI’s CEO, Lakshman Achuthan, to support his February 2012 claim that a recession was imminent. (The ECRI-COg is currently at a level of 2.48%, above the recession trigger line of 2%.) A graph of this indicator can be found in this commentary.

The period for which I analyzed this indicator is from January 1968 to February 2012, totaling 2,298 weeks. There were seven recessions during this period which lasted a total of 360 weeks. An additional 14 weeks (the three-month period prior to each of the seven recessions) were also counted as recession, in order not to penalize for early recession warnings, which a good leading indicator should provide. Therefore the total period with recession was 458 weeks (condition positive) and the total period without recession was 1840 weeks (condition negative).

Figure 1 shows a diagrammatic representation of the input data for the calculation. The actual data for this indicator is in the table 1 below.