We introduce a new index of the business cycle that uses the Mahalanobis distance to measure the statistical similarity of current economic conditions to past episodes of recession and robust growth. Our index has several important features that distinguish it from the Conference Board’s leading, coincident, and lagging indicators. It is efficient because, as a single index, it conveys reliable information about the path of the business cycle. It gives an independent assessment of the state of the economy because it is constructed from variables that are different than those used by the NBER to identify recessions. It is strictly data driven; hence, it is unaffected by human bias or persuasion. It gives an objective assessment of the business cycle because it is expressed in units of statistical likelihood. And it explicitly accounts for the interaction, along with the level, of the economic variables from which it is constructed.
The Mahalanobis distance was introduced in 1927 and modified in 1936 to analyze resemblances in human skulls among castes in India. It was rediscovered in 1999 to measure turbulence in the financial markets, and it has since been applied to diagnose diseases and to detect anomalies in self-driving vehicles. We apply the Mahalanobis distance to measure the similarity of a set of economic variables to past episodes of recession and robust growth.
In this application, we define the Mahalanobis distance as show in the equation below
In the equation, equals the Mahalanobis distance, equals the values of a set of economic variables at each point in time, equals the average values of those variables during past episodes of recession or robust growth, and equals the inverse of the covariance matrix of those values during periods of recession or robust growth. Whereas Mahalanobis sought to determine if a set of dimensions for a skull was more plausibly associated with one caste versus another, we seek to determine if the values for a set of economic variables are more closely associated with the values that prevailed during past recessions or periods of robust growth. We focus on robust growth rather than growth because it is important that the regimes be symmetrically opposite each other and sufficiently separated from each other.
We construct our index using the following economic variables.
Industrial Production (one-year percentage change, measured monthly)
Non-farm Payrolls (one-year percentage change, measured monthly)
Return of the Stock Market (one-year return, measured monthly)
Slope of the Yield Curve (10-year rate minus the Federal Funds Rate)
We define periods as robust growth as months in which the year-over-year percentage change in industrial production ranked above the 75th percentile relative to the prior 10 years. We then proceed as follows.
We isolate two sub-samples from the historical observations starting in January 1926: those that qualify as recessions and those that qualify as robust growth. The data prior to 1956 includes revisions that were not available at the time. For each month starting in January 1956, we only use data (including prior revisions) that were available at that point in time.
We then calculate the means and covariances for each sub-sample.
Next, we calculate the Mahalanobis distance of each month’s observations from each sub-sample.
We convert these distances, , into likelihoods, ,using the multivariate normal probability density function (PDF) as shown. In this formula, the covariance matrix equals that of the sub-sample for which we are measuring distance and likelihood.
We rescale the likelihood of recession by dividing it by the sum of the recession and robust growth likelihoods. We interpret this rescaled likelihood of recession as a probability.
We repeat this process for each month of our sample.
Exhibit 1 presents a time series of our index of the business cycle (solid black line), which we refer to as the KKT Index, beginning in January 1956 and ending in November 2019 (the period for which all observations are out of sample). This line measures how much more likely it is that the conditions at any point in time are associated with recession instead of with robust growth.
The dashed line shows the Conference Board’s Index of Coincident Indicators. This time series begins in January 1982, and its values are indicated by the right axis. A value of 0 indicates neutral economic conditions, whereas large negative values coincide with recessions. (We inverted the scale to coincide with the KKT Index.)
Exhibit 2 presents an event study of the KKT index. The shaded bar represents the events which are either recessions or periods of robust growth that occurred since 1956. The width of the bar is not relevant. These events varied by duration. The left side of the bar represents the beginning of the events while the right side represents the end of the events, irrespective of their duration.
The dark line shows the level of the KKT Index leading up to, during, and following recessions. The light line shows level of the index leading up to, during, and following periods of robust growth. Because our index is constructed as the relative likelihood of recessions, we should expect it to be low during periods of robust growth, which it is.
Exhibit 3 compares the level of the KKT Index to realizations of recessions within various time spans. We are interested in analyzing periods when the probability of recession is rising. Therefore, we require that the standardized shift of the index – defined as its current level minus its average over the past year, divided by its standard deviation over the past year – is greater than 1.
Exhibit 3: KKT Index and Recession Realizations
We highlight the row corresponding to the realization of recessions over the subsequent six months for various levels of the index. We also report the unconditional frequency of recession for the various time spans. Exhibit 6 reveals that when the index exceeded 50%, 54% of the time a recession occurred within the next six months. When it exceeded 60%, a recession occurred 61% of the time within the next six months. When the index exceeded 70% the frequency of recessions was 70%. When it exceeded 80%, recessions occurred 77% of the time. And when it exceeded 90%, recessions followed 91% of the time. The correspondence between the index level and the incidence of recessions is remarkably strong; in fact, the correlation exceeds 99%, and the slope of the relationship equals one. By comparison, the unconditional likelihood of a recession within any six-month period is only 17%.
We next present the same analysis for the yield curve. Specifically, we show the incidence of recessions that occur over varying time spans once the yield curve becomes inverted and its one-year standardized shift is below -1.
Exhibit 4: Yield Curve and Recession Realizations
Exhibit 4 shows the yield curve to be a much less reliable indicator of subsequent recessions than the KKT Index, especially for short horizons. It is only informative for a horizon of 18 months, and even for that horizon, it is less reliable than the KKT Index.
We apply the Mahalanobis distance to construct a new index of the business cycle. Specifically, we measure the statistical similarity of economic conditions each month to economic conditions that prevailed during prior periods of recession and robust growth. We then construct the index as the likelihood of recession relative to the likelihood of robust growth.
We argue that our index is more efficient, more objective, and more informative than the Conference Board’s indexes of leading, coincident, and lagging indicators.
The complete working paper is available on SSRN:
You can also watch the following presentation by Mark Kritzman
Chow, G., E. Jacquier, K. Lowry, and M. Kritzman. 1999. “Optimal Portfolios in Good Times and Bad.” Financial Analysts Journal, vol. 55, no. 3 (May/June): 65–73.
Mahalanobis, P.C. 1927. “Analysis of Race-Mixture in Bengal.” Journal of the Asiatic Society of Bengal, vol. 23: 301–333.
Mahalanobis, P. C. 1936. “On the Generalised Distance in Statistics.” Proceedings of the National Institute of Sciences of India, vol. 2, no. 1: 49–55.
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