Measuring firm performance using nonparametric quantile-type distances

ABSTRACT

When faced with multiple inputs and outputs , traditional quantile regression of Y conditional on X = x for measuring economic efficiency in the output (input) direction is thwarted by the absence of a natural ordering of Euclidean space for dimensions q (p) greater than one. Daouia and Simar (2007) used nonstandard conditional quantiles to address this problem, conditioning on Y ≥ y (X ≤ x) in the output (input) orientation, but the resulting quantiles depend on the a priori chosen direction. This article uses a dimensionless transformation of the (p + q)-dimensional production process to develop an alternative formulation of distance from a realization of (X, Y) to the efficient support boundary, motivating a new, unconditional quantile frontier lying inside the joint support of (X, Y), but near the full, efficient frontier. The interpretation is analogous to univariate quantiles and corrects some of the disappointing properties of the conditional quantile-based approach. By contrast with the latter, our approach determines a unique partial-quantile frontier independent of the chosen orientation (input, output, hyperbolic, or directional distance). We prove that both the resulting efficiency score and its estimator share desirable monotonicity properties. Simple arguments from extreme-value theory are used to derive the asymptotic distributional properties of the corresponding empirical efficiency scores (both full and partial). The usefulness of the quantile-type estimator is shown from an infinitesimal and global robustness theory viewpoints via a comparison with the previous conditional quantile-based approach. A diagnostic tool is developed to find the appropriate quantile-order; in the literature to date, this trimming order has been fixed a priori. The methodology is used to analyze the performance of U.S. credit unions, where outliers are likely to affect traditional approaches.

KEYWORDS:

• 62G32
• 62G35
• 62G10
• 62H12
• 62H15

JEL CLASSIFICATION:

• Efficiency
• production
• extreme value theory
• benchmarking
• quantile

Notes

Note that some elements of (g_x, g_y) could be defined as zero for nondiscretionary inputs or outputs (see Simar and Vanhems, 2012, for details on handling such situations).

DEA estimators are based on using either the conical or convex hulls of the FDH of sample observations; see Simar and Wilson (2013) for a recent survey and discussion.

This drawback has been addressed by Daouia and Simar (2005) for the case q = 1 by isotonizing the resulting estimate of the production function.

Of course, the distance to the unconditional order-α frontier will depend on the chosen orientation.

In the econometric literature on efficiency analysis it is common to assume that the density of (X, Y) has a jump at the frontier; e.g., see Park et al. (2000), Kneip et al. (1998), Kneip et al. (2008) for nonparametric models and Aigner et al. (1977), Meeusen and van den Broeck (1977), Battese and Corra (1977), and Stevenson (1980) for parametric models.

For example, one might use a kernel density estimator, but this would introduce a nonparametric rate (n^1/5) of convergence. Moreover, standard kernel density estimators, without some modification, are biased and inconsistent near support boundaries.

If one plots, in an application, D_n(α(k)) for say, k = 0, 1, …, 100 and finds a large jump near k = 100, the range of values of k over which D_n(α(k)) is plotted might be increased.

See Berger (2003) for details and analysis of the effects of new technology, including advances in IT, on productivity growth in the banking industry, and on the structure of the banking industry.

Spong (2000) provides a summary of current U.S. banking regulations.

Average assets held by U.S. credit unions amounted to $84.6 million in 2006, ($50.6 million in constant 1985 dollars) as opposed to $7.8 million in 1985.

See Wheelock and Wilson (2011) and references cited therein for additional details on U.S. credit unions.

Call report data for individual credit unions are available from the National Credit Union Administration (www.ncua.gov).

In Y1, Y2, and X1, we use the (constant) dollar amounts of loans, investments, and shares and deposits. Although one might wish to consider the number of credit union members that are served, data for the number of loans, investments, shares, or deposits are not available in the call report data.

Our specification of credit unions’ inputs and outputs follows the lines of other studies such as Frame and Coelli (2001), Frame et al. (2003), Wheelock and Wilson (2011, 2012), and others. In particular, our input-output specification reflects the view that credit unions are similar to small community banks, with the additional mandate to provide “service” in the form of favorable interest rates to depositors and borrowers. We treat deposits as an input, as do the studies listed above, because credit unions necessarily must borrow from depositors in order to lend to borrowers. We estimate technical efficiency in the output direction so that our results can be interpreted as a measure of how well credit unions produce loans and other outputs given their observed level of deposits and other inputs.

We omitted observations where either loans or investments were negative, interest rates were outside the range (0, 1), or where inputs were negative. Such observations reflect obviously incorrect values.

Interest rate data are from series MPRIME and MORTG, not seasonally adjusted, St. Louis Federal Reserve Bank FRED database, http://www.research.stouisfed.org/fred2/.

Given the large sample sizes in both years, it is perhaps not surprising that the diagnostic D_n(α) would lead to choosing large values for α. Note, however, that the diagnostic does not return α = 1, which would lead to estimation of the full-frontier. Instead, the diagnostic indicates that a quantile lying perhaps “very close” to the frontier should be the benchmark. To further examine the performance of the diagnostic in (3.51), we repeated the exercise using the data for 2006, but with only the first 200, 500, 1,000, and then 2,000 observations. The corresponding values of α chosen by the diagnostic exercise were 0.9000, 0.9260, 0.9710, and 0.9755. This seems reasonable; in smaller samples, one is necessarily less certain than in larger samples whether a particular extreme observation should be classified as an outlier. Our diagnostic procedure reflects this, and chooses quantiles closer to the full frontier as the sample size becomes larger.