U.S. Banking in the Post-Crisis Era: New Results from New Methods

Abstract

This paper examines the performance of U.S. bank holding companies before, during, and after the 2007–2012 financial crisis. Fully nonparametric methods are used to estimate technical, cost, and input allocative efficiencies. Recently developed statistical results are used to test for changes in efficiencies as well as productivity over time, and to test for changes in technology over time. I find evidence of non-convexity of banks’ production set is found. In addition, the data reveal that mean technical efficiency declined during the financial crisis, but recovered in the years after, ending higher in 2016 than in 2006, while both cost and input allocative efficiencies declined from 2006 to 2016. Statistical tests indicate that technology shifted downward throughout the period 2006–2016.

Appendix: Technical Details

As discussed in Sect. 5, with dimension reduction productivity can be measured by simple ratios. In addition, since neither FDH nor DEA estimators are involved, the usual Lindeberg-Feller CLT can be used to make inference about differences in mean productivity between two periods. However, the samples in each period are unbalanced, and care must be taken to properly account for covariance.

Suppose banks are observed in two periods t ∈{1, 2}. Let n _t be the number of banks observed only in period t, and let n ₀ be the number of banks observed in both periods (here, n ₁ and n ₂ are defined differently than in the discussion about testing differences in mean technical efficiency in Sect. 5). Then the total numbers of observations in periods 1 and 2 are given by (n ₁ + n ₀) and (n ₀ + n ₁). Let P _jti denote productivity (measured by output/input or output/cost after reducing dimensionality to p = q = 1) for bank i in group j ∈{0, 1, 2} in period t ∈{1 2}. Group 0 consists of banks observed in both periods, while groups 1 and 2 consist of banks observed only in periods 1 and 2 (respectively). We have sample means $\widehat \mu _1$, $\widehat \mu _2$ for periods 1 and 2, with

$$\displaystyle \begin{aligned} \widehat\mu_t=(n_0+n_t)^{-1}\left[ \sum_{i=1}^{n_1}P_{11i}+\sum_{i=1}^{n_0}P_{01i}\right] \end{aligned} $$

(6.1)

for t = 1 or 2.

Due to Assumption 24, the Ps are independent within a given period, but may be dependent across periods. Hence any covariance between $\widehat \mu _1$ and $\widehat \mu _2$ can result only from the n ₀ banks observed in both periods. Let

$$\displaystyle \begin{aligned} \sigma_t^2:=\mbox{VAR}(P_{11i})=\mbox{VAR}(P_{01i}) \end{aligned} $$

(6.2)

and

$$\displaystyle \begin{aligned} \sigma_{12}:=\mbox{COV}(P_{01i},P_{02i}) \end{aligned} $$

(6.3)

for all i. Then

$$\displaystyle \begin{aligned} \mbox{VAR}(\widehat\mu_2-\widehat\mu_1)= \frac{\sigma_1^2}{n_1+n_0}+ \frac{\sigma_2^2}{n_0+n_2}- \frac{2n_0\sigma_{12}}{(n_1+n_0)(n_0+n_2)}. \end{aligned} $$

(6.4)

The variances $\sigma _t^2$ and covariance σ ₁₂ can be estimated by the corresponding sample moments, i.e.,

$$\displaystyle \begin{aligned} \widehat\sigma_t^2=(n_0+n_2)^{-1}\left[ \sum_{i=1}^{n_0}(P_{0ti}-\widehat\mu_t)^2+ \sum_{i=1}^{n_t}(P_{tti}-\widehat\mu_t)^2\right] \end{aligned} $$

(6.5)

for t = 1 or 2 and

$$\displaystyle \begin{aligned} \widehat\sigma_{12}=n_0^{-1}\left[ \sum_{i=1}^{n_0}(P_{01i}-\widehat\mu_1)(P_{02i}-\widehat\mu_2)\right]. \end{aligned} $$

(6.6)

Then the test statistic

$$\displaystyle \begin{aligned} \widehat\tau:=\frac{\widehat\mu_2-\widehat\mu_1} {\left[ \frac{\widehat\sigma_1^2}{(n_1+n_0)}+ \frac{\widehat\sigma_2^2}{(n_0+n_2)}- \frac{2n_0\widehat\sigma_{12}}{(n_1+n_0)(n_0+n_2)}\right]^{1/2}} {\ \overset{ d } \longrightarrow \ } N(0,1) \end{aligned} $$

(6.7)

as (n ₁ + n ₀ →∞ and (n ₀ + n ₂) →∞ by the Lindeberg-Levy CLT. For a two-sided test of size α, the null hypothesis H ₀: μ ₁ = μ ₂ is rejected in favor of H ₁: μ ₁≠μ ₂ whenever $|\widehat \tau |>\Phi ^{-1}(1-\frac {\alpha }{2})$ where Φ⁻¹(⋅) is the standard normal quantile function.