Double Bootstrap Confidence Intervals in the Two Stage DEA approach. Essex Business School University of Essex

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1 Double Bootstrap Confidence Intervals in the Two Stage DEA approach D.K. Chronopoulos, C. Girardone and J.C. Nankervis Essex Business School University of Essex 1

2 Determinants of efficiency DEA can be a useful tool in the hands of managers identify best practices. Efficiency levels might reflect not only the ability of the management, but the effects of contextual factors on firm s performance, as well. A second stage regression analysis on efficiency estimates can help quantify these effects. Understanding these relationships can be of help to: Managers improve firm s performance. Policy makers better assess cost of regulation. 2

3 Second Stage Regression: The problem The dependency problem: Efficiency measures estimated with DEA are dependent on each other by definition. (The estimator has a convergence rate of 2 p q 1 n + + ). This dependency disappears asymptotically, but generally at a rate slower than the usual n achieved by the truncated or censored MLE. Conventional inference procedures are invalid, when dimensionality of production is greater than 3 ( p + q > 3) (Xue and Harker 1999; Simar and Wilson 2007). The suggested solution: Bootstrap confidence intervals (Simar and Wilson 2007) 3

4 Aims Examine the convergence properties of the coverage rates of the alternative bootstrap confidence intervals estimators. Investigate the coverage accuracy of double bootstrap confidence intervals. Provide a less computationally demanding algorithm for constructing double bootstrap confidence intervals. 4

5 Data Generating Process A firm faces an environmental variable Z ~ N (2,4). Given Z, the production efficiency level δ is drawn from f( δ / Z). The conditioning operates through this mechanism δ = Zβ + ε [1], where ε ~ truncatedn(0,1), with left truncation at 1 Zβ. The input(s) are distributed as x U ) y P 1 3/4 xp p 1 P 1 3/ 4 xp p 1 = δ. = p ~ (6,16. We distinguish between single and multi output technologies: Single output: Multi output: ζ = δ. = If 2 then draw α U ) l 1 If Q 2 then additionally draw α ~ U (0,1 α ), for each Q = 1 ~ (0,1. l = 2,..., Q 1. l k1 K Then the output mix is given by yq = αζ q and q= 1,..., Q 1. Q Q 1 = (1 ) k= 1 k, for y α ζ 5

6 Step 1: Step 2: Step 3: Bootstrap Confidence Intervals Estimate the efficiency levels ˆ δ. Regress ˆ δ on the environmental variable Z using the truncated regression model to obtain ˆβ and σ ˆε estimates. * * Construct pseudo ˆ δ by drawing ε from the parametric distribution of the * errors truncated N 0, ˆ σ ) such that ˆ* δ = Z ˆ β + ε. A bootstrap estimate ( ε * of the parameter of interest is obtained by regressing ˆ δ on Z and denoted ˆ* β. Repeat the procedure J times. The basic bootstrap CI is given by: ˆ β ( ˆ β ˆ β), ˆ β ( ˆ β ˆ β) * * (1 α)( J+ 1) ( α( J+ 1)) The percentile bootstrap CI is given by: * * ˆ ˆ β( α( J 1), β + (1 α)( J+ 1) 6

7 Double Bootstrap Confidence Intervals Frequently the nominal coverage probability of the bootstrap CI differs from the true one. Step 4: Step 5: For each set of single bootstrap estimates construct a double bootstrap ** * ** ** sample ˆk δ = Zβ + ε k. Again use the truncated regression to obtain ˆ β k. Repeat the process K times. ** * Compute the statistic: U #( ˆ 2 ˆ ˆ = β k β β) K for the basic CI or ˆ** U = #( β ˆ β) K for the percentile CI. k The basic double bootstrap CI is given by: ˆ β ( ˆ β ˆ β), ˆ β ( ˆ β ˆ β) * * ( U ( J+ 1)) ( U ( J+ 1)) (1 a)( J+ 1) ( α ( J+ 1) The percentile double bootstrap CI is given by: ˆ β, α J β * * ( U ( J+ 1)) ( U ( J+ 1) ) ( ( + 1) ((1 α)( J+ 1) 7

8 The 25 th and 26 th values are the upper and lower bounds of and respectively. Stopping rules for double bootstrap Suppose J = 999 then U (25) and U(975) are required. Start with calculating 50 U and sort them in an increasing order. If the is greater than the current bound of and smaller than U then not all U 51 U (25) (975) K double bootstrap estimations are required. U (25) U(975) 8

9 Monte Carlo evidence - Single bootstrap Table 1. Estimated coverages of confidence intervals generated by conventio single bootstrap methods n Basic Boot. Alg.- Nominal significance Percentile Boot. Alg.- Nominal significance Asympt. Normal Apr.- Nominal significance p = q = Notes: Results based on 1,000 Monte Carlo trials p = q = 2 p = q = 3 9

10 Monte Carlo evidence - Double bootstrap Table 2. Estimated coverages of confidence intervals generated by percentile single and double bootstrap methods n Percentile Single Boot. Nominal significance Percentile Double Boot. Nominal significance p = q = p = q = p = q = Notes: Results based on 1,000 Monte Carlo trials 10

11 Conclusions Correlation of efficiency estimates disappears, but not fast enough. Need for alternative inference making method. Bootstrap offers a good alternative, but single bootstrap CIs do not have good coverage rates (the dimensionality problem of the efficiency estimator carries over to the second stage regression). Double bootstrap offers a significant improvement but at a considerable computational cost. This computational burden can be reduced by adopting deterministic stopping rules (in the spirit of Nankervis(2005)). 11