A Comparison of Resampling Techniques when Parameters are on a

Boundary:

The Bootstrap, Subsample Bootstrap, and Subsample Jackknife

by:

Christiana E. Hilmer and Matthew T. Holt1

Selected Paper

American Agricultural Economics Association Annual Meeting

Tampa, Florida
July 30-August 2, 2000

Abstract:
This paper compares the finite sample performance of subsample bootstrap and
subsample jackknife techniques to the traditional bootstrap method when parameters
are constrained to be on some boundary. To assess how these three methods perform
in an empirical application, a negative semi-definite translog cost function is estimated
using U.S. manufacturing data.

Copyright 2000 by Christiana E. Hilmer and Matthew T. Holt. All rights reserved. Readers
may make verbatim copies of the document for non-commercial purposes by any
means, provided that this copyright notice appears on all such copies.

1
Christiana E. Hilmer is a graduate student in the Department of Agricultural and Resource
Economics and the Department of Statistics, North Carolina State University. Matthew T.
Holt is the Professor in the Department of Agricultural and Resource Economics and holds
the Cardon Chair in Agribusiness at the University of Arizona.
1. Introduction

Empirical economists frequently estimate models in which parameters are constrained

to be on a boundary of the parameter space. The need to do so usually arises when a priori

theoretical restrictions require a certain estimated parameter to be of a specific sign.

Examples of these types of restrictions include traditional demand analysis where the income

effect for a normal good is constrained to be positive while the own-price effect is constrained

to be negative; cost function analysis where curvature constraints imply that second-order

price terms satisfy concavity conditions; and time series models for conditional

heteroskedasticity where the GARCH parameters are constrained to be non-negative. Such

cases are potentially problematic for the researcher, as standard error estimates and

confidence bounds are difficult to compute using classical statistical inference. In short,

traditional distributional assumptions no longer apply when the parameter space is truncated

because the asymptotic distribution is no longer normal.

Two primary strategies have been utilized in order to account for the problems

presented by models in which parameters are constrained to be on a boundary. The first

strategy is to simply refuse to compute standard errors or confidence bounds because the

procedure required to do so would be too complicated (Diewert and Wales [1988], O’Donnell

et al. [1999], Barnett and Yue [1988]). The second strategy is to impose the inequality

constraints implied by the a priori theoretical restrictions and use traditional methods such as

the delta method and the Wald test to compute standard errors and test hypotheses

(Appelbaum and Ulah [1997], Ryan and Wales [1999]). This approach is not recommended

considering that a common regularity condition for use of the Wald and Likelihood Ratio tests

is that the true parameter must be in the interior of the parameter space (Demos and Sentana,

1998).2 Therefore, any conclusions researchers draw from this type of analysis are, at best,
2
questionable. Indeed, both of these strategies are unsatisfactory, as the first amounts to not

having a measure of confidence in the reported estimates and the second amounts to reporting

incorrect estimates of standard errors or test statistics.

The reason researchers have relied heavily on these two unsatisfactory approaches is that most

proposed classical methods for dealing with the problem have proven to be exceedingly complex. For

instance, Goureiroux, et al. (1982), Self and Liang (1987), and Wolak (1989a, 1989b, 1991) find that

for models with inequality constraints the correct testing procedures involve Kuhn-Tucker-like tests in

which the asymptotic distribution of the null hypothesis is a weighted mixture of chi-square

distributions. Specifically, the appropriate distribution for a model with q restrictions is a weighted

average of q+1 chi-square distributions, χ 02 , χ12 , χ 22 ,… χ q2 , which is more concentrated toward the

origin than the traditional tabled χ q2 distribution. See Demos and Sentana (1998) for additional

details. More recently, Andrews (1998) developed a weighted average power optimality criterion that

indexes the amount of weight that is placed on alternatives depending on their distance from the null

hypothesis. These tests involve a weighted combination of truncated normal distributions. The

advantage of this testing approach over that of Goureiroux et al. (1982) is that it can also be applied to

nonlinear regression problems; however, the practical problem with utilizing an approach that involves

a mixture of distributions is determining how much weight to put on each distribution. For example,

Self and Liang (1987) show that the test statistic in a linear model with one linear inequality restriction

is an average of the χ 02 and χ12 distributions. Because the correct weighting factors depend on the

structure of the inverse information matrix and the number of restrictions, they will differ greatly

across applications.

Alternatives to classical approaches include mathematical programming and Bayesian

methods. Gallant and Golub (1984) use a mathematical programming routine to impose curvature

constraints directly during estimation in a flexible functional form. These curvature constraints are

3
imposed on the eigenvalues of the matrix of Allen elasticities of substitution. An approach adopted by

Chalfant and White (1988), Chalfant, Gray, and White (1994), Koop, Osiewalski and Steel (1994), and

Terrell (1996) is to rely on Bayesian techniques to impose curvature restrictions. Bayesian

methodology treats parameters not as fixed values but as items that can be continually updated as

one’s subjective views of the world change.

It is evident that each of the methods listed above are either quite complicated to compute,

apply to fairly specific cases, or rely on Bayesian methodology. There is clearly a place in the

classical statistics literature for a test that applies in both linear and nonlinear applications and is easy

to implement when parameters are constrained to be on a boundary. A recent theoretical work by

Andrews (1999) explores the use of re-sampling techniques to calculate standard errors and confidence

intervals when a parameter is on a boundary. Since being developed by Efron (1979), the bootstrap

has become a popular method to calculate standard errors. As demonstrated by Andrews (1999),

however, this procedure is not asymptotically correct to the first order when parameters are on a

boundary. For this reason, Andrews (1999) proposes using subsample bootstrap or subsample

jackknife methods, in lieu of the traditional bootstrap. This dissertation is the first attempt to

empirically address the important theoretical work of Andrews. We start by comparing the finite

sample performance of the subsample bootstrap and subsample jackknife methods to the traditional

bootstrap method preferred by previous researchers. An important concern when using subsample

techniques is the correct subsample size to use for different sample sizes. Because previous theoretical

work does not address this issue, Monte Carlo simulations are conducted for various sample and

subsample sizes to help determine the relationship between the two. In addition to comparing the

finite sample performance of the different techniques, it is necessary to examine how the three

methods perform in actual empirical situations where parameters are inequality constrained. To this

end, we estimate a translog cost function, applied to Berndt-Woods manufacturing data, in which

concavity requires the matrix of parameter estimates to be negative semi-definite.

4
2. Theory

Re-sampling techniques, such as the bootstrap or the jackknife, are often used when the

underlying distribution of the parameter is either unknown, mathematically difficult to compute, or has

no analytic distribution. These techniques rely on computer power instead of distributional

assumptions or mathematical rigor. The jackknife is the older of the two techniques, and was first

used by Quenouille (1949) to reduce bias in estimators and later by Tukey (1958) as a method for

estimating variances. The standard (or delete one) jackknife estimate is computed by deleting one

observation from the sample data set, calculating the parameter of interest for the remaining n-1

observations, and repeating this process until each of the n observations has been deleted. The

jackknife estimator is then the average across all of the parameter estimates from the n different

jackknife samples. A shortcoming of this technique is that it fails if the parameter of interest is

discontinuous, such as the median. For example, Miller (1974) demonstrated that jackknife estimates

of sample quantiles do not perform well in finite samples. Because of this shortcoming, Efron (1979)

developed the bootstrap as an alternative. The goal of the bootstrap is to use the sample data to mimic

the overall population distribution and to use resampling to calculate an estimated sampling

distribution. For an initial sample data set of size n, this is accomplished by drawing a random sample

of size n with replacement from the initial sample data set, calculating the parameter of interest for the

sample drawn, and repeating the process many times. The bootstrap estimator is then the average

across all of the parameter estimates from the different bootstrap samples. The intuition behind this

methodology is that the distribution of the parameter estimates from the bootstrap samples mimics the

traditional asymptotic sampling distribution of parameter estimates for samples drawn from the

population distribution. Hence, the bootstrap technique allows the researcher to generate an estimated

sampling distribution in cases in which she/he only has access to a single sample rather than the entire

population.

These types of re-sampling techniques hold promise in our quest to derive a comparatively

easy and general method for conducting statistical inference when parameters are on a boundary of a
5
parameter space. This is because the bootstrap doesn’t require excessive mathematical rigor and

modern computer power makes re-sampling relatively painless. In order for the bootstrap to provide

consistent estimates of the asymptotic distribution of a parameter, however, certain conditions must be

met. Bickel and Freedman (1981) define these necessary conditions as:

(1) weak convergence of the statistic when the sample data has distribution G, for all G in a

neighborhood of the true distribution, F.

(2) the convergence of G to F is uniform in a neighborhood of F.

(3) continuity of the mapping from G to the asymptotic distribution of the statistic.

Unfortunately, when parameters are on a boundary of the parameter space, the mapping from the

sample data to the true distribution is no longer continuous, thereby violating condition (3). Hence, as

Andrews (1997a) demonstrates, the bootstrap method for obtaining confidence intervals and standard

errors when a parameter is on a boundary is theoretically invalid. Intuitively, this is because the

bootstrap yields an estimated sampling distribution that puts too much mass below the cutoff for the

parameter and therefore does a poor job of mimicking the true population distribution.

Because the traditional bootstrap yields inconsistent standard error estimates when parameters

are on a boundary, alternative techniques must be developed. Andrews (1999) offers three different

possibilities for obtaining consistent estimators of the entire asymptotic distribution in such cases: the

law of the iterated logarithm; the subsample jackknife; and the subsample bootstrap. For discussion

purposes assume we are trying to estimate NT( θ̂ - θ ) where θ̂ is an estimator that maximizes a

function over the parameter space, θ is the true value of the parameter, and NT is a normalizing

constant.3

The first method, the law of the iterated logarithm, is based on a rule that decides whether or

not any of the inequality constraints are binding. This rule involves comparing the parameter estimate

to a cutoff point derived by multiplying the critical value from a Wald test by the estimated standard

error obtained by assuming the parameter is not on the boundary. If the parameter is above the cutoff

6
point, one concludes that the true parameter is not on the boundary, implying the distribution is not

truncated. It is therefore possible to proceed by using standard asymptotic results. If the estimated

parameter is below the cutoff point, however, the inequality constraint is binding and the asymptotic

distribution must be simulated by a truncated distribution. This method is unattractive because, among

other things, it is highly complicated and computationally much more difficult to employ than the

latter two methods. Consequently, it will not be addressed in this dissertation.

The second method, the subsample jackknife, differs from the standard jackknife by deleting

more than one observation. Specifically, to perform the subsample jackknife, d (greater than 1)

observations are dropped, the parameters are calculated using the remaining m (where m = n-d)

observations, and the process is repeated until all possible samples of size m have been drawn. Hence,

æn ö
to cover all possible subsets of the data with d observations deleted, çç statistics should be
èm

æn ö
computed. Simple inspection reveals that as d increases, çç ÷÷ increases. This presents a practical
èm

problem for the researcher in that the potential number of subsamples to be drawn is likely far too

large to allow for efficient calculation of each of the possible subsamples. Instead, the researcher will

only want to take a random sample of the possible subsamples. Accordingly, for the subsample

jackknife, the asymptotic distribution of NT( θ̂ - θ ) is approximated by m (θˆ sj − θˆ ) , where θ̂ sj is

the subsample jackknife estimator and θ̂ is some maximizer of the objective function.

The third method, the subsample bootstrap, corrects the problems inherent in the standard

bootstrap by drawing, with replacement, repeated samples of size q (where q is less than n) from the

initial sample of size n. Andrews (1999) demonstrates that basing the bootstrap on these smaller

samples yields a consistent asymptotic distribution. For the subsample bootstrap NT( θ̂ - θ ) is

approximated by q (θˆ sb − θˆ ) , where θ̂sb is the subsample bootstrap estimator and θ̂ is again some

maximizer of the objective function.

7
 Although subsampling techniques have not been applied to economic problems where a

parameter is on a boundary, the literature does contain examples where they have been applied to other

situations. The subsample jackknife was first proposed by Wu (1989) to rectify the deficiencies of the

standard jackknife for non-smooth estimators. Those results were extended by Politis and Romano

(1994) to show that neither asymptotic normality nor iid data are needed to accurately estimate a

sampling distribution. The latter work is the basis of many papers (e.g., Li and Maddala [1997],

Politis et al. [1997]) that extend re-sampling techniques to stationary time series data. The subsample

bootstrap was first used in applied economic research by Ziari et al. (1997) who utilized a

mathematical programming model to produce estimates without any statistical properties. Comparing

bootstrap, subsample bootstrap, and subsample jackknife techniques to identify statistical parameter

estimates, those authors found that the subsample jackknife method performed best within their

application. A final application of the subsample bootstrap method appears in a paper by Bickel and

Ren (1996) who set critical values for Cramer-von Mises goodness-of-fit tests with doubly censored

data.4 They first prove that the standard bootstrap fails to estimate the null distribution of the test

statistic and then proceed to show that the subsample bootstrap has the correct asymptotic power

function and is asymptotically consistent.

3. Simulation Study

The theoretical work of Andrews demonstrates that the subsample bootstrap and subsample

jackknife provide consistent standard error estimates in a model where parameters are constrained to

be on a boundary. In practice, however, researchers do not have access to infinite samples. It is

therefore important to consider the finite sample performance of the various re-sampling techniques.

To this end, we perform Monte Carlo studies to compare how each method performs for samples of

different sizes. In addition, we conduct traditional bootstrap estimation to assess whether there is any

finite sample gain in using the re-sampling techniques over the traditional bootstrap methodology. In

other words, even though the traditional bootstrap in theoretically inconsistent in the limit, we would

like to know the degree to which a researcher is in error if she/he uses the traditional bootstrap in lieu
8
of one of the subsampling methods in finite samples. A collateral concern when using the two

subsample techniques is that previous theoretical work has given no indication as to the optimal

subsample size. Consequently, within our Monte Carlo studies we explore the effects that different

subsample sizes have on the relative performance of the estimators.

To replicate situations in which economists may use these techniques, Monte Carlo studies are

conducted for one and two equation systems. This paper presents results from sample sizes of 50 for

both the one equation and two equation systems in order to determine how subsampling techniques

behave as sample size changes. As we are also interested in how results vary with different subsample

sizes, results are calculated for a number of different subsampling sizes, ranging from n to n- n .

3.1 One-equation Model

The first application we consider is a simple one-equation model in which we wish to estimate

two parameters. The model is specified as

(1) y t = α + βx t + ε t

where a priori theoretical restrictions constrain β to be non-negative and the usual Gauss Markov

assumptions hold. A simplistic macroeconomic version of equation (1) is the principles-level

consumption function that relates consumption to disposable income with y t and x t representing

consumption and disposable income respectively in period t. In this formulation, the intercept α is

autonomous spending and β is the marginal propensity to consume. Basic macroeconomics

principles suggest that β is positive, meaning that if disposable income increases then consumption

increases by some fraction of that amount. Therefore, due to a priori restrictions, β is constrained to

be greater than or equal to zero.

To examine the finite sample performance of the bootstrap, subsample bootstrap, and

subsample jackknife techniques Monte Carlo studies are conducted using a nested loop. The first step

in the nested loop is to assign values to the coefficients in equation (1). Because we are trying to

9
assess finite sample performance when a parameter is on a boundary, we set β equal to the lowest

value under the constraint, which is zero. α is set equal to two. Following Godfrey and Veall (1998),

we randomly construct the independent variable, which is fixed in repeated samples, using the model

x t = .75x t −1 +η t , where η t is a normal random variable with mean zero and variance two. The

error term in (1) is randomly generated from a normal distribution with mean zero and variance one.

The Monte Carlo strategy we employ nests an inner loop within an outer loop. The outer loop, which

we will call a Monte Carlo run, begins by creating dependent variables, yt, using our fully specified

model in equation (1). After generating the data, we employ maximum likelihood estimation.

In this outer loop, or Monte Carlo run, we denote the estimates of α and β from equation (1) as Aml

and Bml. To impose the non-negativity constraint on β , we use the technique of Barnett and Yue

(1988) to actually estimate c, where by definition cc=B. The immediate implication of this construct

is that even though c may be negative, an estimate of β that is less than zero will never be obtained.

The inner loop, or bootstrap run begins at this point. Using one of the three resampling

techniques, we obtain a new set of data by taking random draws from the residuals of the Monte Carlo

estimation to acquire a set of resampled residuals, e*i . A new dependent variable, yt*, is then obtained

by combining Aml, Bml, xt, and e*i as prescribed by equation (1). The model is then re-estimated using

the new data obtained from the re-sampling methods. We denote the resulting estimates as Ars, crs, and

Brs = crscrs, where rs indicates that a re-sampling estimator has been used. An asymptotic estimate of

the distribution NT(Brs- Bml) is obtained by subtracting the Monte Carlo estimate from the resampling

estimate and multiplying by the weighting factor NT. The inner loop, or bootstrap run, is performed

1000 times and the outer loop, or Monte Carlo run, is also performed 1000 times. Therefore, equation

(1) is estimated a total of 1,000,000 times for each of the five different sample sizes. At the end of

each inner loop, both 90 and 95 percent confidence intervals of the asymptotic distribution are

obtained.

10
 It is instructive to carefully consider the method used for calculating confidence intervals.

Figures 1 depicts a histogram of the asymptotic distribution resulting from a single Monte Carlo run

for the subsample bootstrap for a sample size of size fifty with a subsample size of forty. These

figures are consistent with the other sample sizes and re-sampling methods. As illustrated by this

figure, the re-sampling distribution is skewed towards the left. When constructing confidence

intervals, it is therefore necessary to account for skewness of the distributions. The usual approach to

calculating confidence intervals for symmetric distributions is the percentile method, under which the

lower limit of a 95% confidence interval with 1000 bootstrap runs is the average of the 25th and 26th

ordered observations and the upper limit is the average of the 975th and 976th observations. This

method places a total of 5 percent of the distribution equally in the two exterior tales. This method is

not appropriate for skewed distributions, however, as it leads to confidence intervals that are wider

than desired. The wider intervals result in a loss of precision because they place less of the mass of

the confidence interval in the tightest part of the distribution. To calculate more precise intervals for

our skewed distributions, we use the modified percentile method as detailed in Davidson and

MacKinnon (1993, pp. 766). The modified percentile method creates narrower confidence intervals

by minimizing the length of the confidence interval. The intuition behind this method is that the

confidence interval is moved away from the side of the distribution that has the least amount of

information.5 In analyzing the precision of our estimators we will be interested in those with the

shortest average confidence interval length as those will contain the most information.

To measure the accuracy of our re-sampling estimators, we calculate the coverage probability,

which is the number of times the true value of the estimate, n ( Bml - 0 ), falls within the estimated

confidence intervals, NT(Brs -Bml), for each of the 1,000 Monte Carlo runs. Recall that NT(Brs -Bml) is

the estimate of the asymptotic distribution, where Brs is the re-sampling estimator from the inner loop,

Bml is the Monte Carlo estimator from the outer loop, and 0 is the true value of the parameter. We

then compare the percentage of times that the estimated confidence intervals contain the true value of

11
the estimate to the nominal confidence level of either 90 or 95 percent. Perfect coverage probabilities

would be ones that contain n (Bml- 0 ) 90 and 95 percent of the time, respectively. We would

therefore conclude that the best-performing estimator would be the one that provides coverage

probabilities closest to those percentages.

Table 1 presents the estimated coverage probabilities for the 90 and 95 percent confidence

intervals for the sample size of fifty and subsample sizes of 40, 25, and 10 in the one equation system.

As discussed above, to explore the issue of appropriate subsample size, the Monte Carlo studies are

run for a variety of different subsample sizes. Recall that in analyzing the performance over

estimators, we are looking for the shortest average confidence lengths and for coverage probabilities

that are closest to the nominal confidence level. The bootstrap, subsample jackknife, and subsample

bootstrap provide virtually identical coverage probabilities for each of the three subsample sizes. The

coverage probabilities generally lie a little below the nominal coverage levels, except for the

subsample jackknife for the subsample size of 10. In this case, the coverage probability is just above

the nominal level. The average lengths for the various subsample sizes are fairly similar. The

subsample jackknife is the most precise for the subsample sizes of 40 and 25 whereas the subsample

bootstrap is the most precise for the subsample size of 10. These results suggest that the three re-

sampling methods perform similarly for the sample size of 50. The Monte Carlo results suggest that

coverage probabilities and levels of precision are very similar for the bootstrap, subsample bootstrap,

and subsample jackknife. Thus, even though the bootstrap is asymptotically inconsistent, it seems to

perform well in finite samples.

3.2 Two-equation System

The next step in assessing the relative performance of the three estimators is to examine a

more advanced two-equation system that might be more relevant to empirical economists. For

example, in production or demand analysis a researcher might want to estimate the following model

(2) y1t = A1 + B11 x 1t + B12 x 2 t + µ1t

12
(3) y 2 t = A 2 + B12 x 1t + B 22 x 2 t + µ 2 t .

where the error terms µ1t and µ2t are assumed to be correlated. A production analogy to (2) and (3)

might have representing share equations from a standard three input cost function model (one share

equation is dropped due to adding up restrictions) with the ususal a priori assumption that the model is

concave. This implies that the matrix of parameter estimates, B11, B12, B21, and B22 is negative semi-

definite. We impose this constraint using the Cholesky decomposition and assuming symmetry

between the cross-partials. Hence, we end up estimating c11, c12, and c22, where B11= - (c11 c11), B12= -

(c11c12), and B22= - (c12c12 + c22c22). Once again, we are interested in evaluating the performance of the

three estimators when the parameter is on a boundary. Therefore, when constructing the data, the

values for each of the Bij are set to zero, A1 is set to 2, and A2 is set to 3. The xit are created in a

manner similar to the method detailed above, except that the error terms now have a bivariate normal

æ0ö æ1 .5 ö
distribution with mean çç and covariance matrix çç .
è0 è .5 1

Because we have a two-equation system with correlated error terms, the model is estimated

using seemingly unrelated regression. The process for the two-equation Monte Carlo study is very

similar to that of the one-equation Monte Carlo study detailed above. The only difference is that

instead of estimating the two parameters, A and B, from the one-equation system we are now

estimating the five parameters, A1, A2, B11, B12, and B22, from the two-equation system. Therefore, we

are interested in the asymptotic distribution of the three parameters that lie on the boundary, B11, B12,

and B22.

Once again, we ultimately want to calculate coverage probabilities and measure of precision to

compare and contrast how well the three re-sampling estimators perform in finite samples. These

results, contained in Table 2 for a sample size of 50 are fairly similar to the one-equation results. The

main difference between the one and two equation results is the dramatic increase in the coverage

probabilities for the two subsampling methods as the subsample size decreases. The three methods in

13
Table 2 have similar coverage probabilities for the subsample size of 40 but for the subsample size of

10 the two re-sampling methods over cover.

General conclusions can be drawn about the relative performances of the three techniques

after examining the results in the one and two equation simulation studies. The main implication of

these results is that even though the bootstrap is asymptotically inconsistent, it performs very similarly

to the two consistent techniques in finite samples. These results also suggest that the subsampling

techniques perform similarly for various subsample sizes for the one-equation model and tend to have

increased coverage probabilities as sample sizes decrease for the two-equation model. Now that we

have analyzed these methods in a Monte Carlo study, we will now discuss how to implement them

into economic applications.

4. Empirical Application – Translog Cost Function

Inequality constraints are used extensively in the economics literature in supply and demand

analysis to impose well-known microeconomic conditions such as convexity, concavity, and quasi-

convexity. The specific classes of supply and demand specifications we focus on in this dissertation

are those with flexible functional forms. The idea behind these specifications is to assign to a

consumer, firm, or industry a function that requires relatively few restrictions to attain arbitrary

elasticities at any given point. These functional forms are usually second-order (or Diewert) flexible

and are local approximations to any arbitrary twice continuously differentiable function at any given

point in the data space. Examples of local and global flexible functional forms include the translog,

generalized Leontief, Box-Cox, Almost Ideal, Asymptotically Ideal, Laurent, Muntz-Szatz, and

Fourier flexible forms. In this paper, we focus on the translog model.

The translog model was proposed by Christensen, Jorgenson, and Lau (1975) as a model that

is a second-order locally flexible functional form. Here we plan to estimate a translog cost function

similar to that used by Diewert and Wales (1987):

L L L L
(4) ln C(p, y, t ) = B 0 + A y ln y + Bi ln p i + .5 D ij ln p i ln p j + D iy ln p i ln y
i =1 i =1 j=1 i =1

14
 L
+.5 Ayy ln y ln y + D it ln p i t + Att + .5Attt2 + Ayt ln y t
i =1

where C is cost, p is price, y is output, t is time, and L is the number of inputs. We assume that the

cost function is symmetric and linearly homogeneous in factor prices.

In this study we are interested in imposing proper curvature on equation (4). A priori

restrictions constrain this function to be concave in prices, which is equivalent to requiring the Hessian

matrix to be negative semi-definite. A procedure due to Moschini (1998) Ryan and Wales (1998) is to

impose concavity at a reference point using the Cholesky decomposition.6 This procedure entails

redefining the Hessian matrix, H, so that it is equal to B + C, where B is a matrix of parameters of the

same order as H and C is a matrix of some or all of the other parameters in the model. After B and C

are determined, the Hessian matrix is set equal to -AA', where A is a lower triangular matrix. The

equation is then solved for B = -AA' – C. The elements of A and C, rather than the matrix B, are used

in estimation. For well-behaved data, this procedure guarantees that the Hessian matrix is negative

semi-definite and the model is concave at the reference point. Applying this technique to equation (5),

yields Dij = (-AA')ij – BiBj + δ ij Bi, where δ ij equals one if i=j, zero otherwise, and

éa11 0 0 ù
(5)
ê
A = a12 a 22 0
ê
êëa13 a 23 a 33

so that (-AA')11 = - a11a11, (-AA')12= - a11a12, and so on. The aij’s are used in estimation in lieu of the

Dij’s.

Our application of the translog cost model uses the familiar 1947-71 U.S. manufacturing data

from Berndt and Wood (1975). These data contain price and input information about capital (K),

labor (L), energy (E), and manufacturing (M) and output of U.S. manufacturing. Therefore, we are

interested in estimating the share equations derived form the cost function found in equation (4):

L
(6) s i (p, y, t ) = B i + D ij ln p j + D iy ln y + D it t .
j=1

15
Due to adding up restrictions (shares sum to one) one share equation is dropped in estimation,

manufacturing, to prevent a singular covariance matrix. Because the error terms may be

contemporaneously correlated, we will employ seemingly unrelated estimation.

The elasticities derived from the estimation of the above model contain the economic

information of interest. The price-elasticities derived from the translog cost function specified in

equation (4) are

D ij
(7) ε ij = + ŝ j − δ ij where δ ij = 1 if i=j and 0 otherwise
ŝ i

where ŝ i is the predicted share from input i.

As mentioned above, a similar model has previously been estimated by Diewert and Wales

(1987). However, those authors reported only the elasticities and chose not to report confidence

intervals or standard errors. Our analysis corrects this major shortcoming by employing the resampling

techniques in order to obtain consistent standard errors and to conduct hypothesis tests on the

estimated elasticities in the model.

Estimated price elasticities are contained in Tables 3 for the bootstrap, subsample bootstrap,

and subsample jackknife with a sample size of 25 and a subsample size of 18. Point estimates of the

elasticities as well as 95% confidence intervals are presented. Elasticities are the same across the three

methods because they are computed from the initial data set. It is worth noting that these elasticities

have the same sign and are very similar in magnitude to those obtained by Diewert and Wales (1987).

Confidence intervals are computed using the modified percentile method described in the simulation

study. As is typical in studies that use bootstrap or jackknife methodologies, hypothesis tests are

conducted using the empirical distribution generated from the resampling methods rather than relying

on asymptotic theory such as t-tests. In the present case confidence intervals are used to perform

hypothesis tests in the following standard manner. If the confidence interval contains zero then we

conclude that the elasticity is not significantly different from zero. Conversely, if the estimated

confidence interval does not contain zero, we conclude that the estimated elasticity is statistically
16
significant at the 5% level and denote this with an asterisk in the table. The bootstrap and subsample

bootstrap arrive at the same conclusions concerning the statistical significance of the elasticities except

in the case of the elasticity of labor with resect to materials. The subsample jackknife differs of two

occasions from the bootstrap. Specifically, the own-price capital and own-price materials elasticities

are insignificant using the subsample bootstrap where they are significant at the 5% method with the

other two methods. This implies that conclusions about significance will differ depending on the re-

sampling techniques that are used.

5. Conclusion

We have explored two relatively simple methods, the subsample bootstrap and subsample

jackknife, to calculate measures of precision within a classical statistics framework when parameters

are inequality constrained. Monte Carlo studies were conducted for one and two equation models to

gauge finite sample performance of the bootstrap, subsample bootstrap, and sample jackknife

techniques. The three methods were then applied to an inequality constrained cost function with U.S.

manufacturing data. Results suggest that the subsample bootstrap, subsample jackknife, and

traditional bootstrap perform similarly in both the Monte Carlo and empirical applications.

Future research will entail applying the subsampling methods to additional empirical applications

in which the models are curvature constrained such as the Asymptotically Ideal Model as the ultimate

goal of this research is to utilize the proposed estimation techniques in meaningful empirical

applications. The subsampling techniques will facilitate the computation of standard errors and, as

well, will provide a direct way of performing hypothesis tests on point estimates. An implication of

these applications is that we will be able to perform statistical tests on the elasticities from a model

that conforms to economic theory, something that heretofore has proved challenging if not impossible

to do.

17
 Frequency
-0
.2
-0 826

0
50
100
150
200
250
.0
98 021
78
0.
08 369
5 9
0. 034
26 7
88 01
0. 5
45 310
2 2
0. 671
63 50
6 3
0. 489
82 9
03 03
1. 0
00 830
4 4

18
1. 126
18 7
79 05
1. 4
Distribution.

37 510
1 5
1. 763
55 50
5 6
1. 581
73 9
94 06
0
Sample size – 50, Subsample size - 40

1.
92 030
32 7
2. 18
10 7
70 08
2. 3
29 71
08 08
2. 555
47 0
46 9
73
91
Figure 1. Representative Subsample Bootstrap Histogram of the Estimated Asymptotic
 Table 1. Hypothesis Tests Results – Single Equation Model
Non-Symmetric Interval
Sample Size 50

Subsample Size: 40
90% 95%

Coverage Avg. Length Coverage Avg. Length

Bootstrap 88.4 1.2971 93.8 1.5867

Subsample 89.3 1.2943 93.8 1.5848

Jackknife

Subsample 89.6 1.3056 94.1 1.6024

Bootstrap

Subsample Size: 25
90% 95%

Coverage Avg. Length Coverage Avg. Length

Bootstrap 88.4 1.2971 93.8 1.5867

Subsample 89.2 1.2552 93.9 1.5488

Jackknife

Subsample 89.3 1.2551 94.3 1.5513

Bootstrap

Subsample Size: 10
90% 95%

Coverage Avg. Length Coverage Avg. Length

Bootstrap 88.4 1.2971 93.8 1.5867

Subsample 90.6 1.2293 95.5 1.5598

Jackknife

Subsample 87.7 1.1024 93.0 1.3757

Bootstrap

19
 Table 2. Hypothesis Tests Results – Two Equation Model
Non-Symmetric Interval
Sample Size 50

Subsample Size: 40
90% 95%

B11 B12 B22 B11 B12 B22

Bootstrap 89.0 90.2 88.2 94.2 95.6 94.6

Subsample 90.2 91.6 90.6 94.8 96.8 96.0

Jackknife

Subsample 89.0 90.4 88.6 94.8 95.4 93.4

Bootstrap

Subsample Size: 25
90% 95%

B11 B12 B22 B11 B12 B22

Bootstrap 89.0 90.2 88.2 94.2 95.6 94.6

Subsample 90.2 91.6 90.6 94.8 96.8 96.0

Jackknife

Subsample 89.0 90.4 88.6 94.8 95.4 93.4

Bootstrap

Subsample Size: 10
90% 95%

B11 B12 B22 B11 B12 B22

Bootstrap 89.0 90.2 88.2 94.2 95.6 94.6

Subsample 92.8 95.0 95.6 96.8 98.6 99.2

Jackknife

Subsample 92.8 98.4 99.2 97.2 100.0 99.8

Bootstrap

20
 Table 3. Translog Cost Function Estimated Price Elasticities and Confidence Intervals.

Bootstrap Subsample Subsample

Bootstrap Jackknife

ε KK Estimate -0.315* -0.315* -0.315

C. I. (-3.613, -0.304) (-1.781, -0.304) (-3.384, 0.176)

ε KL Estimate 0.550 0.550 0.550

C. I. (-1.509, 0.941) (-1.816, 0.793) (-5.973, 0.956)

ε KE Estimate -0.102 -0.102 -0.102

C. I. (-1.600, 0.616) (-0.879, 0.428) (-0.947, 2.668)

ε KM Estimate -0.133 -0.133 -0.133

C. I. (-0.352, 5.894) (-0.460, 3.509) (-0.677, 8.906)

ε LK Estimate 0.107 0.107 0.107

C. I. (-0.264, 0.150) (-0.360, 0.177) (-0.997, 0.171)

ε LL Estimate -0.359* -0.359* -0.359*

C. I. (-5.339, -0.292) (-7.064, -0.222) (-4.972, -0.36)

ε LE Estimate 0.052 0.052 0.052

C. I. (-0.515, 0.569) (-0.585, 1.115) (-0.090, 1.779)

ε LM Estimate 0.200* 0.200 0.200*

C. I. (0.151, 5.064) (-0.015, 6.188) (0.110, 4.862)

ε EK Estimate -0.122 -0.122 -0.122

C. I. (-2.085, 0.574) (-0.913, 0.494) (-1.168, 3.158)

ε EL Estimate 0.318 0.318 0.318

C. I. (-3.671, 4.482) (-2.614, 6.490) (-0.879, 12.464)

ε EE Estimate -0.592* -0.592* -0.592*

C. I. (-5.656, -0.102) (-4.280, -0.161) (-12.167, -1.186)

ε EM Estimate 0.396 0.396 0.396

C. I. (-0.965, 7.536) (-2.227, 4.543) (-4.780, 4.304)

Notes: * denotes significance at the 5% level. The interval is at 95% confidence.

21
Table 3 Cont. Translog Cost Function Estimated Price Elasticities and Confidence Intervals.

Bootstrap Subsample Subsample

Bootstrap Jackknife

ε MK Estimate -0.011 -0.011 -0.011

C. I. (-0.031, 0.535) (-0.050, 0.306) (-0.050, 0.637)

ε ML Estimate 0.087* 0.087* 0.087*

C. I. (0.061, 2.259) (0.005, 3.036) (0.043, 1.722)

ε ME Estimate 0.028 0.028 0.028

C. I. (-0.088, 0.552) (-0.176, 0.408) (-0.235, 0.279)

ε MM Estimate -0.104* -0.104* -0.104

C. I. (-2.627, -0.108) (-3.286, -0.062) (-2.080, -0.034)

Notes: * denotes significance at the 5% level. The interval is at 95% confidence.

22
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2
Note that this regularity condition does not apply to the Lagrange multiplier test because it is
not affected by the fact that the parameters of interest lie at a boundary of the parameter space
under the null hypothesis. However, it is obvious that this is not the optimal test as it does not
take into account the directional nature of the hypothesis, which implies that the Lagrange
multiplier test is not the most powerful option.
3
Andrews (1999) provides assumptions and sufficient conditions under which the asymptotic
results hold. The proof of the validity of these methods is an extension of a method pioneered
by Chernoff (1954) that approximates the parameter space by a cone in the context of
likelihood ratio tests for iid data. The cone approximation allows a boundary to be linear,
curved, or kinked. Andrews extends Chernoff’s results to allow for trends in the data. The
results are general enough to encompass a variety of objective functions, including log-
likelihood, quasi-log likelihood, least squares, and GMM estimators. The results hold
whether or not the true parameter lies on a boundary of the parameter space.
4
Doubly censored data occur in studies of age-dependent, life-threatening diseases. The
Cramer-von Mises goodness-of-fit test is conducted in order to gauge whether the
assumptions that are made on the underlying lifetime distributions are legitimate.
5
The modified percentile method for a 1- α confidence interval, with B bootstrap replications,
and µ* as the bootstrap distribution, is computed by minimizing .5(µ*(l+(1-α)B) + µ*(l+(1-
α)B+1))-.5(µ*(l-1)+ µ*(l)) with respect to l which is less than .05B. This method finds the
shortest interval that includes (1- α)% of the observations.
6
Lau (1978) proposed the use of the Cholesky decomposition method to impose and test for
curvature constraints in flexible functional forms.

26