bootstrapping demand and supply elasticities: the indian casebdm25/jasian.pdf · bootstrapping...

Bootstrapping Demand and Supply Elasticities: The Indian Case

H. D. VINOD and B. D. MCCULLOUGH

We discuss bootstrap confidence intervals for short-run and long-run demand and supply elasticities for imports and exports. We use Koshal, Shukla and Koirla’s (1992) simultaneous equation demand and supply model for Indian exports based on 1960-1986 data. Although the long run price elasticity of supply is large, (about 25) and significant, the asymptotic as well as bootstrap confidence intervals remain wide. Hence price reduction by devaluation of Indian Rupee alone may not have been enough to spur the major export growth experienced in the early 1990s. We discuss the potential uses of relatively simple bootstrap methods in the context of the delta method for standard errors of long run elasticities, pivoting transform, nonlinearities and nonnormality. (EL F14, C12, C15, C30)

I. INTRODUCTION

In a recent article in this journal, Koshal, Shukla, and Koirla (KSK, 1992) discuss a simultaneous equation demand and supply model for Indian exports based on 1960- 1986 data. A controversial policy issue in the 1970’s and 1980’s was whether reducing prices of Indian exports by devaluation of the Indian Rupee will promote additional exports, or will merely turn the terms-of-trade against India. The latter view prevailed at that time, along with a wide discrepancy between the unofficial (Hawala) open market price and the official price of the Rupee. In international economics, both short-run (SR) and long-run (LR) demand and supply elasticities for imports and exports are often estimated. Although KSK noted that the long run price elasticity of supply is larger than 25, they did not concern themselves with the policy debate mentioned above. They attributed the large estimate to the ease of a shift from a large

H. D. Vinod l Fordham University, Economics Department, Bronx, NY 10458-5158. e-mail:[email protected]

B. D. McCullough l Fordham University, Economics Department, Bronx, NY 10458.5158.

Journal of Asian Economics, Vol. 5. No. 3, 1994, pp. 367-379. Copyright 0 1994 by JAI Press. Inc.

ISSN: 1049.0078 All rights of reproduction in any form reserved.

368 JOURNAL OF ASIAN ECONOMICS (5)3,1YY4

domestic market to foreign markets. Of course, if their large elasticity estimate holds true, it would tend to support devaluation. The Indian experience in the 1990’s suggests that a major devaluation did indeed promote strong growth in exports and improvement in India’s foreign currency reserves. This paper uses relatively newer statistical techniques to reestimate KSK’s model and briefly consider this policy debate and related issues.

Accurate estimation entails not only a point estimate but a measure of the variability of the point estimate and some notion of its sampling distribution. For certain models standard errors for an elasticity are produced by regression software packages, especially in the SR case. Confidence intervals can easily be obtained in the usual fashion. LR elasticities typically are ratios of estimated coefficients, and so confidence intervals are not produced by regression packages. Rather than calculate approximate variances using the delta method (e.g., Greene, 1992, 34.4.4) when appropriate, many researchers interpret long run elasticities as ‘elastic’ or ‘inelastic’ without benefit of formal statistical tests. Such an approach to measurement amounts to casual empiricism, at best.

Perhaps the fact that several econometrics texts published in the 1970’s do not mention the delta method leads many researchers to believe that such confidence intervals cannot be derived. Indeed, in many cases the delta method can be difficult to implement and in some cases cannot be applied at all. Another problem common in international economics, especially for less developed countries (LDC’s), is that data are limited and sample sizes are not large. This implies that the appeal to a central limit theorem necessary to justify such confidence intervals might not apply. Moreover, these confidence intervals produced by ‘plus or minus’ some number of standard deviations are symmetric when, for small samples, the distributions are likely to be asymmetric. One purpose of this article is to suggest that reporting point estimates without confidence intervals should cease. To this end, we offer an intuitive and easily implemented method for determining such confidence intervals which offers several advantages over the traditional asymptotic approach. This method does not assume a symmetric sampling distribution of the parameter, or normal errors. It can yield superior convergence rates to traditional asymptotic theory, and can be implemented in situations where the delta method fails.

Efron’s ( 1979, 1982) ‘bootstrap” generalization of Quenouille’s (1949) ‘jackknife’ enables researchers to obtain more reliable statistical tests and confidence intervals than traditional asymptotic approaches. The bootstrap, like the jackknife, originally was designed to treat the problem of inference in small samples. The bootstrap has been shown to be at least as good as traditional asymptotics under a wide variety of circumstances (Singh, 198 l), and often is better (Navidi, 1989), regardless of sample size. A review of the bootstrap literature appears in Vinod (1993). In recent econometric applications, some researchers have eschewed use of the null bootstrap distribution and its asymmetric confidence intervals in favor of the (simpler but less reliable) standard error of the bootstrap distribution (e.g., Krinsky and Robb, 1986). Some who have used the bootstrap have failed to employ the pivoting transformation which

Bootstrapping Demand and Supply Elasticities 369

enhances the reliability of bootstrap confidence intervals ( e.g., Dorfman, Kling and Sexton, 1990). An additional appeal of the bootstrap is that one can produce plots for estimated sampling distributions superimposed with the closest theoretical distribution. Vinod and McCullough (1994) have suggested these plots as data analytic tools valuable for statistical inference. In formal statistical inference, an applied researcher is often uncomfortable about the rule-of-thumb 95% interval and the normality assumption. Our plots are claimed to be more informative due to faster convergence (traditional asymptotics is G-consistent, while the bootstrap is n-consistent or n2

depending upon appropriate transformations) and because they do not assume normality.

In this article we explain the bootstrap method and the pivoting transformation, and apply these methods to update some recently reported results. In addition, we compare the confidence intervals produced by a number of different methods. Section two briefly outlines the model. Section three introduces the bootstrap, including two-stage least squares (2SLS) bootstrap with a lagged dependent variable (LDV) as applied by Freedman and Peters (1984). Section four presents the estimation results, including graphs comparing the traditional asymptotic distributions and bootstrap distributions. Section five offers the conclusions.

2. THE MODEL

KSK (1992) estimate demand and supply elasticities for the elasticities of Indian exports using a system of multiplicative simultaneous equations. Since KSK present the details regarding both the model and data, we only sketch their model. They posit a multiplicative form for the desired demand for Indian exports (DX,) in year t as a function of the price of exports (PJ, world income (Y,,), and the price of goods in the rest of the world (P,,). Similarly, the desired supply of exports (SX,) from India also is given a multiplicative form as a function of export price (P,J and the domestic price index (Pdt). An adjustment process is introduced to each equation by lagging DX, one period with an adjustment parameter h. On taking logs of both sides of each equation and appending error terms U, to the demand equation and V, to the supply equation, the estimable forms of the equations are given as follows:

lnDX, = hlnh + ha,lnP,, - ha,lnP,, + ha,lnY,, + (1 - h)lnDX,_, + U, (1)

lnSX, = 61r& + GP,lnP,, - GP,lnP,, + (1 - h)lnSX,_, + V, (2)

Equilibrium is achieved by setting lnDX,= InSX,. The model has two endogenous variables: lnDX, (= lnSX, ) and InP,,. The four exogenous/predetermined variables are: Pwf, Y,,, Pd,, and DX,,(= SX,,). To make the relative price model assumption valid, the absolute value of the coefficient of lnP,, must equal the coefficient of InP,,,, in Equation 1 and the absolute value of the coefficient of lnP,, must equal to the coefficient of lnP,, in Equation 2. These restrictions are embodied in the presentation

370 JOURNAL OF ASIAN ECONOMICS (5)3,1994

TABLE 1. Summary of KSK Results, standard errors in parentheses

Elasticity Short run Long run

Demand price -0.95 -2.90

(0.486) (M.)

Demand income 1.31 4.01 (0.543) (n.r.)

Supply price 1.91 25.50

(0.676) (nx.)

Norc: n.r. signifies ‘not reported’

of the model, since a, is the coefficient of lnP,, and --a, is the coefficient of lnP,, in

Equation 1 and similarly for p, in Equation 2. The data, 27 annual observations over

1960-86, for each series, are discussed in KSK. Their estimation method was 2SLS. It is convenient to devise a four character notation scheme for elasticities. The short

run and long run are denoted by SR and LR respectively and are always the first two

characters. The elasticity is always that of a dependent variable with respect to (wrt) a

regressor variable. Our third character is the dependent variable of Equations 1 or 2,

such as quantity demanded (D), or quantity supplied (S). The fourth character refers

to the regressor variable which in the demand Equation 1 is either relative price (P) or

income (Y) and only relative price (P) for supply Equation 2. KSK’s results are

summarized in Table 1.

KSK give the following interpretations to their results. Since -0.95/0.486 = -1.96

they reject the null that the short run elasticity of demand with wrt price (SRDP) equals

zero. Without benefit of formal statistical tests, they assert that both SRDY and SRSP

are ‘elastic.’ Again without benefit of formal statistical tests, they conclude that all the

LR elasticities (LRDP, LRDY and LRSP) are ‘high,’ and that the LR supply curve is

almost perfectly elastic. While the signs of KSK’s elasticities are plausible, none of the above conclusions

is supported even by traditional methods. For example, the t-statistic on SRSY, -1.96,

is blot significant at the five percent level. The appropriate critical value is not -1.96

from a table for the standard normal, but -2.074 from a t-distribution with 27 - 5 = 22

degrees of freedom. From their own reported coefficients and standard errors, a 95%

confidence interval for SRDY is [O. 184,2.436] and for SRSP it is [OSOS, 3.3 11, neither of which suggests an ‘elastic’ response.

As far as the long run elasticities are concerned, KSK report no standard errors and

thus cannot make assertions about the confidence intervals for these elasticities. The traditional statistics would employ the delta method to yield a large-sample approxi- mation to the variance of the LR elasticities. These are obtained as follows. Suppose

we are interested in estimating the large-sample variance of LRDP = $ = bZ/( 1 - b4) where the f3i are from the two stage least squares regressions (these pl are not to be


confused with those in Equation 2). Then assuming that the variance exists, the delta method yields:

(3)

In the same fashion, asymptotic standard errors can be estimated for the other supply

and demand long run elasticities. We note here an important advantage of the bootstrap. If p5 is not bounded away from unity, then the delta method will fail. Moreover, if 6 5 is close to unity in the sample, then estimated standard error will be unstable. The bootstrap is not plagued by either of these problems.

The asymptotic variances thus estimated for the LR elasticities are as follows: LRDY = 0.84945, LRDP = 1.14949, LRSP = 1140.3. The 95% confidence intervals

for the long run elasticities are: LRDP: [-5.000, -0.800]; LRDY: [2.204,5.816]; LRSP: [-40.980,91.380]. To contradict the conclusions of KSK, it is readily seen that LRDP is not different from unity, LRDY is elastic, and nothing can be said about LRSP except there is no evidence that the LR supply curve is flat.

Each of the above confidence intervals, for both SR and LR elasticities, is symmetric and based on traditional l/G asymptotics which are not appropriate for n = 27. This is especially evident given that the t-distribution and not the normal distribution is relevant for traditional hypothesis testing. Furthermore, the use of the

t-distribution implies an assumption of normal errors, which is not supported by diagnostic testing. An examination of the residuals from each regression rejects the null of normality, so neither the asymptotic standard normal (if the sample were large

enough) nor the t-table (and an implicit assumption of exact normality) can be reliably employed. Non-normal errors, however, are not a problem for the bootstrap.

Bera and Jarque (1980a, 1980b) have shown that the following statistic has a x2 distribution with two degrees of freedom:

+ (kurtosis- 3)2

24 1 where skewness and kurtosis are, in the present case, calculated from the residuals of the supply and demand regressions. For supply, L = 13.516 and for demand L = 8.776, while a 5% test for a x2 with two degrees of freedom has critical value 5.99 and, thus, normality of the errors is rejected. Thus, there is no alternative but to employ a testing procedure which does not depend upon normality. Such a procedure is the bootstrap which can be shown to have l/n convergence rate and does not depend on an assumption of normal errors. In the present case bootstrap confidence intervals and bootstrap hypothesis tests are to be preferred to traditional asymptotic confidence intervals and tests.


3. THE BOOTSTRAP

Consider the k-variable linear model with y1 observations, y = Xl3 + E, with the usual assumptions (excluding normality of the error). Here we only sketch the bootstrap method; a more detailed exposition can be found in McCullough and Vinod (1993). Bootstrap parameter estimates are obtained by running an initial OLS regression to obtain 6, the estimate of p, and the residual vector e, which estimates E. After resealing the residuals by a factor of 6&?& they are randomly sampled with replacement from a uniform distribution to produce e*, a vector of pseudo-errors. A pseudo-y vector is formed as y* = Xb + e*. Regressing y* on X yields bootstrap parameter estimates 6*. This procedure is repeated a large number of times J = 1000 say, to yield b;, j = 1, . . , J.

On an element-by-element basis these may be sorted to yield the bootstrap distribution of each parameter. In the case of elasticities, on each iterationj, form the bootstrap estimate of the elasticity q;, = as a ratio of parameter estimates. This yields I$, j= 1, . . . , J, which may be sorted to yield the bootstrap distribution of the elasticity. A 95% confidence interval may be obtained by selecting order statistics 25 and 975 from the sorted distribution when, say, J = 1000.

This method, known as the ‘percentile method,’ can be improved upon by a simple transformation because b* -b often is not pivotal (Hinkley, 1988, $4). A more accurate method is the ‘percentile-t method’ which forms a studentized statistic at each bootstrap iteration j. In the present case, a vector of such statistics is T = (by - b)/se(b;) where se(b,T) is the vector of standard errors of the bootstrap coefficient estimates. On an element-by-element basis these may be sorted to form confidence intervals for the pivotal statistic (b - P,J/se(b) to yield hypothesis tests that 0 = B0 on a coefficient-by- coefficient basis. Alternatively, the statistic T can be inverse transformed to yield confidence intervals for each coefficient, as follows.

For an individual coefficient, form the J x 2 matrix (c, bj*) ,j = 1, . . , J. Sort the matrix according to T from low to high. If J=lOOO, say, a 95% confidence interval for p is given by &, b& . This method is more accurate than simply sorting the by and selecting entries 25 and 975, because the pivoting transformation enhances the rapidity of convergence.

A problem arises in pivoting LR elasticities, because there are no stable standard error estimates upon which to pivot. The asymptotic standard error produced by the delta method needed for the denominator of the pivot is too unstable for reliable pivoting. This is not unusual; the percentile-t method does not work well when the parameter of interest is a correlation coefficient or a ratio of means. Therefore, we have no recourse other than the percentile method. However, the percentile method can still be expected to produce more reliable confidence intervals than traditional asymptotics in small samples.

One final complication must be mentioned. In the lagged dependent variable regression case further care is needed in forming the bootstrap resamples y*. As in Freedman and Peters (1984) we fix the initial value of y so that y*” = yO. Successive values of y* are constructed recursively, y: = yT_i + X,b + ef, where the errors are


obtained by a bootstrap shuffle. For dependent data, moving block and other methods of generating errors described in Vinod (1993) are possible, but not attempted here.

4. BOOTSTRAP IMPLICATIONS AND RESULTS

The bootstrap procedure is straightforward.’ For each equation, we run an initial 2SLS regression with the relative price model restrictions imposed (these initial estimates are the same as those of KSK) and save the residuals, which are resealed. For each of j=l,..., 1000 bootstrap iterations, we form y,? as in the above paragraph and run a 2SLS regression, again imposing the restrictions of the relative price model, saving the appropriate coefficients and forming the appropriate elasticities, pivoting when necessary (as in the case of the SR elasticities).’

Table 2 compares the asymptotic and bootstrap confidence intervals for the six elasticities. Percentile confidence intervals are shown for all elasticities, while percentile-t confidence intervals also are shown for the SR elasticities. With respect to the percentile confidence intervals, the only qualitative difference is for SRDP, where the asymptotic confidence interval contains the origin whereas the bootstrap interval does not. To the extent that the asymptotic interval is inconsistent with economic theory, the advantage of using the bootstrap is clear here. However, for the percentile-t confidence interval, another change is noted: SRSP does not include the origin and is seen to be greater than unity, suggesting an elastic response whereas both the asymptotic and bootstrap percentile intervals suggest that SRSP is not significantly different from zero.

Table 3 presents the mean, standard deviation, and traditional skewness and kurtosis tests for the six bootstrap distributions. Clearly, non-normality and hence an incompatibility with the traditional asymptotic approach is evident in this small sample case. More revealing, however, is a comparison between the empirical bootstrap distributions and their theoretical asymptotic counterparts,

Figures la, lb, and lc show the bootstrap percentile distributions for SRDP, SRDY, and SRSP, respectively, together with the asymptotic normal distribution with mean equal to the parameter estimate and standard deviation equal to the standard error of the parameter estimate. The bootstrap SRDP distribution is clearly skew left, while the bootstrap SRDY is skew right. In contrast, the bootstrap SRSP is decidedly

TABLE 2. Comparison of 95 % Confidence Intervals

Elasticity Asymptoptic Percentile Percentile-t

SRDP [-1.959,0.054] [-2.177, -0.300] [-1.658, -0.3661

LRDP [-5.000, -0.800] [-4.905, -0.6781 ma.

SRDY [O. 189, 2.4391 [0.819,2.983] [0.976, 2.5221

LRDY [2.204,5.816] [2.002,5.565] n.a.

SRSP [-0.593,4.414] [-0.152,5.021] [2.533, 3.2471

LRSP [-40.98,91.38] [-145.4, 169.71 ma.

Note: “.a. slgnities ‘not applicable.’


t *)

8 .I

Bootstrapping Demand and Supply Elasticities

TABLE 3. Summary Statistics on Bootstrap Percentile Distributions

375

elasticity mean st. dev. skewness kurtosis

SRDP -1.129 0.482 -0.897 2.622

LRDP -2.5736 1.045 -0.596 1.608

SRDY 1.646 0.572 1.460 5.564

LRDY 3.671 0.924 0.677 2.381

SRSP 2.777 1.227 -0.858 3.674

LRSP 25.963 649.9 18.22 544.8

leptokurtic. Clearly, there can be marked discrepancies between the bootstrap distri-

bution and the asymptotic distribution. Since the bootstrap distribution, by virtue of its

faster convergence rate, can be expected to be closer to the true, exact, finite sample

distribution than the asymptotic distribution, these graphs clearly show the inadequacy of the asymptotic approach when dealing with small samples.

Additional insight into the bootstrap is gained from Figures 2a, 2b, and 2c, which

show the percentile-t (pivoted) distributions for SRDP, SRDY, and SRSP, respec-

tively, together with normal distributions which have the same mean and variance as

the bootstrap distributions.3 Note that the discrepancy between the bootstrap distribu-

tion and the normal distribution is substantially less, though still skew, reflecting the

fact that the pivoted statistic can converge more quickly than the unpivoted statistic.

Thus, these figures show that confidence intervals produced via pivoting are likely to

be more reliable than unpivoted confidence intervals, as demonstrated theoretically by Beran (1987, 1988).

Figures 3a, 3b, and 3c show the percentile distributions for LRDP, LRDY, and

LRSP, respectively. Again, a substantial discrepancy between the bootstrap distribu-

tion and the asymptotic distribution may be observed, especially in the case of LRSP.

These graphs show the unreliability of the asymptotic delta method, when the sample size is small. The LRSP, which was mentioned in the Introduction in the context of a

policy debate is indeed not only large in magnitude, but has positive skewness.

5. CONCLUSIONS

Using the annual data for 196&1986 we report confidence intervals on demand and

supply elasticities with respect to price and income. The long run price elasticity of

supply (LRSP) has large mean (>25) and has positive skewness. Since the bootstrap and asymptotic confidence intervals for LRSP are wide, we may speculate that the recent strong export performance of the Indian economy may not have been obtained by the Rupee devaluation alone, without being accompanied by market reforms started by the Finance Minister Mr. Singh.


One aim of this paper has been to suggest that point estimates no longer be reported without confidence intervals, and we have offered a simple and intuitive method for deriving such confidence intervals in situations where the output of a standard regres-

sion output is insufficient to the task. Another aim of this paper has been to draw the attention of researchers using data from Asian countries to the computer-intensive bootstrap, which is shown to be a powerful tool in the context of their empirical research. The bootstrap enables reliable formal statistical inference despite the data limitations, ubiquitous nonlinearities (arising from explosive growth in population, stock market variables, prices, etc.) and nonnormalities of errors often present in empirical models for Asian countries.

We have briefly explained the bootstrap and how it can be used to obtain better confidence intervals for estimated parameters (such as short-run elasticities) and nonlinear functions of estimated parameters (such as long-run elasticities). We have also explained the pivoting transformation, of which many bootstrap practioners are unaware, as a means of improving the reliability of bootstrap confidence intervals when a stable pivot exists. Our data-analytic graphs reveal marked discrepancies between the traditional asymptotic theory and the bootstrap approach to determining confidence intervals, and can be an important tool in assessing model reliability. Finally, we have

updated recently reported results on the supply and demand elasticities for Indian exports.

Acknowledgments: We thank D. Kwiatkoski for comments and M. Shults for computational

assistance.

NOTES

I. We initially attempted a more ambitious double-bootstrap, but failed for two reasons. First, the double-bootstrap requires pivoting at each stage, and here we can’t pivot for the long run elasticities. Second, while we could pivot for the short-run elasticities, collinearity in the data caused singular matrices to arise at the second stage of the double bootstrap. The double-bootstrap is not always superior to the single-bootstrap. Vinod (1994) discusses a modification of the single bootstrap for multicollinear data.

2. For LR elasticities it is really necessary. but not feasible.

3. In traditional OLS, as opposed to 2SLS with restrictions imposed, the pivoted statistic converges to a standard normal. Hence wed; not compare the pivoted distribution to the standard normal. Based on our experience, we suggest three possible reasons for the failure of the pivoted distributions to converge to standard normals: (I) The restrictions of the relative price model are invalid; (2) The model is misspecified; (3) The Freedman and Peters method of treating LDV is incorrect. However, WC do not pursue these matters.

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Received April 1993; Revised June 1994

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