- Email: [email protected]
Bootstrapping in multiplicative models
Journal
of Econometrics
42 (1989) 287-297.
BOOTSTRAPPING
North-Holland
IN MULTIPLICATIVE
MODELS*
M.S. SRIVASTAVA Univewity of Toronto, Toronto. Cunuda MSS IA1
Balvir SINGHt Concordia Uniuersif~ Received
October
1985, final version received September
1987
This paper develops the bootstrap theory to obtain a confidence interval for the constant term of the Cobb-Douglas multiplicative model. This also proposes a simple and easily computable estimator of the constant term along with its mean square error.
1. Introduction Multiplicative models of the Cobb/Douglas/translog type have been very extensively used in economics and other behavioural sciences. Among other advantages, these models provide a useful framework for situations where the variables are measured in different units or where the data involve wide fluctuations. In the empirical implementation of such models it has been customary to assume a multiplicative lognormal disturbance term so that an apparently nonlinear model could be linearized (in logarithms) and estimated by classical estimation techniques. However, the problem of estimating the constant term and finding a confidence interval for it has always presented some problems. Bradu and Mundlak (1970) gave an uniformly minimum variance unbiased estimator (UMVUE), but it uses a g-function, introduced by Finney (1941) which requires extensive tables and yields unacceptable negative values for some values of its arguments [see Teeken and Korets (1972, p. 804)]. In this paper, we propose another estimator which, though biased, gives results almost identical with the UMVUE. *Presented at the Fifth World Congress of the Econometric Society held in Boston. August 1985. Thanks are due to two referees and Prof. T. Amemiya for their helpful suggestions. Thanks are also due to Dr. Ah Born Sim of Concordia University and Dr. K.J. Keen of the University of Toronto for carrying out the calculations in this paper. The research of the first author was supported by the Natural Sciences and Engineering Research Council of Canada. ‘Deceased.
0304-4076/X9/$3.5001989,
El.aevier Science Publishers
B.V. (North-Holland)
288
M.S. Srivasiuvu and B. Singh, Bootstrapping
in multiplicutive
models
In order to find a confidence interval for the constant term, we need the distributions of its estimators, which even under the assumption of lognormality is difficult to obtain and is not available in the literature. Alternatively we may apply the asymptotic theory and use normal or t-tables. However, it has been demonstrated by Beran and Srivastava (1985) and Efron (1979,1985) that the confidence intervals obtained by the asymptotic theory are unsatisfactory when compared with the exact results. On the other hand, the bootstrap method of Efron usually gives improved confidence intervals in many numerical examples considered in the literature. We thus develop the bootstrap theory to obtain a confidence interval for the constant term. The organization of this paper is as follows. In the next section, we discuss the model and classical estimators. In section 3 we shall briefly outline the bootstrap method, while the theory is given in section 4. Section 5 gives numerical results. 2. The model and the classical estimation methods Consider
a stochastic
multiplicative
model,
K
i=l
q=llBx$q,
k=2
‘...,
n,
(1)
where Yj is the ith observation on the dependent variable, X,, is the corresponding observation on the k th (k = 2,. . . , K) nonstochastic independent variable, and B and pk (k = 2,. . . , k) are the unknown parameters. Let U, = log ui. It is assumed that ui’s are i.i.d. (independent and identically a*), where u > 0 and is unknown. This implies that distributed) N( - $J *, U, > 0 and E( u;) = 1, as required by situations in economics where such models are used. Letting y, = log q, x,~ = log X;k, E; = U, + $a*, and & = log B - ;a’,
(2)
the model (1) after taking logarithms notation as
on both sides can be rewritten
in matrix
y=xp+&,
(3)
where
.II Yl
3
i
El E=
)
i- Et,i
x= i
1
x1*
. .
. .
i
x,,
***
XIK
. . .
X,K
\
I
M.S. Srivustava and B. Singh, Bootstrapping in multiplicutive models
Hence from the Gauss-Markov theorem, with or without the assumption normality, the best linear unbiased estimator of j3 is given by p^=
(xx-‘X’y,
and an unbiased
an estimator
of
(4)
estimator
of a* is given by
s2=(n-K)-‘y’[1-X(X’X)-‘X’]4’. Thus,
289
(5)
of B may be given by
j=,B,+Y.
(6)
Under the assumption of lognormality, this estimator is a modified maximum likelihood estimator since the divisor in s2 is n - K instead of n. However, this is a biased estimator. Using Finney’s (1941) results for estimating the parameters of a lognormal distribution, Bradu and Mundlak (1970) gave a uniformly minimum variance unbiased estimator of B. This UMVUE is given by
(7) where J tJ
j!’ m=n-K,
y=l-h,
h = a’( XT-%, The variance
a’= (1,o )...) 0).
of B, is given by
var( i,)
= B2[e2(h-:)sZG,(
&q2)
- 11,
where
’
y=l-h.
290
Bradu
M.S. Srivustavu and B. Singh, Bootstrupprng in multiplicutive
and Mundlak
(1970) suggest estimating
models
it by
in which E( jf) has been replaced by &f; from the numerical results in tables 1 and 2 it appears that this estimator needs some improvement. Although, Bradu and Mundlak (1970) provide a table for Lhe g-function, more extensive tabulation is needed to be able to compute B, for all cases. Also, it may sometimes give an unacceptable negative estimator. To overcome this difficulty several alternative but biased estimators have been proposed in the literature. Since, under normality assumption,
we propose
the following
of B:
estimator
&, = ,h + w,
(8)
which has 2
E(&)
‘[e--Iro’]
&I!!_
=B
[
m
-:m
I
’
MSE( j,)
An estimate
of MSE(&)
is given by
-- i ni =
which,
+
e21jl
for large m, can be approximated Ea(
$,)
= e28,+(t-2h)s2 [eh.r’ _
by
11.
,(I
2h).s?
M.S. Srivustuva
and B. Singh. Bootstrupping
in mulliplicuiwe
models
291
Table 1 ,I
10
20
40
a*
h
0.50
0.2 0.4 0.6 0.8
0.5959e-02 0.3315e-02 O.l457e-02 0.3606e-03
0.1210 0.2316 0.3550 0.4933
0.1179 0.2293 0.3537 0.4929
1.0
0.2 0.4 0.6 0.8
0.2507e-01 O.l374e-01 0.5959e-02 O.l457e-02
0.3110 0.5537 0.8559 1.236
0.2782 0.5306 0.8430 1.232
2.0
0.2 0.4 0.6 0.8
0.1144 0.5991e-01 0.2507e-01 0.5959e-02
1.327 1.794 1.650 4.065
0.7848 1.464 2.475 4.010
0.50
0.2 0.4 0.6 0.8
0.2393e-02 O.l340e-02 0.5931e-03 O.l476e-03
1.1112 0.2254 0.3519 0.4924
0.1104 0.2246 0.3514 0.4923
1.0
0.2 0.4 0.6 0.8
0.9765e-02 O.O437e-02 0.2393e-02 0.5931e-03
0.2520 0.5145 0.8351 1.230
0.2446 0.5077 0.8307 1.228
2.0
0.2 0.4 0.6 0.8
0.4101e-01 0.2248e-01 0.9765e-02 0.2393e-02
0.6909 1.399 2.437 3.997
0.6080 1.322 2.383 3.976
0.2 0.4 0.6 0.8
O.l497e-02 0.8399e-03 0.3723e-03 0.9283e-04
0.1089 0.2239 0.3511 0.4922
0.1085 0.2234 0.3509 0.4921
0.2 0.4 0.6 0.8
0.6064e-02 0.3389e-02 O.l497e-02 0.3723e-03
0.2398 0.5057 0.8302 1.228
0.2360 0.5018 0.8275 1.227
0.2 0.4 0.6 0.8
0.2499e-01 O.l384e-01 0.6064e-02 O.l497e-02
0.6038 1.328 2.391 3.980
0.5641 1.286 2.360 3.96X
0.50
0.2 0.4 0.6 0.8
O.l090e-02 O.h116e-03 0.2713e-03 0.6769e-04
0.1079 0.2232 0.3508 0.4921
0.1076 0.2229 0.3506 0.4920
1.0
0.2 0.4 0.6 0.8
0.4397e-02 0.2462e-02 O.l090e-02 0.2713e-03
9.2346 0.5018 0.8280 1.227
0.2320 0.4991 0.X261 1.227
2.0
0.2 0.4 0.6 0.8
O.l797e-01 0.995e-02 0.4397c-02 O.l090e-02
0.5656 1.29X 2.371 3.973
0.5443 1.269 ?.349 3.964
Bias (&)/B
MSE ( & ),‘B*
var( 8, )/B*
292
M.S. Srivusruva and B. Singh, Bootstrupping
In many
econometric
studies,
in multiplicutive
models
an estimator
has been used to estimate B. In their example of Israfli agriculture, Bradu and Mundlak (1970) have shown that the bias@ using B, as an estimator of B is nearly 12%. Thu:, we do, not consider B, in this paper and compare the performance of B, with BiAonly. For variou: values of n, h, and u2, table 1 records Bias( B2)/B, MSE( B2)/B2, and var( B1)/B2. It is clear from this table that the bias ratio is negligible even forAa sample size asAsmall as 10, with almost negligible increase in the MSE(B,)/B’ over var(B,)/B2. Thus, the performance of the proposed estimator B, is almost identical to that of Bradu-Mundlak’s UMVU estimator B,. Given the inherently complicated nature of the g-function and the need of its extensive tabulation for the computation which may sometimes give unacceptable negative values, the proposed estimator may be preferred. In this paper we give the bootstrap theory to obtain a confidence interval for B. The results are, however, applicable to more general situations such as
where
c and 11 can be chosen appropriately. f(B,a’)=B = Y,
for
~‘=a’,
q=$,
for
c’=xi,
?I=:.
For example,
We suggest using (8) as an estimator with appropriate estimating Y,, y will change to 1 - xi( X’X)-lxk. 3. Bootstrap
y. For example,
in
method for obtaining confidence interval
First, we remove the assumption of normality of E in (3). Here, it would suffice to assume that the n components of the disturbance vector E’ = E,)’ are i.i.d. with unknown cumulative distribution function (cdf), F (q,..., (say), mean zero and variance u 2( F). We obtain the OLS estimator same as (4) and the residual vector c=M&=My,
(10)
where M=I-P
and
P=X(X’X)-‘X’.
(11)
M.S. Srivastava and B. Singh, Bootstrapping
293
in multiplicative models
Let us now take a random sample with replacement from &, . . . , S,, denoted The bootstrap OLS estimate B* of p by, say, Z: ,..., El,*.Let E* =(E: ,..., E:)‘. is then given by p* =p1+
(x’x)-‘x’;*.
(12)
Since the regression model (3) has an intercept2 CEI, = 0. Hence for the above sampling scheme, E*(EI*) = 0 and E*/?* = p, where E* denotes the conditional expectation given E^r,. . . , i,. The bootstrap estimate of a*(F) is given by
It will be shown in section 4 that the asymptotic distribution of c’( p^ - p) + $y(S* - 02(F)) is the same as that of c’(p^* - /?) + ~Y(S*~ - (n - K) x n ~‘S*). The distribution of the latter is approximated by Monte Carlo method by obtaining L independent samples of size n with replacement from $, . . . , El, and calculating the empirical cumulative distribution function, where L is chosen large, usually between 200 to 1000. In other words the L values of c$* + $s*’ are ordered, and if dz,* and dI*_,,, denote the +L~I and (1 - (u/2)L points of these ordered values, the bootstrap (1 - a) x 100% confidence interval for ee, B = c’p + 1ze*, is given by edzjz, ed,*_@. Thus, by choosing c’ = a’ = (l,O, . . . ,O), we get the confidence interval for B. This is the percentile method of Efron (1982). 4. Bootstrap theory To develop the elements
the K
(Cl)
max x;( 1ci5n
a Cramer-Wold following:
first given method
1. asymptotically
theory for n matrix
shall assume that
= 0, Srivastava verifying the
Under Srivastava ‘s condition
[(G - v)‘, S* - a*(F)] where
bootstrap method = (x,, . . xn) are
(1968,
1972). Using condition, we
(Cl ) on the matrix
- N(K+l) [O, Q(F)]
1
X,
we
the
have
294
M.S. Srivustavcr and B. Singh, Bootstrapping in multiphcutive models
Corollary 1. Let g be a continuously nonvanishing derivative at (q’, a’(F)); cal& g(f/,
differentiable function of (p’, S2) with a g: RKf ’ -+ R’. Then we have asymptoti-
s2) - g(c 0’) - N,(O,S(Fk’)~
where the 2 is the r X (K + 1)
of
of
evaluated
at q and
We shall now consider the model (3) with K 3. In case the elements of the first column of the X matrix are all ones /3i is the constant of the For the theory hold we shall assume that the elements of the X are such
-
a2
_
1_
nEiCxf,-
[
(
CxZi)(
2X*X3CX2iX3r Xxii)
-
+ ‘:Cx5i
(CX2iX3r)*
1
has a finite limit. Clearly, if n-‘( X’X) tends to be a positive definite matrix, a2 will have a finite limit. However, this condition is not necessary as shown in Srivastava and Srivastava (1986). Corollary
2.
Suppose
K = 3 and a2 has a jinite
limit for the model (3).
Then
h{ [(&+ is’> - (P,+ :u’)]} -+
N(0,a2a2( F) + jd2(F)).
We shall now proceed to give the bootstrap theory. Let F,, be the empirical cdf of ei,. . . , E,. Then it is well known that F, + F as. However, we are sampling from .Zi,. . , ZN. Let 8” be its empirical cdf. By following Freedman (1981) or Srivastava (1985) it can be shown that e, - F, -+ 0 in probability and hence i” -+ F in probability. The mean of the distribution e,$ p( 6,) = nP1xEnr = 0 since the regression equation has a constant term. Similarly, the variance of the distribution E’,,, p2( gn) = n -‘~:=, Ci, -+ u’(F) a.s., and the [see Srivasfourth moment of in, p4( &) = n-lx’._ ,_1 .?p-+ p4( F) in probability tava (1985)]. Also, (n - 1))%*‘[ I - n - ‘ll’]e* -+ u2( F) in probability. Thus, we have: Theorem 2. Let cl,. . . , .?N be:he residuals with empirica! cdf 6,. Then 6, -+ F in probability, p( F,) = 0, a’( F,) + u2( F) a.s., and p4( F,,) -+ pLq(F) in probability.
M.S. Srivastuva
and B. Singh, Bootstruppmg
in multiplicuiive
295
models
Noting that Corollary 2 was obtained for any class of distributions finite fourth moment, we get the following result by applying the central theorem to triangular arrays as in Beran and Srivastava (1985). Theorem
3.
Under the conditions of Corollary 2,
&{(&+ -
with limit
N(0,
$s**- (j&-t a*u*(F)
f(n-Iqn-‘9))
IE^;,}
+ id*(F)).
5. Empirical results In this section, we shall report some empirical results to demonstrate the performance of the bootstrap method in comparison to the two estimators given in (7) and (8). In this connection, we first consider the actual example of Israeli agriculture investigated by Bradu and Mundlak (1970) and then present a simulation experiment which uses the same x-variables.
5.1. The Bradu-Mundlak Let us consider
example
the trivariate
of Israeli agriculture
Bradu-Mundlak
model:
where Y, stands for the output of ground nuts relative to that of cotton in year i, X2 i_2 for the price of ground nuts relative to that of cotton in year i - 2, and x3; for the trend term. Linearizing by the logarithms, we get the model (3) with K = 3. The regression has been estimated for the period 1953-63. Table 2
Table 2 Comparison
between
the proposed
and the Bradu-Mundlak
estimators
of B.
Proposed
B-M
Estimates
34.159
34.146
E( B)/=(B) Monte Carlo confidence limits” Confidence limitsh Bootstrap confidence
16.29
16.68
(4.771.110.70) ( - 4.367,73.67) (17.35,80.45)
(4.744.108.70) (- S.303.73.59)
limits’
a Based on 1000 simulations. ‘Confidence limits are obtained using r-values ‘Based on 200 bootstrap samples.
296
M.S. Sriuastava and B. Singh, Bootstrapping
in multiplicative
models
provides a comparative picture of the proposed and the Bradu-Mundlak estimators. The least squares estimators of pi, &, and & are 3.590, 2.840, and -0.355, respectively, and that of u* is 0.138. It is to be noted here that the estimate of B given above differs from that reported by Bradu and Mundlak (1970) who_ for some unknown reasons apply the g-correction to eP1+ is2 rather than to epl+ ~ys*. As is evident, the proposed estimator of B reported above is almost equal to Bradu-Mundlak’s UMVU estimator. Accordingly, to save computing time, we obtain the bootstrap confidence interval for the proposed estimator only. The theory, however, is applicable to the confidence interval based on the Bradu-Mundlak estimator. The bootstrap confidence interval for B, based on 200 bootstrap samples, is given by (17.35, 80.45) and is similar to the one obtained by large-sample theory using t-values. The confidence interval by the Monte Carlo method, based on 1000 simulations, seems to favour the one obtained by the bootstrap method.
5.2. Some simulations We recognize that the results based on real data may always be questioned because of the possible difference between the real world and the assumed theoretical distribution of the error term E. Consequently, we resort to some simulation exercises where such problems do not arise. Here, we take the same values of x-variables as in Bradu-Mundlak, and the parameter values are nearly equal to their estimates. In other words, we take B = 32.00, fi2 = 2.80, & = -0.365, and a2 = 0.14, and we assume that the E~‘S are i.i.d. normal variables with mean zero and variance 0.14. In this experiment, we make 50 Monte Carlo trials and (as above) 200 bootstrap replications. The results of this experiment are presented in table 3. These results are similar to table 2.
Table 3 Comparison
of the proposed
and the Bradu-Mundlak estimators B = 32.00. Proposed
Estimates Bias -MSE/var Confidence limit? Bootstrap confidence
32.931 0.937
limits
(simulation
result);
true value of
B-M 32.921 0.921
289.388
307.894
(- 5.890,71.765) (9.988,71.001)
( - 8.463,74.303)
“Confidence limits are obtained using t-values ‘Based on 200 bootstrap samples.
M.S. Srivustuva and B. Singh, Bootstrupping
m multiplicative
models
291
References Beran, R. and M.S. Srivastava, 1985, Bootstrap tests and confidence regions for functions of a covariance matrix, Annals of Statistics 13, 95-115. Bradu, D. and Y. Mundlak, 1970, Estimation in lognormal linear models, Journal of the American Statistical Association 65, no. 329, 198-211. Efron. B., 1979. Bootstrap methods: Another look at the jackknife, Annals of Statistics 7, l-26. Efron, B., 1982, The jackknife, the bootstrap and other resampling plans (Society for Industrial and Applied Mathematics, Philadelphia, PA). Efron, B., 1985. Bootstrap confidence intervals for a class of parametric problems, Biometrika 72, 45-58. Finney. D.J., 1941, On the distribution of a variate whose logarithm is normally distributed. Journal of the Royal Statistical Society, Suppl. I, 7, 155-161. Freedman, D., 1981, Bootstrapping regression models, Annals of Statistics 11, 569-576. Srivastava, M.S., 1968, On a class of nonparametric tests for regression parameters, Annals of Mathematical Statistics 39, 697. Srivastava, M.S., 1971, On fixed-width confidence bounds for regression parameters, Annals of Mathematical Statistics 42, 1403-1411. Srivastava, M.S., 1972, Asymptotically most powerful rank tests for regression parameters in MANOVA. Annals of the Institute of Statistical Mathematics 24. 285-297. Srivastava, M.S., 1985, Bootstrapping Durbin-Watson statistic, Technical report no. 5 (University of Toronto, Toronto). Srivastava, M.S. and V.K. Srivastava, 1986, Asymptotic distribution of least squares estimator and a test statistic in linear regression models, Economics Letters 21, 173-176. Teeken, R. and J. Korets. 1972, Some statistical implication of the log transformations of multiplicative model, Econometrika 40, 793-819.
of Econometrics
42 (1989) 287-297.
BOOTSTRAPPING
North-Holland
IN MULTIPLICATIVE
MODELS*
M.S. SRIVASTAVA Univewity of Toronto, Toronto. Cunuda MSS IA1
Balvir SINGHt Concordia Uniuersif~ Received
October
1985, final version received September
1987
This paper develops the bootstrap theory to obtain a confidence interval for the constant term of the Cobb-Douglas multiplicative model. This also proposes a simple and easily computable estimator of the constant term along with its mean square error.
1. Introduction Multiplicative models of the Cobb/Douglas/translog type have been very extensively used in economics and other behavioural sciences. Among other advantages, these models provide a useful framework for situations where the variables are measured in different units or where the data involve wide fluctuations. In the empirical implementation of such models it has been customary to assume a multiplicative lognormal disturbance term so that an apparently nonlinear model could be linearized (in logarithms) and estimated by classical estimation techniques. However, the problem of estimating the constant term and finding a confidence interval for it has always presented some problems. Bradu and Mundlak (1970) gave an uniformly minimum variance unbiased estimator (UMVUE), but it uses a g-function, introduced by Finney (1941) which requires extensive tables and yields unacceptable negative values for some values of its arguments [see Teeken and Korets (1972, p. 804)]. In this paper, we propose another estimator which, though biased, gives results almost identical with the UMVUE. *Presented at the Fifth World Congress of the Econometric Society held in Boston. August 1985. Thanks are due to two referees and Prof. T. Amemiya for their helpful suggestions. Thanks are also due to Dr. Ah Born Sim of Concordia University and Dr. K.J. Keen of the University of Toronto for carrying out the calculations in this paper. The research of the first author was supported by the Natural Sciences and Engineering Research Council of Canada. ‘Deceased.
0304-4076/X9/$3.5001989,
El.aevier Science Publishers
B.V. (North-Holland)
288
M.S. Srivasiuvu and B. Singh, Bootstrapping
in multiplicutive
models
In order to find a confidence interval for the constant term, we need the distributions of its estimators, which even under the assumption of lognormality is difficult to obtain and is not available in the literature. Alternatively we may apply the asymptotic theory and use normal or t-tables. However, it has been demonstrated by Beran and Srivastava (1985) and Efron (1979,1985) that the confidence intervals obtained by the asymptotic theory are unsatisfactory when compared with the exact results. On the other hand, the bootstrap method of Efron usually gives improved confidence intervals in many numerical examples considered in the literature. We thus develop the bootstrap theory to obtain a confidence interval for the constant term. The organization of this paper is as follows. In the next section, we discuss the model and classical estimators. In section 3 we shall briefly outline the bootstrap method, while the theory is given in section 4. Section 5 gives numerical results. 2. The model and the classical estimation methods Consider
a stochastic
multiplicative
model,
K
i=l
q=llBx$q,
k=2
‘...,
n,
(1)
where Yj is the ith observation on the dependent variable, X,, is the corresponding observation on the k th (k = 2,. . . , K) nonstochastic independent variable, and B and pk (k = 2,. . . , k) are the unknown parameters. Let U, = log ui. It is assumed that ui’s are i.i.d. (independent and identically a*), where u > 0 and is unknown. This implies that distributed) N( - $J *, U, > 0 and E( u;) = 1, as required by situations in economics where such models are used. Letting y, = log q, x,~ = log X;k, E; = U, + $a*, and & = log B - ;a’,
(2)
the model (1) after taking logarithms notation as
on both sides can be rewritten
in matrix
y=xp+&,
(3)
where
.II Yl
3
i
El E=
)
i- Et,i
x= i
1
x1*
. .
. .
i
x,,
***
XIK
. . .
X,K
\
I
M.S. Srivustava and B. Singh, Bootstrapping in multiplicutive models
Hence from the Gauss-Markov theorem, with or without the assumption normality, the best linear unbiased estimator of j3 is given by p^=
(xx-‘X’y,
and an unbiased
an estimator
of
(4)
estimator
of a* is given by
s2=(n-K)-‘y’[1-X(X’X)-‘X’]4’. Thus,
289
(5)
of B may be given by
j=,B,+Y.
(6)
Under the assumption of lognormality, this estimator is a modified maximum likelihood estimator since the divisor in s2 is n - K instead of n. However, this is a biased estimator. Using Finney’s (1941) results for estimating the parameters of a lognormal distribution, Bradu and Mundlak (1970) gave a uniformly minimum variance unbiased estimator of B. This UMVUE is given by
(7) where J tJ
j!’ m=n-K,
y=l-h,
h = a’( XT-%, The variance
a’= (1,o )...) 0).
of B, is given by
var( i,)
= B2[e2(h-:)sZG,(
&q2)
- 11,
where
’
y=l-h.
290
Bradu
M.S. Srivustavu and B. Singh, Bootstrupprng in multiplicutive
and Mundlak
(1970) suggest estimating
models
it by
in which E( jf) has been replaced by &f; from the numerical results in tables 1 and 2 it appears that this estimator needs some improvement. Although, Bradu and Mundlak (1970) provide a table for Lhe g-function, more extensive tabulation is needed to be able to compute B, for all cases. Also, it may sometimes give an unacceptable negative estimator. To overcome this difficulty several alternative but biased estimators have been proposed in the literature. Since, under normality assumption,
we propose
the following
of B:
estimator
&, = ,h + w,
(8)
which has 2
E(&)
‘[e--Iro’]
&I!!_
=B
[
m
-:m
I
’
MSE( j,)
An estimate
of MSE(&)
is given by
-- i ni =
which,
+
e21jl
for large m, can be approximated Ea(
$,)
= e28,+(t-2h)s2 [eh.r’ _
by
11.
,(I
2h).s?
M.S. Srivustuva
and B. Singh. Bootstrupping
in mulliplicuiwe
models
291
Table 1 ,I
10
20
40
a*
h
0.50
0.2 0.4 0.6 0.8
0.5959e-02 0.3315e-02 O.l457e-02 0.3606e-03
0.1210 0.2316 0.3550 0.4933
0.1179 0.2293 0.3537 0.4929
1.0
0.2 0.4 0.6 0.8
0.2507e-01 O.l374e-01 0.5959e-02 O.l457e-02
0.3110 0.5537 0.8559 1.236
0.2782 0.5306 0.8430 1.232
2.0
0.2 0.4 0.6 0.8
0.1144 0.5991e-01 0.2507e-01 0.5959e-02
1.327 1.794 1.650 4.065
0.7848 1.464 2.475 4.010
0.50
0.2 0.4 0.6 0.8
0.2393e-02 O.l340e-02 0.5931e-03 O.l476e-03
1.1112 0.2254 0.3519 0.4924
0.1104 0.2246 0.3514 0.4923
1.0
0.2 0.4 0.6 0.8
0.9765e-02 O.O437e-02 0.2393e-02 0.5931e-03
0.2520 0.5145 0.8351 1.230
0.2446 0.5077 0.8307 1.228
2.0
0.2 0.4 0.6 0.8
0.4101e-01 0.2248e-01 0.9765e-02 0.2393e-02
0.6909 1.399 2.437 3.997
0.6080 1.322 2.383 3.976
0.2 0.4 0.6 0.8
O.l497e-02 0.8399e-03 0.3723e-03 0.9283e-04
0.1089 0.2239 0.3511 0.4922
0.1085 0.2234 0.3509 0.4921
0.2 0.4 0.6 0.8
0.6064e-02 0.3389e-02 O.l497e-02 0.3723e-03
0.2398 0.5057 0.8302 1.228
0.2360 0.5018 0.8275 1.227
0.2 0.4 0.6 0.8
0.2499e-01 O.l384e-01 0.6064e-02 O.l497e-02
0.6038 1.328 2.391 3.980
0.5641 1.286 2.360 3.96X
0.50
0.2 0.4 0.6 0.8
O.l090e-02 O.h116e-03 0.2713e-03 0.6769e-04
0.1079 0.2232 0.3508 0.4921
0.1076 0.2229 0.3506 0.4920
1.0
0.2 0.4 0.6 0.8
0.4397e-02 0.2462e-02 O.l090e-02 0.2713e-03
9.2346 0.5018 0.8280 1.227
0.2320 0.4991 0.X261 1.227
2.0
0.2 0.4 0.6 0.8
O.l797e-01 0.995e-02 0.4397c-02 O.l090e-02
0.5656 1.29X 2.371 3.973
0.5443 1.269 ?.349 3.964
Bias (&)/B
MSE ( & ),‘B*
var( 8, )/B*
292
M.S. Srivusruva and B. Singh, Bootstrupping
In many
econometric
studies,
in multiplicutive
models
an estimator
has been used to estimate B. In their example of Israfli agriculture, Bradu and Mundlak (1970) have shown that the bias@ using B, as an estimator of B is nearly 12%. Thu:, we do, not consider B, in this paper and compare the performance of B, with BiAonly. For variou: values of n, h, and u2, table 1 records Bias( B2)/B, MSE( B2)/B2, and var( B1)/B2. It is clear from this table that the bias ratio is negligible even forAa sample size asAsmall as 10, with almost negligible increase in the MSE(B,)/B’ over var(B,)/B2. Thus, the performance of the proposed estimator B, is almost identical to that of Bradu-Mundlak’s UMVU estimator B,. Given the inherently complicated nature of the g-function and the need of its extensive tabulation for the computation which may sometimes give unacceptable negative values, the proposed estimator may be preferred. In this paper we give the bootstrap theory to obtain a confidence interval for B. The results are, however, applicable to more general situations such as
where
c and 11 can be chosen appropriately. f(B,a’)=B = Y,
for
~‘=a’,
q=$,
for
c’=xi,
?I=:.
For example,
We suggest using (8) as an estimator with appropriate estimating Y,, y will change to 1 - xi( X’X)-lxk. 3. Bootstrap
y. For example,
in
method for obtaining confidence interval
First, we remove the assumption of normality of E in (3). Here, it would suffice to assume that the n components of the disturbance vector E’ = E,)’ are i.i.d. with unknown cumulative distribution function (cdf), F (q,..., (say), mean zero and variance u 2( F). We obtain the OLS estimator same as (4) and the residual vector c=M&=My,
(10)
where M=I-P
and
P=X(X’X)-‘X’.
(11)
M.S. Srivastava and B. Singh, Bootstrapping
293
in multiplicative models
Let us now take a random sample with replacement from &, . . . , S,, denoted The bootstrap OLS estimate B* of p by, say, Z: ,..., El,*.Let E* =(E: ,..., E:)‘. is then given by p* =p1+
(x’x)-‘x’;*.
(12)
Since the regression model (3) has an intercept2 CEI, = 0. Hence for the above sampling scheme, E*(EI*) = 0 and E*/?* = p, where E* denotes the conditional expectation given E^r,. . . , i,. The bootstrap estimate of a*(F) is given by
It will be shown in section 4 that the asymptotic distribution of c’( p^ - p) + $y(S* - 02(F)) is the same as that of c’(p^* - /?) + ~Y(S*~ - (n - K) x n ~‘S*). The distribution of the latter is approximated by Monte Carlo method by obtaining L independent samples of size n with replacement from $, . . . , El, and calculating the empirical cumulative distribution function, where L is chosen large, usually between 200 to 1000. In other words the L values of c$* + $s*’ are ordered, and if dz,* and dI*_,,, denote the +L~I and (1 - (u/2)L points of these ordered values, the bootstrap (1 - a) x 100% confidence interval for ee, B = c’p + 1ze*, is given by edzjz, ed,*_@. Thus, by choosing c’ = a’ = (l,O, . . . ,O), we get the confidence interval for B. This is the percentile method of Efron (1982). 4. Bootstrap theory To develop the elements
the K
(Cl)
max x;( 1ci5n
a Cramer-Wold following:
first given method
1. asymptotically
theory for n matrix
shall assume that
= 0, Srivastava verifying the
Under Srivastava ‘s condition
[(G - v)‘, S* - a*(F)] where
bootstrap method = (x,, . . xn) are
(1968,
1972). Using condition, we
(Cl ) on the matrix
- N(K+l) [O, Q(F)]
1
X,
we
the
have
294
M.S. Srivustavcr and B. Singh, Bootstrapping in multiphcutive models
Corollary 1. Let g be a continuously nonvanishing derivative at (q’, a’(F)); cal& g(f/,
differentiable function of (p’, S2) with a g: RKf ’ -+ R’. Then we have asymptoti-
s2) - g(c 0’) - N,(O,S(Fk’)~
where the 2 is the r X (K + 1)
of
of
evaluated
at q and
We shall now consider the model (3) with K 3. In case the elements of the first column of the X matrix are all ones /3i is the constant of the For the theory hold we shall assume that the elements of the X are such
-
a2
_
1_
nEiCxf,-
[
(
CxZi)(
2X*X3CX2iX3r Xxii)
-
+ ‘:Cx5i
(CX2iX3r)*
1
has a finite limit. Clearly, if n-‘( X’X) tends to be a positive definite matrix, a2 will have a finite limit. However, this condition is not necessary as shown in Srivastava and Srivastava (1986). Corollary
2.
Suppose
K = 3 and a2 has a jinite
limit for the model (3).
Then
h{ [(&+ is’> - (P,+ :u’)]} -+
N(0,a2a2( F) + jd2(F)).
We shall now proceed to give the bootstrap theory. Let F,, be the empirical cdf of ei,. . . , E,. Then it is well known that F, + F as. However, we are sampling from .Zi,. . , ZN. Let 8” be its empirical cdf. By following Freedman (1981) or Srivastava (1985) it can be shown that e, - F, -+ 0 in probability and hence i” -+ F in probability. The mean of the distribution e,$ p( 6,) = nP1xEnr = 0 since the regression equation has a constant term. Similarly, the variance of the distribution E’,,, p2( gn) = n -‘~:=, Ci, -+ u’(F) a.s., and the [see Srivasfourth moment of in, p4( &) = n-lx’._ ,_1 .?p-+ p4( F) in probability tava (1985)]. Also, (n - 1))%*‘[ I - n - ‘ll’]e* -+ u2( F) in probability. Thus, we have: Theorem 2. Let cl,. . . , .?N be:he residuals with empirica! cdf 6,. Then 6, -+ F in probability, p( F,) = 0, a’( F,) + u2( F) a.s., and p4( F,,) -+ pLq(F) in probability.
M.S. Srivastuva
and B. Singh, Bootstruppmg
in multiplicuiive
295
models
Noting that Corollary 2 was obtained for any class of distributions finite fourth moment, we get the following result by applying the central theorem to triangular arrays as in Beran and Srivastava (1985). Theorem
3.
Under the conditions of Corollary 2,
&{(&+ -
with limit
N(0,
$s**- (j&-t a*u*(F)
f(n-Iqn-‘9))
IE^;,}
+ id*(F)).
5. Empirical results In this section, we shall report some empirical results to demonstrate the performance of the bootstrap method in comparison to the two estimators given in (7) and (8). In this connection, we first consider the actual example of Israeli agriculture investigated by Bradu and Mundlak (1970) and then present a simulation experiment which uses the same x-variables.
5.1. The Bradu-Mundlak Let us consider
example
the trivariate
of Israeli agriculture
Bradu-Mundlak
model:
where Y, stands for the output of ground nuts relative to that of cotton in year i, X2 i_2 for the price of ground nuts relative to that of cotton in year i - 2, and x3; for the trend term. Linearizing by the logarithms, we get the model (3) with K = 3. The regression has been estimated for the period 1953-63. Table 2
Table 2 Comparison
between
the proposed
and the Bradu-Mundlak
estimators
of B.
Proposed
B-M
Estimates
34.159
34.146
E( B)/=(B) Monte Carlo confidence limits” Confidence limitsh Bootstrap confidence
16.29
16.68
(4.771.110.70) ( - 4.367,73.67) (17.35,80.45)
(4.744.108.70) (- S.303.73.59)
limits’
a Based on 1000 simulations. ‘Confidence limits are obtained using r-values ‘Based on 200 bootstrap samples.
296
M.S. Sriuastava and B. Singh, Bootstrapping
in multiplicative
models
provides a comparative picture of the proposed and the Bradu-Mundlak estimators. The least squares estimators of pi, &, and & are 3.590, 2.840, and -0.355, respectively, and that of u* is 0.138. It is to be noted here that the estimate of B given above differs from that reported by Bradu and Mundlak (1970) who_ for some unknown reasons apply the g-correction to eP1+ is2 rather than to epl+ ~ys*. As is evident, the proposed estimator of B reported above is almost equal to Bradu-Mundlak’s UMVU estimator. Accordingly, to save computing time, we obtain the bootstrap confidence interval for the proposed estimator only. The theory, however, is applicable to the confidence interval based on the Bradu-Mundlak estimator. The bootstrap confidence interval for B, based on 200 bootstrap samples, is given by (17.35, 80.45) and is similar to the one obtained by large-sample theory using t-values. The confidence interval by the Monte Carlo method, based on 1000 simulations, seems to favour the one obtained by the bootstrap method.
5.2. Some simulations We recognize that the results based on real data may always be questioned because of the possible difference between the real world and the assumed theoretical distribution of the error term E. Consequently, we resort to some simulation exercises where such problems do not arise. Here, we take the same values of x-variables as in Bradu-Mundlak, and the parameter values are nearly equal to their estimates. In other words, we take B = 32.00, fi2 = 2.80, & = -0.365, and a2 = 0.14, and we assume that the E~‘S are i.i.d. normal variables with mean zero and variance 0.14. In this experiment, we make 50 Monte Carlo trials and (as above) 200 bootstrap replications. The results of this experiment are presented in table 3. These results are similar to table 2.
Table 3 Comparison
of the proposed
and the Bradu-Mundlak estimators B = 32.00. Proposed
Estimates Bias -MSE/var Confidence limit? Bootstrap confidence
32.931 0.937
limits
(simulation
result);
true value of
B-M 32.921 0.921
289.388
307.894
(- 5.890,71.765) (9.988,71.001)
( - 8.463,74.303)
“Confidence limits are obtained using t-values ‘Based on 200 bootstrap samples.
M.S. Srivustuva and B. Singh, Bootstrupping
m multiplicative
models
291
References Beran, R. and M.S. Srivastava, 1985, Bootstrap tests and confidence regions for functions of a covariance matrix, Annals of Statistics 13, 95-115. Bradu, D. and Y. Mundlak, 1970, Estimation in lognormal linear models, Journal of the American Statistical Association 65, no. 329, 198-211. Efron. B., 1979. Bootstrap methods: Another look at the jackknife, Annals of Statistics 7, l-26. Efron, B., 1982, The jackknife, the bootstrap and other resampling plans (Society for Industrial and Applied Mathematics, Philadelphia, PA). Efron, B., 1985. Bootstrap confidence intervals for a class of parametric problems, Biometrika 72, 45-58. Finney. D.J., 1941, On the distribution of a variate whose logarithm is normally distributed. Journal of the Royal Statistical Society, Suppl. I, 7, 155-161. Freedman, D., 1981, Bootstrapping regression models, Annals of Statistics 11, 569-576. Srivastava, M.S., 1968, On a class of nonparametric tests for regression parameters, Annals of Mathematical Statistics 39, 697. Srivastava, M.S., 1971, On fixed-width confidence bounds for regression parameters, Annals of Mathematical Statistics 42, 1403-1411. Srivastava, M.S., 1972, Asymptotically most powerful rank tests for regression parameters in MANOVA. Annals of the Institute of Statistical Mathematics 24. 285-297. Srivastava, M.S., 1985, Bootstrapping Durbin-Watson statistic, Technical report no. 5 (University of Toronto, Toronto). Srivastava, M.S. and V.K. Srivastava, 1986, Asymptotic distribution of least squares estimator and a test statistic in linear regression models, Economics Letters 21, 173-176. Teeken, R. and J. Korets. 1972, Some statistical implication of the log transformations of multiplicative model, Econometrika 40, 793-819.