Bootstrapping for standard errors of instrumental variable estimates

Economics Letters North-Holland

14 (1984) 297-301

297

BOOTSTRAPPING FOR STANDARD ERRORS OF INSTRUMENTAL VARIABLE ESTIMATES * Renate

FINKE

and Henri THEIL

Unruersrty of Florida, Gainesurlle, FL 32611. USA Received

1 September

1983

Theil and Finke (1983) used the distance from the equator as an instrumental in the estimation of a cross-country demand system. Here we obtain standard IV estimates using Efron’s (1979) bootstrap technique.

variable (IV) errors of the

Theil and Finke (1983) applied an instrumental variable (IV) technique to the estimation of a system of cross-country demand equations. Here our objective is to compute standard errors of the IV coefficient estimates using Efron’s (1979) bootstrap technique. The Theil-Finke demand system consists of equations for n = 10 goods and is based on data for 30 countries collected by Kravis et al. (1982). The system uses Working’s (1943) model which expresses budget shares as linear functions of the logarithm of total expenditure, supplemented by a substitution term under preference independence with a constant income flexibility $I. The dependent variable of the ith demand equation for country c is

Y,,

=

Y,

(1)

-

where w,, and

p,<

are the budget share and the price, respectively,

* Research supported in part by the McKethan-Matherly sity of Florida, and the Deutsche Forschungsgemeinschaft tion). 0165-1765/84/$3.00

0 1984, Elsevier Science Publishers

of good

Eminent Scholar Chair, Univer(German Research Founda-

B.V. (North-Holland)

Food Beverages Clothing Rent Furnishings Medical Transport Recreation Education Other

P,

Food Beverages Clothing Rent Furnishings Medical Transport Recreation Education Other

a,

(98)

8 (50) -46 (48) 315 (63) 253 (40) 237 (34) 295 (55) 193 (33) -40 (53) 328 (56)

-1543

1620 (121) 558 (64) 839 (58) 1448 (78) 982 (50) 888 (41) 1179 (67) 692 (942) 535 (59) 1260 (68)

ML estimate

(2)

Parameter

for instrumental

(1)

Table 1 Discrete and normal bootstraps

-1520 31 19 378 259 286 174 227 - 128 274

1636 580 904 1512 990 938 1060 726 447 1208

- 1519 34 15 370 260 287 17x 229 -130 276

1631 582 902 1504 990 940 1068 729 445 1210

D (4)

(3)

A.’

Means

-0.5

estimates.

Data based

+=

variable

1635 577 902 1506 994 945 1056 730 443 1212

,

1519 28 16 373 263 292 174 22x -131 275

N (5)

149 80 82 94 59 50 86 46 77 82

164 89 88 103 65 57 96 52 Xl 90

(6)

D

RMSEs N

2 : \ 2 c r: 2 z P

90 83 101 70

79 93 61 48 89 46 72 83

80

144

52 76 90

55 96

5:

3 $ .m

?

153

(7)

$

(98)

8 (50) -46 (48) 315 (63) 253 (40) 237 (34) 295 (55) 193 (33) -40 (53) 328 (56)

-1543

1620 (121) 558 (64) 839 (58) 1448 (78) 982 (50) 888 (41) 1179 (67) 692 (42) 535 (59) 1260 (68)

-1548 29 28 374 249 278 170 212 -66 275

1620 580 911 1504 975 929 1052 707 515 1206

-1546 28 34 378 248 282 167 209 -72 273

1622 583 915 1506 972 931 1050 707 510 1203

-1546 21 26 379 252 274 170 207 -60 276

1623 574 909 1507 978 926 1052 703 522 1206

D

N (10)

D (9)

(8)

152 82 80 92 59 49 84 44 86 82

165 91 86 100 69 53 92 49 86 88

(11)

RMSEs

Means

Data based

$I= -0.6

’ All entries to be multiplied by 10m4 h D is discrete bootstrap, N is normal bootstrap.

P, Food Beverages Clothing Rent Furnishings Medical Transport Recreation Education Other

Food Beverages Clothing Rent Furnishings Medical Transport Recreation Education Other

a,

ML estimate

(2)

Parameter

(1)

N

138 84 81 100 59 52 81 45 88 85

154 94 88 105 64 59 91 52 91 92

(12)

-1579 27 37 366 237 268 164 196 10 274

1604 580 917 1494 958 919 1042 687 598 1201

(13)

Data based

l#l= ~ 0.7

- 1566 25 36 368 237 268 160 194 4 274

1618 574 919 1495 959 923 1039 686 589 1198

(14)

D

Means N

24 36 373 237 266 158 196 1 269

1626 576 917 1500 959 919 1037 685 586 1195

(15)

94 48 112 89

92 47 116 92

86 82 100 57 55 85 42 111 83

88 81 108 60 56 81 42 114 86

147

90 110 62 63

89 118 65 61

149

157 94

(17)

N

168 96

(16)

D

RMSEs

Y :: 2 % & 2 a 3

%

@?

3 2 i= a %

2 ”. \

.5

3 % P

300

R. Fir&e, H. Thetl / Bootstruppwzg for .stundard error\

i in country c, while j?, is the geometric mean of the prices of good i across the 30 countries. Let q, be the logarithm of per capita real income of country c; then the i th demand equation is n-1

.?J,,.=

a, + P!4,.+ G(Y<+P,) z,<.i

c

.I=

by,.+/+,, 1

I

+c,<.,

(2)

where z,~ = log( p,,./j?,) - log( p,,,/j?,,) and c,( is the error term. Columns (1) and (2) of table 1 list the parameters and their ML estimates and, in parentheses, the asymptotic standard errors of these estimates. (All entries in table 1 should be multiplied by 10m4.) Columns (3) (8) and (13) contain the IV estimates conditional on three alternative values of +, with the distance of the country’s capital from the equator used as an instrument for q,,. These five columns are from Theil and Finke (1983); all other columns of table 1 concern bootstrap procedures. The conventional bootstrap procedure uses the coefficient estimates [columns (3), (8) or (13)], the associated residual vectors, and the 30 observations on each of the right-hand variables in eq. (2). An error distribution is constructed which assigns mass l/30 to each residual vector. We draw from this distribution with replacement 30 times. ’ The result is substituted for c,<, in (2) along with the observed values of the q<,‘s and the z,,‘s and the IV estimates of the cr,‘s and /3,‘s, after which the IV technique is applied again to obtain new coefficient estimates. This is repeated 100 times and then 500 times. The results of these two sets of experiments are essentially the same and those for 500 replications are summarized in table 1. Columns (4). (9) and (14) contain the means of the coefficient estimates of the 500 bootstrap replications. Note that these means are quite close to the corresponding data-based IV estimates in columns (3), (8) and (13). Columns (6), (11) and (16) contain the RMSEs of the 500 bootstrap estimates around the corresponding data-based values. These RMSEs are the bootstrap-simulated values of the standard errors of the IV estimates in columns (3), (8) or (13). The bootstrap procedure generates error terms from the discrete distribution which assigns mass l/30 to each of the 30 IV residual vectors. This procedure ignores the knowledge that the true error distribution is continuous rather than discrete. To verify whether continuity We use the IMSL subroutine 30 with replacement.

GGUD

which selects at random

30 numbers

between

1 and

R. Fmke,

H. Theil / Bootstrapping

for standrrrd error.~

301

makes a difference, we substitute pseudo-normal error terms for the e,<‘s in eq. (2). That is, we use the 30 residual vectors associated with the data-based IV coefficient estimates [columns (3) (8) or (13)] in the form of their moment matrix 2 and we draw at random from N(0, 2). ’ For the 500 normal bootstrap replications thus obtained we show the means of the IV coefficient estimates in columns (5) (10) and (15) and their RMSEs in columns (7) (12) and (17). The results for these normal (N) bootstraps are typically close to the corresponding values for the conventional discrete (D) bootstraps. Also note that the bootstrap RMSEs exceed the corresponding asymptotic standard errors in column (2).

References Efron, B., 1979, Bootstrap methods: Another look at the jackknife, Annals of Statistics 7, 1-26. Kravis, LB.. A. Heston and R. Summers, 7982, World product and income (The Johns Hopkins University Press, Baltimore, MD). Theil. H. and R. Finke, 1983, The distance from the equator as an instrumental variable, Economics Letters 14, 357-360. Working, H., 1943. Statistical laws of family expenditure. Journal of the American Statistical Association 38. 43-56.

* We use the IMSL subroutine transform to IJ(O, 9).

GGNML

to obtain

pseudo-normal

(0, 1) variates

which we