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1 This PDF is selection from n out-of-print volume from the Ntionl Bureu of Economic Reserch Volume Title: Price nd Quntity Trends in the Foreign Trde of the United Sttes Volume Author/Editor: Robert E. Lipsey Volume Publisher: Princeton University Press Volume ISBN: Volume URL: Publiction Dte: 1963 hpter Title: Appendix E: Dt on Vribility, Smpling Error, nd overge hpter Author: Robert E. Lipsey hpter URL: hpter pges in book: (p )

2 Appendix E Dt on Vribility, Smpling Error, nd overge The first prt of this Appendix presents the bsic dt on the vribility of price movements within nd mong minor clsses nd estimtes of smpling error for minor nd mjor clsses. The second contins detiled informtion on coverge nd shifts in coverge. VARIABILITY AND SAMPLING ERRORS Two ssumptions must be kept in mind s the bsis for these mesurements nd their interprettion. One is tht the commodities which re covered in the minor clss indexes re completely covered the prices for individul commodities re ssumed to be known precisely nd not subject to smpling error. The other is tht commodities nd minor clsses hve been selected for the smple rndomly, either with equl probbilities or with probbility proportionl to size. Tble E-l gives weighted nd unweighted stndrd devitions for minor clsses, where they were computed. They could not be clculted for uncovered clsses or for clsses in which there ws only one covered commodity. The ltter re divided into two groups: those with only one commodity, for which, by our ssumption, there is no vrince, nd those with one covered commodity nd one or more uncovered ones, for which we cnnot mesure the vrince. These stndrd devitions re mesures of the homogeneity of clsses rther thn of the ccurcy of the indexes, lthough homogeneity does, of course, ffect ccurcy. They re descriptive of the covered commodities within ech clss nd do not require ny ssumption of rndomness in the smpling procedure. A lrge stndrd devition implies heterogeneous strtum but it my not, if coverge is high, imply inccurcy in the estimtion of the men. The two stndrd devitions in Tble E-1 re: fpi\2 Unweighted: ; = / I /p\2 ILTP Q \ Weighted: 1 ) = 2 EPQ The is pproprite for n unweighted index or for n ssumption tht the commodity weights mong the covered items re irrelevnt to the uncovered ones; tht is, ech commodity, no mtter how lrge, is only 378

3 APPENDIX E single observtion of the men. The weighted stndrd devition is pproprite for use with weighted index nd, in generl, for the ssumption tht the importnce of different price behvior ptterns in the uncovered items would mtch tht mong the covered commodities. Equlity of the two stndrd devitions implies no correltion between the weight or importnce of commodity nd its distnce from the men. The usul cse nmely tht is greter thn mens tht the correltion is negtive; nd greter thn implies tht the importnce of commodity is positively correlted with exceptionl behvior (distnce from the men). Stndrd errors of the men (the men being the Lspeyres price index) re given in Tble E-2. These do involve inference from the stndrd devitions. They re mesures of the ccurcy of the minor clss indexes, under the ssumption tht the commodities smpled re representtive of ll commodities in their groups. In other words, smples re treted s if they hd been drwn rndomly. Since only the weighted stndrd errors re shown here, the ssumption implied is tht the smple ws drwn with probbility proportionl to size (vlue) rther thn, s in the unweighted indexes, equl probbility of representtion for ech commodity. Two sets of stndrd errors re computed. The first, with no finite smpling djustment, tkes ccount of the number of commodities drwn from ech clss but not of the proportion of totl vlue covered. It trets the smples s if they included oniy smll prt of the whole clss. It nswers the question, "How ccurte n estimte of the men could be mde with smple of this size from lrge popultion?" The second set tkes ccount not only of the number of items but lso of their shre in the totl vlue of the clss. It mkes use of the fct (or ssumption) tht the men is known precisely for substntil prt of the totl (the smple) nd tht, in effect, the estimtion pplies only to the remining, often smll, frction of the totl vlue. The two mesures of stndrd error cn be described in terms of the stndrd devitions of Tble E-1, where N is the number of covered commodities in the minor clss. Without finite smpling djustment: m (undjusted) With finite smpling djustment (where f is the coverge rtio): = m (djusted) = Vx i / 1 f 379 1

4 oefficients of vrition, presented in Tble E-3, re the rtios of stndrd errors to the mens they pply to. In this cse the mens re the Lspeyres price indexes. Only the weighted mesures re shown, but the reltionship between weighted nd unweighted coefficients would be the sme s in Tble E-1. The vrince of mjor clss men cn be clculted from the vrince within minor clsses (lredy computed) nd the vrince mong minor clsses, s follows: where: n iv. m. = l [(v_ = number of smpled minor clsses number of minor clsses = number of smpled commodities in minor clss i = number of commodities in minor clss i = vlue of commodityj in minor clss i = vlue of smpled commodities in minor clss i E = vlue of ll commodities in minor clss i = Vj, j=1 v = vlue of smpled minor clsses = i=1 V = vlue of ll minor clsses = vlue of price reltive for commodityj in minor clss i V..r.. mens of minor clss z = " '.' =. = mjor clss men = 2 i=1 n = m1 j=1 V. V1 = minor clss vrince 2 The is the squre of the weighted stndrd devition of Tble E-l. 38

5 APPENDiX E These computtions re crried out in Tble E-4 nd the coefficients of vrition derived from these vrinces re shown in hpter 5, Tble 18. MEASURES OF OVERAGE AND HANGE IN OVERAGE Tbles E-5 through E-8 give bsic dt on coverge for ll mjor nd intermedite clsses. The figures show, for the erliest yer of ech period, the rtio of the vlue of covered commodities to the vlue of ll commodities in the clss. For the lst yer of ech period they show the rtio of the vlue of those commodities which were covered in the first yer to the vlue of ll the commodities which were prt of the clss in the first yer. Thus, for within-period comprisons, chnges in coverge due to incresing vilbility of dt re eliminted. For ech period, the tble revels whether the commodities covered in the initil yer grew in vlue t fster or slower rte thn the uncovered ones. In order to see the trend of coverge s whole (not just tht for fixed groups of commodities) one must follow the movement from the right-hnd column of one period to the right-hnd column of the next. Export lss 115 in Tble E-5 illustrtes the two uses of the tble. By 1889, the commodities covered in 1879 hd fllen from 91 to 89 per cent of the totl vlue of the clss. But those commodities which ctully were covered in 1889 formed 97 per cent of the totl vlue of the clss in tht yer. It should be noted tht these tbles show the proportion of totl vlue in covered commodities, not tht contined in covered minor clsses. These chnges in coverge do not indicte the possibility of bis in the totl or mjor clss indexes, becuse they include the effects of both shifts in the weight or importnce of minor clsses nd shifts within them. Only the ltter, s is pointed out in hpter 5, would suggest bis becuse they indicte tht covered commodities possessed different chrcteristics (possibly different price chnges) from uncovered products. Tbles E-9 through E-12 re intended to revel such shifts. They show the ctul end-yer coverge rtios for commodities covered in the initil yer (first columns of Tbles E-5 through E-8) s percentges of the rtios tht would hve existed if vlues of covered nd uncovered items hd grown t the sme rte within ech minor clss.' Thus, rtio over 1 per cent indictes tht covered commodities grew t more rpid rte thn uncovered commodities. 1 The computtion of the hypotheticl rtios is performed by pplying the initil yer coverge rtio to the end-yer vlue for ech minor clss. 381

6 APPEJS'DIX E TABLE E 1 STANDARD DEVIATIONS FOR MINOR LASS PRIE INDEXES Minor lss Weighted Stndrd Devitions Unweighted Stndrd Devitions A b 4 b b d.75 d d d O b b b.32 b b b d d 23 b b b b 24 d b d d d b d d d b b b d d d.83 d d.12 3b 31 c b b b b 35 b 494 c b b d b , b.9 b.148 b.71 b 43 b b c c b c 47 b b 382

7 Minor lss b b 1 Ob Weighted Stndrd Devitions APPENDIX E TABLE E d d.3 c b b.221 d b b o b b c.98 d ci o b b o c b b b b B. IMPORTS d o o b b b b b.114 b b o b b 46 o ci Unweighted Stndrd Devitions ci o b b.225 ci b b o b b o e.128 ci ci c b b c b b b b c.159 ci c b b b b b.119 b b b b.19 c d.13

8 APPENDIX E TABLE E 1 Minor lss Weighted Stndrd Devitions Unweighted Stndrd Devitions b c c.222 b U b c U b o b b c b b c c b b c 24 b b b b d d d d d b b.137 b b b b.28 b b b c c b d.16 b b b b b b 4 b 35 c c 36 b b 4.13 b b d b b b.271 b b b d d U b 46 b b b b b b b b 49 b b b b b b 5 b b o d b b b U b b U c b b U U b b b d.113 d U d c b b.3 c b b

9 APPEXDIX E TABLE E 1 (concluded) Minor lss Weighted Stndrd Devitions Unweighted Stndrd Devitions b b d d b c c c b b b c c c b d d 41 d b d ci d b b ci b b b d d b b b b b d d ci d d b d d ci c b b ci c e b Uncovered clss. b One-commodity clss, complete coverge. lss not listed seprtely in this yer. 41 One covered commodity, incomplete coverge. TABLE E-2 STANDARD ERRORS OF MEAN FOR WEIGHTED MINOR PRIE INDEXES Minor lss Withou Finite Smpling Adjustment With Smpling Finite Adjustment A. EXPORTS b b b b 4 c c c b c d d d d

10 APPENDIX E TABLE E 2 Minor lss Without Finite Smpling Adjustment With Smpling Finite Adjustment d.56 c c d d.51.5 d d b b b d d d d 23 b b d d d d d. ci b b b d ci. ci.48 ci ci d.18 3 b b b b 31 c c e c b b 35 b b ci d b.9 b 43 b c c b 47 b 48 d d.3 U d 49 c b b ci..11 d

11 APPENDIX E E 2 Minor lss Without Finite Smpling Adjustment With Smpling Finite Adjustment o c b b b b b b d.97.1 b b c o b b b b o.6.35 b b o o c c b b b 9 b b 1 b b 11 c b b d b b 24 b b 25 d 26 b B. IMPORTS b b.57 b b b b b b.23 d b.18 d b c b d o c o d.28.2 d o d

12 APPENDIX E TABLE E 2 lss lss Without Finite Smpling Adjustment With Finite Smpling Adjustment d d & d d 29 b b b e b d b b d b 35 c c 36 b b b b b d d.196 c d d.563 ' ' o 45 b b b b b c b b b c 49 c b b b 5 c b c d.143 c ' b b d b o c b ' ' b o 58 c ' 24 c ' d ' ' c b b b b ' d d ' 68 b 69 ' ' 7 c b b b 71 b 72 c c 73 d ' d ' ' ' '.56.4 '

13 APPENDIX E TABLE E 2 (concluded) Minor lss Without Finite Smpling Adjustment With Finite Smpling Adjustment d d d d 8 c b b d d c b.13 c e Uncovered clss. b One-commodity clss, complete coverge. lss not listed seprtely in this yer. d One covered commodity, incomplete coverge. TABLE E-3 OEFFIIENTS OF VARIATION FOR WEIGHTED MINOR LASS INDEXES Minor lss Without Finite Smpling Adjustment With Smpling Finite Adjustment A. EXPORTS b d d.53 d d d & d.79 d 1 18 d d

14 Minor lss Without Finite Smpling Adjustment APPENDIX E TABLE E With Finite Smpling Adjustment b b b ci d d ci 23 b b 24 ci d b ci ci ci ci d ci b b b ci ' ci ' ci.68 ' ' '.25 3 b b b b 31 c b b 35 c b.226 c c b ci U b b 47 c o c b c c c 48 d ci.6 d d 49 b c b.236 c 55 ci 23 ci b b c c 59 D b b b 6 b b c c.85 c c U d d ' c b b c

15 APPEXDIX E TABLE E 3 Minor lss Without Finite Smpling Adjustment With Smpling Finite Adjustment c b b c c 73 b b c.39 c c 77 B. IMPORTS 1 U U b b b b b b b b 1 b b b b 11 b b c c b b c d.25 U b U.19.1 U 2 U b U b b b 24 b b U d U.154 U U d b U U b U U U d 29 b b b c c b U.13 U b b U b U 35 c c 36 b b U.37 U.8 Ø37 38 b b b d U.14 U U

16 APPENDIX E TABLE E 3 (concluded) Minor lss Without Finite Smpling Adjustment o c o ci ci ci ci 553 b b b b b b b b b b b b b c ci b b d c b b ci ci b c ci With Finite Smpling Adjustment ci ci.84 d ci c d ci c o ci d c d k b b b b ci o c e b c d c b b b b c ci d d ci ci c b d ci ci o ii b c c b ci.4.85 ci d d c ci d d d ci ci ci.7 d ci ci o Uncovered clss. b One-commodity clss, complete coverge. lss not listed seprtely in this yer. ci One covered commodity, incomplete coverge. 392

17 ALULATION APPENDIX E TABLE E-4 OF VARiANE FOR SELETED MAJOR EONOML LASSES Mjor Yer lss = V V v v Si Vr [(1) x (2)]+(3) (3) (4) (5) A. EXPORTS (25) B. IMPORTS

18 TABLE E-5 OVERAGE RATIOS FOR INTERMEDIATE EXPORT LASSES: EARLIEST AND BASE YEARS OF EAH PERIOD (per cent) Export lss E

19 APPENDIX E TABLE E-6 OVERAGE RATIOS FOR MAJOR EXPORT LASSES: EARLIEST AND BASE YEARS or EAU PERIOD (per cent) Export lss TABLE E-7 OVERAGE RATIOS FOR INTERMEDIATE IMPORT LASSES: EARLIEST AND BASE YEARS OF EAn PERIOD (per cent) Import lss

20 APPENDIX E TABLE E 7 (concluded) Import lss

21 APPENDIX E TABLE E-8 OVERAGE RATIOS FOR MAJOR IMPORT LASSES: EARLIEST AND BASE YEARS OF EAH PERIOD (per cent) Import lss [ [ j 79, TABLE E-9 INTERMEDIATE EXPORT LASS OVERAGE AT END OF EAH PERIOD AS PER ENT OF ALULATED OVERAGE ASSUMING No HANGE WITHIN MINOR LASSES Export lss Per ent

22 APPEJVDIX E TABLE E 9 (concluded) Export Per ent lss

23 APPE.WDIX E TABLE E-1 MAJOR EXPORT LASs OVERAGE AT END OF EAH PERIOD AS PER ENT OF ALULATED OVERAGE ASSUMING No WITHIN MINOR LASSES Export lss ent TABLE E 11 INTERMEDIATE IMPORT LASS OVERAGE AT END OF EAH PERiOD AS PER ENT OF ALULATED OVERAGE ASSUMING No HANGE WITHIN MINOR LASSES Import lss Per ent

24 APPENDIX E TABLE E 1 1 (concluded) Import Per ent lss ,

25 APPEXDIX E TABLE E-12 MAJOR IMPORT LASS OVERAGE AT END OF EAH PERIOD AS PER OP ALULATED OVERAGE ASSUMING No HANGE WITHIN MINOR LASSES Import lss Per ent , ,