Comparison of Simulation Methods Using Historical Data in the U.S. International Price Program M.J. Cho, T-C. Chen, P.A. Bobbitt, J.A. Himelein, S.P. Paben,

Comparison of Simulation Methods Using Historical Data in the

U.S. International Price Program

M.J. Cho, T-C. Chen, P.A. Bobbitt, J.A. Himelein,

S.P. Paben, L.R. Ernst, and J.L. Eltinge

U.S. Bureau of Labor Statistics

[email protected]

ICES III Session 14 – June 19, 2007

Disclaimer: The views expressed in this paper are those of the authors and do not necessarily reflect the policies of the BLS

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OutlineI. U.S. International Price Program (IPP)

II. Bootstrap Variance Estimator

III. Adequacy of Approximations in Simulation Studies

IV. Simulation Methods Considered

V. Numerical Results

VI. Conclusions

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I. U.S. International Price Program(IPP)

A. IPP is the nation’s primary source of information on price trends in the

international trade of the U.S. economyExport and Import Price Indexes

B. IPP involves population structure, sample design and estimation methodswith a high degree of complexity

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C. IPP Index Aggregation Tree

Upper Level Strata

Lower Level Strata

Classification Groups

Weight Groups (Company|CG)

Items

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1. Sample frame provided by U.S. Customs Border Protection, and divided into two biannual panels

2. Stratified multistage sampling within panels

a. Within a broad product category (stratum), select establishments proportional to trading dollar value

b. Within establishment, select detailed product categories (CGs) using systematic PPS

c. Within Company|CG, select items

D. IPP Sampling Design (Imported Goods)

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E. IPP Index Formula (modified Laspeyres index)

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tparent

childchild

tchild

childchild

tchild

tparent LTR

WLTR

WLTRLTR

1 ttt LTRSTRLTR

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F. Weights

1.Item weight is weight group’s weight divided by number of items

2.Weight group’s weight is a trade dollar value divided by selection probability

3.CG weight is based on trade weights from the Census Bureau

4.Stratum weight is aggregation of CG weights

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1. Draw PSUs (establishments) with replacement

from the sampled PSUs in each stratum

2. Define bootstrap weights:

II. Bootstrap Variance Estimator

A. Based on Rao, Wu and Yue (1992)

selected is PSU that the timesofnumber theis

and , stratum of PSUin item for the

weightssample original is where

1

*

**

in

hik

w

nn

nww

hi

hik

hih

hhikhik

1hn

hn

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2

1

1 ˆ ˆˆ

where

ˆ is a bootstrap STR estimator and

ˆ is the STR estimator from the original sample

B

BT b fullb

b

full

VB

3. Calculate the price STR using bootstrap weights

4. Repeat steps 1-3 B times

5. Compute the bootstrap variance estimator:

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B. Properties of Under Realistic Approximations to IPP Population Structure,

Sample Design and Estimation Methods

1. Over R replications of the “simulation design” compute

1

1

21

1

ˆ ˆˆ ˆ , ,

ˆ ˆ ˆ 1 -

R

BT r BT rr

R

BT BTBT rr

V R V

V V R V V

BTV̂

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2. Evaluation Criteria

1 ˆˆ ˆ 11

1

21

BT

R

rr VR

21 ˆ 2 ˆ ~

BTBT VVV

Relative Bias =

Degrees of Freedom =

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III. Adequacy of Approximations in Simulation Studies

A. Notation

X Features of population and design( True, Approximation )

ˆ Estimator, with cdf ),ˆ( XF

),ˆ( XFQ Specific functional ofe.g. bias or MSE

),ˆ( XF

TX AX

B. Adequacy of Simulation & Approximation

)},ˆ{{ˆAXFQ for true

1. Components to include in a. Population structure b. Sampling steps c. Nonresponsed. Weighting e. Variance estimation

2. Approximations for each component in (1)

ˆ{ { , )}TQ F X ?

AX

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*

2

*

ˆ ˆˆ{ { , )} { { , )}

ˆ{ { , )}

ˆ1 { { , )}

2

(Replication Error)

where

(1 ) for 0,1

T

A T

A T

X X

A T A T

X X

A T

Q F X Q F X

Q F XX X

X

Q F XX X X X

X X

X X X

1. Taylor expansion: under conditions

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2. Four Cases

Case 1: Small first derivatives, small

Case 2: Small first derivatives, large

Case 3: Large first derivative

Case 4: Large second derivative

TA XX

TA XX

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IV. Simulation Methods Considered

A. Use a Single Method for Selection of Sample Units and Related Weights:

Same sample units and weights for each month

B. Three Methods to Construct Population of Item-Level STRs

1. Resampling Method2. Fixed-One-Rate Method3. CDF Interpolation Method

Stratum Stratum Description

P07 Edible vegetables, roots, and tubers

P08 Edible fruit and nuts; peel of citrus fruit or melons

P09 Coffee, tea, mate and spices

P22 Beverages, spirits, and vinegar

P61 Articles of apparel and clothing accessories

P74 Copper and articles thereof

P90 Optical, photographic, measuring and medical instruments

V. Numerical ResultsA. Selected Strata

Item STR1 of Design Strata

Original Variance of STR (P90)

Relative Bias of STR (P90)

Degrees of Freedom of STR (P90)

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VI. Conclusions

A. For Complex Establishment Surveys Like IPP, Simulation-Based Evaluations of Est Properties Require Consideration of

1. Approximations to the true population, design and estimation procedures

2. Adequacy of the resulting approximations

to the true properties

AX

ˆ{ ( , )}AQ F Xˆ{ ( , )}TQ F X

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B. Future Work

1. For IPP: Other features of the pop and design

e.g. Independence of STRs from sample units

2. Consider generalized variance estimator to improve stability

3. Explore the surface defined by

in “neighborhoods” of the true

TX

ˆ( , )Q F X