Economic Measurement Group Workshop

Preliminary Working Paper

30th November – 2nd December 2011

Revisiting the Estimation of Substitution Biases and the Construction of Spatial

Price Indexes using Scanner Data

Cristian Rotaru, Shaun McNaughton, Franklin Soriano

Analytical Services Branch

Australian Bureau of Statistics

Abstract

Understanding how scanner data can be used to improve current procedures, has led price

statisticians to assess different aspects of price index construction. One current area of

research is to examine the effects of substitution bias in the context of fixed basket indexes.

Substitution bias refers to the failure of the index to adjust for the changes in consumer

behaviour in response to relative price changes. By providing up-to-date information on

expenditures and quantities, scanner data provides the potential to construct superlative

indexes, and therefore assess the magnitude of these substitution biases.

Scanner data also allows for the construction of fine level spatial price indexes, as the

volume of information contained in the data makes it more likely for spatial price

comparisons to be made on the same product. Knowing how scanner data can affect spatial

price estimates can help inform statistical agencies about the best ways to utilise the data.

The focus of this paper is to apply and understand the implications of the different methods

used to calculate the relative magnitudes of outlet and item biases, to investigate the effects

of different aggregations on price indexes, and to examine the construction of spatial price

indexes.

The views expressed in this report are those of the authors and do not necessarily reflect

those of the Australian Bureau of Statistics. Where quoted or used, they should be attributed

to the authors.

1. Introduction

Electronic point-of-sale data, also known as scanner data, has received considerable

attention from statistical agencies during recent years. It has the potential to increase the

efficiency of current price collection techniques by providing timely consumer expenditure

and quantity information.

Understanding how scanner data can be used to improve current procedures, has led price

statisticians to assess different aspects of price index construction. One current area of

research is to examine the magnitude of substitution bias in the context of fixed basket

indexes. Substitution bias refers to the failure of the index to adjust for the changes in

consumer behaviour in response to relative price changes (Boskin et al., 1996). By providing

up-to-date information on expenditures and quantities, scanner data provides the potential

to construct superlative indexes, and therefore assess the magnitude of these substitution

biases.

Scanner data also allows for the construction of fine level spatial price indexes. These

indexes permit the comparison of prices across areas. The volume of information contained

in scanner data makes it more likely for spatial price comparisons to be made on the same

product. Knowing how scanner data can affect spatial price estimates can help inform

statistical agencies about the best ways to utilise the data.

This aim of this paper is to examine substitution bias and spatial price indexes using the

Information Resources Incorporated (IRI) Academic Data Set1 by focusing on several US

markets. Note that the indexes constructed in this paper are not comparable with any official

price index and that the results reported should not be generalised to draw conclusions

about indexes computed by official statistical agencies. In this report, we have used scanner

data relating to supermarket sales of just five commodities, covering the geographical areas

of six US markets and spanning a period of three years. Our main focus was to apply and

understand the implications of the different methods used to calculate the relative

magnitudes of outlet and item biases, to investigate the effects of different aggregations on

price indexes, and to examine the construction of spatial price indexes using scanner data.

The structure of the paper is as follows. Section 2 provides the conceptual framework of

substitution bias. The framework for the decomposition of the substitution bias is presented

in sections 3. Section 4 follows by briefly discussing spatial price indexes. Section 5 describes

the data. The results are presented in Section 6. Finally, the conclusion and some future

work are presented in section 7.

1 All estimates and analyses in this paper based on SymphonyIRI Group, Inc. data are by the author(s) and not by SymphonyIRI Group, Inc.

2. Substitution bias

Fixed weight price indexes have been widely used by statistical agencies to inform the

public of changes to the price of a fixed basket of goods over time. As these indexes use

fixed weights, they do not take into account the substitution between items or outlets, which

can result in substitution bias.

One of the major studies that examined the effects of substitution bias on fixed weight price

indexes was the Boskin Commission report (1996). The report outlined the concept of bias

and suggested the use of the Cost of Living (COL) indexes to measure changes in prices.

These indexes are free from substitution bias as they allow the expenditure weights to vary

over time. In spite of their attractiveness, one of the major drawbacks of the COL indexes is

that they require timely data on both quantities and prices. The recent move to utilise

scanner data for price index computation has allowed statistical agencies to estimate these

indexes.

While several possible sources of bias exist, this report focuses on two types of substitution

bias:

Item substitution bias, which can be defined as the failure of the index to adjust for

the consumers’ behaviour of substituting from relative expensive goods to relatively

cheaper ones;

Outlet substitution bias, which can be defined as the failure of the index to adjust for

the consumers’ behaviour of substituting between relative expensive outlets to

relatively cheaper ones.

At this point it is also important to differentiate between upper and lower level substitution

bias, that is, to identify when substitution occurs within a commodity or across commodities.

For example, lower level substitution would consist of changing expenditure within a

similar item type (e.g. between red and green apples), while upper level substitution would

be across different categories (e.g. between apples and oranges) as price relatives change. In

this study we only considered substitution biases within commodities, restricting ourselves

to the lower level substitution effects.

In estimating substitution bias, we followed the approach described by Jain & Abello (2001)

and Ivancic (2007), in which a measure of bias can be estimated by calculating the difference

between a fixed basket Laspeyres index and a superlative index2 , which is used to

approximate a COL index.

- t t tSubstitution Bias Laspeyres Superlative (1)

2 For further information on the properties of superlative indexes see Diewert (1976)

3. Aggregation and the decomposition of substitution bias

An important element in the calculation of indexes is the level at which aggregation occurs.

Aggregation is often done in order to make the data set more manageable and to reduce the

high volatility of price changes (Ivancic & Fox, 2010). In the case of scanner data,

aggregating across different dimensions, such as stores or chains, plays an important role,

due to the large volume of data.

There are several different combinations of dimensions over which data can be aggregated,

including time, space and outlet dimensions (Hawkes and Piotrowski, 2003). Within each

dimension the level at which data can be aggregated can also differ. For example, in the case

of time dimension, one can calculate monthly, weekly or quarterly aggregates.

Deciding on the level of aggregation is particularly important when using unit values as a

measure of price. Unit values represent the average price paid for an item over a particular

aggregate (see Appendix 1). Scanner data offers large possibilities to examine the effects of

aggregation on indexes and allows us to investigate different types of bias being captured.

In calculating these indexes, we used two levels of aggregation, across items and across

outlets. The difference between an index using fixed weights at both levels and a superlative

index can be thought of as the combined effect of the item and outlet substitution bias. It

should be noted that it is almost impossible to fully isolate the two types of biases as they

interact with each other. We first define the estimation of total substitution bias as follows:

-t t tTotal Substitution Bias L F (2)

While there are a number of superlative indexes that could have been used, only Fisher

indexes were considered in this study. By expanding on the concepts outlined in Jain &

Abello (2001), we used two different methods for estimating the outlet and item substitution

bias3.

j

t t tOutlet Substitution Bias L LF (3)

-i

t t tOutlet Substitution Bias LF F (4)

Where j

tLF is an index that is computed by firstly using a Fisher index to aggregate across

stores, then aggregating across items using a Laspeyres index. Similarly

i

tLF is an index that

is computed by firstly using a Fisher index to aggregate across items, then aggregating

across stores using a Laspeyres index.

3 It should be noted that the Fisher index does not have a “natural” additive decomposition (Balk

2008). As a two-stage Fisher index calculated by firstly aggregating across items and then across outlets will differ from a single-stage Fisher index aggregating over items and outlets simultaneously.

Item substitution can be similarly estimated using a similar method by switching the order

of aggregation.

i

t t tItem Substitution Bias L LF (5)

j

t t tItem Substitution Bias LF F (6)

4. Spatial Price Indexes

The concept of spatial price indexes is simple, in that it measures the change in price

between different areas. Unfortunately, standard price indexes cannot be used to compare

different regions as they suffer from the failure of the circularity test. Circularity, otherwise

known as transitivity, ensures that a consistent result will be achieved if prices are compared

directly between two cities or indirectly through other cities.

The Gini-Elteto-Koves-Szulc (GEKS) method satisfies this criterion, so it was adopted for

producing estimates of the differences in price levels between spatial areas (see Appendix 1

for the GEKS formula). One of the benefits of scanner data is its large volume of information,

which allows for the construction of detailed spatial price comparisons to be made on the

same product.

5. The Data

The scanner data consists of weekly aggregated data provided by the SymphonyIRI Group

Inc. The full scanner data set includes 31 commodities, 2093 stores, covering 50 geographical

markets and spanning a period of seven years.

This study focused only on several item categories namely, Soft Drinks, Butter, Facial

Tissues, Paper Towels and Coffee. We also limited our studies to three years of data and six

markets: New York, Washington, Chicago, Dallas, Portland and San Francisco.

Table 1. IRI Academic Data descriptive statistics for selected items (2005-2007)

Commodity Number of Observations Number of Items Number of Stores

Soft Drinks 15,048,211 4,651 468

Coffee 6,204,194 3,421 468

Butter 2,781,841 367 469

Tissues 1,558,677 461 469

Paper Towels 1,728,300 775 469

For our study unit values and quantities were calculated after aggregating over weekly,

monthly or quarterly periods depending on the time frame of interest. These aggregates do

not correspond to the seasonal months or quarters but to 4-week months and 12-week

quarters.

Matching of items and stores over a specified time frame was done to keep only the items

and stores which were present in the base period (the first 3 months) and each of the other

weeks. The time window used in the aggregation was also varied, in order to check the

effects of shrinking the time frame. In the case of soft drinks, this can be seen in Table 2 and

Chart 1.

Table 2. Matching of items across stores and time using weekly aggregates for soft drinks

(2005-2007)

Total number of items Total number of stores

Timeframe

Before

matching

After

matching

% Remaining

(After/Before)

Before

matching

After

matching

% Remaining

(After/Before)

1 years 2784 1336 48.0 373 268 71.8

2 years 3009 930 30.9 421 156 37.1

3 years 4651 393 8.4 468 104 22.2

Chart 1. Laspeyres and Fisher indexes for soft drinks (after matching)

90.0

95.0

100.0

105.0

110.0

115.0

120.0

1 6 11 16 21 26 31 36

3 Year Laspeyre

2 Year Laspeyre

1 Year Laspeyre

3 Year Fisher

2 Year Fisher

1 Year Fisher

Soft Drinks

Month

6. Preliminary results

Overall, the results of estimating substitution biases were different depending on the

method and type of aggregation used. In most cases item substitution bias was larger than

outlet substitution bias, which reflects the findings of Jain & Abello (2001). We also found

that the effect of aggregating over larger time frames reduced the average level of

substitution bias, mirroring the findings of Ivancic (2007).

Most surprising however, was the difference in magnitude of the bias between the two

methods used to estimate the outlet and item substitution bias. While there are unaccounted

effects between the two measures, the magnitude of the difference between the two methods

was at times large.

It should be noted that only direct indexes were considered as the drift from chaining

indexes was expected to be substantial in the case of high frequency data (Ivancic, 2007).

Note also that although these calculations were replicated on five commodities, only the

results for soft drinks will be presented in the following sections. The bias estimates for the

other commodities are included in Appendix 2.

6.1 Direct price index estimates over time

It can be seen from Chart 2, that the magnitude of substitution bias is relatively constant

across 3 years, fluctuating around some average level. However, as expected, the j

tLF and

i

tLF indexes lie between the Laspeyres and Fisher indexes as shown in Chart 3.

Chart 2. Price indexes over time

90.0

95.0

100.0

105.0

110.0

115.0

120.0

1 6 11 16 21 26 31 36

Month

Soft Drinks

Laspeyre

Paasche

Fisher

Chart 3. j

tLF and i

tLF indexes relative to Fisher and Laspeyres indexes

6.2 Time and area aggregation effects on substitution bias estimation

Table 3, on the following page, summarises the effect of aggregation on the substitution bias

estimates. It can be seen that aggregating over a larger time frame reduces the level of

substitution bias. In particular it can be observed that the bias estimates are halved when

changing from a weekly to a monthly time frame. It can also be noted that different areas

have substantially differing levels of substitution bias, with New York having the largest

average outlet substitution bias effect for soft drinks in both measures. However, the largest

average item substitution bias estimate is either Chicago or New York depending on the

method used.

90.0

95.0

100.0

105.0

110.0

115.0

120.0

1 6 11 16 21 26 31 36

Month

Soft Drinks

Laspeyre

LF(j)

LF(i)

Fisher

Table 3. Indicative average bias estimates for soft drinks under different aggregation

methods (2005 – 2007)

Time Aggregation

Average

Total Bias

(2)

Average

Outlet Bias

(3)

Average

Outlet Bias

(4)

Average

Item Bias

(5)

Average

Item Bias

(6)

Weekly L-F L-LF(j) LF(i)-F L-LF(i) LF(j)-F

Overall 7.01 5.69 0.93 6.08 1.32

New York 9.15 6.44 1.29 7.85 2.71

Chicago 6.87 3.67 0.29 6.58 3.20

Dallas 5.52 3.91 0.61 4.91 1.61

Washington 7.14 5.01 0.63 6.52 2.13

San Francisco 2.86 1.43 0.50 2.37 1.43

Portland 4.88 2.68 0.98 3.90 2.20

Monthly

Overall 3.39 2.66 0.45 2.94 0.72

New York 3.43 2.16 0.45 2.98 1.28

Chicago 3.85 2.12 0.12 3.73 1.73

Dallas 3.67 2.60 0.35 3.32 1.08

Washington 2.36 1.56 0.30 2.06 0.80

San Francisco 1.83 0.94 0.29 1.54 0.89

Portland 3.36 2.05 0.79 2.57 1.31

Quarterly

Overall 2.39 1.86 0.31 2.08 0.53

New York 2.13 1.23 0.28 1.85 0.90

Chicago 2.92 1.50 0.05 2.87 1.42

Dallas 2.88 2.11 0.28 2.60 0.77

Washington 1.39 0.88 0.19 1.20 0.51

San Francisco 1.36 0.70 0.19 1.17 0.66

Portland 2.45 1.71 0.65 1.80 0.74

6.3 Spatial Price Indexes

For the computation of spatial price indexes we only considered the 2007 calendar year.

Note that a similar approach could be used to construct spatial price indexes for the other

years. In order to ensure consistency, we only used the items that were present in every

month and in each geographical area.

Table 4. Annual and quarterly spatial price indexes (2007)

Time Period New York

(Base) Washington Dallas

San Francisco

Portland Chicago

Jan – Dec (Annual)

100 107 103 113 109 101

Jan – March (Qtr 1)

100 104 101 112 108 99

Apr – June (Qtr 2)

100 108 101 115 114 101

Jul – Sept (Qtr 3)

100 109 105 114 110 101

Oct - Dec (Qtr 4)

100 107 105 112 102 102

Table 4 provides the annual and quarterly spatial price indexes. The price relationships

appear to change over time, with the annual price measure providing an average of the price

change. A monthly breakdown shown in Chart 4 reveals that quarter 3 masked a substantial

change occurring during August. An annual or quarterly spatial index did not capture this

particular event as the surrounding periods masked the change. As these are not spatial-

temporal indexes, one cannot immediately compare price levels between months, so these

are only indicative of the relationship that exists for that month.

Chart 4. Spatial price indexes over a 12 month (calendar) period, with New York as the base

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Month

Spatial Price Indexes (Soft Drinks)

New York (Base)

Washington

Dallas

San Francisco

Portland

Chicago

7. Conclusions and Future Work

This paper examined the relative magnitude of substitution bias over different aggregation

methods. Note that the methods provide only indicative measures of the outlet and item

substitution bias. While item substitution bias is generally larger than the outlet substitution

bias, the amount differs depending on where the market is located, the time frame used and

the method chosen to estimate the bias. Spatial price indexes also show that aggregation can

hide markets events that may be averaged or smoothed out.

However, more work can be done in this field in order to improve our understanding of the

way aggregation impacts on price index construction. Several suggested ideas for future

work are given below:

Examining the use of the alternative superlative indexes (Balk, p.101, 2008); this is

because the Fisher index is not consistent in aggregation.

Examining hedonic regression methods as a way of investigating the homogeneity of

outlets.

Estimating spatial-temporal price indexes that are calculated using a Country

Product Dummy method. These could then be compared to the price indexes

computed using the GEKS method.

Acknowledgements

This research has been undertaken with assistance from the Macroeconomic Research

Section, the Prices Branch and the Analytical Services Unit of the Australian Bureau of

Statistics. We would especially like to thank Ruel Abello, Bert Balk, Lewis Conn, Derick

Cullen, and Stephen Frost for their guidance and advice.

8. References

Balk, B.M. 2008 ‘Price and Quantity Index Numbers: Models for Measuring Aggregate Change and

Difference’, Cambridge University Press, New York.

Boskin, M.J. Dulberger, E.R. Gordon, R.J. Griliches, Z. and Jorgenson, D. 1996 ‘Toward a More

Accurate Measure of the Cost of Living’, Final report from the Advisory Commission to study

the Consumer Price Index, United States Senate Finance Committee, Washington, D.C; U.S

Government Printing Office.

Bronnenberg, B. J. Kruger, M.W. Mela, C.F. 2008 ‘Database paper: The IRI marketing data set’,

Marketing Science, 27(4) 745-748.

Diewert, W.E. 1976, ‘Exact and Superlative Indexes’, Journal of economics, Vol. 44, no. 3.

Hawkes, W.J.& Piotrowski, F.W. 2003 ‘Using Scanner Data to Improve the Quality of

measurement in the Consumer price index’, Scanner Data and Price Indexes, University of

Chicago Press, Chicago.

Ivancic, L. Fox, K. 2010 ‘Understanding Price Variation Across Stores and Supermarket Chains:

Some Implications for CPI Aggregation Methods’, paper presented to the Ottawa Group, 2011.

Ivancic, L. 2007 ‘Scanner Data and the Construction of Price Indices’, PhD thesis, University of

New South Wales, Sydney.

Jain, M. & Abello, R. 2001 ‘Construction of price indexes and exploration of biases using scanner

data’, paper presented at the Sixth Meeting of the International Working Group on Price

Indices, Canberra, 2nd - 6th April, 2001.

Waschka, A. Milne, W. Khoo, J. Quirey, T. Zhao, S. 2003 ‘Comparing Living Costs in Australian

Capital Cities, A Progress Report on Developing Experimental Spatial Price Indexes for Australia’,

paper presented at the 32nd Conference of Economists, Canberra, 29th September – 1st

October, 2003.

Appendix 1. Formulae used in calculations

Unit values

Unit values are used as a measure of price paid in the case of multiple transactions for any

given period t .

,

,

tij tij

i j

t

tij

i j

P Q

UV

Q

Where i and j correspond to item and outlet respectively.

Fixed basket price indexes

The two general types of fixed basket indexes commonly used by statistical agencies are the

Laspeyres and Paasche type indexes. These are defined below.

Laspeyres Index

0

,

0 0

,

0 0

0

t

tij ij

i j

ij ij

i j

tij

i ij

i j ij

P Q

L

P Q

Pw w

P

Where 0iw and 0ijw are given as follows:

0 0

0

0 0

,

ij ij

j

i

ij ij

i j

P Q

w

P Q

0 0

0

0 0

ij ij

ij

ij ij

j

P Qw

P Q

Paasche Index

,

0

,

1

0

t

tij tij

i j

ij tij

i j

ij

ti tij

i j tij

P Q

Pa

P Q

Pw w

P

Where tiw and tijw are given as follows:

,

tij tij

j

ti

tij tij

i j

P Q

w

P Q

tij tij

tij

tij tij

j

P Qw

P Q

Superlative indexes

The Fisher index is the geometric mean of the Laspeyres and Paasche indexes. It also

provides an approximation of a Cost of Living index under certain consumer behaviour

assumptions.

Fisher Index

1/ 2

0

, ,

0 0 0

, ,

*

tij ij tij tij

i j i j

t

ij ij ij tij

i j i j

P Q P Q

F

P Q P Q

In order to estimate the outlet and item substitution bias we used different aggregation

strategies and computed related indexes, namely j

tLF and i

tLF , where they are defined as

follows:

0

0

*

*

j j

t i ti

i

i i

t j tj

j

LF w F

LF w F

j

tiFi

tjF are defined as:

1/ 2

0

0 0 0

1/ 2

0

0 0 0

*

*

tij ij tij tij

j jj

ti

ij ij ij tij

j j

tij ij tij tiji i i

tj

ij ij ij tij

i i

P Q P Q

F

P Q P Q

P Q P Q

F

P Q P Q

Spatial Price Indexes

Gini-Elteto-Koves-Szulc index is an index that satisfies the transitivity property that many

standard price indexes do not pass. Transitivity ensures that the indexes are internally

consistent across the different locations under investigation. The GEKS index can be written

in the following form

1/

, , ,

1

/

KK

b c b k c k

k

GEKS F F

Where K is a vector of spatial areas.

Appendix 2. Substitution bias results for commodities

Coffee

Time Aggregation

Average

Total Bias

(2)

Average

Outlet Bias

(3)

Average

Outlet Bias

(4)

Average

Item Bias

(5)

Average

Item Bias

(6)

Weekly L-F L-LF(j) LF(i)-F L-LF(i) LF(j)-F

Overall 9.20 5.77 3.81 5.38 3.42

New York 12.86 7.93 4.69 8.17 4.93

Chicago 2.98 0.44 1.41 1.58 2.54

Dallas 5.86 4.21 3.04 2.81 1.64

Washington 4.60 2.25 1.43 3.18 2.36

San Francisco 2.51 0.00 -0.02 2.53 2.51

Portland 8.27 4.39 2.05 6.21 3.87

Monthly

Overall 5.47 3.98 1.92 3.55 1.50

New York 7.81 5.89 2.36 5.45 1.93

Chicago 1.30 0.20 0.50 0.80 1.10

Dallas 3.93 2.83 1.74 2.19 1.10

Washington 2.78 1.60 0.70 2.09 1.18

San Francisco 1.17 0.02 -0.05 1.22 1.15

Portland 6.23 4.17 1.78 4.45 2.06

Quarterly

Overall 3.31 2.30 1.27 2.04 1.01

New York 4.52 3.30 1.51 3.01 1.22

Chicago 0.75 0.13 0.33 0.42 0.62

Dallas 2.52 1.77 1.03 1.49 0.75

Washington 1.82 1.15 0.40 1.42 0.67

San Francisco 0.58 0.02 -0.08 0.66 0.56

Portland 4.36 3.01 1.26 3.10 1.34

Paper Towels

Time Aggregation

Average

Total Bias

(2)

Average

Outlet Bias

(3)

Average

Outlet Bias

(4)

Average

Item Bias

(5)

Average

Item Bias

(6)

Weekly L-F L-LF(j) LF(i)-F L-LF(i) LF(j)-F

Overall 4.00 2.30 1.79 2.22 1.71

New York 2.41 1.33 1.65 0.75 1.07

Chicago 2.83 0.68 1.09 1.73 2.14

Dallas 3.25 0.73 1.86 1.39 2.53

Washington 3.19 1.24 1.68 1.51 1.95

San Francisco 6.29 2.33 1.54 4.75 3.97

Portland 0.37 0.00 0.37 0.00 0.37

Monthly

Overall 2.81 1.81 1.10 1.71 1.00

New York 1.78 1.21 1.16 0.62 0.56

Chicago 1.93 0.57 0.62 1.32 1.36

Dallas 2.57 0.65 1.46 1.11 1.92

Washington 2.23 1.06 1.14 1.09 1.18

San Francisco 4.58 1.95 0.91 3.67 2.63

Portland 0.28 0.00 0.28 0.00 0.28

Quarterly

Overall 1.69 1.11 0.67 1.02 0.58

New York 1.17 0.91 0.80 0.37 0.26

Chicago 1.01 0.31 0.28 0.72 0.70

Dallas 1.80 0.42 1.14 0.66 1.38

Washington 1.57 0.79 0.81 0.76 0.78

San Francisco 2.57 0.98 0.37 2.20 1.58

Portland 0.04 0.00 0.04 0.00 0.04

Facial Tissues

Time Aggregation

Average

Total Bias

(2)

Average

Outlet Bias

(3)

Average

Outlet Bias

(4)

Average

Item Bias

(5)

Average

Item Bias

(6)

Weekly L-F L-LF(j) LF(i)-F L-LF(i) LF(j)-F

Overall 4.65 1.94 3.06 1.60 2.71

New York 5.68 1.94 3.23 2.45 3.74

Chicago . . . . .

Dallas 2.03 0.61 2.03 0.00 1.42

Washington 2.37 0.43 1.36 1.01 1.94

San Francisco . . . . .

Portland . . . . .

Monthly

Overall 3.79 1.65 2.26 1.53 2.14

New York 4.81 1.78 2.42 2.39 3.03

Chicago . . . . .

Dallas 2.08 0.69 2.08 0.00 1.39

Washington 1.92 0.42 1.08 0.84 1.51

San Francisco . . . . .

Portland . . . . .

Quarterly

Overall 2.53 1.21 1.48 1.06 1.32

New York 3.03 1.34 1.40 1.62 1.69

Chicago . . . . .

Dallas 1.95 0.69 1.95 0.00 1.26

Washington 1.57 0.37 0.91 0.66 1.20

San Francisco . . . . .

Portland . . . . .

Butter

Time Aggregation

Average

Total Bias

(2)

Average

Outlet Bias

(3)

Average

Outlet Bias

(4)

Average

Item Bias

(5)

Average

Item Bias

(6)

Weekly L-F L-LF(j) LF(i)-F L-LF(i) LF(j)-F

Overall 3.22 2.32 0.81 2.41 0.90

New York 3.42 1.97 0.80 2.62 1.46

Chicago 3.13 1.38 0.52 2.62 1.76

Dallas 2.07 1.21 0.65 1.42 0.86

Washington 2.18 1.16 0.40 1.78 1.02

San Francisco 4.64 1.93 0.73 3.91 2.72

Portland 3.12 1.30 0.47 2.65 1.83

Monthly

Overall 1.95 1.49 0.41 1.55 0.47

New York 2.03 1.38 0.35 1.68 0.65

Chicago 1.65 0.83 0.15 1.50 0.82

Dallas 1.64 0.98 0.50 1.14 0.66

Washington 1.24 0.75 0.15 1.09 0.50

San Francisco 3.02 1.46 0.50 2.52 1.56

Portland 2.17 0.89 0.23 1.93 1.28

Quarterly

Overall 1.15 0.90 0.23 0.92 0.26

New York 1.17 0.84 0.12 1.04 0.33

Chicago 0.72 0.36 0.04 0.68 0.36

Dallas 1.22 0.76 0.40 0.82 0.46

Washington 0.73 0.43 0.06 0.67 0.30

San Francisco 1.80 1.09 0.37 1.43 0.70

Portland 1.41 0.54 0.09 1.32 0.87