topic 3:index numbers and times seriesbettyj/245/topic3webpp_2015.pdf · 1/18/2019 · an index...

Topic 3-Econ 245

Topic 3:Index Numbers and Times Series

Part I: Index Numbers:

_____ number?

______Indices

____

Reference: Chapter 16

We need to develop a measure that summarizes the

characteristics of large data sets and is useful for period to period

comparisons.

Topic 3-Econ 245

An index number accomplishes this by aggregating information

into a _____ measure that permits easy comparisons.

Definition: An INDEX NUMBER is a number which summarizes

pieces of information in a _______ way, and forms the

basis for comparative judgements.

An index number is a ratio of 2 numbers expressed as a

percentage.

Used if we want to keep track of “overall” price changes over time

or need to get an _________measure of output– different items

have the own units.

Examples: CPI (Consumer Price Index);

PPI (Producer Price Index)

Topic 3-Econ 245

Basic Considerations for a “____” Index Number:

(i) Include __________ on all items.

(ii) Combine in way which recognizes r______ importance of each

item.

(iii) Need a “bench mark” value and period to provide reference

point for comparisons. *Base period required

Topic 3-Econ 245

Price and Quantity Indices

Price Indices

Two broad types: (and there are several variants of each)

(i) Aggregative Indices

(ii) ___________Indices

Topic 3-Econ 245

Looking at (i): Aggregative Indices

Example: Bundle of Two Goods

Item 1990 2010

Price ($)

(P)

Quantity

(Q)

Price ($)

(P)

Quantity

(Q)

Good 1 0.75 1.80

Good 2 0.50 2.70

We want to summarize overall price change from 1990 to 2010.

(Base Year =1990.)

Must be the same goods in each year! (Assumption)

Topic 3-Econ 245

P

P

P

t

S

it

i

n

i

i

n0

1

0

1

100

Type of Price Index

Price of the “ith” item

in the year of interest.

Price of the

“ith” item in

the base year.

Base Year Year of

interest

The 100

indicates that

we have a

base value of

100 in the

base year.

Topic 3-Econ 245

Simple Aggregative Index:

Let Pit= Price of the ith item at time “t”.

(i= 1, 2,....,n; t=0, 1, 2,....,T)

Construction of the S_____ Price Index:

P

P

Pt

Sit

i

n

ii

n0

1

01

100

*

The simple price index from base year ‘0' to year ‘t’ for

items 1 to n.

Topic 3-Econ 245

If t=0, index=____ (Base year value).

Example: See previous data for 1990 and 2010:

Base Year =1990

UnitlessP

P

P

P

S

t

n

i

i

n

i

it

S

t

$$360100

50.075.0

70.280.1

100

2000,19900

1

0

1

0

Prices rose, overall, to be 3.6 times ________in 2010 than in

1990.

Topic 3-Econ 245

This type of index only tells about changes in ______.

Individual price falls can be offset by other individual rises.

With the simple index, all goods are given ______weight.

So......

Topic 3-Econ 245

Weighted Aggregative Index:

P

P W

P W

t

W

it i

i

n

i i

i

n0

1

0

1

100

where Wi=(fixed) weight given to the ith price.

(Limitation of these indices: ______ changes: Assumption of identical

items.)

Topic 3-Econ 245

How to Choose the Weights:

__________of goods purchased. (Which quantity?)

Percentage of ____________on good. (piqi)

Both measure “importance”, but may differ over time.

So, when should they be measured?

Topic 3-Econ 245

Possibilities: (I) Laspeyres’ Index:

P

p q

p q

t

L

it i

i

n

i i

i

n0

0

1

0 0

1

100

____ period quantities= weights

Disadvantage:

Weights get _________over time.

Tends to ____state changes in price for the price index.

Topic 3-Econ 245

(II) Paasche’s Index:

P

p q

p q

t

P

it it

i

n

i it

i

n0

1

0

1

100

Current period quantities=weights

Disadvantage:

Weights constantly changing, so index change may be partly due

to _________ changes. (Tends to ______state changes.)

Topic 3-Econ 245

Some Compromises:

(III) Marshall-Edgeworth Index:

P

pq q

pq q

t

ME

it

i it

i

n

i

i it

i

n0

0

1

0

0

1

2

2

100

Average of base period and current period _________ are the

weights.

Topic 3-Econ 245

(IV) Fisher’s “Ideal” Index:

P P Pt

F

t

L

t

P

0 0 0

(Geometric Mean of the Laspeyres’ and Paasche indices)

Example: Calculations from earlier data:

Year Pt

S

0 Pt

L

0 Pt

P

0 Pt

ME

0 Pt

F

0

1990 100 100 100 100 100

2010

Topic 3-Econ 245

Calculations:(from data on page 2)

If the base year is 1990, with a value of 100, then for t=2010:

Simple Aggregative Index

P

t

S

0

1 80 2 70

0 75 0 50100 360

. .

. .

Laspeyres’:

Pt

L

0

180 1 2 70 4

0 75 1 0 50 4100 458 18

. ( ) . ( )

. ( ) . ( ).

Topic 3-Econ 245

Paasche: P

t

P

0

180 2 2 70 3

0 75 2 0 50 3100 390 1

. ( ) . ( )

. ( ) . ( ).

Marshall-Edgeworth:

P

pq q

pq qt

ME

it

i it

i

n

i

i it

i

n0

0

1

0

0

1

2

2

100

1 801 2

22 70

4 3

2

0 751 2

20 50

4 3

2

100 422 61

. .

. .

.

Fisher: P P Pt

F

t

L

t

P

0 0 0458 390 422 7 .

Topic 3-Econ 245

Changing The Base:

Often one may want to change the ____period and /or base value

of an index after it has been constructed.

Suppose we have an index with a base value of 100 in 1970. But,

we desire to have a base of 100 in 1972:

Year Old Index New Index

1970 Adjust 83.33

1971 each 91.67

1972 entry 100

1973 by 108.33

1974 dividing 118.33

by 1994 1.2 246.66

Topic 3-Econ 245

What numerical value multiplied by 120 equals 100, such that

the relatives are left unaltered? (Or equivalently 120 divided by

what numerical value equals 100?)

120100 1 2

120 100 0 833

X

or

X

solving for X: 120

100

solving for X:100

120

.

.

Base period is changed by proportional scaling– ________

unaltered.

To create the new index, divide each value of the “old” index by 1.2

(or multiply each value of the “old” index by 0.833).

Overall % changes do ____ change.

Topic 3-Econ 245

Quantity Indices: Same idea as price indices.

Reverse the roles of prices and quantities. Prices are now acting as

the ______.

Simple Quantity Index: Q

q

qt

S it

i

0

0

100

Laspeyres’ Quantity Index:

Q

q p

q p

t

L

it i

i

n

i i

i

n0

0

1

0 0

1

100

Base period prices = weight

Topic 3-Econ 245

Paasche Quantity Index:

Q

q p

q p

t

P

it it

i

n

i it

i

n0

1

0

1

100

Current period prices are the weights.

Marshall Edgeworth:

Q

qp p

qp p

t

ME

it

i it

i

n

i

i it

i

n0

0

1

0

0

1

2

2

100

Fisher: Q Q Q

t

F

t

L

t

P

0 0 0

Topic 3-Econ 245

Choosing Between Indices: 3 TESTS

(1) Time Reversal Test:

Pit=price of the ith good at time t.

For the single good:

P

P PP

it

i i

it

0 0

1

I.e. Information about _____ change is the same, regardless of

reference date.

Topic 3-Econ 245

Example:

Gum: Price in 2000 is $1.15.

Price in 2005 is $1.75.

1 75

1 15

1

1 151 75

1 75 1 151 75

1 15

1 52173913 1 52173913

.

. ..

. ..

.

. .

Topic 3-Econ 245

This motivates the idea of the ______Reversal Test for indices.

If it passes:

( * )

:

P P

where

P

P

t t

t

t

0 0

0

0

1

is base at 0 to time t and

is base at t to time 0.

Topic 3-Econ 245

Example: Laspeyres’ Index:

P

p q

p q

P

p q

p q

P P

P P

L

i i

i

n

i i

i

n

L

i i

i

n

i i

i

n

L L

L L

01

1 0

1

0 0

1

10

0 1

1

1 1

1

01 10

01 10

1

1

(assume the base value is 1 not 100)

Test:

Fails if:

Laspeyres’ generally ______ the Time Reversal Test.

Topic 3-Econ 245

Using the data from the previous example:

P

p q

p q

P

p q

p q

L

i i

i

n

i ii

i

n

L

i i

i

n

i i

i

n

1990 2010

2010 1990

1

1990 1990

1

2010 1990

1990 2010

1

2010 2010

1

4 5818

0 75 2 0 50 3

1 80 2

,

,

.

( . )( ) ( . )( )

( . )(

=(1.80)(1) + (2.70)(4)

(0.75)(1) + (0.50)(4)

12.6

2.75

) ( . )( ) ..

( . )( . ) ., ,

2 70 3

3

11 70 2564

4 5818 0 2564 1 1748 11990 2010 2010 1990

Test: P PL L

Laspeyres’ Price index _____ the Time Reversal test.

Generally: Time Reversal Test >1 for Laspeyres’ and < 1 for

Paasche.

Topic 3-Econ 245

(2) _______Reversal Test:

For a single good, price ratio is p

p

1

0

and quantity ratio is

q

q

1

0

.

The change in expenditure is the product: p q

p q

p

p

q

q

1 1

0 0

1

0

1

0

.

The F_______Reversal test for an index checks:

P Qp q

p q

Where P

Q

i i

i i

01 01

1 1

0 0

01

01

*

price index

= quantity index

Topic 3-Econ 245

Suppose P01 shows a 20% rise and expenditure has risen by 50%,

then Q01 should show a 25% rise:

P Qp q

p q

i i

i i

01 01

1 1

0 0

1 20 1 25 150

( . ) ( . ) .

Both Laspeyres’ and Paasche’ Indices _____the Factor Reversal

Test.

Topic 3-Econ 245

Example: Using the previous example:

P

p q

p q

Q

q p

q p

The test Q

p q

p q

L

i i

i

n

i i

i

n

L

i i

i

n

i i

i

n

L

i i

i

n

i i

i

n

01

1 0

1

0 0

1

01

1 0

1

0 0

1

01

1 1

1

0 0

1

: P01

L

Topic 3-Econ 245

P

p q

p q

Q

q p

q p

L

i i

i

n

i ii

i

n

L

i i

i

n

i i

i

n

1990 2010

2010 1990

1

1990 1990

1

1990 2010

2010 1990

1

1990 1990

1

4 5818

2 0 75 3 0 5

1 0 75

,

,

.

( )( . ) ( )( . )

( )( .

=(1.80)(1) + (2.70)(4)

(0.75)(1) + (0.50)(4)

12.6

2.75

) ( )( . ) ..

( . )( . ) ., ,

4 0 5

3

2 751 0909

4 5818 1 0909 4 9981990 2010 1990 2010

LHS P QL L

Topic 3-Econ 245

p q

p q

RHSp q

p q

it it

i i

it it

i i

( . )( ) ( . )( ) .

( . )( ) ( . )( ) .

.

..

1 8 2 2 7 3 11 7

0 75 1 0 5 4 2 75

11 7

2 754 2545

0 0

0 0

Since, 4 998 4 2545. . , fails test.

Marshall-Edgeworth

_______ time-reversal test

______factor-reversal test

Fisher’s Ideal

passes time reversal test

passes factor reversal test

Topic 3-Econ 245

(3) Time Period Sequences:

So far we have considered only 2 points in time.

How do we measure price changes from period 1 to period

2?

Either: (1) Form P12 directly (using period 1 as the base year)

Or (2) take PP

02

01

=P12 directly.

Should get the same result either way.

Topic 3-Econ 245

Note:

PP

P

PP

P

PP

PP

P

P

P

P

P

PP

02

2

0

01

1

0

2

0

1

0

2

0

0

1

2

1

12

rearranging: *

Suppose: Prices rise by 20% from period 0 to 1.

Prices rise by 10% from period 1 to 2.

Overall rise from period 0 to period 2 is (1.2 × 1.1) =1.32 (32%

rise)

Index Values: 100 120 ___

(0) (1) (2)

Topic 3-Econ 245

This motivates the Circularity Test:

Index passes this test if: P P P

02 01 12 ( )

For example, consider the Laspeyres’:

P

p q

p q

P

p q

p q

But

p q

p q

P P

L

i i

i

n

i i

i

n

L

i i

i

n

i i

i

n

i i

i

n

i i

i

n

L L

01

1 0

1

0 0

1

12

2 1

1

1 1

1

2 0

1

0 0

1

01 12

( )

( )

( )

( )

:

( )

( )

P02

L

Topic 3-Econ 245

Laspreyres’ index ____ the circularity test.

Is this important? YES!!–the ____is essentially a Laspreyres’-

type of index. Fails tests and tends to o___-state price rises.

All indices mentioned so far ____ this test, except for the

weighted aggregative index with fixed weights.

Topic 3-Econ 245

We have only considered Aggregative Indices up to this point.

Now consider:

(II) Averages of Relatives Indices

A price _________ is: P

Pit

i 0

.

The assumption is that each commodity is used equally as much.

But, an index should take into account the differing quantities of

the ____ used.

Weighted price indices permit the consideration of the relative

importance of the commodities in the basket of goods. I.e. use

quantities as weights.

Topic 3-Econ 245

A weighted average of price relatives index weights the various

price relatives in a market basket by the total amount s____ on

that commodity.

Weighted Arithmetic Mean of Price Relatives

P

vP

P

v

v p q P P

i e base year weights enditure

v

t

wmi

it

i

i

i i i t

wm

t

L

i

0

0

0 0 0 0

100

*

,

( . . exp )

Note: if then

represents the weight.

Pp q

p qP

t

wm it i

i i

t

L

0

0

0 0

0

.

Topic 3-Econ 245

Geometric Mean of Price Relatives

P

p

pt

G it

i

n

0

0

1

can also be weighted.

Harmonic “Mean of Price Relatives”

P

vP

P

v

v p q P P

i e current year weights enditure

t

H

i

it

i

i

i it it t

H

t

P

0

0

11

0 0

Note: if then ,

( . . exp )

Topic 3-Econ 245

Topic 3-Econ 245

Using Price indices: “Price Deflators”

It is common practice to distinguish between nominal and ____

values for a series.

I.e. current dollar terms versus constant dollar terms.

Recall: the ____ value is obtained by:

Real Value

Nominal Value

Index Number 100

Example: invest money for a year at 7% p.a. interest. Now

suppose prices also rise by 2%.

Topic 3-Econ 245

Real Rate of Interest: 1 07 1 02

1 02

1 07

1 021 5%

. .

.

.

.. .

p a

You can purchase approximately 5% more goods with each dollar

than was possible a year ago.

Convert a nominal variable into a “real” variable by deflating –

divide “nominal” value by price index to get “real” value.

Price Index used as a price __________.

Must choose the appropriate price index for the purpose of price

deflator.

Real income is commonly referred to as the purchasing power of

the money income.

Topic 3-Econ 245

In comparing incomes, wages, rents, GNP, and personal income

per capita of different countries, the use of an appropriate deflator

is common practice. Hence, the real value is more easily

recognized.

Real ExportsNominal Exports

Export Price Index 100

Real ConsumptionNominal Consumption

CPI 100

Real PayrollNominal Value of Payroll

Wage Rate Index 100

Obviously, increases/decreases in “nominal” series may be in the

same/ opposite direction to those in corresponding “real” series.

Topic 3-Econ 245

Topic 3-Econ 245

Linking (“Splicing”) Indices

Suppose we have a Laspeyres’ price index, Pt

L

0 for the period

t=1963,...,1973. The weights have become outdated over this time

period.

You decide to form new weights in 1973 and construct a separate

Pt

L

0 series for t=1973,1974,1975,...,1990.

Topic 3-Econ 245

Although these are different indices, you may need a continuous price

time series, so combine the two series into one linked series:

Year Series

1

Series

2

Linked

1963 100

1964 120

1973 160 100 (100) × 1.6 =

1974 110

1990 185 (185) × 1.6

To make the 100 in year 1973 be equal to 160, simply multiply

(100)(1.6)=160.

Relative prices are preserved over time.

Topic 3-Econ 245

We could link with any other base year:

Year Series

1

Series 2 Linked

1963 100 ÷ 1.6

1964 120 ÷ 1.6

1973 160 (160) ÷ 1.6 = 100

1974 110

1990 185

In each case, the linked series suggests prices are 1.96 times higher

in 1990 than in 1963.

185 62 5

62 51 96

296 100

1001 96

.

..

.

Catalogue no. 62F0014M ISBN 978-0-660-29452-0

by Taylor Mitchell

An Analysis of the 2019 Consumer Price Index Basket Update, Based on 2017 Expenditures

Release date: February 27, 2019

How to obtain more informationFor information about this product or the wide range of services and data available from Statistics Canada, visit our website, www.statcan.gc.ca. You can also contact us by Email at [email protected] Telephone, from Monday to Friday, 8:30 a.m. to 4:30 p.m., at the following numbers:

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Standards of service to the publicStatistics Canada is committed to serving its clients in a prompt, reliable and courteous manner. To this end, Statistics Canada has developed standards of service that its employees observe. To obtain a copy of these service standards, please contact Statistics Canada toll-free at 1-800-263-1136. The service standards are also published on www.statcan.gc.ca under “Contact us” > “Standards of service to the public.”

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Statistics Canada – Catalogue no. 62F0014M 3



by Taylor Mitchell

Introduction

The Consumer Price Index (CPI) is one of the most widely-known, quoted and utilized economic indicators in Canada and is of interest to a wide range of users. It can be used to compare, through time, the cost of a fixed basket of goods and services purchased by consumers. The CPI is used for economic analysis and provides insight into overall economic conditions.

Private and public pension programs, income tax deductions, and some government social payments are adjusted using the CPI. The index is used as a deflator of various economic aggregates to obtain estimates at constant prices. The CPI is also a tool for setting and monitoring economic policy. For example, the Bank of Canada uses the CPI and special aggregates of the CPI for this purpose.

As a Laspeyres-type1 price index, the CPI basket quantities are those of the reference period of the basket weights, which are used to estimate quantities consumed for upper level aggregation. The larger the basket weight of a given aggregate in the CPI basket, the more a price change in that aggregate will impact the all-items CPI.

Basket weights are derived primarily from household expenditures reported in Statistics Canada’s Survey of Household Spending (SHS)2 and are updated every two years. The January 2019 CPI marks the introduction of updated basket weights in the calculation of the index. As of its release on February 27 2019, the basket weights used in the aggregation of the CPI were updated based on consumer spending patterns from the 2017 SHS, replacing those from the 2015 SHS. In addition to updates to the classification structure of the basket, these changes enhance the quality of the CPI.

Composition of the CPI Basket

The Consumer Price Index is a weighted average of the price changes of a fixed basket of goods and services, based on the expenditures of a target population3 in a certain reference period. Each good or service in the basket is representative of consumer spending patterns, and price movements are assigned a basket share that is proportional to the consumption expenditure for which they account. For example, 26.92% of total consumer expenditures in scope of the CPI accounted for shelter-related goods and services. This corresponds with the 26.92% weight assigned to shelter in the 2017 CPI basket. The weights of goods and services are fixed during the life of a given basket and play an important role in determining the impact of a given item’s price change on the CPI. For instance, Canadians on average spend a much larger share of their total expenditures on gasoline than on milk. As a result, a 10% price increase in gasoline prices will have a greater impact on the all-items CPI than an equivalent increase in the price of milk.4

The Consumer Price Index classification of goods and services is organized according to a top-down hierarchical structure (see diagram below). At the top of the structure is the all-items CPI, which contains eight major components. Below the eight major components are intermediate level aggregates, such as owned accommodation and operation of passenger vehicles, which, along with the major components, provide insight into the sources of monthly and annual price change. There are 177 basic aggregates.5 These basic aggregates are typically the result of aggregating one or more elementary aggregates, many of which are unpublished.

1. The Laspeyres formula is a basic method for calculating price indices and is consistent with the CPI’s fixed basket concept. It expresses the change in the cost between period 0 and period t of buying a fixed basket of products, by aggregating the prices of products in the basket using quantities consumed from the price reference period 0 as weights.

2. CPI basket shares may differ from SHS shares of total expenditure due to differences in item classification. Further, some consumer expenses considered to be in scope of the CPI are not included in the SHS. In addition to the SHS, other expenditure data sources such as national accounts data, retail sales data and census data enters into the calculation of the CPI basket. Users can consult the Survey of Household Spending for further information.

3. The target population of the CPI consists of families and individuals living in urban and rural private households in Canada.4. Gasoline comprises 3.34% of the 2017 CPI basket, whereas fresh milk comprises 0.37%.5. While indices are published for 177 basic aggregates at the Canada level, not all are published at the provincial level and below.

http://www23.statcan.gc.ca/imdb/p2SV.pl%3FFunction%3DgetSurvey%26SDDS%3D3508

Statistics Canada – Catalogue no. 62F0014M4


Figure 1Classification Structure of the Consumer Price Index

All-items CPI

Eight major components:

Food; Shelter; Household operations, furnishing and equipment; Clothing and footwear; Transportation;

Health and personal care; Recreation, education and reading; Alcoholic beverages, tobacco products and

recreational cannabis

Intermediate level aggregates

177 basic aggregates

556 elementary aggregates

Elementary aggregates are added or deleted from the basket as consumption patterns change over time. The aggregate for DVD rentals,6 for instance, was deleted from the basket as they became less popular with consumers and subsequently commanded a lower share of overall expenditures. At the elementary aggregate level, the classification includes a sample of items that are chosen to characterize all products in that class. Representative products are chosen with emphasis on items that are widely available and known to be among the most popular with consumers, ensuring that the items selected are representative of the purchases consumers actually make. The number of representative products assigned to an elementary aggregate can vary based on the basket weight of the aggregate, as well as the price variability and heterogeneity of products in that class. For instance, when pricing certain dry grocery products, representative products typically include both brand-name and store-brand items. At the same time, there is only one representative product priced under the bananas aggregate.

Importance of Updating the Consumer Price Index Basket Weights

If the fixed-quantity basket of goods and services was kept unchanged for an extensive period of time, it would gradually lose accuracy and relevance as a reflection of consumer spending. This is partly due to the nature of consumption patterns, which have a tendency to evolve in response to shifts in relative prices. For example, if the price of chicken increased between basket updates, consumers may opt away from chicken and substitute other meats. A Laspeyres-type price index cannot reflect this expenditure change until the basket weights are updated. This can lead to an overstatement of the importance of changes in the price of chicken in the index and

6. DVD rentals were deleted from the basket in the 2017 basket update, based on 2015 expenditure shares.



a subsequent upward bias in the CPI. Typically, the longer a fixed set of basket weights is used, the greater this upward bias.

Consumer spending patterns are also influenced by factors such as variations in the level and distribution of household income, demographics (such as an aging population), evolving habits and technological advances. New products and services are introduced to the market and existing ones may be modified or become obsolete. As a result, the basket needs to be revised periodically to reflect changes in consumers’ spending patterns. For example, the significant increase of the basket share for Internet access services from 0.53% in 2005 to 1.06% in 2017 reflects the growing importance of the Internet in the daily lives of Canadians.7 In the same time period, the basket share of cigarettes fell from 1.27% to 0.82% as consumer spending patterns shifted away from cigarettes.

In addition to the review of the expenditure weights, a basket update is also an opportunity to review and update other aspects of the indices. This includes changing the CPI classification to make it more representative of consumer spending and the products and services available for purchase. It is also an opportunity to review and update the sample of prices collected, review price index estimation methodologies, and update documentation and dissemination products, although these activities are not limited to basket updates.8

Overview of the 2017 Basket Update

The basket is updated periodically to reflect changes in consumer spending patterns. The continuity of the CPI series is maintained by chain linking9 the corresponding indices obtained from consecutive baskets. This is done separately for each aggregate series, which is defined as the intersection of a commodity and a geographic area.

With the introduction of the 2017 basket weights, new product classes were added to reflect the evolving consumption patterns of Canadians, with emphasis on incorporating services that have gained in popularity amid the increasing digitization of the economy. With respect to the rise of the sharing economy,10 this includes the introduction of an elementary aggregate for ride sharing. The aggregate for other traveller accommodation was expanded to include prices for online vacation property rentals. These additions reflect evolving transportation and accommodation preferences among Canadians. In the previous basket, the basic aggregate video and audio subscription services contained two elementary aggregates: broadcasting subscription services and online subscription services. With the introduction of the 2017 basket weights, the latter aggregate now contains two lower-level indices: video streaming services and audio streaming services.

Amid steadily increasing use of public transportation by Canadians in their daily commutes,11 an elementary aggregate for commuter trains was also introduced to the 2017 basket.

At the basic aggregate level, a number of indices were added to the 2017 basket. The alcoholic beverages and tobacco products major component has been renamed alcoholic beverages, tobacco products and recreational cannabis to reflect the consumption of recreational cannabis by Canadian consumers.12 Additionally, medicinal cannabis has been added as a basic aggregate under the intermediate aggregate medicinal and pharmaceutical products.

7. Data from the 2016 General Social Survey reveals that 91% of Canadians aged 15 and older used the Internet during the previous month, up from 86% in 2013. Source: Statistics Canada (2016). General Social Survey. Microdata file (45-25-0001) Last updated June 13, 2018.

8. According to the Canadian Community Health Survey, the percentage of Canadians who identify as daily or occasional smokers has declined from 21.8% in 2005 to 16.2% in 2017. Source: Statistics Canada. No date. Table 13-10-0096-01 Canadian Community Health Survey, Canadian health characteristics, annual estimates. Last updated June 26, 2018.

9. Chain linking is the process of chaining a fixed-basket index to another fixed-basket index in order to create a continuous time series. This process ensures that period-over-period percentage changes in a chain-linked index will only reflect price change and not changes in the fixed quantities.

10. An estimated 9.5% of persons (or 2.7 million people) aged 18 or older living in Canada participated in the sharing economy between November 2015 and October 2016 by using peer-to-peer ride services or private accommodation services. Source: Statistics Canada (2017). The sharing economy in Canada. Last updated February 28, 2017.

11. In Canada, the proportion of people commuting to work by public transit has edged up with every census since collection of this information began in 1996, rising from 10.1% that year to 12.4% in 2016. Source: Statistics Canada (2017). Journey to work: Key results from the 2016 Census. Last updated November 29, 2017.

12. According to the National Cannabis Survey, 14% of Canadians aged 15 years or older reported some use of cannabis products for medical or non-medical purposes in the first quarter of 2018. Source: Statistics Canada (2018). National Cannabis Survey, first quarter 2018. Last updated April 4, 2018.

https://www150.statcan.gc.ca/n1/en/catalogue/45250001

https://www150.statcan.gc.ca/t1/tbl1/en/tv.action%3Fpid%3D1310009601

https://www150.statcan.gc.ca/n1/daily-quotidien/170228/dq170228b-eng.htm

https://www150.statcan.gc.ca/n1/daily-quotidien/171129/dq171129c-eng.htm

https://www150.statcan.gc.ca/n1/daily-quotidien/180418/dq180418b-eng.htm



Table 1Selected product classes added or deleted from the 2017 CPI basketProduct class Parent index Type of aggregate Added Deleted

Broadband and other Internet access services Internet access services Elementary

Area rugs and mats Household textiles Basic

Commuter trains City bus and subway transportation Elementary

Ride sharing Taxi and other local and commuter transportation Elementary

Medicinal cannabis Medicinal and pharmaceutical products Basic

Blank CDs and DVDs Purchase of digital media Elementary

Portable drives Purchase of digital media Elementary

Other traveller accommodation Traveller accommodation Elementary

Audio streaming services Audio and video subscription services Elementary

Cannabis Recreational cannabis Elementary

While one basic aggregate, area rugs and mats, has been deleted from the 2017 basket, the classification structure of the remaining basic aggregates remains intact. Elementary aggregates for broadband and other Internet access services, blank CDs and DVDs and portable drives were deleted from the basic aggregates Internet access services and purchase of digital media respectively.

As alternative data sources have become available, minor changes were made to the classification of the food component at lower level aggregates to better facilitate the use of scanner data.13 This includes the re-allocation of representative products and the introduction of new elementary aggregates. Similar changes were made in the clothing and footwear component, reducing unnecessary detail at the lowest levels of the clothing aggregate.

The index reference period or index base period is the period in which the index is set to equal 100. For the CPI, the index base period is usually a calendar year expressed as “index year=100”. The current index base period for the CPI remains 2002=100 in the 2017 basket.

Update to Basket Weights

Considerations

Between periods, the basket share for a given good or service may either increase, decrease or remain unchanged. Each potential outcome corresponds with the change in the proportion of overall expenditures (increase, decrease or no change) between the 2015 reference year and the 2017 reference year.

It is important to note, however, that expenditure share is a relative measure. Basket weight changes are a function of the expenditure growth within a given category, as well as the growth rate of all other expenditures in scope of the CPI. A basket share that is declining, for instance, indicates that the rate of growth in that category was less than the rate of growth of overall expenditures.

For the following analysis of basket weight changes, the effect of rounding on basket weights should also be taken into consideration. Given that basket weights are rounded and published at the second decimal place, minute changes in expenditure share may not be apparent. Basket share changes of smaller magnitudes are, however, accounted for in the CPI, as full-precision expenditure weights are used in its calculation.

13. In 2018, the Canadian CPI first introduced scanner data as a method of collection for grocery products. The use of scanner data allows the CPI to capture actual transaction prices paid by consumers, increasing the relevance of the data captured. Scanner data will be further incorporated into the CPI as data sources become available.



Analysis of Basket Weights

The chart below shows the evolution from 199214 to 2017 of CPI basket weights for Canada by major component. Over the past 25 years, the basket share of the transportation component has increased the most, while the respective basket shares of the food component and the alcoholic beverages, tobacco products and recreational cannabis component posted the largest declines. A consistently declining trend was observed in the latter since its weight was first published in 1986, with the basket shares for alcoholic beverages and tobacco products both steadily decreasing over time. The 2017 basket, however, recorded the first increase in the basket share of the alcoholic beverages, tobacco products and recreational cannabis component as its weight increased from 2.58% to 3.16%. This increase reflects, in part, the introduction of recreational cannabis to the CPI basket.

Notes: Weights are expressed as proportions of total household consumption expenditures in scope of the CPI in the weight reference period dollars. Additional weights from CPI basket updates in 1996, 2001, 2005, 2009, 2011 and 2013 not shown.Source: Statistics Canada Table 18-10-0007-01 Basket Weights of the Consumer Price Index.

Chart 1Basket share by major component, Canada, 1992 to 2017

0 5 10 15 20 25 30

Alcoholic beverages, tobacco products and recreational cannabis

Recreation, education and reading

Health and personal care

Transportation

Clothing and footwear

Household operations, furnishings and equipment

Shelter

Food

1992 2015 2017

% of CPI basket

Shifting basket shares often result from sustained price changes, which require consumers to reallocate their expenditures. For example, the basket share of the shelter component grew the most in Ontario and British Columbia, reflecting the increasing cost of housing in Canada’s highest priced housing markets.15 At the national level, the basket share of mortgage interest cost has declined with every basket since 2009, reflecting the increasingly low interest rates offered by commercial banks between 2009 and 2017.16

In 2017, consumers allocated a smaller share of spending to clothing and footwear, as well as several durable household goods, including furniture, audio equipment and video equipment. These declines in basket share are attributable, in part, to lower prices for these commodities in 2017 compared with 2015.17

Shifting basket shares can also result from evolving social and economic factors. For example, Canadian consumers allocated a larger share of spending to health care goods in 2017 than they did in 2005,18 an increase

14. 1992 is the first year where weights for all eight major components were published.15. Data from the New Housing Price Index reveals that the annual average price of a new house rose 14.1% in Toronto and 11.1% in Vancouver between 2015 and 2017, compared with a 6.2%

increase at the Canada level. Source: Statistics Canada. No date. Table 18-10-0205-01 New Housing Price Index, monthly (2016=100). Last updated December 13, 2018.16. Canadian Mortgage and Housing Corporation data reveals that the average 5-year residential mortgage rate quoted by institutional lenders over the course of 2009 was 5.05% and declined

to 3.79% in 2017. Source: Statistics Canada. No date. Table 34-10-0145-01 Canada Mortgage and Housing Corporation, conventional mortgage lending rate, 5-year term. Last updated December 10, 2018.

17. Source: Statistics Canada. No date. Table 18-10-0005-01 Consumer Price Index, annual average (2002=100). Last updated January 18, 2019.18. The basket share of health care goods rose to 1.70% in 2017 from 1.39% in 2005, the year its weight was first published.






that coincided with the aging of the Canadian population.19 Similarly, the basket share of cigarettes continued to decline in the 2017 basket, falling to 0.82%, reflecting the changing smoking habits of Canadians.20

While the basket share of food purchased from stores declined between 2015 and 2017, food purchased from restaurants grew in importance from 4.92% in the 2015 basket to 5.03% in 2017. This greater emphasis on food consumed outside the home reflects changing lifestyles and increasing preferences for quick meal solutions, as does the growth in basket share of the other food products and non-alcoholic beverages index, which includes prepackaged foods.

The operation of passenger vehicles index decreased in basket share from 9.45% in 2015 to 9.14% in 2017. Canadians allocated a smaller percentage of their budgets to gasoline in 2017, as expenditures grew more slowly between 2015 and 2017 compared with other commodities. Higher consumer prices for gasoline during the same time period were moderated by reduced growth in gasoline consumption,21 which contributed to a decline in basket share from 3.49% in 2015 to 3.34% in 2017. Other driving-related expenses, including passenger vehicle insurance premiums and passenger vehicle maintenance and repairs, also declined in basket share.

The importance of cellular and Internet services continued to increase in the 2017 basket, which is consistent with the increasing popularity of both technologies, particularly among older Canadians.22 A greater share of consumer expenditures was allocated to multipurpose digital devices, including tablets and smartphones, in 2017 than in 2015. At the same time, the basket share of newspapers continued to decline, reflecting widespread access to online digital media that provides much of the same content. Similarly, the importance of the purchase of digital media index, which includes CDs and DVDs, declined in the 2017 basket.

Conclusion

With the 2019 basket update, important changes aimed at improving the quality of the Canadian Consumer Price Index (CPI) were introduced. The contents included in the basket were updated to reflect changing consumer behaviours and ensure the CPI remains a reliable indicator of consumer price change.

19. The median age in Canada increased from 38.6 years in 2005 to 40.8 years in 2017 and the proportion of Canadians over the age of 65 increased from 13.1% in 2005 to 16.8% in 2017. Source: Statistics Canada. No date. Table 17-10-0005-01 Population by Age and Sex, annual. Last updated June 11, 2018.

20. Data from the Canadian Community Health Survey reveal that the percentage of Canadians who identify as daily or occasional smokers declined from 17.7% in 2015 to 16.2% in 2017. Source: Statistics Canada. No date. Table 13-10-0096-01 Canadian Community Health Survey, Canadian health characteristics, annual estimates. Last updated June 26, 2018.

21. Gross sales of gasoline increased 0.8% between 2015 and 2017, after increasing 3.9% between 2013 and 2015. Source: Statistics Canada. No date. Table 23-10-0066-01 Motor Vehicle Fuel Sales. Last updated October 4, 2018.

22. While people aged 15 to 44 had similarly high Internet usage rates, individuals aged 45 and older increased their Internet use substantially from 2013 to 2016. Among 65- to 74-year-olds, Internet use rose from 65% to 81% while it rose from 35% to 50% in those over the age of 75. Source: Statistics Canada (2016). General Social Survey. Microdata file (45-25-0001) Last updated June 13, 2018.




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Appendix

Table 2Basket Share (%) by Consumer Price Index Component, 2005 – 2017

Major components, selected product groups

Basket reference year2005 2009 2011 2013 2015 2017

percent

Food 16.85 16.05 16.35 16.07 16.45 16.31Food purchased from stores 11.72 11.22 11.48 11.36 11.54 11.28

Other food products and non-alcoholic beverages 3.01 2.75 2.77 2.77 2.71 2.79Food purchased from restaurants 5.13 4.83 4.88 4.71 4.92 5.03

Shelter 25.71 27.52 25.86 26.19 26.79 26.92Rent 5.34 6.04 5.76 5.67 6.20 6.24Mortgage interest cost 5.16 5.81 4.13 4.03 3.50 3.30Homeowners' replacement cost 3.04 4.05 4.25 4.52 4.80 5.18

Household operations, furnishings and equipment 11.41 11.84 12.57 12.92 13.01 12.98Telephone services 2.35 2.20 2.37 2.42 2.37 2.39Internet access services 0.53 0.70 0.78 0.89 0.97 1.06Furniture 1.66 1.42 1.17 1.05 1.23 1.19

Clothing and footwear 5.58 5.61 6.20 6.25 5.68 5.39Transportation 19.60 19.25 20.05 20.01 19.48 19.72

Purchase, leasing and rental of passenger vehicles 7.94 7.75 7.64 7.38 7.97 8.55Purchase of passenger vehicles 6.44 6.57 6.64 6.68 7.08 7.60

Operation of passenger vehicles 9.84 9.61 10.42 10.47 9.45 9.14Gasoline 4.49 4.42 4.85 4.77 3.49 3.34Passenger vehicle maintenance and repairs 1.19 1.12 1.08 1.17 1.60 1.54Passenger vehicle insurance premiums 2.94 2.74 2.92 2.88 2.80 2.68

Public transportation 1.82 1.89 1.99 2.16 2.06 2.02Air transportation 0.96 1.04 1.14 1.25 1.19 1.14

Health and personal care 4.83 4.95 4.95 4.75 4.98 4.86Health care goods 1.39 1.47 1.85 1.64 1.67 1.70

Recreation, education and reading 12.95 11.79 11.26 11.07 11.02 10.66Multipurpose digital devices .. 0.04 0.11 0.22 0.23 0.29Audio equipment 0.21 0.20 0.18 0.10 0.08 0.05Video equipment 0.62 0.76 0.46 0.33 0.30 0.26Purchase of digital media 0.31 0.22 0.18 0.15 0.12 0.07Traveller accommodation 1.20 1.28 1.33 1.12 1.16 1.13Travel tours 1.03 0.96 0.95 1.00 1.25 1.31Newspapers 0.20 0.15 0.06 0.04 0.03 0.02

Alcoholic beverages, tobacco products and recreational cannabis 3.07 2.97 2.76 2.74 2.58 3.16Alcoholic beverages 1.74 1.79 1.60 1.63 1.66 1.74Tobacco products and smokers' supplies 1.33 1.18 1.16 1.11 0.92 0.87

Cigarettes 1.27 1.14 1.10 1.04 0.89 0.82Recreational cannabis .. .. .. .. .. 0.55

.. not available for a specific reference periodNotes: Figures may not add to 100% due to rounding. Weights are expressed as proportions of All-items CPI in the weight reference period dollars. Basket weights are at basket weight reference year price.Source: Statistics Canada, Consumer Prices Division.

Topic 3-Econ 245

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Constructing the CPI

The CPI is based on “price ________” for each item in each

location. Then they are aggregated to get regional indices and

indices for groups. Eventually, you get to an “All groups”

index for Canada.

The construction of the CPI involves the “weighted average of

price relatives” approach.

The weights are (base period) expenditures, so this is the

__________ Index.

Base year updated regularly -- used to be every 4 years. Now,

every ___ years.

Topic 3-Econ 245

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Weights revised to reflect changing expenditure patterns (2013 is

the latest.)

Topic 3-Econ 245

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Note:

Laspeyres: weighted aggregative index, with base period

quantities as weights.

CPI: weighted average of price relatives, with base period

expenditures as weights.

Topic 3-Econ 245

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Topic 3-Econ 245

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Topic 3-Econ 245

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Topic 3-Econ 245

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The Consumer Price Index

The CPI measures changes in consumer prices facing the Canadian

consumer.

It is derived by comparing the cost of a _____ basket of

commodities purchased by consumers over time.

The “basket” is assumed to contain commodities of unchanging

quantity and _____ and hence is believed to reflect only “pure

price movements.”

Topic 3-Econ 245

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Between 1913 and1926 both the Department of Labour and the

Dominion Bureau of Statistics (Statistics Canada) each produced

separate CPI’s. The present CPI series descended from these

sources.

Some component indexes in the CPI begin with the date they were

introduced into the CPI.

The CPI is widely used as an indicator of the changes in the rate

of inflation. Hence, consumers can monitor changes in their

personal income.

Topic 3-Econ 245

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The CPI is generally employed in four ways:

1) Evaluate and escalate the purchasing _____ over time of:

-wages -rents

-leases -child or spousal support allowance

-private and public pension programs (Old Age Security and

Can. Pension)

-personal income tax deductions

-government social payments

2) Used to ______ current dollar estimates to obtain constant

dollar estimates.

3) Setting and evaluating economic policies

- Bank of Canada’s monetary policies

- Assessment of public policy– food prices, etc.

Topic 3-Econ 245

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4) Economic analysis and research by economists

- explore effects of inflation

- causes of inflation

- regional disparities in price movements

CPI is based upon prices (including taxes) paid by consumers in

private retail outlets, government stores, offices and other

consumer service establishments.

CPI considers a constant “basket” of goods over time – prices in

cities greater than or equal to 30,000.

“Price movements of the goods and services represented in the CPI

are weighted according to the relative importance of commodities

in the total expenditure of consumers.”

Topic 3-Econ 245

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CPI basket weights are collected from multiple types of surveys

relating to a specific year. The weights are currently based on

____ consumer expenditure data. The current time base of the

index is ____ = 100.

Note: The CPI does not measure changes in the “cost of living” –

the latter is the change in income needed to keep a consumer as

“well off” – depends on individual preferences.

Topic 3-Econ 245

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Topic 3-Econ 245

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Highlights of the Canadian CPI

(i) Relates to prices in urban centres.

(ii) Based on prices of over 300 items.

(iii) 8- “Groups” of goods:

Food

Shelter

Household Operations and Furnishings

Clothing and Footwear

Transportation

Health and Personal Care

Recreation, Education and Reading

Tobacco and Alcoholic beverages

Topic 3-Econ 245

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There are separate indices plus an “All Groups” category.

(iv) Index computed monthly –regional indices also available.

(v) Weight given to each individual price change is based on

expenditure as % of total household expenditure. The primary

sources of expenditure data on consumer goods and services are

from the Survey of Household Spending and the Food

Expenditure Survey.

(vi) Prices of each group/service sampled at several outlets:

110,000 price quotations each month. The All-items index at the

Canada level is based on an annual sample of over 500,000

prices quotes.

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(vii) Some prices quoted twice/month; some once/quarter; some

twice/year; some annual. EG. Food: 2 / month; Haircuts: 1 / 3

months; Car insurance: bi-annual

(viii) The extent and quality of the sample is constrained by

budgets and the information available on which to base the

sampling prices.

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Cost of Living

It is reasonable to measure the change in the cost of living of an

individual between two periods as the change in their money

income which will be necessary for the individual to maintain his

or her original standard of living – no more or no less.

It must be pointed out that if all prices change in the same

proportion, that proportion will measure the change in the cost of

living and there will be no problem of measurement.

However, all prices do not change in the same

_____________.

Topic 3-Econ 245

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Suppose in period 0 an individual spends his income by

purchasing q0 of various commodities at price p0. Assuming

he saves nothing, his total income equals his total

expenditure: p q0 0 .

In period 1, prices change to p1. How much income does he

need in period 1 to make him as ____ ___ as he was in

period 0?

If the quantities of goods which he would need to buy to

leave him exactly as well off as he was in period 0 are q1 , then

with an income of p q1 1 he would be exactly as well off.

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It follows that the change in the cost of maintaining his

original (period 0) standard of living will be given by

C

p q

p q01

0 1 1

0 0

.

The superscript (0) indicates that the change in his cost of

living is being measured in terms of his period 0 standard of

living.

Unfortunately we do not know the q1 ’s because they are not

the quantities he actually buys in period 1, but only what he

would need to buy to be as well off as before.

Topic 3-Econ 245

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Consequently, we do not know the aggregate p q1 1 .

However, we do know the aggregate p q1 0 . This is the

amount of income he would require in period 1 to enable him

to purchase the quantities he purchased in period 0.

It can be shown that p q1 0 will be greater than p q

1 1 ,

provided his ______ have remained unchanged.

Topic 3-Econ 245

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If he did have the income p q1 0 in period 1, he would

not buy the same quantities as he bought in period 0, (the

q0 quantities) because he would take advantage of the

changes in _______ prices and would alter his allocation

of income, buying relatively more of those goods whose

prices had fallen relatively more or risen relatively less.

The fact that he would do this in preference to buying the

q0 quantities indicates that an income of p q1 0 would

make him better off than the original income of

p q0 0 and, hence better off than an income of p q

1 1

which is its equivalent.

Topic 3-Econ 245

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We have accordingly, p q1 0 > p q

1 1 .

This assumption that his ______ must have remained

unchanged is vital because if they have changed, we cannot

conclude that the quantities he would buy in period 1 with an

income of p q1 0 are preferred to those he actually did buy

in period 0.

This can be illustrated, by means of indifference

curves, in the case of an individual spending his

income on two goods.

Topic 3-Econ 245

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We call the two goods A and B. Let the price of A in period

0 be pA

0

and of B be pB

0

. If the individual’s money income

in period 0 is M0, we shall have:

M p q p q

A A B B0

0 0 0 0

where q

A

0

and q

B

0

are the quantities of A and B

respectively which he can purchase under these

circumstances.

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Topic 3-Econ 245

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On the diagram, qA

0

is measured on the X-axis and qB

0

on

the Y-axis. The individual can take up any position on the

line – his budget line, M0. It has a slope given by the price of

A relative to that of B.

Slope:

p

p

A

B

0

0

The individual will move to the point Q0, where quantities of

A and B, qA

0

and qB

0

, meet on the budget line, at which point

he reaches his highest indifference curve. The individual will

purchase qA

0

and qB

0

in period 0.

Topic 3-Econ 245

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Suppose, in period 1 prices begin to change. (Price of A

increases.)

How much income will the individual now need to be as well

off as he was before?

The individual will need sufficient to enable him to take up a

position just on the indifference curve. We now draw the

line M1 with a slope given by

p

p

A

B

1

1 such that it just

touches the indifference curve.

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With an

Topic 3-Econ 245

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With an income corresponding to this line, he would take up

the position Q1 , purchasing q

A

1

of A and qB

1

of B.

This would require income: M p q p qA A B B1

1 1 1 1 , as against

the old income of M p q p qA A B B0

0 0 0 0 .

But the individual is indifferent between the positions Q0

and Q1 , so that the ratio M

M1

0 measures the change in his

old income necessary to make him as well off in period 1 as

he was in period 0.

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Suppose that the individual is given income in period 1

sufficient to enable him to purchase the same quantities as he

did in period 0.

This income must be sufficient to make the budget line for the

new prices pass through Q0 and must be equal to

M p q p qA A B B1

1 0 1 0 .

This is shown as the broken line in the diagram, and it must

lie parallel to but to the right of M1, so that:

M p q p qA A B B1

1 0 1 0 > M p q p q

A A B B1

1 1 1 1 .

Topic 3-Econ 245

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Topic 3-Econ 245

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Price Indices and the “Cost of Living”

Can a price index, such as the CPI, measure a change in the

“cost of living”?

Individuals are affected differently by price changes.

Definition: A change in the cost of living is the change in ($)

income needed to maintain original ‘standard of living.’

This depends on individual p____________.

Let qi1 be the quantity of good i that I need in period “1" to

maintain the standard of living I enjoyed in period “0.”

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Cost of Living Index is:

Cp q

p q

i i

i i

01

0 1 1

0 0

Compare C01

0 with price indices we have used.

Assume:

(1) _____ individual

(2) _______preferences

Topic 3-Econ 245

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qb

q A

Q0

Q1

M p q p qA A B B1 1 1 1 1

M p q p qA A B B0 0 0 0 0

M p q p qA A B B1 1 0 1 0

*

qA1

qA 0

qB 0

qB1

Topic 3-Econ 245

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The price of A increases:

Relative prices change: “A” is now more expensive.

Budget line shifts slope increases

Consumer is indifferent between:

q , q

q , q

A0 B0

A1 B1

So, C

p q

p q

M

M

i i

i i

01

0 1 1

0 0

1

0

.

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To purchase the original bundle, consumer needs income of:

( )p q p qA A B B1 0 1 0

(the ______line). M1*

Note: p q p qi i i i1 0 1 1

( M1* ) ( M

1 )

If we divide both sides by :

p qi i0 0

base period expendiure :

Topic 3-Econ 245

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p q

p q

p q

p q

i i

i i

i i

i i

1 0

0 0

1 1

0 0

or

P C01

L

01

0

Laspeyres’ Price Index ________ rises in true cost-of-living,

and understates falls in the true cost-of-living.

Topic 3-Econ 245

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Can also define an alternative indicator of the change in

cost of living:

“the individual’s income in period “1" as a ratio of what

would have been needed in period “0", at period “0" prices

to be as well off”:

C

p q

p q

M

M

i i

i i

01

1 1 1

0 0

1

0

where qi 0 = quantity of good i in period “0" needed to

maintain “welfare” experienced in period “1".

the superscript on C.O.L. index refers to the utility base period.

Topic 3-Econ 245

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qb

q A

Q0

Q1

M p q p qA A B B1 1 1 1 1

M p q p qA A B B0 0 1 0 1

*

M p q p qA A B B0 0 0 0 0

qA1

qA 0

qB 0

qB1

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Start at the blue line: M p q p qA A B B1 1 1 1 1

Suppose price is lower is the previous time period:

M p q p q

M p q p q

Now p q p q p q p q

or

A A B B

A A B B

A A B B A A B B

1 1 1 1 1

0 0 0 0 0

0 1 0 1 0 0 0 0

( )

( )

: ( ) ( )

green line black line

M M

p q p q

0

*

0

i0 i1 i0 i0

Topic 3-Econ 245

Page 87

If we divide by p qi i1 1 :

p q

p q

p q

p qinvert both sides

reverse the inequality sign

Paasche price indexp q

p q

p q

p q

understates rises & overstates falls in true C.O. L. index

i0 i1

i1 i1

i0 i0

i1 i1

i1 i1

i0 i1

i1 i1

i0 i0

&

.

C01

1

P CP

01 01

1

A Paasche price index u____states rises and overstates _____

in the true cost of living index!

Topic 3-Econ 245

Page 88

Note:

2 different C.O.L. indices.

Each index involves “hypothetical” quantities.

Cannot tell extent of under/over-statement.

Each individual has own p_____________.

A _____ index takes no account of preferences –

cannot measure changes in true C.O.L. (Or of

welfare”).

Topic 3-Econ 245

Page 89

Topic 3 – Part II

Analysis of Economic Time Series Data

Recording observations of a variable that is a function of time,

results in a set of numbers called a time series.

Topic 3-Econ 245

Page 90

Time Series Data: a sequence of data values, gather over time –

usually at some fixed interval.

Since data have a natural order, we can analyse a time series of

data in terms of 4 basic components:

(i) Trend: Long term “__________” of the series, over many

years.

Direction and shape of the series is important to us.

Topic 3-Econ 245

Page 91

(ii) Cycle: Represents a pattern repeated over time periods of

different _______, usually longer than a year.

“Wave-like” pattern – (3-5 duration)

eg. Business cycles.

Topic 3-Econ 245

Page 92

(iii) Seasonal: represents fluctuations that repeat themselves

within a fixed period of ____ year.

Same pattern each year.

Topic 3-Econ 245

Page 93

(iv) Irregular: fluctuation that is unpredictable, or takes

place by ________or randomly.

Random “noise” in the data.

Non-systematic.

Topic 3-Econ 245

Page 94

The Time Series Model:

It is convenient to “model” a time series using these

components.

Time series models fall into 2 major categories:

(A) Additive: Y=T+C+S+I assume components are

independent of one another.

(B) Multiplicative: Y=T*C*S*I 4 components are

related to each other, yet result from different basic causes.

**(Multiplicative model is more important and connection between the

two.)

Topic 3-Econ 245

Page 95

Focus first on the decomposition of Y into components.

Why?

(I) Components are useful in own right.

(II) Aids in forecasting of Y.

For the purpose of estimating each of the components of

a time series, models normally treat S, C, and I as deviations

from the ________.

Trend is usually estimated in units of Y, and the other

components are measured in _____ form, with values greater

than 100 indicating a deviation above the trend value and

values below100 indicating a movement below the trend. I.e. Represent cycle and seasonal as fraction of trend:

> 100% above trend; < 100% below trend

Topic 3-Econ 245

Page 96

The Steps of Decomposition of a time series into its

components for a multiplicative model:

(1) Isolate ________ component by the ratio-to-moving

average method.

Determine seasonal index with a base value of 100.

(2) Remove __________ from Y, leaving (T×C×I).

Divide the values of Y by the seasonal index S, and

multiply by 100 to obtain:

100

Y

ST C I

Topic 3-Econ 245

Page 97

(3) Determine _____ component, and remove, leaving (C×I)

divide T×C×I by the trend value Y to obtain (C×I).

(4) “Smooth” remainder to eliminate ________ component

leaves cycle.

Once all of the components of a time series are

identified, forecasts of the value of the time series can be

made by first estimating the value of the trend component

at that point in time in the future, and then modifying this

trend value by an adjustment that takes into account S

and C components.

Topic 3-Econ 245

Page 98

Seasonality

We are interested in the seasonal component and in eliminating it

from Y (i.e. Seasonally adjusting the data).

Basic procedure involves “smoothing” the data using moving

averages:

Yt

8 2 6 1 10 4 12 2 8

2 Period MA:

8 + 2

2

2 + 6

2

6 + 1

2

1 + 10

25 4 3 5 5 5 7 8 7 5

. .

More terms in average ____________

Choose terms to span exactly one year.

Quarterly data 4 terms; Monthly data 12 terms

Need to take account of “timing” of data.

Topic 3-Econ 245

Page 99

Year Quarter Yit 4-Qtr.

M.A.

Centred M.A.

(Tit*Cit ) Ratio:S IY

(T C )i t

it

it it

1994 Q1 40

Q2 60

45.0

Q3 20

47.5 0.432

Q4 60

50.0 1.231

1995 Q1 50

52.5 0.976

Q2 70

57.5 1.273

Q3 30

55.0 0.533

Q4 80

55.0 1.455

1996 Q1 40

50.0 0.762

Q2 70

52.5

1.366

Q3 10 ---- ----

Q4 90 ---- ----

Example: How To determine A Seasonally Adjusted Series

(Multiplicative Model)

Topic 3-Econ 245

Page 100

Obtain Seasonal Indices: (Average)

%2.86100)762.0976.0( 21

1S

S2

121 273 1 366 100 131 9 . . . %

S3

120 432 0 533 100 48 0 . . . %

S4

121 231 1 455 100 133 8 . . . %

Topic 3-Econ 245

Page 101

Obtain the geometric mean of Si’s:

Si

GM 86 2 131 9 48 133 8 92 44 100

14. . . . %

Since the geometric mean does not equal 100%, we must divide

each Si by ________:

S1

86 2

92 44100 93 2

* .

..

S2

131 9

92 44100 142 7

* .

..

S3

48

92 44100 51 9

*

..

S4

133 8

92 44100 144 7

* .

..

Topic 3-Econ 245

Page 102

S

GM

i

GM*. . . . %

( . )

93 2 142 7 51 9 144 7 100

99 97

14

Apply same Si’s to each year. To seasonally adjust the data:

YY

Sit

S it

i

*:100

Yit = Tit * Cit *Sit* Iit

YY

Sit

S it

i

( )

( )*

*100

Topic 3-Econ 245

Page 103

Year Quarter Yit Si* Seasonally Adjusted

Series

1994 Q1 40 93.2 42.918

Q2 60 142.7 42.046

Q3 20 51.9 38.536

Q4 60 144.7 41.465

1995 Q1 50 93.2 53.648

Q2 70 142.7 49.054

Q3 30 51.9 57.803

Q4 80 144.7 55.287

1996 Q1 40 93.2 42.918

Q2 70 142.7 49.054

Q3 10 51.7 19.268

Q4 90 144.7 62.198

Topic 3-Econ 245

Page 104

Differences Between Additive and Multiplicative Models

Multiplicative Model Additive Model

1) Seasonal factors should average

to 1 or 100%:

GM si

i

1

1

Example:

GM s s s s ( )1 2 3 4

14

1) Seasonal factors should average

to 0:

s

i

i 0

Example:

xs s s s

1 2 3 4

40

2)When adjusting for seasonality:

Divide by index:

yy

s

s

i

*

2)When adjusting for seasonality:

Always subtract:

y

s y s

i

*

Topic 3-Econ 245

Page 105

Seasonality and Additive Model:

There are only two differences:

(1) Always ________ instead of divide: Y Y Sit

S

it i ( )

*

(2) Seasonal factors should average to ____ (not 100).

S S S S1 2 3 4

40

Topic 3-Econ 245

Page 106

Multiplicative or Additive?

Multiplicative: assume fixed proportion of Yit is seasonal.

Example: retail trade turnover (December Versus September)

(generally reasonable for trended data.)

Additive: assumes fixed amount of Yit is seasonal.

(May be appropriate if no trend; real terms.)

Topic 3-Econ 245

Page 107

Refinements to Ratio-to-Moving-Average” Method

(1) Trading Day Adjustments Different months have different number of trading days.

May differ year after year.

(2) O________: May need to “trim” data to avoid distorting seasonal factors.

Example: introduction of G.S.T.

(3) “Evolving” Seasonal Pattern Si

* ’s usually allowed to change cyclically over years.

(4) E__ Point Problems

Calculation of Si* ’s may be sensitive toY

it values at end (s) of

series.

Topic 3-Econ 245

Page 108

The Trend –T

An important step in analyzing a time series is an estimation

of the trend component, T.

One possibility is to use a “moving average” to smooth out (Tt Ct)

to leave trend.

Not favoured because the length of M.A. (and choice of

“weights”) arbitrary. Can be distortive.

☺The preferred approach is to identify a time series’ trend,

and then “model” the trend components:

{Increasing / Decreasing}

{Linear / Non-linear}

Topic 3-Econ 245

Page 109

One principle we can use to fit a trend line to de-seasonalized

data is the L____ Squares principle.

Let: Yi=ith observation on seasonally adjusted series.

Xi=ith observation on “time.”

Example: X = 1, 2, 3, 4, ....

X= 100, 110, 120, 130,...

X=-3, -2, -1, 0, 1, 2, 3, ....

Each observation for data gives us an (Xi, Yi) point (plot Yi

against Xi).

Try to fit a line through points so it is “close” to as many data

points as possible.

Topic 3-Econ 245

Page 110

(Use all data.)

b slope

Xi

Error of fit

Yi

Yi

X=time

Y=time series

Y a bX

a

0

Topic 3-Econ 245

Page 111

Relationship is Y Xi i

(Not exact due to other components.)

Fitted line is Y a bXi i

Error of fit= Y Yi i (+/-)

Place the line so as to m________: Y Y

i i

i

T

1

2

(Choose a and b).

Sums to Zero!!

Topic 3-Econ 245

Page 112

That is: Min Y a bX

a bi i

i

T

,

2

1

Solution:

a

b

Y bX

(X X)(Y Y)

(X X)

X Y TX Y

X TX

i i

i 1

T

i

2

i 1

T

i i

i 1

T

i

2

i 1

T

2

where T=# of observations.

Topic 3-Econ 245

Page 113

Example: T=7

Yi 10 12 12 13 16 17 16

Xi

X 0

Y 13.7143

X 28

X Y 32

b32 0

28 01.1429

a 13.7143 (1.1429)(0) 13.7143

Y = 13.7143 1.1429X

i

2

i i

i i

Topic 3-Econ 245

Page 114

Xi Yi Yi

Y Yi i

10 10.2856 -0.2856

12 11.4285 0.5715

12 12.5714 -0.5714

13 13.7143 -0.7143

16 14.8572 1.1428

17 16.0001 0.9999

16 17.1430 -1.1430

Topic 3-Econ 245

Page 115

Some Properties of Least Squares

(1) The predicted line passes through the _____of X and Y, (X, Y) :

Y a bX a bX

but,

a Y bX

so,

Y Y bX bX Y

when

X X.

i i

i

i

Y

Y

X

Topic 3-Econ 245

Page 116

(2) Scaling each Xi by a ________ leaves Yi unchanged.

Begin with:

Y a bX

Form:

X CX .

Then:

b(X X )(Y Y)

(X X ),

where:

X CX;

So:

b(CX CX)(Y Y)

(CX CX)

1

Cb

i i

i

*

i

* i

* *

i

i

* * 2

*

* i i

i

2

Topic 3-Econ 245

Page 117

Similarly:

a Y b X

Y1

Cb C(X)

Y bX a

So:

Y a b X

a1

Cb C(X ) a bX

Y same fitted line.

* * *

i

* * *

i

*

i i

i

Topic 3-Econ 245

Page 118

(3) Adding a ________ to each Xi leaves Yi unchanged.

Begin with:

Y a bX

Then form:

X (X k); for some ' k' .

bX X Y Y

X X

where:

X (X k)

So,

bX k X k Y Y

X k X k

bX X Y Y

X X

b

i i

i

*

i

* i

* *

i

i

* *2

*

* i i

i

2

* i i

i

2

Topic 3-Econ 245

Page 119

a Y b X

Y bX

Y b(X + k)

Y bX) - bk

(a - bk)

So:

Y a b X

(a bk) b(X k)

a bk bX bk

(a bX ) Y same fitted line.

* * *

*

i

* * *

i

*

i

i

i i

=

(

We are free to

choose the Xi

values just as we

wish – the fitted

line is unaffected

by this choice.

Topic 3-Econ 245

Page 120

Often, a ______ trend is clearly not appropriate. There are

several non-linear possibilities.

Consider an Exponential Trend:

Y b 1

0 1 b

0 X

Y abi

Xi

Topic 3-Econ 245

Page 121

To make this trend linear in parameters, take the natural ___:

lnY a X ln b

or

Y a b X linear in parameters

i i

i

* * *

i

ln

Obtain a* and b* by least squares, using Y Yi i

*( ln ) and Xi data.

Then, the fitted trend line is:

Exp(Y ) Exp ln(Y ) Yi

*

i i

Topic 3-Econ 245

Page 122

Example: T=7

Xi Yi Y Yi i

*( ln )

1 0

3 1.0958

4 1.3863

6 1.7918

9 2.1972

8 2.0794

12 2.4849

Topic 3-Econ 245

Page 123

X 0

Y 1.5769

X 28; X Y 10.2272

bX Y TX Y

X TX

10.2272

280.36526

a Y b X 1.5769

*

i

2

i i

*

* i i

*

i

2 2

* * *

Topic 3-Econ 245

Page 124

So,

Y 1.5769 0.36526X

or

Y Exp

Y (4.8399) (1.4409)

i

*

i

i

Y

ia b

Xi

i*

( . . )

1 5769 0 36526 Xi

Xi Yi Yi ( )Y Y

i i

1 1.6178 -0.6178

3 2.3311 0.6689

4 3.3589 0.6411

6 4.8399=a 1.1601

9 6.9738 2.0262

8 10.0489 -2.0486

12 14.4790 -2.4790

Topic 3-Econ 245

Page 125

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