Producer Price Index Manual: Theory and Practice ... - METAC

Producer Price Index Manual
D. The Stochastic Approach
1.80 The stochastic approach treats the observed
price relatives as if they were a random sample
drawn from a defined universe whose mean can be
interpreted as the general rate of inflation. However,
there can be no single unique rate of inflation.
There are many possible universes that can be defined,
depending on which particular sets of industries,
products, or transactions the user is interested
in. Clearly, the sample mean depends on the choice
of universe from which the sample is drawn. The
stochastic approach does not help decide on the
choice of universe. It addresses issues such as the
appropriate form of average to take and the most efficient
way to estimate it from a sample of price
relatives, once the universe has been defined.
1.81 The stochastic approach becomes particularly
useful when the universe is reduced to a single
type of product. When there are market imperfections,
there may be considerable variation within a
country in the prices at which a single product is
sold in different establishments and also in their
movements over time. In practice, statistical offices
have to estimate the average price change for a single
product from a sample of price observations.
Important methodological issues are raised, which
are discussed in some detail in Chapter 5 on sampling
issues and Chapter 20 on elementary indices.
The main points are summarized in Section I below.
D.1 The unweighted stochastic approach
1.82 In Section C.2 of Chapter 16, the unweighted
stochastic approach to index number theory
is explained. If simple random sampling has
been used to collect prices, equal weight may be
given to each sampled price relative. Suppose each
price relative can be treated as the sum of two components:
a common inflation rate and a random disturbance
with a zero mean. Using least-squares or
maximum likelihood estimators, the best estimate
of the common inflation rate is the unweighted
arithmetic mean of price relatives, an index formula
known as the Carli index. This index can be regarded
as the unweighted version of the Young index.
This index is discussed further in Section I below
on elementary price indices.
1.83 If the random component is multiplicative,
not additive, the best estimate of the common inflation
rate is given by the unweighted geometric
mean of price relatives, known as the Jevons index.
The Jevons index may be preferred to the Carli on
the grounds that it satisfies the time reversal test,
whereas the Carli does not. As explained later, this
fact may be decisive when deciding on the formula
to be used to estimate the elementary indices compiled
in the early stages of PPI calculations.
D.2 The weighted stochastic approach
1.84 As explained in Section F of Chapter 16, a
weighted stochastic approach can be applied at an
aggregative level covering sets of different products.
Because the products may be of differing economic
importance, equal weight should not be
given to each type of product. The products may be
weighted on the basis of their share in the total
value of output, or other transactions, in some period
or periods. In this case, the index (or its logarithm)
is the expected value of a random sample of
price relatives (or their logarithms) with the probability
of any individual sampled product being selected
being proportional to the output of that type
of product in some period or periods. Different indices
are obtained depending on which revenue
weights are used and whether the price relatives or
their logarithms are used.
1.85 Suppose a sample of price relatives is randomly
selected, with the probability of selecting
any particular type of product being proportional to
the revenue of that type of product in period 0. The
expected price change is then the Laspeyres price
index for the universe. However, other indices may
also be obtained using the weighted stochastic approach.
Suppose both periods are treated symmetrically,
and the probabilities of selection are made
proportional to the arithmetic mean revenue shares
in both periods 0 and t. When these weights are applied
to the logarithms of the price relatives, the expected
value of the logarithms is the Törnqvist index.
From an axiomatic viewpoint, the choice of a
symmetric average of the revenue shares ensures
that the time reversal test is satisfied, while the
choice of the arithmetic mean, as distinct from
some other symmetric average, may be justified on
the grounds that the fundamental proportionality in
current prices test, T5, is thereby satisfied.
1.86 The examples of the Laspeyres and Törnqvist
indices just given show that the stochastic ap-
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