Inflation and relative price variability: new evidence for the United States.
Becker, Sascha S. ; Nautz, Dieter
1. Introduction
Various economic theories predict that inflation increases relative
price variability (RPV) and thus, impedes the efficient allocation of
resources. In fact, recent macroeconomic models put much emphasis on the
distorting impact of inflation on relative prices, yet the empirical
relationship between inflation and RPV seems underresearched. (1) In
particular, recent theoretical and empirical contributions suggest that
the impact of expected inflation on RPV may depend on the level of
inflation. This article reexamines the empirical relationship between
U.S. inflation and RPV in order to shed more light on the role of
expected inflation during the recent low inflation period.
Since the seminal study by Parks (1978), the empirical evidence on
inflation's impact on RPV has been mixed and elusive. While most
studies (see Jaramillo 1999) find a significant positive impact of
inflation on RPV, the relationship has broken down, according to
Lastrapes (2006), while Reinsdorf (1994) concludes that RPV decreases
with inflation. Bick and Nautz (2008) partly reconcile this
contradicting evidence by allowing for inflation thresholds where the
marginal impact of inflation on RPV varies with the inflation regime.
Many empirical studies on the inflation-RPV nexus do not account
for the different effects of expected and unexpected inflation
emphasized by the theoretical literature. For example, menu cost models
imply that RPV is increased only by expected inflation. An early attempt
to account for the implications of economic theories relating inflation
and RPV is provided by Aarstol (1999). Using U.S. producer price data
from 1973 to 1997, he finds that both expected and unexpected inflation
significantly increase RPV. Yet recent theoretical contributions
question the stability of the empirical relationship between inflation
and RPV. In particular, the monetary search model introduced by Head and
Kumar (2005) suggests that the influence of expected inflation on RPV
may have changed during the recent low inflation period. In order to
investigate the empirical relevance of this prediction, the focus of our
empirical analysis is on the (changing) role of expected inflation for
the U.S. inflation-RPV nexus. However, to ensure that our results
concerning expected inflation are not driven by further instabilities in
the empirical relationship between inflation and RPV, we will also
account for breaks in the role of unexpected inflation and inflation
uncertainty.
Adopting the empirical framework of Aarstol (1999), we find that
the effect of expected inflation on RPV becomes insignificant if the
sample includes the recent low-inflation period. The instability of the
relationship between inflation and RPV can be confirmed for different
price indices, disaggregation levels, and RPV measures. In order to shed
more light on the changing role of expected inflation for RPV, we employ
endogenous break-point tests to identify the timing and to test for the
significance of a structural break. In line with recent evidence
obtained for Germany (Nautz and Scharff 2005) and the Euro area (Nautz
and Scharff 2006), our results indicate that the influence of expected
inflation on RPV has already disappeared since the early 1990s, when
U.S. monetary policy made interest rates more responsive to inflation
and, thereby, stabilized inflation expectations on a lower level (see,
e.g., Judd and Trehan 1995; Mankiw 2001).
This article is organized as follows: Section 2 reviews theory and
empirical evidence on the relationship between inflation and RPV.
Section 3 provides first results suggesting a changing role of expected
inflation for the U.S. inflation-RPV nexus. Section 4 uses endogenous
breakpoint tests to assess the timing and significance of the structural
break in the relationship between expected inflation and RPV, where we
controlled for possible changes in the effects of unexpected inflation
and inflation uncertainty on RPV. Section 5 concludes.
2. Inflation and Relative Price Variability: Theory and Evidence
Theoretical Literature
The theoretical literature on the relationship between inflation
and RPV consists mainly of three types of models: menu cost models,
signal extraction models, and monetary search models. Interestingly, the
implications of these models concerning the role of expected and
unexpected inflation are very different.
Menu Cost Models
Menu cost models assume that nominal price changes are subject to
price adjustment costs (Sheshinski and Weiss 1977; Rotemberg 1983;
Benabou 1992). In this case, it can be shown that firms set prices
discontinuously according to an (S, s) pricing rule. Because of
inflation, the firm's real price begins at S and then falls to s
over time. At that point, the firm raises its nominal price so that the
real price once again equals S. In case of deflation, a firm decreases
its nominal price accordingly. Since the width of the (S, s) band
depends on the size of its menu costs, firm-specific menu costs lead to
staggered price setting, distorted relative prices, and an inefficient
increase of RPV. The crucial point is that only the anticipated part of
inflation affects the width of the (S, s) band. Therefore, increases in
expected inflation amplify the distorting effect of menu costs on
relative prices. Because of the symmetry in firms' pricing
strategy, menu cost models typically imply that RPV is increasing in the
absolute value of expected inflation.
Signal Extraction Models
Signal extraction models share the assumption that inflation is not
always anticipated correctly. As a consequence, firms and households
confuse absolute and relative price changes. For example, according to
Lucas (1973), Barro (1976), and Hercowitz (1981), higher inflation
uncertainty makes aggregate demand shocks harder to predict. Solving the
implied signal extraction problem, firms adjust output less in response
to all shocks, including idiosyncratic real demand shocks. As a result,
increases in unexpected inflation and inflation uncertainty will raise
RPV.
Monetary Search Models
Monetary search models emphasize that buyers have only incomplete
information about the prices offered by different sellers. In these
models, the overall effect of inflation on RPV is not always obvious
(Reinsdorf 1994; Peterson and Shi 2004). On the one hand, higher
expected inflation lowers the value of flat money, which increases
sellers' market power and thereby, the dispersion of prices. On the
other hand, higher expected inflation also raises the gains of search,
which lowers sellers' market power and, thus, RPV. As inflation
rises, the RPV increasing effect will eventually dominate. Yet there
will be a region within which small changes in expected inflation have
little effect on RPV. Head and Kumar (2005) showed that expected
inflation may increase RPV only if it exceeds a critical value.
Empirical Literature
The early empirical evidence on the relationship between inflation
and relative price variability is typically based on linear regressions
of RPV on inflation. In line with menu cost and signal extraction
models, most empirical contributions find a significant positive
coefficient of expected inflation, unexpected inflation, or inflation
uncertainty (Grier and Perry 1996; Parsley 1996; Debelle and Lamont
1997; Aarstol 1999; Jaramillo 1999). Yet there are notable exceptions.
In particular, according to Lastrapes (2006) the relationship between
U.S. inflation and RPV broke down in the mid-1980s, while Reinsdorf
(1994) demonstrates that the relationship is negative even during the
disinflationary early 1980s. Similarly, Fielding and Mizen (2000) and
Silver and Ioannidis (2001) show for several European countries that RPV
decreases in inflation.
In accordance with the implications of monetary search models, more
recent evidence suggests that the relationship between inflation and RPV
might be more complex. In particular, several studies have found that
the impact of inflation on RPV is different for high- and low-inflation
periods and countries with different inflationary contexts (Caglayan and
Filiztekin 2003; Caraballo, Dabfls, and Usabiaga 2006). Using
nonparametric methods, Fielding and Mizen (2008) find that the U.S.
inflation-RPV linkage is nonlinear. Nautz and Scharff (2006) apply panel
threshold models to price data of Euro-area countries. In line with Head
and Kumar (2005), they find evidence in favor of threshold effects in
the European link between expected inflation and RPV. Similar threshold
effects are found by Bick and Nautz (2008) using price data from U.S.
cities, although they do not differentiate between expected and
unexpected inflation. Finally, analyzing price observations from
bazaars, convenience stores, and supermarkets in Turkey, Caglayan,
Filiztekin, and Rauh (2008) show that the relationship between RPV and
expected inflation confirms the predictions of monetary search models.
In particular, expected inflation increases RPV only if it exceeds a
certain threshold.
Given the overall decline of U.S. inflation and inflation
expectations over the past decades, the focus of our analysis is on the
impact of expected inflation on RPV in the United States. In light of
the recent theoretical and empirical literature, a changing role of
expected inflation should be reflected in a structural break of the
traditional inflation-RPV nexus.
3. The Empirical Relation between Inflation and RPV
Data and Variables
Our benchmark measures of inflation ([[pi].sup.PPI]) and relative
price variability (RPV) use monthly price data of the U.S. Producer
Price Index (PPI). At the two-digit disaggregation level, the
corresponding RPV measure, [RPV.sub.PPI-2], is based on the prices of
the complete set of 15 subcategories. In order to check the robustness
of our results, we additionally employ four alternative inflation and
RPV measures typically applied in the empirical literature.
Specifically, we consider [RPV.sub.core] as a second RPV measure, where
food and energy prices are excluded to control for supply shocks. More
precisely, we eliminated the prices of "farm products,"
"processed foods and feeds," and "fuels and related
products and power," that is, 3 out of the 15 PPI subcomponents
(compare, e.g., Aarstol 1999). Our results should not depend on the
aggregation level of the price index. Therefore, the third RPV measure,
[RPV.sub.PPI-3], is based on the three-digit PPI disaggregation level,
that is, on the prices of 77 subcategories. Fourth, we consider
[RPV.sub.Abs] = [square root of ([RPV.sub.PPI-2])] since it should not
be important whether one measures RPV by the variance or the standard
deviation of relative prices. And, finally, we define inflation
([[pi].sup.CPI]) and RPV ([RPV.sub.CPI-2]) with respect to the eight
subcategories of the two-digit Consumer Price Index (CPI) to guarantee
that the following empirical results are robust with respect to the
choice of the price index. The definitions of the various RPV measures
are summarized in Table 1.
Following Aarstol (1999), we define each RPV measure via the
unweighted variance of subcategory-specific inflation rates around the
corresponding rate of inflation. (2) It is worth noting, however, that
the use of weighted RPV measures that account for the importance of
subcomponents in the price index does not affect our main results. More
detailed information on the price indices and the corresponding
subcategories is presented in the Appendix (see Tables A1 and A2). All
data run from January 1973 to December 2007 and are provided by the
Bureau of Labor Statistics. Unit root tests clearly indicate that all
inflation and RPV measures are stationary. (3)
Inflation Forecasts
The theories on the relationship between inflation and RPV
presented in section 2 highlight the different roles of expected
inflation, unexpected inflation, and inflation uncertainty. It is a
general problem of any such decomposition that the empirical results
might depend on the accuracy of the expected inflation measure. Many
measures of inflation expectations exist, including the forecasts of
professional economists, results from consumer surveys, or information
extracted from financial markets. Despite the increasing importance and
quality of this kind of data, survey data are not available over the
whole sample period and on a monthly basis. In particular, there are no
surveys on expectations about PPI inflation. In view of these problems,
we follow the bulk of the empirical literature and base our measure of
expected inflation on a time-series representation of inflation. Note,
however, that beating the forecasting performance of univariate time
series models of inflation is not an easy task, particularly over a
monthly forecast horizon (see, e.g., Elliott and Timmermann 2008).
Allowing for time-varying inflation uncertainty (CVAR), the
forecast equations for overall U.S. producer price inflation
([[pi].sup.PPI]) and consumer price inflation ([[pi].sup.CPI]) are
specified as GARCH models, where the corresponding mean equations follow
an ARMA process. (4) Expected inflation (EI) is derived as the
one-period-ahead inflation forecast, while unexpected inflation (UI) is
the resulting forecast error (UI = [pi] - EI). The GARCH equations
provide us with time series for inflation uncertainty (CVAR). Using
maximum-likelihood estimation, we applied a standard information
criterion (Bayesian information criterion) to determine the optimal lag
structure. Detailed results of the estimated inflation forecast
equations are shown in the Appendix (see Table A3). It is worth noting
that alternative specification strategies for obtaining the inflation
forecast equations lead to very similar results. In particular, using
inflation forecasts based on a simple AR(12) mean model will not affect
the following outcomes.
Aarstol (1999) finds that the impact of unexpected inflation on RPV
depends on the sign of the inflation forecast error. In order to control
for this effect, we define positive unexpected inflation as UIP = UI if
UI [greater than or equal to] 0 and UIP = 0 otherwise and negative
unexpected inflation as UIN accordingly.
The (Changing) Impact of Expected Inflation on RPV
After these preliminaries, let us now estimate the impact of
expected inflation (E/), unexpected inflation (UIP, UIN), and inflation
uncertainty (CVAR) on RPV. Using the various inflation and RPV measures,
we estimate the relationship between inflation and RPV based on two
specifications, typically applied in the empirical literature. Following
Aarstol (1999), Equation 1 contains squared terms of inflation and is
applied to the four RPV measures based on the variance of relative
prices ([RPV.sub.PPI-2], [RPV.sub.Core], [RPV.sub.CPI-2], and
[RPV.sub.PPI-3]):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
Accordingly, Equation 2, which explains the standard deviation of
relative prices, [RPV.sub.Abs], includes the absolute value of the
inflation terms:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
Table 2 summarizes the results for the U.S. inflation-RPV nexus for
all RPV measures. For the sake of comparability, the upper part of Table
2 presents the estimates for the sample period used by Aarstol (1999),
ranging from January 1973 until May 1997. Since he used the two-digit
PPI, the results shown in the first row exactly replicate his findings.
Specifically, there is a significant positive impact of expected
inflation ([[gamma].sub.1]) on RPV. According to Wald tests of parameter
equality, the effect of unexpected inflation ([[gamma].sub.2],
[[gamma].sub.3]) depends on the sign of the inflation forecast error.
And, finally, the coefficient of inflation uncertainty ([[gamma].sub.4])
is significant and plausibly signed. During the first sample period,
most of these conclusions remain valid with respect to different RPV and
inflation measures. Although the absolute size of the inflation
coefficients changes with the underlying RPV measure, the relative size
of the inflation coefficients and their statistical significance remain
merely unaffected. The only exception refers to the coefficient of
inflation uncertainty, where the evidence is more elusive.
The overall impression of structural stability of the U.S.
inflation-RPV linkage changes, however, if the sample period is extended
by more recent data (January 1973-December 2007; see the middle part of
Table 2). In particular, both magnitude and significance of the impact
of expected inflation on RPV have decreased regardless of the underlying
measures of inflation and RPV. The evidence in favor of a structural
break stirred by a changing role of expected inflation gets even more
striking if the inflation-RPV equations are estimated for the recent
period separately (see the lower part of Table 2).
Table 2 indicates that the impact of expected inflation on RPV has
become insignificant in the United States over the past years. The
implied instability of the inflation-RPV nexus may explain the breakdown
of the traditional linear relation between RPV and U.S. inflation found
by Lastrapes (2006). However, before we have a closer look at the
changing role of expected inflation, two remarks are in order. First, it
is worth mentioning that Equations 1 and 2 involve generated regressors
such that the appropriateness of an ordinary least squares (OLS)
estimation and the validity of standard t-statistics is not obvious.
Pagan (1984) has shown that OLS estimation is consistent and does not
necessarily lead to efficiency losses if generated regressors (EI) as
well as forecast errors (UIP and UIN) enter the equation. The only
problem concerns the OLS-generated t-statistic of the coefficient of EI
([[gamma].sub.1]), which tends to be overstated. Since the acceptance of
the relevant null hypothesis (no influence of expected inflation) with
the overstated t-statistic must lead to the acceptance with the correct
one, only those EI coefficients require further investigation for which
the null hypothesis is rejected. Therefore, we reinvestigated the
significance of EI in the early sample period (January 1973-May 1997)
using Pagan's corrected t-statistics. In line with Silver and
Ioannidis (2001), however, the corrected t-statistics had no
quantitative effect on the significance of expected inflation (see Table
A4).
Second, Table 2 further suggests that the vanishing influence of
expected inflation might not be the only source of instability in the
relationship between inflation and RPV. Therefore, we have to ensure
that the results concerning the changing role of expected inflation are
not driven by further instabilities. To that aim, the following
endogenous breakpoint analysis of the empirical relationship between
expected inflation and RPV will also control for the effect of
instabilities in the role of unexpected inflation and inflation
uncertainty (see section 4).
In the United States, average inflation has significantly decreased
over the past two decades. Therefore, in line with the predictions of
the monetary search model introduced by Head and Kumar (2005), our
empirical results may indicate that the impact of expected inflation on
RPV has been reduced because inflation expectations have been stabilized
on a low level. This interpretation of our empirical results obtained
for recent U.S. data would be in line with evidence for Germany and the
Euro area, two textbook examples for low-inflation currency areas (see
Nautz and Scharff 2005, 2006). Finally, note that our findings are
compatible with the thresholds effects of U.S. inflation established by
Bick and Nautz (2008) and the nonlinear relationship between expected
inflation and RPV found by Fielding and Mizen (2008).
4. Structural Break Tests for the U.S. Inflation-RPV Nexus
Endogenous Break-Point Tests
This section sheds more light on the changing role of expected
inflation for the inflation-RPV nexus. In particular, we investigate the
timing and the significance of the structural instability in the
relationship between expected inflation and RPV using endogenous
break-point tests. Specifically, we apply the testing procedure by
Andrews (1993) and Andrews and Ploberger (1994), which is designed to
detect a structural break even if the break-point is unknown.
The endogenous break-point tests are implemented as follows: Having
defined a sequence of dummy variables, D(j), that equal 0 if t < j
and equal 1 otherwise, we estimate for each j a break-augmented RPV
equation that allows a shift in the marginal impact of expected
inflation at date j. For example, for the four variance-based RPV
measures, we obtain the following test equations (5):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
In a first step, we derive for each j [member of] [[T.sub.1],
[T.sub.2]] the likelihood ratio statistic, LR(j), corresponding to the
null hypothesis that [[delta].sub.j], the coefficient of the dummy
variable, is zero. In a second step, we compute the test statistics
ave-LR and sup-LR for the unknown break-point defined as the average and
the maximum of all LR(j)-statistics, respectively. The date j that
corresponds to sup-LR serves as the estimate of the break date.
Andrews (1993) showed that the asymptotic distributions of the test
statistics are nonstandard and depend on the number of coefficients that
are allowed to break and on the fraction of the sample that is examined.
(6) Below, we use the approximate asymptotic p-values provided by Hansen
(1997).
Test Results on the Changing Role of Expected Inflation
Table 3 summarizes the test results obtained for the various
measures of inflation and RPV. With the exception of [RP.sub.Abs], the
LR-statistics clearly indicate a structural break in the impact of
expected inflation on RPV. The estimated break dates implied by the
maximum of the LR-statistics are August 1990 for [RPV.sub.PPI-2] and
[RPV.sub.Abs] and September 1990 for [RPV.sub.CPI-2]. According to the
sup-LR statistics, the breaks for the other two RPV measures,
[RPV.sub.PPI-3] and [RPV.sub.Core], occur in December 1992 and June
1995, that is, about two and five years later.
However, a closer inspection of the underlying sequence of LR-test
statistics reveals that even for these RPV measures, the instability of
the inflation-RPV nexus has already started around December 1990, very
close to the break-date estimates of the other RPV measures (compare to
Figure 1). In both cases, [RPV.sub.PPI-3] and [RPV.sub.Core], the
enormous jump in the LR-test values at the end of 1990, strongly suggest
that the instability in the relationship between expected inflation and
RPV has started before the LR-statistics eventually reached their
maximum. This break date is confirmed by the Chow-type break-point tests
[LR.sup.12/1990] (also presented in Table 3), which for both RPV
measures clearly indicate that the relationship between expected
inflation and RPV was already unstable in December 1990.
Next, we revisit the inflation-RPV equation for all RPV measures,
taking into account the insights of the endogenous break-point tests.
Assuming the break point in 1990, as suggested by the behavior of
LR-statistics, we reestimate the RPV equation for the two resulting
sample periods. For the early subsample, the results presented in Table
4 confirm the significant impact of expected inflation on RPV
established by Aarstol (1999) and others. (7) However, the results look
very different for the more recent subperiod. The former significant
impact of expected inflation on RPV has disappeared in the recent
low-inflation period, independent of the price index, the disaggregation
level, and the RPV measure.
[FIGURE 1 OMITTED]
Sensitivity Analysis: The Role of Further Instabilities in the
Inflation-RPV Nexus
The evidence in favor of a structural break in the relationship
between expected inflation and RPV might be affected by further
instabilities in the empirical inflation-RPV nexus. In fact, according
to Table 4, there seem to be considerable movements in the coefficients
of both unexpected inflation and inflation uncertainty that may distort
the estimated relationship between expected inflation and RPV.
Therefore, this section reexamines the stability of the EI-RPV
relationship, taking into account possible breaks in the coefficients of
unexpected inflation and inflation uncertainty.
Testing for Further Instabilities in the Inflation-RPV Nexus
In a first step, we test for the presence of additional structural
breaks in the inflation-RPV nexus related to unexpected inflation or
inflation uncertainty. In a second step, in case of a significant break
in one of the coefficients of UIP, UIN, or CVAR, we rerun the endogenous
break-point test for the relationship between expected inflation and RPV
based on an augmented test equation that takes this further instability
into account. If the augmented test equation confirms the changing role
of expected inflation, we can be confident that the vanishing impact of
expected inflation on RPV is a robust result and not a statistical
artifact stirred by the instability of other variables.
For example, the test for an additional break in the RPV equation
corresponding to a changing role of inflation uncertainty is based on
the following test equation (8):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
where the step dummy variable, [D.sub.90], is defined in accordance
with the changing coefficient of expected inflation (see Table 4).
Table A5 summarizes the test results for the various empirical
specifications of the inflation-RPV nexus. Overall, the results confirm
the conclusion of modest instability in unexpected inflation and
inflation uncertainty already suggested by the estimation results
obtained for the different sample periods (see Table 4). In accordance
with Table 4, the evidence in favor of a structural break is strongest
for the variable UIN, that is, for the impact of negative unexpected
inflation.
The Changing Role of Expected Inflation in the Presence of Further
Instabilities
Let us now investigate how the results on the changing role of
expected inflation are affected by these additional breaks in the
relationship between inflation and RPV. To that aim, we augment the
original test equations for the EI-RPV relationship, Equations 3 and 4,
by the break dummies that were found to be significant for unexpected
inflation or inflation uncertainty. For example, in the case of the RPV
measure [RPV.sub.PPI-2], we found significant breaks in the coefficients
of UIP and UIN in August 1990 and August 1985, respectively, while the
coefficient of CVAR remained stable over the whole sample period (see
Table A5). As a consequence, the augmented test equation for
[RPV.sub.PPI-2] is obtained as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where [D.sub.07/90] and [D.sub.07/85] are the step dummy variables
indicating the corresponding break dates of UIP and UIN.
The results for the augmented break-point tests do not differ
qualitatively from those of the previous section (see Table A6).
Therefore, regardless of further, less theory-related structural breaks
in the inflation-RPV nexus due to unexpected inflation and inflation
uncertainty, the results confirm the evidence in favor of a changing
role of expected inflation for RPV. In particular, in line with our
previous results, the various breaks in the impact of EI can all be
dated around 1991. Moreover, confirming the results of Table 4, pre- and
after-break estimations of the augmented inflation-RPV equations reveal
that for all RPV and inflation measures the impact of expected inflation
is significant before the break but small and insignificant thereafter.
(9)
5. Conclusions
This article provided new evidence on the empirical relationship
between inflation and RPV in the United States. Reconciling the mixed
results offered by earlier contributions, we found that the impact of
expected inflation on RPV has declined significantly since the 1990s.
Endogenous break-point tests confirmed the timing and statistical
significance of the changing role of expected inflation for RPV
regardless of the inflation and RPV measure.
Our results support the implications of recent monetary search
models that predict that the inflation-RPV nexus depends on the level of
expected inflation (see Head and Kumar 2005; Caglayan, Filiztekin, and
Rauh 2008). In accordance with the evidence obtained by Nautz and
Scharff (2005, 2006) for Germany and the Euro area, our results suggest
that the impact of expected inflation on RPV broke down in the United
States because inflation expectations had been stabilized on a low
level.
According to recent macroeconomic theory, the impact of expected
inflation on RPV is a major channel for real effects of inflation (see
Woodford 2003). The current study demonstrated that the empirical
analysis of the relation between inflation and RPV can be largely
improved by paying more attention to the predictions of theoretical
models. In addition to the different role of expected and unexpected
inflation implied by well-established menu cost and signal extraction
models, our results suggest that recent monetary search models provide
particularly useful insights on the functional form of the inflation-RPV
nexus.
Appendix
Table A1. Subcategories of the U.S. Consumer Price Index
(Two-Digit)
Food and beverages
Housing
Apparel
Transportation
Medical care
Recreation
Education and communication
Other goods and services
Source: Bureau of Labor Statistics. Series IDs:
CUSR0000SAA-CUSR0000SAT. Note that in January 1998, the
subcategory "Entertainment" was replaced by the
subcategories "Recreation" and "Education and Communication."
Table A2. Subcategories of the U.S. Producer Price Index (Two Digit
and Three Digit)
Two Digit Three Digit
Farm products Fruits and melons, fresh/dry
vegetables and nuts
Slaughter livestock
Chicken eggs
Plant and animal fibers
Processed foods and feeds Cereal and bakery products
Processed fruits and vegetables
Fats and oils
Sugar and confectionery
Miscellaneous processed foods
Textile products and apparel
Hides, skins, leathers, and Hides and skins, including
related products cattle
Footwear
Fuels and related products and Coal
power Petroleum arid coal products
Electric power
Chemical and allied products Industrial chemicals
Fats and oils, inedible
Drugs and pharmaceuticals
Rubber and plastic products Rubber and rubber products
Lumber and wood products Lumber
Plywood
Pulp, paper, and allied products Pulp, paper, and products,
excluding building paper
Metal and metal products Iron and steel
Hardware
Fabricated structural metal
products
Metal containers
Machinery and equipment Agricultural machinery and
equipment
General purpose machinery and
equipment
Metalworking machinery and
equipment
Miscellaneous machinery
Furniture and household durables Household furniture
Household appliances
Floor coverings
Nonmetallic mineral products Glass
Clay construction products,
excluding refractories
Gypsum products
Concrete products
Other nonmetallic minerals
Transportation equipment Motor vehicles and equipment
Miscellaneous products Toys, sporting goods, small arms,
and so on
Photographic equipment and
supplies
Other miscellaneous products
Two Digit Three Digit
Farm products Grains
Slaughter poultry
Hay, hayseeds, and oilseeds
Fluid milk
Processed foods and feeds Meats, poultry, and fish
Beverages and beverage materials
Prepared animal feeds
Dairy products
Textile products and apparel
Hides, skins, leathers, and Leather
related products Other leather and related
products
Fuels and related products and Gas fuels
power Petroleum products, refined
Chemical and allied products Plastic resins and materials
Agricultural chemicals and
chemical products
Other chemicals and allied
products
Rubber and plastic products Plastic products
Lumber and wood products Millwork
Other wood products
Pulp, paper, and allied products Building paper and building
board mill products
Metal and metal products Nonferrous metals
Plumbing fixtures and fittings
Heating equipment
Miscellaneous metal products
Machinery and equipment Construction machinery and
equipment
Special industry machinery and
equipment
Electrical machinery and
equipment
Furniture and household durables Commercial furniture
Home electronic equipment
Other household durable goods
Nonmetallic mineral products Concrete ingredients and
related products
Refractories
Glass containers
Asphalt felts and coatings
Transportation equipment Railroad equipment
Miscellaneous products Tobacco products, including
stemmed and redried
Notions
Source: Bureau of Labor Statistics. Series IDs (two digit):
WPU01-WPU15. Series IDs (three digit): WPU011-WPU159.
Table A3. The Inflation Forecast Equations
(January 1973-December 2007)
Producer Price Index
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Adjusted [R.sup.2]=0.23 Q(12)=5.42 [0.49] ARCH(12)=7.65 [0.81]
Consumer Price Index
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Adjusted [R.sup.2]=0.30 Q(12)=4.23 [0.64] ARCH(12)=5.50 [0.93]
For both equations, the optimal lag structure is determined using
Bayesian information criteria. Alternative specification strategies
lead to very similar estimates of expected inflation (EI, UI) and
inflation uncertainty (CVAR). Q(12) denotes the Ljung-Box statistic
testing for serial correlation in the residuals; ARCH(12) denotes
the LM-statistic testing for ARCH effects.
p-Values are given in square brackets, t-Statistics are given in
parentheses.
Table A4. The Impact of Expected Inflation on RPV (the Role of the
Generated Regressor Problem)
Sample January 1973-May 1997
[[??].sub.1]
[RPV.sub.PPI-2] 0.897 ***
(5.40)
[4.11]
[RPV.sub.Abs] 0.254 *
(2.10)
[1.91]
[RPV.sub.PPI-3] 3.897 ***
(3.53)
[3.04]
[RPV.sub.Core] 0.445 **
(2.46)
[1.98]
[RPV.sub.CPI-2] 0.118 *
(1.97)
[1.82]
The results of this table demonstrate that the significant impact of
expected inflation on RPV found by Aarstol (1999) and others is not an
artifact of the generated regressor problem (see Pagan 1984). Numbers
in parentheses are t statistics, ignoring that expected inflation is a
generated regressor, as in Table 2. Numbers in square brackets are t
statistics corrected for the generated regressor problem (see Silver
and Ioannidis 2001). *, **, and *** indicate significance at the 10%,
5%a, and 1% levels, respectively (based on the corrected t-statistics).
See Table 2 for further explanations.
Table A5. Test for Unknown Break Point in the U.S. Inflation-RPV
Nexus: The Case of Unexpected Inflation and Inflation Uncertainty
[H.sub.0]: No Break in the Coefficient on
UIP ([[gamma].sub.2])
Model ave-LR sup-LR
[RPV.sub.PPI-2] 3.48 6.67 (August 1990)
[0.05] [0.06]
[RPV.sub.Abs] 1.13 3.65
[0.28] [0.23]
[RPV.sub.PPI-3] 0.16 0.36
[0.89] [1.00]
[RPV.sub.Core] 0.02 0.10
[1.00] [1.00]
[RPV.sub.CPI-2] 14.43 25.97 (July 1994)
[0.00] [0.00]
[H.sub.0]: No Break in the Coefficient
on UIN ([[gamma].sub.3])
Model ave-LR sup-LR
[RPV.sub.PPI-2] 12.48 23.08 (August 1985)
[0.00] [0.00]
[RPV.sub.Abs] 4.89 12.37 (August 1985)
[0.02] [0.00]
[RPV.sub.PPI-3] 0.31 0.60
[0.70] [0.96]
[RPV.sub.Core] 11.96 19.33 (April 1985)
[0.00] [0.00]
[RPV.sub.CPI-2] 16.65 26.45 (March 1986)
[0.00] [0.00]
[H.sub.0]: No Break in the Coefficient
on CVAR ([[gamma].sub.4])
Model ave-LR sup-LR
[RPV.sub.PPI-2] 0.04 0.23
[1.00] [1.00]
[RPV.sub.Abs] 0.34 0.68
[0.66] [0.94]
[RPV.sub.PPI-3] 0.43 1.10
[0.59] [0.77]
[RPV.sub.Core] 2.03 4.47
[0.15] [0.16]
[RPV.sub.CPI-2] 11.01 15.25 (March 1986)
[0.00] [0.00]
Test equations assume a break in the coefficient of expected inflation
as it is indicated by the results of the endogenous break-point tests
(see Table 3 and Equation 5). p-values of the ave-LR and sup-LR
statistics according to Hansen (1997) in square brackets, and estimated
break dates are in parentheses. Feasible range of break points is
September 1984-April 1996. See section 4 for further explanations.
Table A6. Test for Unknown Break Point in the EI-RPV Relationship
(Accounting for Structural Breaks of UIP, UIN, and CVAR)
[H.sub.0]: No Break in the Role of Expected Inflation for RPV
Model Statistic Value
[RPV.sub.PPI-2] ave-LR statistic (November 1990) 5.17
sup-LR statistic 8.05
[RPV.sub.Abs] ave-LR statistic (November 1990) 1.39
sup-LR statistic 2.33
[RPV.sub.PPI-3] ave-LR statistic (June 1995) 11.15
sup-LR statistic 12.22
[LR.sup.12/1990 11.54
statistic
[RPV.sub.Core] ave-LR statistic (December 1990) 6.10
sup-LR statistic 8.61
[RPV.sub.CPI-2] ave-LR statistic (March 1991) 9.02
sup-LR statistic 12.46
[H.sub.0]: No Break in the Role of Expected Inflation for RPV
Model Statistic Probability
[RPV.sub.PPI-2] ave-LR statistic 0.01
sup-LR statistic 0.02
[RPV.sub.Abs] ave-LR statistic 0.23
sup-LR statistic 0.51
[RPV.sub.PPI-3] ave-LR statistic 0.00
sup-LR statistic 0.00
[LR.sup.12/1990 0.00
statistic
[RPV.sub.Core] ave-LR statistic 0.00
sup-LR statistic 0.02
[RPV.sub.CPI-2] ave-LR statistic 0.00
sup-LR statistic 0.00
p-values of ave-LR and sup-LR according to Hansen (1997), and estimated
break dates are in parentheses. Feasible range of break points is
September 1984-April 1996. [LR.sup.12/1990] refers to a standard Chow
break-point test. See section 4 for further explanations.
Received August 2008; accepted November 2008.
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(1) For example, standard new Keynesian dynamic stochastic general
equilibrium models support price stability as an outcome of optimal
monetary policy only because inflation increases RPV; see Woodford
(2003).
(2) [[pi].sub.it] = ln([P.sub.it]/[P.sub,it-1] is the inflation
rate, and [P.sub.it] is the price index of the ith subcategory in period
t. rot is the aggregate inflation rate.
(3) Results of ADF and KPSS tests are not presented but are
available on request.
(4) Preliminary investigations indicate that the forecast errors of
the best-fitting ARMA model are heteroscedastic.
(5) Accordingly, in case of RPV = [RPV.sub.abs], the test equations
are obtained as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(6) Note that the distributions become degenerate as the first
period tested approaches the beginning of the equation sample, or the
end period approaches the end of the equation sample. To compensate for
this behavior, it is generally suggested that the start/end of the
equation sample should not be included in the testing procedure. In
accordance with Andrews (1993), the sample range where breaks are
considered is defined by [T.sub.1] = 1/3 * T and [T.sub.2] =2/3 * T.
Therefore, possible break dates range from September 1984 to April 1996.
(7) Note that this result is not driven by the generated regressor
problem because using corrected t-statistics does not affect the
significance of [[??].sub.i] (see section 3).
(8) In the case of RPV = [RPV.sub.Abs], test equations are obtained
by replacing all squared terms by the corresponding absolute values and
the CVAR term by its square root.
(9) For brevity, the results of the augmented pre- and after-break
regressions, which are very similar to those shown in Table 4, are not
presented but are available on request.
Sascha S. Becker, Department of Economics, Freie Universitat
Berlin, Boltzmannstr. 20, D-14195, Berlin, Germany; E-mail
sascha.becker@fu-berlin.de.
Dieter Nautz, Department of Economics, Freie Universitat Berlin,
Boltzmannstr. 20, D-14195, Berlin, Germany; E-mail
dieter.nautz@fu-berlin.de; corresponding author.
We thank Alex Bick and two anonymous referees for their helpful
comments and suggestions. Financial support by the Monetary Stability
Foundation is gratefully acknowledged.
Table 1. RPV Measures
Variable Measure
[RPV.sub.PPI-2] 1/15 [[SIGMA].sup.15.sub.i=1] [([[pi].sub.it] -
[[pi].sub.t]).sup.2]
[RPV.sub.Core] 1/12 [[SIGMA].sup.12.sub.i=1] [([[pi].sub.it] -
[[pi].sub.t]).sup.2]
[RPV.sub.PPI-3] 1/77 [[SIGMA].sup.77.sub.i=1] [([[pi].sub.it] -
[[pi].sub.t]).sup.2]
[RPV.sub.Abs] [square root of 1/15 [[SIGMA].sup.15.sub.i=1]
[([[pi].sub.it] - [[pi].sub.t]).sup.2]]
[RPV.sub.CPI-2] 1/8 [[SIGMA].sup.8.sub.i=1] [([[pi].sub.it] -
[[pi].sub.t]).sup.2]
Variable Data
[RPV.sub.PPI-2] PPI, two digit
15 subcategories
[RPV.sub.Core] PPI two digit
12 subcategories
[RPV.sub.PPI-3] PPI, three di t
77 subcategories
[RPV.sub.Abs] PPI, two digit
15 subcategories
[RPV.sub.CPI-2] CPI two digit
8 subcategories
Price data provided by the Bureau of Labor Statistics. The variable
[RPV.sub.PPI-2] accounts for the change in the composition of the CPI
subcategories in January 1998.
Table 2. The Changing Impact of Expected Inflation on RPV
[RPV.sub.t] = [[gamma].sub.0] + [[gamma].sub.1] [EI.sup.2.sub.t] +
[[gamma].sub.2] [UIP.sup.2.sub.t]; + [[gamma].sub.3]
[UIN.sup.2.sub1] + [[gamma].sub.4] [CVAR.sub.t] + [v.sub.t]
Sample January 1973-May 1997
[[??].sub.1] [[??].sub.2]
[RPV.sub.PPI-2] 0.897 *** (5.40) 1.276 *** (9.78)
[RPV.sub.Abs] 0.254 ** (2.10) 0.844 *** (6.23)
[RPV.sub.PPI-3] 3.897 *** (3.53) 5.520 *** (11.77)
[RPV.sub.Core] 0.445 ** (2.46) 0.867 *** (12.58)
[RPV.sub.CPI-2] 0.118 * (1.97) 0.647 *** (3.58)
Sample January 1973-December 2007
[[??].sub.1] [[??].sub.2]
[RPV.sub.PPI-2] 0.161 (0.35) 1.438 *** (20.59)
[RPV.sub.Abs] 0.233 * (1.76) 0.911 *** (10.19)
[RPV.sub.PPI-3] 2.294 ** (2.31) 5.541 *** (13.49)
[RPV.sub.Core] 0.324 (1.33) 0.886 *** (16.32)
[RPV.sub.CPI-2] -0.012 (-0.21) 0.935 *** (6.88)
Sample June 1997-December 2007
[[??].sub.1] [[??].sub.2]
[RPV.sub.PPI-2] 0.107 (0.15) 1.638 *** (6.21)
[RPV.sub.Abs] 0.189 (1.15) 0.996 *** (10.10)
[RPV.sub.PPI-3] 2.041 (1.26) 5.248 *** (3.81)
[RPV.sub.Core] 0.318 (0.86) 0.927 *** (10.96)
[RPV.sub.CPI-2] -0.207 (-0.58) 1.611 *** (9.85)
[[??].sub.3] [[??].sub.4]
[RPV.sub.PPI-2] 0.210 (1.40) 0.316 ** (1.84)
[RPV.sub.Abs] 0.594 *** (3.88) 0.805 (1.62)
[RPV.sub.PPI-3] 0.668 ** (2.07) 1.455 ** (2.12)
[RPV.sub.Core] 0.166 * (1.91) 0.226 (1.50)
[RPV.sub.CPI-2] 0.578 *** (3.21) 0.202 (0.74)
[[??].sub.3] [[??].sub.4]
[RPV.sub.PPI-2] 0.949 *** (5.00) 0.171 (1.17)
[RPV.sub.Abs] 0.737 *** (8.54) 0.515 (1.37)
[RPV.sub.PPI-3] 1.112 *** (5.75) 1.221 ** (2.48)
[RPV.sub.Core] 0.429 *** (6.64) 0.119 (1.36)
[RPV.sub.CPI-2] 0.932 *** (5.02) 0.282 (1.03)
[[??].sub.3] [[??].sub.4]
[RPV.sub.PPI-2] 1.189 *** (17.24) 0.087 (0.61)
[RPV.sub.Abs] 0.833 *** (12.08) 0.129 (0.25)
[RPV.sub.PPI-3] 1.102 *** (4.46) -0.384 (-0.54)
[RPV.sub.Core] 0.528 *** (10.78) -0.008 (-0.15)
[RPV.sub.CPI-2] 1.513 *** (5.92) 0.306 (0.68)
Estimation results of [RPV.sub. Equations 1 and 2 using different
inflation and [RPV.sub. measures for various sample periods. The
inflation forecast equations implying expected inflation (En,
unexpected (UIP, UIN) inflation, and inflation uncertainty (CVAR) are
shown in Table A3. r-Statistics (Newey-West standard errors) in
parentheses. *, **, and *** indicate significance at the 10%, 5%, and
1% significance levels, respectively.
Table 3. Test for Unknown Break Point in the U.S. Inflation-RPV Nexus:
The Case ofExpected Inflation
[H.sub.0]: No Break in the Role of Expected Inflation for
RPV
Model Statistic Value
[RPV.sub.PPI.-2] ave-LR statistic (August 1990) 10.56
sup-LR statistic 11.36
[RPV.sub.Abs] ave-LR statistic (August 1990) 2.07
sup-LR statistic 2.41
[RPV.sub.PPI-3] ave-LR statistic (June 1995) 11.15
sup-LR statistic 12.22
[LR.sup.12/1990] 11.54
statistic
[RPV.sub.Core] ave-LR statistic (December 1992) 8.62
sup-LR statistic 8.83
[LR.sup.12/1990] 8.74
statistic
[RPV.sub.CPI-2] ave-LR statistic (September 1990) 5.05
sup-LR statistic 7.23
Model Statistic Probability
[RPV.sub.PPI.-2] ave-LR statistic 0.00
sup-LR statistic 0.00
[RPV.sub.Abs] ave-LR statistic 0.14
sup-LR statistic 0.47
[RPV.sub.PPI-3] ave-LR statistic 0.00
sup-LR statistic 0.00
[LR.sup.12/1990] 0.00
statistic
[RPV.sub.Core] ave-LR statistic 0.00
sup-LR statistic 0.01
[LR.sup.12/1990] 0.00
statistic
[RPV.sub.CPI-2] ave-LR statistic 0.00
sup-LR statistic 0.04
Table 4. The Inflation-RPV Nexus in the United States
Before the Break
[[??].sub.1] [[??].sub.2]
[RPV.sub.PPI-2] 0.902 ** (2.34) 1.349 *** (13.67)
January 1973-July 1990
[RPV.sub.Abs] 0.374 ** (2.11) 0.771 *** (4.08)
January 1973-July 1990
[RPV.sub.PPI-3] 3.823 *** (2.97) 5.385 *** (9.84)
January 1973-November 1990
[RPV.sub.Core] 0.557 *** (2.74) 0.905 *** (12.23)
January 1973-November 1990
[RPV.sub.CPI-2] 0.109 ** (2.01) 0.667 *** (3.60)
January 1973-August 1990
After the Break
[[??].sub.1] [[??].sub.2]
[RPV.sub.PPI-2] -0.102 (-0.17) 1.859 *** (5.96)
August 1990-December 2007
[RPV.sub.Abs] 0.078 (0.56) 1.054 *** (9.97)
August 1990-December 2007
[RPV.sub.PPI-3] 1.794 (0.85) 5.648 *** (6.41)
December1990-December 2007
[RPV.sub.Core] 0.351 (0.78) 0.932 *** (11.37)
December 1990-December 2007
[RPV.sub.CPI-2] -0.094 (-0.50) 1.468 *** (8.34)
September 1990-December 2007
[[??].sub.3] [[??].sub.4]
[RPV.sub.PPI-2] 0.178 (1.06) 0.307 (1.43)
January 1973-July 1990
[RPV.sub.Abs] 0.472 *** (2.68) 0.932 * (1.82)
January 1973-July 1990
[RPV.sub.PPI-3] 0.673 (1.37) 1.254 * (1.91)
January 1973-November 1990
[RPV.sub.Core] 0.145 (1.46) 0.207 (1.28)
January 1973-November 1990
[RPV.sub.CPI-2] 0.597 *** (3.09) 0.157 (0.65)
January 1973-August 1990
[[??].sub.3] [[??].sub.4]
[RPV.sub.PPI-2] 1.215 *** (18.00) 0.102 (0.78)
August 1990-December 2007
[RPV.sub.Abs] 0.864 *** (14.08) -0.056 (-0.12)
August 1990-December 2007
[RPV.sub.PPI-3] 1.138 *** (4.87) 0.627 (0.86)
December1990-December 2007
[RPV.sub.Core] 0.537 *** (10.73) 0.005 (0.11)
December 1990-December 2007
[RPV.sub.CPI-2] 1.362 *** (5.39) 0.520 (1.12)
September 1990-December 2007
t-Statistics (Newey-West standard errors) in parentheses. *, **,
and *** indicate significance at the 10%, 5%, and 1% significance
levels, respectively. See Table 2 for further explanations.
