Inflation Expectations: Category Beliefs and Spending Plans
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Inflation Expectations: Category Beliefs and Spending Plans
Alexander M. Dietrich, Edward S. Knotek, Kristian Ove R. Myrseth,
Robert W. Rich, Raphael S. Schoenle and Michael Weber∗
This version: December 2021
—Preliminary and incomplete—
Abstract
We explore the distinction between explicit beliefs, such as those articulated in surveys,
and tacit beliefs, on which individuals act. Drawing on a large daily-tracking survey, we
find that novel measures of inflation expectations based on Personal Consumption Expen-
ditures categories predict spending plans better, both in the cross-sectional and the time
series dimension, than aggregate inflation expectations elicited conventionally. In partic-
ular, non-linear heuristic operators applied to consumption categories improve model fit
substantially in the time series dimension. Our results suggest that tacit inflation expec-
tations are not best represented by explicit, aggregate inflation expectations; aggregated
category-based expectations hold promise as more economically informative measures.
Keywords: Household expectations, Survey, Sectoral expectations
JEL-Codes: C83, E31, E52
∗
Dietrich: University of Tübingen, Email: alexander.dietrich@uni-tuebingen.de; Knotek: Federal Re-
serve Bank of Cleveland, Email: edward.knotek@clev.frb.org; Myrseth: University of York, Email: kris-
tian.myrseth@york.ac.uk; Rich: Federal Reserve Bank of Cleveland, Email: robert.rich@clev.frb.org; Schoenle:
Brandeis University, CEPR and CESifo, Email: schoenle@brandeis.edu; Weber: University of Chicago Booth
School of Business, Email: michael.weber@chicagobooth.edu. The views stated in this paper are those of the
authors and are not necessarily those of the Federal Reserve Bank of Cleveland or the Board of Governors of
the Federal Reserve System.1 Introduction
Inflation expectations impact both the price and production decisions of firms, as well as the
real interest rate of households. Inflation expectations are thus central to monetary policy,
and so is their measurement. In this paper, we draw a conceptual distinction between explicit
beliefs, which are articulated by respondents in surveys, and tacit beliefs, which underlie
economic behavior. To explore this distinction empirically, we utilize a large, nationally rep-
resentative daily tracking survey to elicit an aggregate measure of inflation expectations as
well as a novel, aggregated measure of price expectations based on decomposed consumption
categories. The aggregate measure is similar to the approach used in canonical household
surveys, such as the Survey of Consumer Expectations and the University of Michigan Sur-
veys of Consumers. The aggregated measure, in contrast, elicits inflation expectations for
each of 11 different Personal Consumption Expenditures (PCEs) categories. We find that
non-linear, heuristic strategies applied to the category inflation expectations consistently out-
perform explicit, aggregate inflation expectations in predicting household spending plans.
Linear combinations of the category expectations also tend to outperform the conventional
measure of inflation expectations. These findings segue with classic work in psychology on
heuristics and human judgment; human intuition is notoriously weak at integrating discrete
pieces of information in a consistent fashion, and therefore may struggle to explicitly estimate
‘inflation’ as an aggregate, abstract concept. Our results thus suggest that there is scope to
obtain more informative measures of inflation expectations by mechanically aggreating in-
flation expectations for consumption categories; this may better represent the tacit inflation
concept than would the explicitly articulated aggregate inflation measure itself.
On the basis of data collected as part of the Cleveland Fed Consumers and COVID-19
daily tracking survey (Knotek et al. (2020)), we document three important facts about our
aggregated, category-based inflation measure. Compared to a conventional aggregate mea-
sure, it appears: (i) less overstated and less dispersed both in the (ii) cross-section and (iii)
across time. In a horse race among models to match movements in the aggregate measure, a
heuristic ‘take-the-second-highest-category’ approach comes closest, suggesting that individ-
uals rely on a nonlinear cognitive procedure to report the aggregate measure. Importantly,
multiple variations of the aggregated measure, including the one relying on self-reported ex-
penditure weights, outperform the conventional measure in predicting individual spending
plans; this indicates that the aggregated measure may provide a better proxy for the tacit
inflation concept on which individuals act and thus prove a more informative measure for
policymakers.
11.1 Measuring Inflation Expectations
Before proceeding with the theoretical discussion of human forecasting that motivates our
empirical study, we outline the current practice in measuring inflation expectations. Broadly
speaking, there are two basic approaches: surveys and market-based measures.
Surveys on Inflation Expectations Perhaps the most straightforward way to understand
how people expect prices to evolve is to ask them directly–as a large number of surveys do,
with influential examples including the University of Michigan’s Surveys of Consumers (SoC)
and the Federal Reserve Bank of New York’s Survey of Consumer Expectations (SCE). While
the former has collected data on household inflation expectations since 1978, the latter started
in 2013. Both ask about aggregate inflation or the change of aggregate prices directly, at a
monthly frequency, and they include some kind of panel structure; while the SoC asks a
subset of participants to answer the survey again, half a year later, the SCE has a rolling
panel structure, with respondents answering 12 consecutive monthly surveys.
In addition to point predictions of inflation, the SCE elicits density forecasts inflation in
the form of a histogram at the individual level. Here, respondents assign probabilities to 10
bins of potential outcomes, with support ranging from ”deflation, more than 10%” to ”more
than 10% inflation”. This method allows computation of individual uncertainty measures, in
addition to mean forecasts. Notably, the correlation between individual-level point predictions
and expectations derived from the distribution question is only 0.32, evincing pronounced
individual-level inconsistency in reported beliefs across question types.
A second type of surveys target businesses. The Federal Reserve Bank of Atlanta’s
Business Inflation Expectations Survey (BIE) measures expectations of businesses, also at
a monthly frequency, asking specifically about the their unit costs. Firms assign probabilities
to five verbal options, ranging from ”down (< −1%)” to ”up very significantly (> 5%)”.
A quarterly survey collects data on inflation for longer time horizons, relying on the same
method.
Rather than elicit firm or household expectations, a third type of surveys gauges the out-
look among professional forecasters, usually economists or research institutions. A prominent
example is the Survey of Professional Forecasters, administered by the Federal Reserve Bank
of Philadelphia. Questions on inflation are posed quarterly, both for a point prediction and
a distribution measure (similar to the SCE, though with a smaller support). Another is the
Blue Chip Economic Indicators, which surveys “more than 50 leading business economists”
twice a month about CPI inflation for each quarter of the financial year (Aguinaldo et al.
(2021)).
2To summarize, surveys have elicited inflation expectations by asking respondents to report
point predictions or density forecasts. Our empirical investigation is focused on the former–a
natural point of departure as it is used both in the SoC and the SCE, and it has been in use
for more than four decades.
Market-Based Measures Market-based measures infer inflation expectations from market
prices. Usually, the break-even inflation rate is used, namely the difference in yields on similar
securities (typically treasury bonds) with and without inflation indexation. The advantage of
the market-based approach is that expectations may be computed at any frequency, but also
for every horizon (bond maturity) upon which a suitable pair of securities is available.
2 Human forecasts and inflation expectations
When household surveys ask respondents to report their inflation expectations, they are in
effect asking for forecasts of an uncertain, abstract variable. The canonical work on heuristics
and biases by Tversky and Kahneman (1974), however, demonstrates that human judgment
is ill-equipped for the task; judgments of uncertain events tend to rely on heuristics–simple
rules of thumb–which are vulnerable to predictable discrepancies from rational norms, such as
the salience bias. For example, following recent exposure to media coverage of a major airline
accident, individuals may overestimate the likelihood that their next flight crashes. Simi-
larly, consumers exposed to price spikes in their grocery bundles may report higher inflation
expectations (D’Acunto et al. (2021)). By asking respondents about their inflation expecta-
tions for each of the 11 PCE categories, we can estimate the time-series relationship between
respondents’ category-specific beliefs and their aggregate inflation expectations, allowing us
to gauge the extent to which salient cases of the former account for the latter. One way to
do this, is to assume that the category with the greatest expected inflation also is the most
salient and then check whether this ’max operator’ tracks aggregate inflation expectations.
Human judgment is also notoriously inconsistent; even experts struggle to incorporate mul-
tiple cues into a reliable forecast. Starting with the influential work of Meehl (1954), psycholo-
gists discovered that clinical expert forecasts–that is, forecasts based on expert intuition–were
surprisingly unreliable across a wide range of domains and were consistently outperformed
by rudimentary statistical models. Subsequent work by Dawes (1979) found that linear mod-
els with arbitrary weights–including equal weights–outperformed expert human judgment; as
long as linear models include the relevant predictor variables, with coefficients set in the cor-
rect direction, they prove surprisingly robust (Einhorn and Hogarth (1975)). These findings
have held up over time (Dawes et al. (1989)), and they bear directly on the conventional
3practices for eliciting inflation expectations; inflation is a linear combination of price changes
across consumption categories, and conventional elicitation techniques ask both experts and
lay respondents to report their aggregate estimates, explicitly. The problem here, of course,
is that the human mind is poorly equipped to process this question, and if the literature
has found that expert judgment struggles to incorporate cues in a consistent and balanced
fashion, then there is little reason to think that lay respondents would perform better.
By asking respondents to forecast inflation for specific PCE categories, we have the op-
portunity to combine category-specific forecasts mechanically into a ’bottom-up’, aggregated
inflation estimate. This is likely to be more internally consistent than any intuitive com-
bination performed by the respondents, themselves, including that reported for the conven-
tional measure of inflation expectations. Moreover, there is a possibility that respondents’
tacit inflation beliefs–which they may not necessarily articulate explicitly, but act as if they
hold–are better represented by a mechanical aggregation principle of PCE categories than by
an intuitively formed and explicitly articulated aggregate inflation concept; the mere act of
articulating and reporting the abstract inflation concept might involve further cognitive dis-
tortion from heuristics and biases. That is, asking respondents to use what they more likely
know–price dynamics for individual PCE categories–to articulate something they don’t really
know–the aggregate inflation concept–might exacerbate bias. To test whether tacit inflation
beliefs are better represented by a mechanical aggregation of PCE category beliefs than by
conventionally reported aggregate inflation expectations, we compare a series of estimations
in which we regress spending plans on competing measures of inflation expectations.
This papers makes a contribution to the literature by demonstrating that expectations
of aggregate inflation, conventionally elicited, is in fact the weakest predictor of individual
spending plans; various linear combinations of PCE category beliefs–including a specification
setting equal weights on categories–outperform expected aggregate inflation. A second-max
heuristic, which takes the respondent’s second-highest PCE category belief, appears to be the
strongest predictor.
2.1 Model of Inflation Expectations
To formalise our thinking about the process by which inflation expectations form, we present
a simple model. The model posits that an economic agent uses her set of information St to
(1) form expectations and (2) to make economic decisions. Crucially, we allow her to hold
inconsistent expectation formation processes; the beliefs on which she acts–tacit expectations–
need not be consistent with aggregate inflation expectations elicited, nor with the category-
based expectations aggregated. The intuition behind this is that explicit elaboration of a
4question posed in a survey, such as a inflation expectations, may trigger processes entirely
distinct from those lurking implicitly when the agent makes consumption decisions. Put
differently, the agent doesn’t necessarily ask herself what her one-year inflation forecast is
prior to each consumption decision. Yet, she will act as if she has some belief, and the
question we ask in this paper is whether the aggregate inflation expectation is the most
accurate representation or whether some aggregation principle applied to the category-based
beliefs can offer improvement.
Moreover, variations of the same question may trigger different expectation formation
processes. That is, the framing of the question may influence the answer even in the face of
the same information St ; the way the agent thinks about the aggregate inflation concept need
not match how she considers inflation for each category, separately. The latter may prove
more intuitive than the former, which may be more difficult to grasp, leading to differential
cognitive heuristics used. However, it is also possible that an aggregation principle applied to
the category-based inflation expectations may inform us about the process by which aggregate
inflation expectations form. For example, we could test whether a linear combination, such
as weighting by self-reported expenditure importance, tracks aggregate inflation expectations
better than does a non-linear combination, such as a heuristic max operator.
In sum, two empirically testable questions stand out in our model:
1. Does the aggregate inflation expectation, compared to aggregated category-based ex-
pectations, better represent the tacit inflation expectation, on which an agent acts?
2. Can we find an aggregation principle, applied to category-based expectations, that
tracks aggregate expectations?
Formally, we define two separate layers of inflation expectations: the mental sphere, where
the agent forms expectations and the ‘survey’ layer, where he tries to express them to the
world. Figure 2.1 shows this graphically. In the mental layer, economic agents use their
set of information St about the world to form expectations about future prices for different
products or categories, πi . Those individual expectations are then combined into an aggregate
expectation π̃t,t+1 = M (π̃i,t,t+1 ), using a mental model, described by the aggregation function
M (·). Subsequently, this aggregate inflation forecast may be used to determine aggregate
consumption plans for the future. Crucially, this mental layer does not allow us to observe
any expectation - neither the category-level of the aggregate nor the mental model. Should
we want to learn about these tacit expectations of the consumer, we would have to rely on a
method that incentivizes the agent to reveal its expectation numerically, for example a survey.
5Figure 2.1: Inflation Expectations - Formation and Revelation
J(·)
Mental
M (·) I(·)
S π̃i,t,t+1 π̃t,t+1 C̃t,t+1
Revealed
F
πi,t,t+1 = π̃i,t,t+1 + i F F
πt,t+1 = π̃t,t+1 + γ Ct,t+1 = C̃ + δ
i ∼ N (¯, σ ) γ ∼ N (σ̄, σγ ) δ ∼ N (δ̄, σδ )
A(·)
Aggregated
agg PN F
πt,t+1 = i=1 ωi πi,t,t+1
Notes: The figure shows the proposed mechanism for inflation expectations formation as well as the revelation
of those expectations in surveys.
This constitutes the second layer of our model: Here, the agent expresses his expectation
numerically, as a forecast of future inflation. Still, forecasts revealed might for various reasons
deviate from the tacit expectation: it may be difficult for the participant to express his
expectation numerically; he might rely on mental heuristics, or he may only pay limited
attention. We will thus end up with a blurred expression of the true expectation. Formally,
F
we thus add an error term to the expressed forecasts of consumers: πi,t,t+1 = π̃i,t,t+1 + i gives
the category-level forecast. We assume that the reporting error i is normally distributed
with standard deviation σi . Crucially, we allow for the possibility of a non-zero reporting
error mean, that is, E = ¯. A similar equation holds for the aggregate inflation forecast
F
πt,t+1 F
= π̃t,t+1 + γ and the revealed spending plan Ct,t+1 = C̃t,t+1 + δ. γ and δ give reporting
errors relative to the true expectation, again with mean γ̄ as well as δ and standard deviation
σγ and σδ , respectively.
A researcher interested in aggregate inflation expectations of households now has two op-
F
tions: the revealed aggregate forecast, πt,t+1 , or a mechanical, aggregated measure of inflation
agg F
expectations πt,t+1 = A(πi,t,t+1 ) based on the revealed category forecasts.
3 Survey
In this section, we provide background on the survey, followed by basic descriptives results.
63.1 Survey Description
Our survey module is part of a larger daily tracking-survey of consumer expectations hosted
by the Federal Reserve Bank of Cleveland, and administered by Qualtrics Research Services.
It includes a nationally representative sample of 17.888 responses, collected between July 09,
2020 and September 09, 2021. Dietrich et al. (2021) and Knotek et al. (2020) provide further
information about the survey of which our module is a part.
The survey was run in real time, with a daily sampling size of at least 100 respondents. We
required all respondents to be U.S. residents and to speak English as their primary language.
Respondents were representative of the US population according to several key demographic
and socioeconomic characteristics. In terms of demographics, we looked to have respondents
divided equally between males and females. Moreover, approximately one third of respondents
were targeted to be between 18 and 34, another third between ages 35 and 55, and a final
third older than age 55. We also required a distribution across U.S. regions in proportion to
population size, drawing 20% of our sample from the Midwest, 20% from the Northeast, 40%
from the South, and 20% from the West. 66% of the sample were targeted to be non-Hispanic
White, 12% non-Hispanic Black, 12% Hispanic and 10% Asian or other.
We also tried to make the sample nationally representative in terms of socioeconomic
make-up. In particular, we sampled from the income distribution with a goal of 35% of
respondents with a household income of less than 50k, 35% with an income between 50k
and 100k, and the remaining 30% with an income above 100k. We aimed for half to hold a
bachelor’s degree or higher, and half to have some college or less. The survey also includes
filters to eliminate respondents who write gibberish for at least one response, or who complete
the survey in less (more) than five (30) minutes. Table 12 provides a detailed breakdown of
our sample. It shows that our sample was roughly representative of the U.S. population
according to our sampling criteria. In addition, our analysis uses a raking scheme to compute
respondent weights ensuring that our sample is representative of the U.S. population by
gender, age, income, education, ethnicity, and Census region.
Within the survey, we first asked participants about their aggregate inflation expectations
over the next 12 months (Q1 in Table 2). Subsequently, we elicited inflation expectations for
11 PCE categories (Q4 in Table 2). Table 4 in section 3 shows both the PCE categories used in
our survey and some summary statistics. The PCE disaggregation used in our survey is based
on the U.S. national income and product accounts (NIPA) disaggregation, with some small
sectors combined in order to reduce the cognitive burden of answering the survey. Dietrich
(2021) provides more details on categories. Besides inflation expectations within these sectors,
we also collected data on how much survey respondents spent within the respective sector
7Table 1: Survey Respondent Characteristics
pct. (Target) pct. (Target)
Age Race
18-34 33.61% (33.3%) non-Hispanic white 70.55% (66%)
35-55 33.61% (33.3%) non-Hispanic black 12.03% (12%)
older than 55 32.78% (33.3%) Hispanic 7.69% (12%)
Asian or other 9.73% (12%)
Gender
female 49.38% (50%) Household Income
male 50.21% (50%) less than 50k$ 46.23% (30%)
other 0.41% (-%) 50k$ - 100k$ 29.08% (35%)
more than 100k$ 24.69% (30%)
Region
Midwest 19.48% (20%) Education
Northeast 20.03% (20%) some college or less 48.83% (50%)
South 40.74% (40%) bachelors degree or more 51.17% (50%)
West 19.75% (20%)
N=17.888
Notes: This table presents data on the characteristics of participants in the survey administered by Qualtrics.
Appendix B provides a list of all questions.
during the last month (Q3 in Table 2) and how ‘important’ they consider it for aggregate
inflation (Q2 in Table 2). These data allow us to both compute expenditure shares per sector
(relative to total expenditure) as well as a measure of relative importance.
Following category expectations and expenditure shares, participants were asked about
their expected spending relative to that today in one, two, 12, and 24 months’ time. This ques-
tion was also repeated for other, more narrowly defined spending categories, such as services
spending and expenditures on non-durable consumption goods. Additionally, respondents
were asked about their socioeconomic background and consumer habits.
8Table 2: Survey Questions
Aggregate Inflation Question
Q1 What do you expect the rate of inflation I expect [...] to be [positive/negative]
to be over the next 12 months? [...] percent over the next 12 months.
Category Inflation Questions
Q2 Which of the following broad consump- Participants move a slider from 0 (no im-
tion categories matter the most to you portance) to 100 (highest importance),
right now in your daily life? Please move per category.
the slider to indicate the importance for
each of them [...]
Q3 In terms of consumption spending, how Per category, participants enter an ap-
much money did you spend on each of the proximate amount in dollar into a
following broad consumption categories bracket.
during the last month? [...]
Q4 Twelve months from now, what do you I expect the price of [category] to [in-
think will have happened to the price of crease/decrease] by percent.
the following items?
Spending Questions
Q5 Compared with your spending last [up/no change/down] by percent.
month, how do you expect your total
spending to change in the next [time hori-
zon]?
Q6 Compared with your spending on services [up/no change/down] by percent.
[...] last month, how do you expect your
total spending to change in the next [time
horizon]?
Q7 Compared with your spending on non- [up/no change/down] by percent.
durable goods [...] last month, how do
you expect your total spending to change
in the next [time horizon] ?
Notes: List of questions asked in the survey.
94 Aggregate vs. Category Inflation Expectations
Figure 4.1 shows the time series for aggregate and category inflation expectations during the
survey period. Panel (a), to the left, displays category expectations for the durable (red lines)
and non-durable goods (blue lines) sectors, while panel (b) shows services categories (green
lines). All time series are balanced 11-day moving averages. Note that reported aggregate
inflation expectations generally are higher than any individual sectoral inflation expectation.
This feature is noteworthy because a rational agent would weight category expectations using
some type of linear model in order to obtain an aggregate measure. Consequently, there
is no linear combination of categories that could produce the reported aggregate inflation
expectation. This leads us to:
Fact 1. Aggregate inflation expectations reported by consumers are higher than any category
expectations. There exists no possible linear combination of category expectations with non-
negative aggregation weights that maps category expectations into aggregate expectations.
Figure 4.1: Aggregate vs Sectoral Inflation Expectations
(a) Durable and Non-durable Goods (b) Services
12
12
Aggregate Aggregate
Motor vehicles Housing and utilities
Recreational goods Health care
10
10
Other durable goods Transportation services
Food and beverages Food services
Gasoline Other services
percentage points
percentage points
Other nondurable goods
8
8
6
6
4
4
2
2
Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021
Notes: figure shows aggregate inflation (black line) as well as category inflation rates, durable and non-durable
goods inflation in panel (a), services in panel (b). Time series show an 11-day balanced moving average.
Underlying daily observations are Huber robust and survey weighted means.
The time series in figure 4.2 compares aggregate inflation expectations against aggregated
expectations, of which there are two broad types: (1) linear combinations of category infla-
tion expectations, and (2) non-linear, max operators. For the linear combinations, we have
measures that use PCE weights, expenditure and ’importance’ weights reported by respon-
dents at the individual-level, and equal weights (see Qs 2-4, Table 2). Aggregation by PCE,
expenditure, and importance weights would for the rational consumer be expected to yield
10Figure 4.2: Aggregate vs Aggregated Measures
(a) Mean (b) Disagreement
20
15
Aggregate
Expenditure weights
Importance weights
Equal weights
15
PCE weights
Max operator
10
percentage points
percentage points
Second max operator
10
5
5
Aggregate Expenditure weights
Importance weights Equal weights
PCE weights Max operator
Second max operator
0
0
Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021
Notes: figure shows aggregate inflation expectations (black line) as well as measures of aggregated inflation
expectations. Panel (a) daily mean; panel (b) daily standard deviation. Time series show an 11-day balanced
moving average. Underlying daily observations are Huber robust and survey-weighted means.
roughly similar results–and converge with aggregate inflation expectations (Q1, Table 2).
Equal weights are included as a curious point of comparison since they prove surprisingly
robust in human forecasting (Dawes et al. (1989)). The max operator sets for a respondent’s
aggregated inflation expectation the same respondent’s highest inflation expectation for any
category, and the second max the second-highest. The two max operators are thus intended
capture potential non-linear judgment processes by which salient categories may be taken as
heuristics. A visual inspection of panel a (Figure 4.2) indicates that the second-max opera-
tor appears to track aggregate inflation expectations pretty closely, while the max operator
exceeds it consistently and substantially. However, panel b (Figure 4.2) shows that the dis-
agreement in aggregate inflation expectations is best matched by the max operator, with
the second-max disagreement undershooting, until the summer months of 2021. Beyond this
point, the max appears to overshoot while the second max tracks disagreement in aggregate
inflation expectations fairly well.
Figure 4.3 compares aggregate inflation expectations against aggregated expectations in
the cross section, plotting on the horizontal axis (binned) measures of the latter, with the
vertical axis giving the mean of aggregate inflation expectations for each respective bin. Two
features stand out. First, almost all observations are above the 45° line, indicating that
aggregate inflation expectations tend to be higher than aggregated measures. Second, the
relationship is non-linear; beyond a certain threshold, more extreme aggregated expectations
correspond to only slightly more extreme aggregate expectations.
11Figure 4.3: Aggregate vs Aggregated Expectations
20
Aggregate Expectation
10
Expenditure Weights
Importance Weights
0
Equal Weights
PCE Weights
max Operator
second max Operator
-10
-20 0 20 40 60
Aggregated Expectation (binned)
Notes: figure divides aggregated expectations into 15 equal-sized bins and computes mean aggregate inflation
expectations for each bin. Blue circles: expectations aggregated using reported expenditure shares. Red dia-
monds: expectations aggregated using reported importance weights. Black squares: expectations aggregated
using equal weight. Green triangles: expectations aggregated using monthly PCE weights. Orange squares
and pink crosses show the first and second max of the category expectations, respectively.
In table 3, we regress aggregate inflation expectations on aggregated expectations and a
constant. For all measures of aggregated expectations, we find a positive, highly significant
constant, as well as an aggregated-inflation-expectations coefficient smaller than 1. This
indicates that aggregate expectations on average are higher than measures of aggregated
expectations, and that the discrepancy increases with the level of expectations. According
to the R2 , linear aggregated measures explain roughly 20% of the variance in the aggregate
measure, whereas the second-max operator explains about 15%.
Fact 2. Disagreement among households over aggregate inflation expectations is higher than
disagreement for any category.
Figure 4.4 shows disagreement among respondents for aggregate inflation expectations
(black line) and sectoral expectations, where we measure disagreement as the daily standard
deviation of the cross section. The figures display an 11-day moving average, with durable
and non-durable good sectors in panel (a) and services in panel (b). For most of the time
surveyed, disagreement is much higher for aggregate expectations than it is for more narrowly
defined sectoral expectations.
Fact 3. Volatility over time is higher for the reported aggregate inflation expectation than it
is for both individual category expectations and linear combinations of category expectations.
12Table 3: Aggregate vs Aggregated Inflation Expectations
(1) (2) (3) (4) (5) (6)
Expenditure Importance Equal PCE max second max
weight weight weight weight operator operator
βπCat 0.420 0.605 0.643 0.507 0.169 0.335
(36.77) (43.45) (45.93) (43.79) (24.09) (35.08)
Constant 3.265 2.718 2.636 2.746 3.369 3.109
(39.31) (31.81) (31.91) (32.50) (33.88) (35.96)
R2 0.176 0.209 0.223 0.210 0.077 0.151
t statistics in parentheses
∗ ∗∗ ∗∗∗
p < 0.05, p < 0.01, p < 0.001
Figure 4.4: Aggregate vs Sectoral Inflation Expectation Disagreement
(a) Durable and Non-durable Goods (b) Services
18
18
Aggregate Motor vehicles Aggregate Housing and utilities
Recreational goods Other durable goods Health care Transportation services
16
16
Food and beverages Gasoline Food services Other services
Other nondurable goods
14
14
percentage points
percentage points
12
12
10
10
8
8
6
6
4
4
Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021
Notes: figure shows aggregate inflation (black line) as well as category inflation rates, durable and non-durable
goods inflation in panel (a), services in panel (b). Time series show an 11-day balanced moving average.
Underlying daily observations are Huber robust and survey weighted means.
Table 4 shows summary statistics for aggregate inflation expectations, category expecta-
tions, and aggregated expectations. The table reports the mean expectation and the disagree-
ment among households (cross-section standard deviation); the time-series standard deviation
represents the volatility over time–that is, the standard deviation of daily mean estimates.
The mean expectation, disagreement, and volatility over time are all higher for the ag-
gregate measure than for any individual category. They are also higher than for any linear
aggregation of the categories.
Fact 4. Higher socioeconomic status, both in terms of income and education, is associated
13Table 4: Summary Statistics
Standard Deviation
Mean Cross Section Time Series
Aggregate expectation 5.16 7.59 2.86
Category expectations
Motor vehicles 4.56 6.61 1.89
Recreational goods 3.24 6.52 1.81
Other durable goods 3.21 6.05 1.87
Food and beverages 4.91 6.90 1.94
Gasoline 4.58 7.33 2.31
Other nondurable goods 3.57 5.92 1.56
Housing and utilities 4.84 7.02 1.83
Health care 3.19 7.15 1.72
Transportation services 4.29 6.68 1.68
Food services 4.23 7.05 1.72
Other services 3.93 5.76 1.44
Category-based aggregated expectations
Linear Aggregation
Expenditure Weights 4.50 5.19 1.42
Importance Weights 3.97 4.44 1.35
Equal Weights 3.79 4.25 1.33
PCE Weights 4.73 5.33 1.58
Nonlinear Aggregation
First max 10.37 7.54 3.32
Second max 6.64 6.96 2.04
Notes: This table presents summary statistics on the Huber-robust and survey-weighted mean on expecta-
tions, the standard deviation in the cross section, and the time series standard deviation.
with i) lower mean aggregate inflation expectations as well as category expectations and ii)
lower cross-sectional disagreement.
Tables 12 and 11 in the appendix show mean expectations and disagreement among dif-
ferent demographic groups. Across almost all categories and aggregated measures, woman as
well as grocery shoppers display higher inflation expectations and greater disagreement. The
same holds for lower socioeconomic status, proxied by income and education.
145 Aggregation Inconsistency
Next, we examine the relationship at the individual level between aggregate and aggregated
inflation expectations. For this purpose, we define the aggregation inconsistency as the dif-
ference between the aggregate expectation and any aggregated measure.
F agg
Λi = πi,t,t+1 − πi,t,t+1
Λi gives the aggregation error survey participant i as the difference between his aggregate
F agg
forecast πi,t,t+1 and the bottom up measure πi,t,t+1 .
Figure 5.1: Aggregation Errors
Abs. agg. inconsistency
Agg. inconsistency
Notes: figure shows the mean for both the aggregation error as well as the absolute aggregation error across
different aggregation weights. Means are Huber robust and survey weighted.
5.1 Demographics and Aggregation Inconsistency
Figure 5.1 displays the mean aggregation error and the mean absolute aggregation error for
different aggregation measures. While the absolute error seems to be roughly similar for all
linear combinations, the aggregation error is smallest for the PCE weights. Still, all linear
combinations show a positive aggregation error; on average, aggregate expectations are higher
than aggregated expectations. This relates to fact 1, above.
When looking at nonlinear aggregation measures, we find that the max and second-max
operators both yield a large absolute error, but for the latter the error is almost unbiased.
15Table 13 and 14 in the appendix regress aggregation error and absolute aggregation error
on various demographic characteristics. We find that females tend to make larger absolute er-
rors than do males; younger respondents larger absolute errors compared to older respondents;
and less educated respondents larger absolute errors compared to more educated respondents.
The errors are also relatively biased upwards; these groups tend to systematically report ag-
gregate expectations that are higher than the aggregated measures. For females, however, we
do not find this bias. While they have larger errors than do males, there is no difference in
direction. Overall, behavior is consistent across all aggregated measures.
5.2 Uncertainty and Aggregation Errors
Figure 5.2: Aggregation Error and Standard Deviation
30
40
Absolute aggregation error (Expenditure weights)
Absolute aggregation error (Expenditure weights)
25
30
20
20
15
10
10 5
0
0 5 10 15 20 0 10 20 30 40
Beta distribution standard deviation Category expectations standard deviation
Notes: figure shows correlation between the absolute aggregation error based on expenditure shares, abs(Λexp
i )
and the individual standard deviation of aggregate inflation expectations obtained via a beta distribution over
a probabilistic question. Right hand side shows correlation of absolute aggregation error with the standard
deviation across an individuals’ category inflation expectations.
Figure 5.2 shows that the absolute value of the aggregation error increases with a) indi-
viduals uncertainty about aggregate inflation and b) the heterogeneity across category expec-
tations.
6 Inflation Expectation Measures and Spending Plans
We next investigate the relative predictive content of reported aggregate expectations versus
aggregated category-based expectations for households spending plans. To do so, we assume
that consumers follow a standard euler equation, such as
" 1 #
i Ci,t+1 − σ Pt
Qi,t = Et βi (1)
Ci,t Pt+1
16This representation of the household Euler equation is widely used in modern macroeconomics
(see for example Galı́ (2015); Woodford (2003)). We adjust the conventional representative-
agent version by allowing for individual i specific levels of the discount factor βi , as well a
nominal interest rate ri,t = − log(Qi,t ). Eit gives the expectation operator for respondent i. A
log-linearized version of equation (1) reads as:
ci,t = Et ci,t+1 − σ ri,t − Eit πt+1 − ρi
(2)
where πt = pt − pt−1 denotes the inflation rate. While Et ci,t+1 denotes expected log real
consumption, questions Q5 to Q7 of our survey ask respondents about expected expenditure
relative to the last month, that is, Eit ∆si,t+1 = Eit (∆ci,t+1 + πt+1 ). Inserting into equation
(2) yields a version of the Euler equation that links expected spending to expected inflation:
Eit ∆si,t+1 − Eit πt+1 = σ ri,t − Eit πt+1 − ρi
(3)
On the left-hand side, we have the expected change in spending, net of the expected rate of
inflation. Building on the empirical approach by Crump et al. (2021), we can now estimate
this equation in the following form:
Eit ∆si,t+1 = β0 + β1 Eit πt+1 + Di + Tt + i,t (4)
where Di represents demographic fixed effects and Tt represents time fixed effects. The
estimation coefficient β1 is equal to 1 − σ in the model in equation (3). Including both
relies on the assumption that ri,t − ρi may be explained by both variation in time (think,
for example, about changes in the nominal interest rate over time), as well as demographic
factors, which can impact both the rate of time preference rate and the nominal interest rate
faced by households (i.e., specific risk premiums).
Table 5 shows estimation results for an array of inflation expectation measures in the cross
section. Here, we report 1 − β̂1 , which is equal to the intertemporal elasticity of substitution
σ. The fourth column gives the R2 values, the fifth the Akaike Information Criterion, and the
sixth the p-value of a Likelihood Ratio test, which compares the fit of the respective models
to the reported aggregate inflation expectation model.
Coefficients for inflation expectations are highly significant in all models. Notably, the AIC
and the Likelihood ratio test suggest a better fit for the category-based inflation measures
compared to the reported aggregate measure. Moreover, the aggregate model obtains the
lowest R2 . That is, the proportion of variation explained in planned consumption one year
ahead is lower for the aggregate measure than for any other category-based measure.
The same picture is evident in tables 6 and 7, which repeat the estimations for one-year-
ahead non-durable and services spending, respectively. The reported aggregate model for
17Table 5: 1 Year ahead spending plans
σ̂ = 1 − β̂1 t-stat R2 AIC p-val (LR)
Aggregate 0.968∗∗∗ 5.05 0.06 81615 -
Expenditure 0.910∗∗∗ 6.97 0.07 81527 0.000
Importance 0.801∗∗∗ 10.33 0.08 81090 0.000
Equal 0.788∗∗∗ 10.43 0.08 81318 0.000
PCE 0.837∗∗∗ 10.28 0.08 81104 0.000
First max 0.939∗∗∗ 7.81 0.07 81530 0.000
Second max 0.881∗∗∗ 9.62 0.08 81368 0.000
t statistics in parentheses
∗ ∗∗ ∗∗∗
p < 0.05, p < 0.01, p < 0.001
Table 6: 1 Year ahead non-durable spending plans
σ̂ = 1 − β̂1 t-stat R2 AIC p-val (LR)
Aggregate 0.967∗∗∗ 4.16 0.05 37652 -
Expenditure 0.889∗∗∗ 6.12 0.06 37587 0.000
Importance 0.755∗∗∗ 10.24 0.09 37326 0.000
Equal 0.738∗∗∗ 10.21 0.09 37432 0.000
PCE 0.799∗∗∗ 10.23 0.09 37335 0.000
First max 0.920∗∗∗ 6.68 0.06 37578 0.000
Second max 0.859∗∗∗ 8.20 0.08 37482 0.000
t statistics in parentheses
∗ ∗∗ ∗∗∗
p < 0.05, p < 0.01, p < 0.001
Table 7: 1 Year ahead services spending plans
σ̂ = 1 − β̂1 t-stat R2 AIC p-val (LR)
Aggregate 0.978∗∗∗ 4.19 0.06 79434 -
Expenditure 0.931∗∗∗ 6.58 0.06 79359 0.000
Importance 0.827∗∗∗ 10.70 0.08 78917 0.000
Equal 0.814∗∗∗ 10.78 0.08 79118 0.000
PCE 0.858∗∗∗ 10.41 0.08 78930 0.000
First max 0.945∗∗∗ 8.24 0.07 79318 0.000
Second max 0.899∗∗∗ 9.48 0.08 79184 0.000
t statistics in parentheses
∗ ∗∗ ∗∗∗
p < 0.05, p < 0.01, p < 0.001
18Table 8: Time Series: 1 Year ahead spending plans
σ̂ = 1 − β̂1 t-stat R2 AIC p-val (LR)
Aggregate 0.731∗∗ 3.35 0.208 96.62 -
Expenditure 0.366∗∗∗ 4.83 0.358 88.94 0.022
Importance 0.342∗∗∗ 3.63 0.356 92.50 0.128
Equal 0.351∗∗∗ 3.74 0.318 98.29 1.000
PCE 0.546∗∗∗ 3.79 0.284 98.75 1.000
First max 0.759∗∗∗ 4.44 0.420 91.23 0.067
Second max 0.485∗∗∗ 4.61 0.465 85.56 0.004
t statistics in parentheses
∗ ∗∗ ∗∗∗
p < 0.05, p < 0.01, p < 0.001
Table 9: Time Series: 1 Year ahead non-durable spending plans
σ̂ = 1 − β̂1 t-stat R2 AIC p-val (LR)
Aggregate 1.027 -0.18 0.19 25.14 -
Expenditure 1.050 -0.21 0.19 25.02 0.940
Importance 0.745 1.02 0.23 24.18 0.620
Equal 0.751 1.06 0.23 24.13 0.604
PCE 0.477∗∗ 2.71 0.37 17.51 0.022
First max 0.813∗∗ 2.42 0.34 17.58 0.023
Second max 0.594 1.88 0.34 17.92 0.027
t statistics in parentheses
∗ ∗∗ ∗∗∗
p < 0.05, p < 0.01, p < 0.001
Notes: Table shows Euler equation estimates for one year ahead expected spending. Weekly means of cross
section used for expected spending and inflation expectations. We control for weekly mean household income
expectations.
non-durable spending (table 6) obtains the highest AIC and the lowest R2 , while all category-
based models are statistically distinct, according to the Likelihood Ratio test. Similarly, the
reported aggregate model for services spending (table 7) yields the highest AIC and the lowest
R2 , although its performance is matched by the linear model using expenditure weights.
We turn next to time-series estimations, which–though limited to 42 data points–have the
advantage of using averages of individual-level estimates that should act to cancel response
noise. Table 8 gives estimations for one-year-ahead spending plans, table 9 for non-durable
spending plans, and table 10 for services spending plans. For spending plans (table 8), all
seven models yield statistically significant coefficients for inflation expectations, with the
reported aggregate model yielding the third-lowest AIC and the lowest R2 (0.21). Of the
19Table 10: Time Series: 1 Year ahead services spending plans
1 − β̂1 t-stat R2 AIC p-val (LR)
Aggregate 0.750∗∗ 2.81 0.162 71.19 -
Expenditure 0.576∗∗ 3.01 0.174 77.60 1.000
Importance 0.521∗∗∗ 4.83 0.277 74.54 1.000
Equal 0.566∗∗∗ 3.71 0.224 77.13 1.000
PCE 0.614∗∗∗ 4.47 0.266 76.02 1.000
First max 0.815∗∗∗ 11.48 0.642 33.22 0.000
Second max 0.608∗∗∗ 11.04 0.553 51.37 0.000
t statistics in parentheses
∗ ∗∗ ∗∗∗
p < 0.05, p < 0.01, p < 0.001
Notes: Table shows Euler equation estimates for one year ahead expected spending. Weekly means of cross
section used for expected spending and inflation expectations. We control for weekly mean household income
expectations.
six category-based models, only two yield model fit statistically different from the reported
aggregate model: the linear model using expenditure weights (R2 = 0.36) and the second-
max operator (R2 = 0.47). For non-durable spending plans (table 9), only two models yield
significant coefficients for inflation expectations: the linear model using PCE weights and
the first-max operator. These are also statistically different from the reported aggregate
model, and they both yield lower AIC and higher R2 values (0.19, compared to 0.37 and 0.34,
respectively). For services spending (table 10), however, all models yield highly statistically
significant coefficients for inflation expectations, but only two models are statistically different
from the reported aggregate model: the first- and second-max operators. Compared to the
reported aggregate model, the first- and second-max operators both yield lower AIC and
substantially higher R2 values (0.16, compared to 0.64 and 0.55, respectively).
7 Conclusion
This paper draws a conceptual distinction between explicit beliefs, which are expressed in
surveys, and tacit beliefs, which drive economic behavior. We explore this distinction empir-
ically with a large daily-tracking survey that elicits both a conventional aggregate measure
of inflation expectations as well as a novel measure of inflation expectations based on PCE
categories.
A cursory glance at our data reveals four striking facts. The first is that reported aggregate
inflation expectations are higher than any category-based expectations, thereby ruling out
a linear mapping (with non-negative weights) of the category-based expectations into the
20reported aggregate expectations. The second is that disagreement among respondents over
reported aggregate inflation expectations is higher than that for any category. The third
is that volatility over time is higher for the reported aggregate expectations than it is for
individual category expectations. And the fourth is that higher socioeconomic status, both in
terms of income and education, is related to lower mean aggregate and category-level inflation
expectations as well as lower cross-sectional disagreement.
The first fact is consistent with a psychological interpretation of expectation formation:
individuals appear to rely on non-linear cognitive heuristics to express their explicit aggregate
inflation expectations. The remaining facts all suggest that reported aggregated inflation
expectations aren’t quite as ‘well-behaved’ as are the individual category-level expectations,
indicating that the former may not provide the most relevant or accurate measure of the
beliefs on which individuals actually act–that is, tacit beliefs.
We explore this point further with two sets of models that use a variety of inflation
measures to predict spending plans, one in the cross-sectional dimension and the other in
the time series dimension. Our cross-section estimations have the advantage of a very large
number of observations, albeit at the expense of noisy survey measures. Conversely, our time-
series estimations rely on response averages, which should cancel noise, but only provide a
small number of data points. Nevertheless, both sets of models paint a consistent picture:
models with non-linear, second-max operators always yield improved fit over reported models
with reported aggregate inflation expectations. In the time-series models, this improvement is
very pronounced. Moreover, models with linear combinations of category-based expectations
consistently outperform models with reported aggregate expectations in the cross section, and
in some cases also in the time series dimension.
It therefore appears that tacit inflation expectations are not best represented by explicit,
conventionally reported aggregate inflation expectations. Rather, category-based inflation
measures–in particular non-linear max operators–seem more informative.
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22A Additional Tables
A.1 Demographic Summary Statistics
Table 11: Summary Statistics - Mean Demographics
Gender Grocery Education Income
Female Male Yes No High Low High Middle Low
Aggregate expectation 5.63 4.56 5.40 3.54 4.48 5.75 4.41 4.89 5.47
Category expectations
Motor vehicles 4.48 4.53 4.71 3.61 4.65 4.41 4.61 4.57 4.48
Recreational goods 3.54 3.01 3.37 2.42 3.32 3.17 3.36 3.28 3.11
Other durable goods 3.29 3.18 3.33 2.52 3.24 3.20 3.33 3.33 3.09
Food and beverages 5.36 4.42 5.00 4.22 4.74 4.98 4.81 5.03 4.79
Gasoline 4.66 4.50 4.66 4.05 4.41 4.71 4.35 4.84 4.54
Other nondurable 3.77 3.40 3.66 3.01 3.55 3.62 3.73 3.61 3.42
Housing and util. 5.19 4.52 4.92 4.36 5.04 4.68 4.91 5.27 4.44
Health care 3.16 3.22 3.30 2.46 3.23 3.16 3.47 3.25 2.95
Transportation 4.58 3.96 4.42 3.35 4.15 4.36 4.02 4.39 4.31
Food services 4.39 4.08 4.32 3.66 4.29 4.19 4.41 4.27 4.06
Other services 4.24 3.65 4.08 3.31 3.86 3.99 3.92 4.09 3.86
Category based aggregated expectations
Linear Aggregation
Expenditure 4.89 4.19 4.61 3.90 4.47 4.61 4.37 4.72 4.49
Importance 4.27 3.72 4.06 3.43 3.90 4.05 3.93 4.12 3.88
Equal 4.03 3.62 3.87 3.24 3.72 3.88 3.77 3.92 3.71
PCE 5.09 4.44 4.83 4.10 4.63 4.85 4.65 4.92 4.65
Nonlinear Aggregation
First Max 11.58 9.80 10.44 9.97 10.30 11.01 10.10 10.53 11.05
Second Max 6.81 5.97 6.72 6.17 6.31 6.43 6.09 6.83 6.44
Notes: This table presents summary statistics on the Huber robust and survey weighted mean on expectations
across demographics.
A.2 Demographic Regressions Aggregation Error
23Table 12: Summary Statistics - Standard Deviation Demographics
Gender Grocery Education Income
Female Male Yes No High Low High Middle Low
Aggregate expectation 9.39 5.79 7.64 5.97 5.66 9.33 5.67 6.22 8.32
Category expectations
Motor vehicles 7.01 5.56 6.65 5.66 5.68 6.78 6.22 5.72 7.03
Recreational goods 7.18 5.05 6.93 4.94 5.02 7.42 5.75 5.93 8.00
Other durable goods 7.69 4.99 6.81 4.90 5.05 6.90 5.70 4.99 7.23
Food and beverages 7.19 5.88 6.95 5.86 5.84 7.13 6.47 5.93 7.38
Gasoline 7.54 7.11 7.33 7.30 7.22 7.41 7.11 7.32 7.50
Other nondurable goods 6.95 5.66 6.62 5.04 5.72 6.77 5.53 5.83 6.27
Housing and utilities 7.45 6.57 7.05 6.83 6.76 7.20 6.59 6.85 7.45
Health care 8.18 6.05 7.79 6.27 7.04 7.86 7.64 7.04 7.32
Transportation services 7.01 5.63 6.70 4.86 5.72 6.88 5.56 5.78 7.18
Food services 7.39 6.69 7.06 6.93 6.91 7.14 6.72 6.88 7.40
Other services 6.73 4.69 6.45 4.68 4.74 6.61 5.42 5.56 6.88
Category based aggregated expectations
Linear Aggregation
Expenditure 5.89 4.62 5.27 4.64 4.76 5.78 4.63 5.01 6.02
Importance 5.01 3.96 4.47 4.25 4.06 4.80 4.03 4.31 4.88
Equal 4.78 3.85 4.26 3.94 3.90 4.58 3.87 4.11 4.66
PCE 6.02 4.74 5.38 5.04 4.85 5.79 4.80 5.16 5.89
Nonlinear Aggregation
First Max 8.72 7.16 7.53 7.57 7.29 8.55 7.23 7.52 8.66
Second Max 6.48 5.83 6.99 6.80 5.89 6.37 5.81 6.83 6.47
Notes: This table presents summary statistics on the Huber robust and survey weighted standard deviation
on expectations across demographics.
24Table 13: Demographics and Aggregation Inconsistency
(1) (2) (3) (4) (5) (6)
Exp Imp Equ PCE 1max 2max
Female -0.123 0.0759 0.0998 -0.0828 -0.697∗∗∗ -0.249
(-0.79) (0.52) (0.71) (-0.54) (-3.50) (-1.49)
35 to 44 years 0.348 0.172 0.188 -0.0657 0.667∗ 0.547∗
(1.61) (0.85) (0.96) (-0.31) (2.50) (2.42)
45 to 54 years -1.078∗∗∗ -1.110∗∗∗ -1.035∗∗∗ -1.455∗∗∗ 0.208 -0.377
(-4.67) (-5.11) (-4.92) (-6.37) (0.71) (-1.54)
above 55 years -2.126∗∗∗ -2.390∗∗∗ -2.289∗∗∗ -2.824∗∗∗ -2.247∗∗∗ -2.039∗∗∗
(-12.12) (-14.54) (-14.49) (-16.18) (-9.90) (-10.78)
High Educated -0.691∗∗∗ -0.744∗∗∗ -0.767∗∗∗ -0.785∗∗∗ -0.763∗∗∗ -0.862∗∗∗
(-4.18) (-4.81) (-5.17) (-4.76) (-3.54) (-4.82)
Middle Income -0.344 -0.0418 -0.00674 0.0603 0.105 0.0446
(-1.82) (-0.23) (-0.04) (0.32) (0.43) (0.22)
High Income -0.228 -0.249 -0.205 -0.137 0.222 0.0927
(-1.09) (-1.28) (-1.10) (-0.66) (0.82) (0.41)
Constant 2.268∗∗∗ 2.613∗∗∗ 2.707∗∗∗ 2.213∗∗∗ -3.461∗∗∗ 0.452∗
(11.42) (13.94) (14.92) (11.22) (-13.86) (2.11)
N 16245 16112 16115 16243 16822 16301
r2 0.0172 0.0232 0.0231 0.0262 0.0122 0.0144
t statistics in parentheses
∗ ∗∗ ∗∗∗
p < 0.05, p < 0.01, p < 0.001
Notes: This table presents Huber robust and survey weighted regressions of the aggregation error on several
demographic characteristics.
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