Mapping Heterogeneity in Consumer Inflation Expectations: Does the Tone of News Affect Consumer...

7/30/2019 Mapping Heterogeneity in Consumer Inflation Expectations: Does the Tone of News Affect Consumer Forecasts?

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Table of Contents

I. Introductiona.Pre-80s Version Of Expectations --------5

1. Pre-Lucas Model------------------------------------------5This subsection explains the overarching themes of old inflation models.

It explains the shortcomings of homogeneous and homoscedasticinflation-expectations models, and why theoretical flaws in these modelslead to flaws in their predictive capabilities.

2. Creation Of Heterogeneous Modelsa. Lucas Model--------------------------------------------------6Producers in an economy are geographically separated, and each

produces one good. Different information causes different inflationexpectations. Accordingly, forecast errors are differentiated and

unanticipated. Heterogeneity in this model is constant, and unrelated to

any fundamental underlying variables.b. Information-Induced Heteroscedasticity-----------7Shows that price expectations differ across individuals because they

acquire different information about inflation. If information is a normal

good, then the amount acquired will vary across individuals according toincome, education, etc. Different information, therefore, is statisticallyindifferentiabe to the different formation processes.

b.Mankiws Rational Inattention Model----81. Problems with Other Models------------------------8

Following the lead of others, Mankiw creates a model in which sporadic

information updates are responsible for persistent forecast errors. Asidefrom various macroeconomic benefits, this model incorporates the

important concept that the rate of updating is crucial in determining theaccuracy of a forecast. Mankiw differentiates his works from others bydifferentiating consumer and professional forecasts.

II. Laying the Foundationsa. Carrols Epidemiological Model---------10

1. The Model--------------------------------------------------10The model for Carrols theory is based off of the perception that

economic agents base their information off of future expectationscontinuously. He uses an epidemiological model, in which consumers

obtain their source of information from one location, at constant update.

2. The Implications------------------------------------------12Just as with Mankiws model, higher updating frequencies indicate morerational forecasts. Carrolls original model will be the base model whichthis report builds upon. Empirical evidence does, in fact, give statistical

proof for Carrolls model. This section will provide an in depth analysis of

the varying consequences this model assumes.

b.Irrational Contributions------------------131. Lamla and Lein--------------------------------------------13

Lamla and Lein analyze the effect media reports have on inflation

expectations in Europe. Interestingly, they show that the medium of news


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reports has differing effects on expectations; whereas television reports

tend to induce biases, print articles tend to reduce consumer biases.Their paper will serve as a critical backdrop for the theories proposed

within this report.

2. Voters, Information Heterogeneity, andExpectations----------------------------------------------13

Analyzes the extensive amount of information heterogeneity in thegeneral public in regards to economic perceptions. It uses level ofeducation as a means of measuring information differences; in other

words, those with less than a high school diploma have less access to(and are less able to extract) information than college graduates.

3. European Findings---------------------------------------15Other findings within Europe suggest an extreme correlation betweennews biases and forecasts biases for consumer inflation expectations;

this subsection will provide a literature review of these articles.

III. Dataa.Inflation Data---------------------------------15

This subsection will provide a brief analysis of the inflation data beingused for the thesis. It will include various points on the quality of the

data, and will also provide a brief analysis on the trends and tendencies

for each time series variable.

b.News Data---------------------------------------17This subsection will elaborate on the news sources being used within the

thesis; these news sources are the foundation of the tetlock statistictesting the threshold models. This subsection will also provide a rationalefor why certain sources are used for certain subpopulations; this rationale

will be defended with empirical studies that demonstrate the authors

choices.

IV. Methodologya. Testing Carrols Equation----------------21

After explaining the data being used for the experiment, an initial testing

of Carrols equation for specific subpopulations will be done to determine

whether the model holds well for all populations in the United States. Themodel will be shown to hold for most subpopulations (with the exceptionof the high school subpopulation).

b.Foundations for A Threshold-------------221. Subjective vs. Objective-------------------------------22

Explains the distinctions between analyzing news data more

quantitatively or qualitatively (i.e. with or without computer programs);then explains the objective method, and provides a justification for using

such a method with the Carrol model.2. Benefits of a Threshold-------------------------------24

The paper will then explain what information can be obtained with a

statistically significant threshold model, and how such information

supersedes that of Carrols model. More specifically, a statisticallysignificant threshold model will provide a more dynamic insight as to what

the updating tendencies of the population are.

3. What to Expect-----------------------------------------24


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Ultimately, this subsection will hypothesize that all subpopulations will

experience some degree of heterogeneity between the updatingprocesses, and give a brief rationale for this hypothesis. These

expectations will later be shown to be false.

V. Resultsa.Differences in Formation Process------25Taking the initial assumption, that the threshold model should work well

for all consumer data, my findings have shown that the gains of athreshold model are restricted towards the high school subpopulation.

The college subpopulations formation process will be shown to bepredicted more efficiently without a threshold; the some collegesubpopulations results will be less conclusive and more vague.

b.Explaining the Raw Data1. Conclusions----------------------------------------------28

Aside from the individual conclusions from each subpopulation (i.e. the

updating tendencies and traits), this section will establish an overallconclusion as to what the models trends exhibit. Ultimately, this section

will strive to prove that the updating tendencies are not the onlydifferences between models; rather subpopulations exhibit differentqualities of updating as well.

2. Success or Failure?------------------------------------32Whereas the initial hypothesis does not fall in line with the ultimate

conclusion, it is still hard to conclude that testing Carrols equations on

various subpopulations was ineffective. Rather, different subpopulationswere found to exhibit different formation processes, each according to the

equation which best fits its inflation expectations.

c. Future Suggestions-------------------------34This section will briefly suggest future areas of research which may helpaugment current conclusions and/or explore related topics about

consumer inflation expectations.

VI. Appendixa.Part One: Data Tables-----------------------37b.Part Two: C++ Threshold Estimation----43

VII.Bibliography--------------------------------59


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I. Introduction

Throughout the twentieth century, progress in economic theory was attached to

the notion that rational consumers decisions were made with complete information at

hand, that was updated constantly (Hansen and Sargent, 1980)1. Albeit unrealistic, these

assumptions provided a degree of efficiency and simplicity; as a foundation for modern

macroeconomics, however, these assumptions have proven to be quite flawed when

tested empirically. Consumers have been shown to forecast inflation with significant,

persistent biases over the years (Mankiw, Reis &Wolfers, 2003). Furthermore, these

biases might persist when taking into account unpredictable future shocks; in other

words, there is a strong possibility that random future shocks do not account for such

biases. Whereas certain models predictions show degrees of promise, consumer

expectations have proven to be fatally flawed components within past economic models.

Prior notions of the publics inflation expectation formation process assumed that

the public was a homogenous structure, which either formed their expectations based

upon rational, future forecasts, or formed their expectations according to prior rates of

inflation. In recent decades, though, notions of heteroscedastic and heterogeneous

expectations have broken through the barrier of simple expectations models. Multiple

levels of biases and differential expectations have been shown to exist between various

subgroups of the overall population. First, professional forecasters often predict future

1Similar papers posit that consumers and producers have homogenous information, and

therefore, homogenous expectations. For examples, see: Frydman and Phelps

(1983); Pesaran (1987); Radner (1982).


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rates of inflation more accurately than the general public; in addition, the variability of

such forecasts tend to be smaller as well. For example, economists interquartile range of

expectations in 2003 varied from 1.5% to 2.5%, while the publics inter-quartile range of

expected rates of inflation varied from 0% to 5% (Mankiw, Reis & Wolfers, 2003).

Further evidence from the University of Michigans survey of consumer expectations

documents even higher rates of variability, as 5% of the population on average expected

deflation and 10% expected inflation rates of over 10% (Mankiw, Reis & Wolfers, 2003);

rates within the consumer population also differ based upon certain demographic groups

(Krause, 1997). Given this information, recent papers have tried to supplement classical

models to include more dynamic, realistic notions of inflation expectations.

One of the first models to recognize a lack of homogeneity in consumer

expectations was Robert Lucas rational expectations model (1972). The Lucas Model

essentially hypothesized that differential expectations resulted from different information

being purveyed to each consumer; with an initial assumption that producers are

geographically separated from one another, Lucas contended that decisions of production

and price-setting differed as each producer was limited in the information he or she could

receive. Different information, therefore, would lead to different inflation expectations. In

accordance with this assumption, rising prices, be it because of increased demand or

increased costs of production, caused the producer to provide a greater quantity of goods

supplied. Across the board price increases, though, would mislead the producer into

providing a greater quantity of goods, as he or she mistakes general inflation for

increased demand in his or her products. Albeit lacking an explanation for the


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heteroscedastic tendencies of expectations, the Lucas Model provides certain

fundamental, theoretical contributions towards consumer expectations. The notion that

differential information was a key cause of differential expectations was pioneered by

Lucas, and paved the way for future research. Whereas incomplete and imperfect

information within the Lucas Model was due to geographical constraints, differential

expectations for consumers will be shown as an effect of different news sources for

different consumer subpopulations within this paper.

Later papers use Lucas model as a backdrop for producer expectations which

contain degrees of heteroscedasticity. Cukierman and Wachtel (1979) expand upon the

Lucas framework by creating a market with several goods, each with varying equilibria

and different price information. Again, aggregate shocks in inflation affect each producer

differently, and create varying levels of price discrepancies with each product.2

Whereas

each producer is capable of forecasting price changes within his own market, he is

nonetheless dependent upon external sources for information on the aggregate price level.

Thus, differential expectations are caused by varying degrees of demand shocks towards

each individual market. The authors find that data from the Livingston Survey does in

fact support these hypotheses, as periods with large variances in the rate of inflation (and

change in nominal income) tend to be associated with periods of large variance in

inflation expectations within the survey.

2

Interestingly, Cukierman and Wachtel appear to be the first to explicitly state the

possibility of a similar relationship holding between consumers and news sources


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Another study, Information Induced Heteroscedasticity (Fishe and Idson, 1990),

hypothesizes that price expectations differ across individuals because they acquire

different information about inflation. Assuming that price information is a normal good,

the amount of price information acquired will vary across individuals according to

income, education, and other demand-specific variables, causing price expectations to be

heteroscedastic with respect to these variables. The authors note, however, that their work

assumes that the forecasting algorithm of each respondent is the same; by definition,

this assumption forces them to work within a framework in which varying demographics

do not update their information in different ways. That being said, the authors predict

education to be the driving cause of heterogeneous and heteroscedastic expectations, with

variables like income3, age, and sex to have some influence as well. Ultimately, their tests

show that more education/income is usually associated with lower variance of

expectations within an individual population, thereby proving that the demand for

information is a key factor in determining the dynamics of consumer expectations. As

stated before, though, the assumptions that forecasting algorithms and sources of

information do not differ across demographic groups are a crucial difference between

Information Induced Heteroscedasticity and this paper. This assumption may hold to be

untrue, and may therefore give way to biases within the results; as stated within the

article, different models may be necessary to forecast consumer expectations adequately.

3

Income (and, to a lesser degree, the other variables) may theoretically dictate theacquisition of price information through opportunity costs, and can therefore manipulate

the data.


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A later model, provided by Gregory Mankiw and Ricardo Reis (2002), explains

the heterogeneous tendencies of consumer inflation forecasts by applying Keynes

sticky-price theorem to inflationary expectations. Mankiw and Reis argue that

heterogeneous expectations arise simply because agents do not frequently update their

own inflation forecasts. These rationally inattentive agents update forecasts only

around once per year (12.5 months, according to Mankiw and Reis), and base their

forecasts on outdated information. The Mankiw-Reis Model contends that heterogeneity

results from different time updates of inflationary expectations, which is explained by the

following model of current inflation rate, a composite of a dynamic, inconsistent updating

process:

t = [/(1-)]yt + (1-)j Et-1-j(t + gt).

Aside from the unrelated theoretical benefits that this model develops, this model

incorporates the concept of a population updating their information slowly over time.

Differential rates are thus explained by different inflationary forecasts within different

periods. Accordingly, The evolution of the state of the economy over time will

endogenously determine the extent of this disagreement (Mankiw, Reis, and Wolfers

230). In other words, different economic trends occurring during a certain time directly

influence those updating inflationary expectations, and thereby produce differential rates

of inflationary expectations. The degree of rational inattentiveness can thereby explain

the difference in expected rates of inflation between professional forecasts and the

general public. This model explains why differences between forecasts rise when the rate


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of inflation changes sharply, as most rationally inattentive agents fail to account for

recent updates in economic trends. Mankiws main critique on economic thought is that

economic theory posits that everyone shares the same information and that all are

endowed with the same information processing technology. The very concept of a

public whose information is different just by the fact that their updating frequencies are

different is revolutionary towards the rational expectations theory. Mankiw himself,

though, acknowledges that his own model fails to incorporate the heterogeneity of

information contained in the publics forecasts.

II. Laying the Foundations

Building upon Mankiws work, Christopher Carroll, in deriving an equation for

consumer inflation forecasts, writes that e = f+ et, whereas fis an underlying

fundamental inflation rate and et is an error term which reflects unforecastable

transitory inflation shocks (Macroeconomic Expectations, 274). Accordingly, agents

believe that the economy is always experiencing a fundamental, base rate of inflation

which follows a random walk; this fundamental rate is then manipulated by

unforecastable events. Because all agents are able to forecast the fundamental rate

themselves, each agent is dependent upon experts who measure the effects of these

shock events. Carroll transformed the understanding of this process by trying to explain

the source of agents information; given a single source of news (to update forecasts) and

a constant period of update, macroeconomic dynamics represent a weighted average of

all prior professional expectations. More succinctly:


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Et = Pt + (1-){Pt-1 + (1-){Pt-2 + (1-){}}}

= Pt + (1-)Et-1

A higher fraction of the population updating, , will essentially determine the gap

between the current rational forecast for future inflation and the mean expectation of the

public. Given the parameter, , both regression coefficients must be statistically

indifferentiable from a value of one when tested. When testing this equation, in various

forms, Carroll finds these models to hold theoretical weight, and finds an updating

frequency similar to Mankiws. Inspired by epidemiological models which measure the

spread of a disease in situations of common source infections, Carroll assumes a

homogenous public with a common source of information to uniformly update

expectations at a constant rate. The caveat, however, is that agents utilize news stories in

order to update their information; Carroll envisions a single news source which simply

reports current professional (and therefore rational) forecasts. Given this equation, as well

as the theory behind it, much research has been done as to how news sources affect

consumers expectations. Whereas Carroll posits that consumers, as per the rational

expectations theory, can only be influenced by the volume of stories about inflation, other

economists suggest that biases contained within news forecasts can affect consumers

projects and perceptions.

Carroll explains higher rates of accuracy (of forecasts) as a result of more

intensive news coverage, as more coverage on inflationary events increases an agents


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knowledge of his exposure to the shock events that determine et. Carroll uses the New

York Times and The Washington Post to prove that the intensity of news coverage does

in fact increase the accuracy of the publics forecasts. Simply put, if the SPF forecasts are

characterized as expert forecasts of future inflation rates, higher intensity of news

coverage increases agents exposure towards these forecasts. Carroll finds the negative

correlation to be statistically significant at the 5% level. Furthermore, the rate of

absorption during periods of intense news coverage is significantly higher than periods

with low media coverage on inflation, as .7 during periods of intense coverage (as

opposed to

.2 during periods in which new stories on inflation is less than the average

rate of stories). Carroll fails, however, to significantly test this relationship beyond that

point. Accordingly, volume of news could be spuriously correlated with the accuracy of

forecasts as it is also highly correlated with other significant variables (i.e. the rate of

inflation). In other words, high volumes of media coverage may be a pseudo-causal effect

on the accuracy of forecasts as bouts of high inflation influence media coverage of

inflation4. Furthermore, given this relationship, periods of high accuracy may just reflect

the opportunity cost present in failing to forecast future rates of inflation. Given that

Carrolls rate of future inflation is partially dependent upon an initial fundamental rate,

prior rates are likely to influence future rates. That being said, periods of rising inflation

reflect higher costs upon consumers, and a likelihood that these costs will continue to be

high; a failure to forecast future rates, therefore, is riskier during periods of higher

inflation, so periods with higher inflation may result in more accurate forecasts with

current information.

4

Carrolls model may also not hold because of the lack of data, as he only uses the New

York Times and the Washington Post.


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Since Carrolls work on inflation expectations, an extensive amount of work on

the expectation formation process has been done5. Drager (2011) utilizes SVEC models

to describe the relationship between inflation expectations and current perceptions of the

inflation rate. Linking perceptions to expectations, however, could prove to be ineffective

theoretically, as perceptions would in essence be tied to some prior updating date;

implicit within the model are heterogeneous updating dates which cannot be modeled.

Nonetheless, Drager does find that news media affects current inflation perceptions; this

can either be explained by citing an increase in Carrolls

, or rather through a direct

distortion on professional forecasts. Others utilize Mankiws and Carrolls work to create

a sticky information model which is based upon adaptive consumer expectations.

Lanne et al. (2009), for example, utilize Carrolls basic epidemiological model and

hypothesize that current expectations are formed as a weighted average of the prior

periods expectations and the inflation rate of the prior period; again, the heterogeneity

within the model is contained within , but the prior inflation rate is perceived to be a

more rational forecast of the future rate than professional forecasts. Given the strong

relationship between the volume of news coverage on inflation and the opportunity cost

in foregoing a forecast on future rates of inflation, the actual tone of these news forecasts

may be a better predictor of the accuracy of forecasts than the volume of news coverage.

Lamla and Leins The Role of Media for Consumers Inflation Expectation Formation

centers on European (in particular German) inflationary expectations from 1998 to 2006.

Lamla and Lein incorporate bias variables into regressions which map the gap between

5

Most of the work on the formation process has been done in Europe, though. See, for

example, Lamla and Sarferaz (2012), Pfajfar and Santoro (2011), and Drager (2011).


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professional and consumer forecasts. Ultimately, they find that biases can be transfered to

consumers through news sources. Using television as a news source is shown to decrease

the accuracy of an agents forecast, whereas newspapers are shown to have the opposite

effect. Lamla and Lein have thereby proven a negative correlation between certain

variables (i.e. tone) and the accuracy of inflationary expectations (refer to the last page

for graphs). Lamla and Lein ultimately conclude:

Our results suggest that an increase in neutrally toned media reports by one standard

deviation reduces the gap by about 8 basis points. An increase in the unfavorably toned

news by one standard deviation increases the gap by about 11 basis points. This

accounts for 15 and 20 percent of the average expectations gap, respectively. (35)

Still, though, Lamla and Lein make sure to measure the volume of biased news,

rather than the bias of news itself. Doepke (2006) finds that professional forecasters

granger-cause household expectations in Europe (but not vice-versa), a finding that

shows similar results towards similar studies in the US. In contrast to the average US

household (which updates roughly once per year, according to Carroll and Mankiw),

Doepke finds European consumers to update their expectations roughly once every 18

months. Similar to all other papers, Doepke finds that adding a constant to the equation

adds statistical significance to the results. Whereas Doepke concludes that its is doubtful

a priori that one can have a reasonable structural specification of inflation expectations

with a non-zero constant term given that this would show that households predictions

are permanently biased away from experts, this paper will try to show that the inherent


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biases induced by news stories (the ultimate source of a consumers expectations)

permanently affect consumers forecasts at a statistically significant level. Drager (2011)

analyzes the relationships between news article volume and inflation expectations/

perceptions in Sweden. In contrast with prior articles, this discusses the dynamics

between consumer perceptions of the inflation rate under stable inflationary regimes. The

authors conclude that during these regimes, lagged values of inflation have stronger

affects on perceptions than consumer expectations of future rates of inflation. Drager also

provides statistical evidence which shows that expectations are not formed through actual

data, but rather through perceived values of inflation; he essentially links actual inflation

data (lagged values) with perceptions, and then links perceptions with expectations.

III. Data

In measuring inflation expectations, this paper will be using the University of

Michigans Survey of Consumer Expectations; professional forecasts will be measured

through the Survey of Professional Forecasters. Each of these variables will be analyzed

from 1980 Q1 2012 Q4. According to models which will later be elaborated upon, each

of these variables will be first-differenced without changing the original model being

tested. First-differencing the data serves multiple benefits: as the table below shows, the

original data is heavily skewed and therefore biased; the data also exhibits non-stationary


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tendencies, and random walks6. First differencing the data solves both of these problems,

and therefore helps model more accurate results.

Differenced Data

1978 Q1 2012 Q4 High School Some College College

Differenced Mean -.00416 -.00391 -.00636

Differenced Median 0.000 0.000 0.000

Differenced Skew .24918 -.35620 -.73343

Original Data

1978 Q1 2012 Q4 High School Some College College

Original Mean 5.1110 4.5027 4.1537

Original Median 4.6 4.0 3.5

Original Skew 1.888 2.005 2.069

Original Std. Dev. 6.3802 5.1678 4.3246

The skew of the data drops significantly once the data is differenced, as is seen above.

Also worth noting is the tendencies of the data analyzed together; both the mean and the

standard deviation of the data drops as the education-level of the subpopulation increases.

The heteroscedasticity and heterogeneity within the general public is clearly seen above.

P-Values for Mean Comparison Tests

Comparison Standard Deviation Mean

HS vs. SC 1.95E-16 1.68807E-05

SC vs. C 2.22459E-12 0.017914283

HS vs. C 1.20643E-45 6.15E-11

The table above tests the null hypothesis that the mean inflation forecast and standard

deviation of this expectation do not differ for each subpopulation, and rejects this

hypothesis at the 5% significance level for all tests; whereas most tests conclude that each

6

Random walks within the data are theoretically consistent with prior reports on inflation

expectations, including Carrolls paper.


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of the subpopulations statistics differ at the 1% significance level, it is clear that the

subpopulation do in fact contribute to the heteroscedasticity and homogeneity of the

overall population.

As stated before, the original data also exhibits significant non-stationary tendencies7.

Whereas the PACF and ACF of the original data for all subpopulations exhibit strong,

persistent autocorrelations, the differenced data does not; the original data all fail to reject

the null hypothesis of no unit roots at the 1% significance level, as seen below, whereas

the differenced data does not. It is clear, therefore, that differenced data is needed to

compute statistically significant and trustworthy models.

Data ADF Statistic P-Value Order of Integration

High School -3.5213 .04308 I(1)

Differenced HS -5.861


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College Sub-Population will be using Newsweek, the Washington Post, the New York

Times, Businessweek, the Wall Street Journal, and the Economist as its news sources;

ABC, NBC, Fox News, CNN, Bloomberg, and CNBC will be utilized as the High School

Sub-Populations news sources. Each of these sub-populations models utilize data from

the news sources specified above from 1980 Q1 2012 Q48. It is also worth noting that,

given the nature of this analysis (and the datasets from which these news sources are

obtained), tone is also a better variable than volume because of the tendency of the data

throughout the period of analysis. Whereas volume increases secularly from 1980

onwards, the measurement of tone does not differ because of the dataset itself; in other

words, more sources are available in recent times, so the data for volume of news is

skewed towards this period by the very nature of the dataset9.

The matching of certain news sources with certain subpopulations is also based

upon prior empirical studies of different populations news preferences as well as prior

economic theories. According to Pews 2008 Biennial News Consumption Survey, for

example, the high-school subpopulation has a heavy reliance on TV News. In line with

the trend (that lower educational groups rely upon more biased sources) the College

subpopulation is shown within Pews research to rely heavily upon nuanced, technical

news sources across a broad array of areas; across the Net-Newsers and Integrators

audience segments, this subpopulation is highly educated, and relies upon quick online

news sources and other traditional sources. Newswires was chosen as their choice of

8

This paper uses multiple datasets, more so than Carrolls, in order to correct forpotential biases9 The news datasets used will be High Beam Research, Dow Jones, and Lexis Nexis


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news within this paper given that newswires demonstrate heavy coverage of national

news with low biases. Finally, the some-College population is also grouped in with

multiple groups that have a strong tendency to either read print newspapers or online

newspapers. This paper, though, will not analyze trends in online news-sources, given

that the prominence of such a source has arisen only within the past decade, so they do

not have a sufficient amount of data to hold statistical significance within this paper.

In terms of analyzing these news sources, this paper will diverge from similar

papers method in analyzing each individual article as a whole by analyzing the degree of

tone within each article. For example, Lamla and Lein (2007) classify individual articles

as positive, negative, or other labels, and then aggregate these classifications to create the

variables which analyze tone; the degree of tone in these papers, therefore, really measure

the amount of negative stories within a period as compared to the total amount of stories.

This paper, in contrast, will analyze the word count within each article by measuring the

ratio between biased words10 and total words within each article. This approach

establishes certain fluidity in the threshold news variable that the other approach fails to

do. Theoretically, it contends that a certain degree of bias within articles in a period

determines the degree of the updating frequency in a population, rather than the amount

of biased news stories within a period. Accordingly, periods which fail to reach this

degree of bias will fail to have an effect on the expectation formation process; low biases

that are constant throughout a period will not affect expectations if this bias is lower than

a predetermined threshold-bias. When analyzing various potential variables, almost none

10 As measured by the Loughran and McDonald Financial Sentiment Dictionaries


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of these variables exhibit any degree of skew or non-stationarity. As is seen in the table

below, only the percent of uncertain words variable exhibits a clear skew. All other

variables exhibit a skew of less than one; most skews near zero.

Skew for % Tone Variable

Negative Positive Uncertain

College

0.978899793 0.137902163 1.757231046

Some College

0.274550489 0.073734868 0.633341697

High School

0.826319483 -0.21077117 0.687371218

One of the biggest problems of measuring whether volume affects the change in

updating frequencies for a population is that volume could be a pseudo-cause of this

change; rather, the real cause is more likely to be the height of inflation (or inflation

expectations itself); in other words, volume is heavily correlated with another variable,

the height of inflation. When looking at the correlation between the height of inflation

expectations and the tone of news, it is apparent that this correlation is virtually

nonexistent.

Correlation Between Height of Expectations and News Tone

News Statistic High School Some College College

% Negative 0.208753085 0.30371630 0.0087968

% Uncertain 0.091704976 -0.11931011 -0.2228387

Tetlock Negative 0.059948296 -0.09680543 -0.0573599

Tetlock Uncertain 0.061896903 -0.20140965 -0.0150751


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IV. Methodology

Given the heterogeneity of expectations, it is reasonable to contend that

(according to Carrolls theory) each significantly different subpopulation will form its

expectations with different update periods and different news sources11

. Carroll fails to

account for Mankiws first critique of inflation expectation theory; Carroll implicitly

assumes within all of his models that, given that consumers are perfectly rational, they all

consume the same information12

. In contrast, when allowing for differing news forecasts,

the mean population forecast is a larger function of individual subpopulations. For

example, suppose we were to divide the United States into three subpopulations:

College: EtC = Pt + (1-)E

Ct-1

Some College: EtSC

= Pt + (1-)ESC

t-1

High School: EtHS= Pt + (1-)E

HSt-1

As a whole, these three subpopulations can then be combined, yielding the equation:

Mt = S%( Pt + (1-)EC

t-1) + S%(Pt + (1-)ESC

t-1) + S%(Pt + (1-)EHS

t-1)

Whereas S% represents the appropriate population size; rearranging the terms yields:

11

See explanation within Data Description section12

More particularly, Carroll holds that, given the incorruptibility of consumer forecasts,

all information is the same for a consumer (i.e. non-biased).


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Mt = S%HS(Pt Ct-1HS

) + S%SC(Pt Ct-1SC

) + S%C(Pt Ct-1C) + Mt-1

Mt = S%HS(Pt Ct-1HS

) + S%SC(Pt Ct-1SC

) + S%C(Pt Ct-1C)13

Now suppose a model in which the updating frequency is a bit more dynamic; rather than

remaining constant throughout the model, changing updating frequencies which are

dependent upon a third variable are present. Given that other authors have found a

relationship with news tone and forecasts accuracy, this paper suggests this third variable

be one that measures the tone in news forecasts. Here, just as in the Lucas Model,

consumers expectations are differentiated due to segmentation in information. Because

each subpopulation consumes different news sources, their formulations are calculated

with different results and data; unlike prior authors, however, I will later show that the

difference in expectations may be due to a difference in the formation process itself,

rather than just the information which consumers have14

. Whereas Lamla and Lein have

done so through a qualitative, article-by-article analysis, I will attempt to do so using a

more quantitative, repetitive method established by Paul Tetlock15

. Each of the three

subpopulations, accordingly, will use different news sources that represent popular

sources amongst the sub-group. For example, whereas agents with a College degree tend

to read newswire sources, agents with (at the most) a High School degree tend to obtain

their information from television and magazine sources. The reasoning behind this is as

follows: first, prior empirical evidence suggests that biased news sources are inferior

13

This equation (i.e. one with a first difference) may prove to be more effective if the

variables exhibit a random walk.14

Cukierman and Wachtel, as well as most prior authors, assume that the forecasting

algorithm is the same for all consumer subpopulations.15

Tetlock wrote various papers on how tone within certain news sources affect financial

returns for stocks, and whether this gain is demonstrably profitable.


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goods16

; second, the theory behind this paper suggest that stories with higher rates of

biases tend to make forecasts less accurate. Theory therefore suggests that these statistics

should increase as the quality of news forecasts decrease. The statistic, which measures

tone, therefore, will look like:

Tett = ((Nt/Tt) yearly-ratio)/yearly-ratio

Accordingly, tone will be measured through various dictionaries which contain positive,

negative, and uncertain words. These dictionaries have been tested with financial

research before, and help provide objective insight on the trend in news tone for various

subpopulations17

.

Yet, although different tones affect the accuracy of forecasts, certain degrees of

tone will have a negligible affect on the accuracy of forecasts, so that a certain threshold

will affect consumers forecast accuracy, and therefore their updating frequency. So

Carrolls model is thereby augmented with an extra variable which takes media tone into

account:

Mt= b0 + 1tPt+ (1 - 1t)Mt-1 if Tett>

= b0 + 2tPt+ (1 - 2t)Mt-1 if Tett

Where: = bias-threshold

1 < 2

16

See Chyi and Yang (2009).17 Research on the tone of words can be seen in Loughran and McDonald (2011)


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or: Mt= a0 + (a1 + b1It)Pt+ (1-(a1 + b1It))Mt-1

Where: It= 0 if Tett>

It = 1 if Tett

Rearranging the terms in this threshold model, to attain a first-difference model similar to

that within the original first-difference equation, yields:

Mt = a0 + (a1 + Itb1)(Pt Mt-1)

Each subpopulation is expected to have different threshold propensities, and therefore

have bigger differences in updating frequencies; this threshold model can thereby

explain, to a greater degree, the heterogeneity within the total population, along with

many other interesting statistics. For example, the degree of change (of the updating

frequency) may measure the sensitivity to respective news sources; higher changes in

may signify a larger sensitivity to the degree and type of tone exhibited in news forecasts.

The rate of change in may also signify higher rates of variance among a specific

subpopulation; this may help explain the heteroscedasticity present in the total

population. This threshold model may thereby help explain tones effect on the accuracy

of expectations more effectively.

Given the fact that different news sources will affect different consumer

subpopulations, and that these different news sources are expected to incorporate

different tones and biases into consumer forecasts, each of the subpopulation tested


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should have varying degrees of biases. These biases can be measured multiple ways;

generally speaking, the values of the intercepts and regression coefficients will ultimately

determine the level of biases within forecasts. First, higher slope coefficients and lower

intercept coefficients should be indicative of higher levels of accuracy, as higher slope

coefficients indicate a higher updating frequency (and therefore forecasts with more

accurate information), and lower intercepts signify lower constant biases. Accordingly,

subpopulations with higher education levels should have higher slope coefficients

coupled with lower intercept values. In addition, the change in slope coefficients within

the threshold model should show the degree of influence media biases have on the

subpopulation. Therefore, a larger shift in updating frequencies should occur with more

biased news sources, like television news. This would infer that changes in updating

processes are more dramatic with the High School subpopulation than with the other

subpopulations. Finally, basic statistical measures may also test each individual model

itself. The R2, RMSE, and MFE of each model measure each individual model separately;

testing whether the sum of the slope coefficients equal one would also provide theoretical

consistency with Carrolls paper. These findings should suggest consistency with the

theoretical framework provided above.

V. Results

Whereas all of the models hold all appropriate conditions of statistical

significance, the degree of significance varies with the degree of education, as expected.

Each models adjusted-R2 holds values above .700, and all values are individually


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statistically significant for (at least) the 10% level. Interestingly enough, the value of the

intercept (along with its significance) tends to fall as the subpopulations level of

education rises; in addition, tends to rise as the level of education rises, indicating

higher updating frequencies for highly educated individuals. These two factors,

combined, should help explain why more-highly educated subpopulations tend to exhibit

forecasts with fewer biases. In fact, High School expectations exhibit an average bias of

~.51; in other words, High School expectations are roughly 20% higher than overall

professional forecasts even if the updating process was completely continuous! Whereas

Carroll and Mankiw struggle to explain similar findings in accordance with rational

expectations theory, this paper will later explain how negative and uncertain tones in

media forecasts are to explain for the statistically significant intercept values.

MC = .328 + .391PT + .599MT-1

(.138) (.059) (.058)

MSC = .479 + .219PT + .721MT-1

(.165) (.064) (.064)

MHS = .509 + .119PT + .809MT-1

(.185) (.063) (.065)

Although each of the original variables exhibited strong unit root tendencies, the nature

of the model self-corrects for this fact. Since the weighted average of current updating

agents and the prior mean of forecasts is really a first-difference model in disguise, all

models exhibit no unit root as the first differenced values exhibit no unit root as well.


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Model ADF Statistic P-Value Order of Integration

College -3.7182 .02523 I(0)

Some College -5.2057 .01000 I(0)

High School -5.6455 .01000 I(0)

Also, each of the models show no evidence of serial correlation; no p-value for the DW

Test exhibit values under .50.

Model DW Statistic P-Value Serial CorrelationCollege 2.207 0.8578 No

Some College 1.9503 0.3294 No

High School 2.2388 0.8947 No

The RMSFE for each of the models is also generally lower as the level of education of

said sub-population increases. As stated before, the weighted terms should also add up to

one to hold onto Carrolls theoretical foundations. All of the models achieve this criterion

with at least a 10% statistical significance level; that being said, there is not a high degree

of statistical-confidence that the high-school subpopulation passes such a test.

When testing the threshold model, varying results confirmed prior assumptions,

and some brought about surprising conclusions18. First, when comparing different news-

statistics within each subpopulation, there appears to be findings similar to those of

Lamla and Lein (2008); namely, that positive news affects the accuracy of expectations

18

Given the large nature of the models calculated (i.e. thousands of models), only a few

will be discussed to illustrate the larger trend of the results


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positively, and negative/uncertain news affects the accuracy of expectations negatively.

For example, when comparing the College 1980 sub-populations threshold model with

the %-Positive and %-Uncertain news statistics, the results are as follows:

MC = .1479 + .25772PT + .74MC-1 if TPOS .71%

MC = .1479 + .67186PT + .32MC-1 if TPOS > .71%

The threshold-model for positive news illustrates a sharp climb in the updating frequency

of the College subpopulation when the news for the subpopulation exceeds a certain

positive tone. In contrast, when the subpopulations consumption of uncertain news

MC = .18436 + .59099PT + .40MC-1 if TUNCER .61%

MC = .18436 + .24764PT + .75MC-1 if TUNCER> .61%

exceeds the threshold variable, the updating frequency of the subpopulation drops

dramatically. This would theoretically illustrate that the updating frequency is dependent

upon the type of bias being portrayed within the news; uncertain news should decrease

the accuracy of forecasts, given that higher degrees of uncertainty should convey less

reliable news forecasts to consumers. Within the updating-frequency paradigm,

consumers could theoretically become more reluctant to use news sources as a means of

updating their expectations, and thereby lose valuable information during the period. In

turn, positive news would influence consumers within the subpopulation by enticing them

to frequent news sources more often, and thereby increase the accuracy of their forecasts.


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Whereas intra-population comparisons illustrate the varying effects of different

biases within each subpopulation, inter-population comparisons exhibit much larger

differences between the different threshold models. Namely, whereas some threshold-

models explain the dynamics of consumers inflation expectations better than Carrolls

models, others fail to do so. The schism in these findings ultimately lie within the

examination of the fit in each of these models; more explicitly, threshold models better-fit

the data for the High School and Some College subpopulations whereas Carrolls model

has a better fit for the College subpopulation. Whereas almost all of the models exhibit

the theoretical consistency required for the model (i.e. the frequencies drop as they

should), the College subpopulation does not explain the data as well as beforehand.

MC = .22788 +.69686IN83(Pt MC-1) + .30347(1-IN83)( Pt MC-1)

MSC = .30075 +.41998IU83(Pt MSC-1) + .19032(1-IU83)( Pt MSC-1)

MHS = .52717 +.33397IU80(Pt MHS-1) + .11525(1-IU80)( Pt MHS-1)

Notice the same trends as beforehand: across the models, the intercept values increase as

the level of education amongst the population decreases, and the updating frequency

varies inversely with the intercept. Accordingly, the updating frequency varies in such a

way in both states of the threshold model; in other words, higher-education

subpopulations updating frequencies are larger than other subpopulations updating

mechanism during similar states within the threshold model. More interestingly, each

subpopulation exhibits varying degrees of shifts within the updating frequency; the

change within the updating frequency between states for the high-school subpopulation is

much larger than other subpopulations. Just as with other findings, this phenomenon also

correlates with the degree of education that a subpopulation has attained. In addition, the


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larger swings in updating could explain the variation between populations expectations;

the heteroscedasticity of consumer expectations follows a similar path to these threshold

models in that larger swings in updating frequencies occur within populations with larger

degrees of variance in expectations. The data and theoretical underpinnings of these

models thereby match up quite smoothly.

Given that Carrolls model does not hold for the High-School subpopulation, and

that the threshold model does, it is of no use in comparing the two models with one

another19

. However, when comparing the other two subpopulations, where both models

(Threshold and Carroll) have theoretical credence, there is a clear distinction between

forecast efficiency.

Subpopulation Model MSFE RMSFE

College Carroll 0.4967312 0.7047916

Threshold 0.498925269 0.706346423

Some College Carroll 0.5702252 0.7551326

Threshold 0.438304619 0.662045783

When comparing the results of these two subpopulations, the threshold model clearly

shows better results than the Carroll model for the Some-College subpopulation. For the

College subpopulation, this distinction is less clear; in fact, the threshold model has less

predictability than the original model. This trend holds for most models within each

news-statistic framework (i.e. Tetlock, percent-negative, Tetlock uncertain, etc.), and

when comparing each of the best models with each other. So, whereas the College

subpopulations diagnostics are minimally better for Carrolls model, the Some-College

19 The slope coefficients only add up to one within the threshold model.


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population exhibits the reverse much more significantly, and the High-School

subpopulation only exhibits credible results for various threshold models.

Accordingly, each of the threshold models discussed above exhibit no clear unit

root trend. Again, this shows an empirical confirmation of the theoretical basis of an

epidemiological inflation expectation model; as consumers expectations are formed on

the backdrop of unit root tendencies, the differencing of expectations should theoretically

eliminate this unit root tendency.

Model ADF Statistic P-Value Order of Integration

College -5.3635 .01 I(0)

Some College -5.7220 .01 I(0)

High School -4.4424 .01 I(0)

Like before, with the original models, no models tested exhibited serial correlation

amongst the residuals; the p-values for each model were extremely low, so the possibility

of serial correlation can be rejected with high statistical confidence.

Model DW Statistic P-Value Serial Correlation

College 2.3118 0.9501 No



Interestingly, whereas all of the original models (i.e. Carrolls models which were

extended to subpopulations) exhibited degrees of heteroscedasticity, the threshold models

do not test positive for heteroscedastic residuals. Accordingly, one could extrapolate that

some, if not all, of the heteroscedasticity present within consumers inflation expectations


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are due to the tone of the news sources which they consume. Said differently, forecast

errors are shown to be somewhat dynamic in that they are dependent upon the degree of

tone within news forecasts.

Model BP Statistic P-Value Heteroscedastic

College 2.3001 0.3166 No



Ultimately, the threshold models exhibit various benefits over the original models. First,

the High-School subpopulation exhibits theoretical consistency which the original model

does not achieve; it also shows no signs of bias or gross errors in estimation. Whereas the

College subpopulations threshold model does not explain the data better than Carrolls

original model, it does eliminate the heteroscedasticity present within the original model.

In contrast, the Some-College subpopulation exhibits a clear preference for the threshold

model; forecasts errors, which are not heteroscedastic, in this model are clearly smaller

than the original model.

Mapping the evidence of nonlinearity in the data gets more complex than was

originally thought; the results of the analysis are somewhat inconclusive in one sense, but

surprising in another. What is clear is that, theoretically, the College models with added

threshold values hold no statistical significance. On the other hand, High School models

that map tone-bias thresholds (and how they change the updating process of consumers)

yield statistically significant results. These models both contribute a higher R2 value than

Carrolls original equation for the subpopulation, and have theoretically consistent


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estimators. So, interestingly, the bias-tone of news forecasts affects some sub-populations

but not others. Why?

One explanation could be the very nature of each subpopulations forecasts.

Whereas rational inattention could explain the forecasting methods of College graduates,

and thus Carrolls model would suffice, the subpopulations dependence upon common

media sources is certainly questionable. Prior assumptions suggest that this subpopulation

obtains forecasts from news sources and is, accordingly, subjected to their biases. This

has clearly been shown to not be true through this analysis. On the other hand, High

School expectations exhibit a strong dependence upon the negativity of news stories; the

threshold equation yields a change of nearly 50% in the updating percentage of the

population when negativity of news stories reaches a certain value. Rather than attempt to

interpret these specific values too literally, the reader is cautioned not to take the

threshold variables as being in line with actual reality. The threshold models, rather,

attempt to give light to the dynamical nature of updating inflation expectations; whereas

the threshold equation shows that this process is partially dependent upon the tone of

news forecasts, the variability of this process is far less constant in reality. Rather, the

degree of change in slope coefficients illustrates the degree of change that tone brings

about in the updating frequency, rather than an actual change in the updating process for

each of these variables. The threshold serves to illustrate the existence of variability

dependent upon tone, rather than the actual dynamics of the variability. Nonetheless, the

results thereby demonstrate a fundamental difference in the nature of forecasting for each

subpopulation. In addition, High School expectations models which include thresholds


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replace heteroscedastic error terms for homoscedastic ones; this, along with a complete

lack of serial correlation amongst residuals, indicate that threshold models for the High

School subpopulation behave more normally than the original Carroll equation. This is

also indicative of the superiority of the threshold model to the simple, original model.

As a technical side note, low R-square for each of the models, although

problematic, is largely due to the huge variation in the change of inflation forecasts.

These large variations create noise that drown out other explanatory variables, and are

largely erased when including the updating mechanisms within the original equation.

Unfortunately, though, the persistence of the intercept within both the original and

threshold equation indicate some type of shortcoming within the model; one of the

original intents of creating a threshold model was a disagreement with Carrolls

assumption of rationality. Within his original paper, Carroll contends that the intercept

within the original equation is troublesome, as consumers are assumed to be rational;

accordingly, consistent biases contradict that fact. This paper, however, strove to show

that the intercept could be explained through the inherent biases within news sources that

subpopulations consumed; with an intercept still within the model, there are clearly some

inherent biases still present within the expectation formation process that cannot be

explained by the tone of media forecasts.

Interestingly, the Some College subpopulation exhibited less clear results than the

High School or College subpopulations. Whereas both were shown to have clear

favorability to one model over another (i.e. the original vs. the threshold), the Some


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College subpopulation does not delineate a clear choice for the preferred model.

Unfortunately, no model can be clearly chosen as better than the other one. Whereas this

could confirm, in an ironic matter, the differences in the formation process of inflation

expectations for consumers20

, the lack of evidence for this subpopulation, one way or

another, restricts the formation of a conclusion on their behavior process.

Given the nature of the argument, that this paper sought detections of

nonlinearities within the data of consumer inflation expectations, future papers ought to

write more extensively on the actual dynamics of the updating mechanism of consumers.

One means of doing so would be to conduct an intervention analysis to conclude whether

the tone of forecasts affects Carrolls equation; this can also be done by mapping the

dynamic causal effects (or impulse response) of negative news tone to the gap between

professional and consumer forecasts. Such an analysis would give light to the degree and

persistence of consumer biases over time. Large persistence in response to a negative

news shock could, theoretically, partially explain constant biases still present within

threshold models.

Together with mapping the dynamic causal effects of media tone on expectations,

analyzing whether similar threshold models hold forincome-related subpopulation may

provide further insight on the dynamics of inflation expectations. Based upon the theory

that updating mechanisms are constrained and determined by cost concerns, income-

20

The nature of consumers formation process is spectral in one sense, with college andHigh School populations exhibiting distinctly different processes, at opposite ends of the

spectrum, and some college populations landing, inconclusively, in the middle.


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related subgroups may portray a more accurate picture than education-related subgroups.

Although, given that these two variables (education and income) are extremely

correlated, this is only a possibility; some might argue that income is the actual cause (i.e.

explanatory variable), rather than education, yet this may ignore the time aspects of both

variables.21

In addition, a more in depth analysis should be given to the measurement of tone,

and its impact on consumer forecasting; the only true setback of the objective

measurement of tone (versus subjective) is the lack of sophistication in analyzing the

context of a specific word. More specifically, research of different methods in measuring

the tone of news stories may be useful for improving the threshold models.

21

In the scheme of things, education should precede income, generally speaking. Also,

income, like volume of news, may just be a proxy for opportunity costs.


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AppendixI.

Belowistheoveralldataforthispaper,whichincludesbasicdataonthemajor

modelstestedthroughouttheproject,aswellkeytestsonthisdataandits

significance.

1.AugmentedDickey-FullerStatistics

2.Breusch-PaganStatistics

3.Durbin-WatsonStatistics

4.TestedCarrollModelsonSubpopulations

5.ThresholdResults

1.ADFStatistics

Belowaremodelsthattestwheretheresidualsofsaidmodelsexhibitunitroot

tendencies;variableswiththeletterTsignifymodelswithTetlocksstatisticasthe

thresholdvariable,whereasothermodelsincludedifferent%Biasvariablesasthe

threshold.Alltestsrejectedthenullhypothesisofnounitrootatthe1%or5%levels.

TABLEI

ADFStatisticsforModelResiduals(1980-)

Variables College SomeCollege HighSchoolTN -4.7586 -5.1346 -5.1959

TU -5.4327 -5.2134 -4.4424

N% -4.7586 -4.9248 -4.9518

P% -4.6978 -5.1419 -4.7709U% -4.6337 -4.3200 -4.9059

C% -4.7730 -4.8396 -4.8576

TABLEII

ADFStatisticsforModelResiduals(1983-)Variables College SomeCollege HighSchoolTN -5.3635 -6.4601 -6.3187

TU -5.4751 -5.722 -6.4715

N% -5.2359 -6.2815 -5.9172

P% -5.4515 -6.1585 -5.861U% -5.4183 -5.4846 -6.1695


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C% -5.5707 -6.1307 -5.7245

2. BPStatisticsTestingthenullhypothesisofhomoscedasticitywithintheresidualsagainstthe

alternatehypothesisofheteroscedasticity,mosttestsshownoclearheteroscedasticity

withinthemodels,contrarytothetestsdonewithoutathreshold.One,two,andthree

starsindicatestatisticalsignificanceforthealternatehypothesisatthe10%,5%,and

1%levels,respectively.Almostallmodelsfailtoproveheteroscedasticityatthe5%

significancelevel,andnearlyallfailtodosoatthe10%level;thisiscontrarytothe

originalmodeltestedonthesubpopulations.

TABLEIII

Breusch-PaganStatisticsforModels(1983-)

Variables College SomeCollege HighSchool

TN 2.3001(.317) 3.3385(.188) 5.902(.052)*

TU 3.7866(.128) 4.9499(.084)* 3.1799(.204)

N% 4.8434(.089)* 1.748(.413) 0.3756(.829)

P% 4.2185(.133) 4.3768(.056)* 0.5821(.748)

U% 7.5576(.023)** 5.4158(.051)* 0.9667(.617)

C% 8.2869(.016)** 2.1952(.274) 2.4421(.194)

TABLEIV

Breusch-PaganStatisticsforModels(1980-)


TN 4.5015(.105) 1.4302(.489) 5.089(.079)*

TU 2.5667(.277) 3.6107(.135) 0.7466(.689)

N% 4.3391(.114) 0.9841(.611) 0.2061(.902)P% 2.9922(.224) 2.0567(.358) 1.1524(.562)

U% 6.6308(.036)** 5.8801(.052)* 0.8347(.659)

C% 4.1909(.123) 7.3619(.025)** 1.2889(.525)


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3. DWStatistics

Thefollowingtablesshowtestsforautocorrelativetrendswithineachrespectivethresholdmodels.Likebefore,one,two,andthreestarsindicatestatisticalsignificance

forthealternatehypothesisatthe10%,5%,and1%levels,respectively.Asseen

below,nomodelshowsanysignificantamountofautocorrelationatthe10%

significancelevel.

TABLEV

Durbin-WatsonStatisticforModels(1980-)


TN 1.8799(.2295) 1.9208(.3035) 2.1055(.706)

TU 2.2460(.8982) 1.9925(.4273) 2.3434(.9663)N% 1.9022(.2638) 1.9973(.4647) 2.1391(.7649)

P% 1.9298(.3119) 1.7888(.1046) 2.2107(.8689)U% 2.0246(.5261) 1.7910(.103) 2.1375(.7627)

C% 1.9538(.3645) 1.8196(.1321) 2.1925(.8643)

TABLEVI

Durbin-WatsonStatisticforModels(1983-)

Variables College SomeCollege HighSchoolTN 2.3118(.9501) 1.9773(.4271) 2.2898(.936)

TU 2.214(.8637) 1.9416(.3504) 2.2115(.8612)N% 2.1168(.7138) 1.9256(.3095) 2.2801(.9266)

P% 2.0807(.6474) 1.918(.2966) 2.2823(.9274)

U% 2.1842(.8262) 2.0101(.4822) 2.252(.9034)

C% 2.0954(.6754) 1.9314(.3204) 2.399(.643)


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4. TestedCarrollModelsonSubpopulations

BelowareCarrollsepimediologicalmodelsforeachrespectivesignificantsubpopulation;allvariablesexhibitstatisticalsignificanceatthe10%level,atleast,

whereasallmodelsexhibitstatisticalsignificanceaswell.Eachofthemodelsare

describedintheformMt=0+1Pt+2Mt-1,whichresemblesCarrollstheoretical

modelMt=Pt+(1-)Mt-1.

TABLEVII

CarrollModelsforEachSignificantSubpopulation

Model 0 1 2 RMSFE

College 0.32769 0.39077 0.59867 0.7047916

(0.1381) (0.0592) (0.0580) SomeCollege 0.4788 0.21967 0.72113 0.7551326

(0.1651) (0.0645) (0.0650)

HighSchool 0.50876 0.11898 0.80917 0.7421156

(0.1853) (0.0629) (0.0646)


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5. ThresholdModelResultsBelowaretheresultsforeachofthethresholdmodelsfrom1980-2012;thesemodels

areexplainedwithintheformMt=0+(1+It1)(PtMt-1),soastoillustratethe

effectseachoftherespectivethresholdshavehadontheupdatingfrequenciesofeach

subpopulation.AlsoincludedistheRMSFEasabasicmeasureofcomparisonbetween

eachofthemodels.Numbersinparenthesissignifythestandarderrorsofeach

respectivevariable.Eachofthesemodelsvariablesheldstatisticalsignificanceatat

leastthe10%level;allmodelsholdstatisticalsignificanceatthe5%level.

TABLEVIII

CollegeSubpopulationTests(OnlyIncludesOptimalThresholds)

Model 0 1 2 RMSFE

TN 0.20885 0.38018 1.15334 0.6889314

(0.07725) (0.06762) (0.37291)

TU 0.25323 0.32281 0.09103 0.6950549

(0.08388) (0.05422) (0.07185)

N% 0.17653 0.19922 0.47419 0.6848022

(0.07739) (0.10424) (0.07492)

U% 0.18436 0.59099 0.24764 0.6738674

(0.07567) (0.09063) (0.08009)

P% 0.1479 0.25772 0.67186 0.6705606 (0.07675) (0.07671) (0.10527)

C% 0.18176 0.31755 0.66182 0.6836677

(0.07698) (0.07324) (0.12634)


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TABLEIXSomeCollegeSubpopulationTests(OnlyIncludesOptimalThresholds)

Model 0 1 2 RMSFE

TN 0.18492 0.17328 0.50053 0.6964564

(0.08678) (0.06567) (0.09279)

TU 0.28582 0.25392 0.01733 0.6942652

(0.09195) (0.04638) (0.08228)

N% 0.18381 0.53278 0.17911 0.6950679

(0.08662) (0.09884) (0.06459)

U% 0.20322 0.40075 0.19789 0.7129468

(0.08861) (0.08724) (0.06834) P% 0.23507 0.18266 0.32607 0.6944159

(0.08674) (0.04361) (0.06193)

C% 0.17443 0.30208 0.10494 0.7155675

(0.09035) (0.06402) (0.10277)

TABLEX

HighSchoolSubpopulationTests(OnlyIncludesOptimalThresholds)

Model 0 1 2 RMSFE

TN 0.34822 0.18044 0.31077 0.7017414

(0.1135) (0.08789) (0.06852)

TU 0.52717 0.33397 0.11525 0.6679863 (0.11688) (0.056) (0.05357)

N% 0.31371 0.40123 0.21319 0.7009327 (0.11311) (0.08458) (0.06961)

U% 0.31484 0.3793 0.21594 0.6973979

(0.11371) (0.08225) (0.07066) P% 0.3415 0.18157 0.31317 0.6944902

(0.11336) (0.08529) (0.06852) C% 0.31214 0.34264 0.14747 0.6886369

(0.11199) (0.06821) (0.07894)


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AppendixII.C++ProgramforThresholdEstimation:

#include

#include#include

#include //needed to use system() function

using namespace std;

const int T = 128;const int M = 10;

/*

* Consumer Expectations are in the form of current and* lagged values. Both are one-dimensional arrays, with

* elements varying over time.*/

struct Consumer {double current[T];

double lag[T];};

/** SPF Expectations are in the form of current and* lagged values. Both are one-dimensional arrays, with

* elements varying over time.*/

struct Professional {double current[T];

double lag[T];};

/*

* This structure deals with the differenced data that is impt* to the model. Ultimately, this data will be used for the

* regressions, rather than the original data.* mich[] is the differenced dependent variable whereas mich_spf[]

* is the differenced independent variables.*/

struct Differenced {double mich[T];


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double mich_spf[T];};

/*

* The dummy variable is a two dimmensional, (n x n) array* based upon the number of observations over time (n) and* the model being specified (m). The models (m) vary because

* each one has a different threshold value. The interactive* variable has two dimmensions (n x n) as each of the data

* observations (varying over n) is multiplied with each dummy* in the varying models. The threshold and dummies are dependent

* upon the tetlock statistic array.*/

struct News {int dummy2[T][T]; //For second regime

int dummy1[T][T]; //For first regimedouble tetlock_stat[T];

double interactive_new[T][T];double interactive_old[T][T];

};

/** The Data structure has professional and consumer structure,

* as well as the News structure.*/

struct Data {Professional spf;

Consumer michigan;News news;

};

/** This structure contains the results of the regression.

* The beta array is 2-dimmensional to account for the different* beta values in each model, along with the varying models.

* Residuals (n x m), like dummies, varies over observation* and models. SSR is an 1-d array which just varies over all models.

*/struct Results {

double beta[M];};

void readarray(int n, double stats[]);

void multiplying_matrices(int first, int second, int inside, double x[T][T],double y[T][T], double z[T][T]);


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double average(double data[T], int n);void dummyarray(int n, double news_stat[], int dummy1[T][T], int

dummy2[T][T]);

void interactivevariables(int n, double spf[],int dummy2[T][T],

int dummy1[T][T], double new_spf[T][T], doubleold_spf[T][T]);void mypause(void);

void olsregression(int n, int m, double y[T][T], double X[T][T],double beta[T]);

void matrix_vector_multiplication(double matrix[T][T], double vector[T],double new_vector[T], int m, int n);

void creating_matrices(double differenced_independent[],double interactive_spf[], int n, double matrix[T][T]);

double calculate_ssr(int n, double differenced_michigan[],double differenced_independent[],

double dummy_spf[], double beta[]);void threshold_regressions(double differenced_michigan[], double

differenced_independent[][T],int m, int n, double beta[], double new_spf[][T]);

void differenced_data(double first[], double second[], double new_one[], int n);void invert_matrix(double matrix[][T], double inv_matrix[][T], int m);

ofstream outfile("Data.txt");

ifstream infile("Expectations.txt");

int main()

{Data high_school, some_college, college;

Differenced HS, SC, C;Results hs, sc, c;

int n, m;double avg_hs, avg_c, avg_sc;

cout


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outfile


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//FINAL REGRESSIONthreshold_regressions(C.mich, college.news.interactive_old, m, n, c.beta,

college.news.interactive_new);return 0;

}

/* Function ReadArray()* I:

* n - observations* stats[]

* P:* Reads and prints values for specific array

* O:* Prints array stats[]

*/void readarray(int n, double stats[])

{ for (int count = 0; count < n; count++)

{infile >> stats[count];

}

//prints out data that was read into programfor (int print = 0; print < n; print++)

{outfile.width(12);

outfile.setf(ios::left);if (print % 5 == 0)

outfile


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* P:* Uses three for loops; the first to adjust for values in rows

* of z[][]; the second to adjust columns in z[][]; the third to* multiply x[][] and y[][] inner values. Prints and stores values

* to z[][].

* O:* Prints the matrix z[][]*/

void multiplying_matrices(int first, int second, int inside, double x[T][T],double y[T][T], double z[T][T])

{for(int reset = 0; reset < 125; reset++)

for (int another = 0; another < 125; another++)z[reset][another] = 0;

for(int row = 0; row < first; row++)

for(int column = 0; column < second; column++)for(int inner = 0; inner < inside; inner++)

z[row][column] += x[row][inner]*y[inner][column];

cout


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*/void dummyarray(int n, double news_stat[], int dummy1[T][T], int

dummy2[T][T]){

double threshold;

for(int a = 0; a < n; a++){

threshold = news_stat[a];

for(int i = 0; i < n; i++){

if (news_stat[i] > threshold){

dummy2[i][a] = 0;dummy1[i][a] = 1;

}if (news_stat[i]


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/* Function InteractiveVariables()

* I:* n - number of observations

* spf[]

* dummy[][] - dummy values dependent upon threshold 't'* new_spf[][] - product of spf[] and dummy[][]* P:

* Multiplies each spf value with respective dummy value (1 or 0)* then prints and stores results.

* O:* new_spf[][] values

*/void interactivevariables(int n, double spf[], int dummy2[T][T],

int dummy1[T][T], double new_spf[T][T], double old_spf[T][T]){

for (int a = 0; a < n; a++)for (int i = 0; i < n; i++)

{new_spf[i][a] = spf[i]*dummy2[i][a];

old_spf[i][a] = spf[i]*dummy1[i][a];}

outfile


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outfile.width(5);outfile.setf(ios::left);

outfile


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//multiplies inverse of X'X with X'y

matrix_vector_multiplication(iX,XtY, beta, m, m);

return;

}

//Function mypause()

void mypause(void){

system("pause");return;

}

/* Function Matrix_Vector_Multiplication()* I:

* matrix[][] - a matrix* vector[] - a vector

* new_vector[] - product of matrix and vector* m - the bounds for the resultant row

* n - the bounds for inner column and outer row (i.e. vector)* P:

* initializes new_vector's elements to 0. Then multiplies the* matrix by the vector, like multiplying_matrices, by

* multiplying [m][n] X [n][1]. Prints resulting values and stores* to new_vector

* O:* new_vector values

*/void matrix_vector_multiplication(double matrix[T][T],double vector[T],

double new_vector[T], int m, int n)

{//sets old values to zero for new vector

for (int starting = 0; starting < n; starting++){

new_vector[starting] = 0;}

//gets values for new vector

for (int begin = 0; begin < m; begin++){

for (int looping = 0; looping < n; looping++)new_vector[begin] += (matrix[begin][looping] * vector[looping]);

}


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outfile


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* mich_lag[]* m - number of independent variables

* n - number of observations* beta[]

* new_spf[][] - a matrix of dummy_spf values for each specific threshold

* who is the product of dummy[][] * spf[]. Its values are* in the form of: observation X model.* (Ex: new_spf[1][6] = observation 1, model 6)

* P:* For each model in new_spf, this function runs a new regression.

* First, it sets interactive_spf[] to the values in new_spf[][model].* Then, it combines this new vector with the others needed in regression

* to form the matrix "combined matrix" (the 'X' in the OLS equation).* Then, it calls the olsregression() function for this model, prints the

* beta[] values, calculates the ssr for the model, and prints that also.* O:

* Regression results.*/

void threshold_regressions(double differenced_michigan[],double differenced_independent[][T],

int m, int n, double beta[], double new_spf[][T]){

double interactive_spf2[T];double interactive_spf1[T];

double ssr;double combined_matrix[T][T];

double final_ssr[T];

for (int model = 0; model < n; model++){

outfile


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{interactive_spf2[i] = new_spf[i][model];

outfile


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* spf[]* mich_lag[]

* dummy_spf[]* beta[]

* P:

* Uses for loop to calculate residual of observation 'k',* squares this residual, and then adds to total sum.* O:

* Returns the sum of the squared residuals.*/

double calculate_ssr(int n, double differenced_michigan[],double differenced_independent[],

double dummy_spf[], double beta[]){

double sum, residual;double ssr;

sum = 0;for (int k = 0; k < n; k++)

{residual = (differenced_michigan[k] - (beta[0] +

beta[1]*differenced_independent[k] + beta[2]*dummy_spf[k]));sum+= residual*residual;

}ssr = sum;

return ssr;}

/* Function: Differnced Data

* I:* first[]

* second[]* new_one[] = first - second

* P:* Differences two arrays

* O:* Nothing

*/void differenced_data(double first[], double second[], double new_one[], int n)

{for (int z = 0; z < n; z++)

{new_one[z] = first[z] - second[z];

}

return;}


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/* Function Invert_Matrix

* I:* matrix[][] - old matrix

******************************

* a b c ** matrix[][] = d e f ** g h k *

******************************* inv_matrix[][] - new, inverted matrix

********************************** A B C *

* inv_matrix[][] = D E F ** G H K *

********************************** m - number of rows/columns in matrix

* P:* Calculates inverse of matrix manually

* O:* nothing

*/void invert_matrix(double matrix[T][T], double inv_matrix[T][T], int m)

{double determinant;

double A, B, C, D, E, F, G, H, K;double a, b, c, d, e, f, g, h, k;

a = matrix[0][0];

b = matrix[0][1];c = matrix[0][2];

d = matrix[1][0];e = matrix[1][1];

f = matrix[1][2];g = matrix[2][0];

h = matrix[2][1]k = matrix[2][2];

A = (e*k) - (f*h);

B = (f*g) - (d*k);C = (d*h) - (e*g);

D = (c*h) - (b*k);E = (a*k) - (c*g);

F = (g*b) - (a*h);G = (b*f) - (c*e);

H = (c*d) - (a*f);K = (a*e) - (b*d);


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determinant = (a*A) + (b*B) + (c*C);

inv_matrix[0][0] = A;inv_matrix[0][1] = B;

inv_matrix[0][2] = C;

inv_matrix[1][0] = D;inv_matrix[1][1] = E;inv_matrix[1][2] = F;

inv_matrix[2][0] = G;inv_matrix[2][1] = H;

inv_matrix[2][2] = K;

for (int q = 0; q < m; q++)for (int b = 0; b < m; b++)

inv_matrix[q][b] = inv_matrix[q][b]/(determinant);

Mapping Heterogeneity in Consumer Inflation Expectations: Does the Tone of News Affect Consumer...

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