Essay: Macroeconomics

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1 Introduction
As important determinants of investment and consumption decisions, expectations are an important field of research in macroeconomics. Especially in the attempt to un- derstand the dynamics and fluctuations in economies, various theories of expectation formation have been developed. As a first model, the theory of adaptive expectations assumed businesses and individuals to form their expectations based on past realizations while also taking into account their previous errors.
With the publication of “Rational Expectations and the Theory of Price Movements” (Muth 1961) the more complex concept of rational expectations gained popularity in macroeconomics. The theory assumed agents to form expectations based on correctly anticipating the future development of a specific economic variable based on all infor- mation that is currently available. Economists like Lucas argued in favor of integrating rational expectations in macroeconomic models (e.g. Lucas 1972).
Even though there is statistical evidence against the full-information rational expec- tations (FIRE) hypothesis, it still underlies most modern macroeconomic models. To provide an economically interpretable assessment of the FIRE hypothesis, Coibion and Gorodnichenko develop a new approach to test it. The new test which is presented in “Information Rigidity and the Expectations Formation Process: A Simple Framework and New Facts” (Coibion and Gorodnichenko 2015) provides results that are meaningful in appraising the FIRE hypothesis and modern models of rational expectations which assume information rigidities.
The paper will be structured as follows: Section 2 will deal with the FIRE hypoth- esis and the traditional tests of it, as well as with modern rational expectation models which incorporate information rigidities. Section 3 will present Coibion and Gorod- nichenko’s new approach to test the FIRE hypothesis which is subsequently discussed and compared to other tests in section 4. Section 5 concludes.
2 Models of Expectation Formation
2.1 The Full-Information Rational Expectation Hypothesis
The theory of rational expectations that is outlined by Muth (1961) models agents as if they know the underlying model. This means that agents’ expectations are equal to the predictions of the underlying theory. Put formally, agents’ subjective probability
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density function of outcomes, fi, is identical to the probability density function of outcomes according to the model, f (Muth 1961, p. 316):
fi(xt+h|Ωit) = f(xt+h|Ψt) (1)
where xt+h is the variable of interest in period t + h. On the left hand side this variable is forecasted conditional to the complete information set, Ωit. On the right hand side it is predicted conditional to the public information set Ψt ⊆ Ωit which is a subset of the complete information set. Private information plays no role in Muth’s theory (Pesaran and Weale 2006, p. 721).
Since Coibion and Gorodnichenko (2015) focus on developing a new test for market rationality rather than rationality on the individual level, we can relax the previous definition of rational expectations and rewrite equation (1):
̄
f(xt+h|Ωit) = f(xt+h|Ψt) (2)
where f ̄ is the average subjective probability density function of outcomes. This def- inition only requires that agents form rational expectations on average. This allows for irrational expectation formation on the individual level, as well as for heterogenous information among agents (Pesaran and Weale 2006, p. 722).
Further, the rational expectation hypothesis asserts that all information is used efficiently (Muth 1961, p. 316). This feature is commonly called the orthogonality property and has various testable implications. In its basic form the orthogonality condition states that the rational forecast error is uncorrelated with any information from the public information set:
E(vt+h,t|St) = 0 (3) where the rational forecast error vt+h,t is defined as the difference between the realization
of the variable of interest and the rational forecast of it formed in period t,
vt+h,t = xt+h − Et(xt+h), (4)
and St is a subset of the public information set (Pesaran and Weale 2006, p. 721). The idea behind the orthogonality condition is that the forecast error cannot be predicted by taking public information into account. This means that agents cannot improve their forecasts when using all the information that is available to them efficiently.
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However, in the context of the FIRE hypothesis, this condition cannot be tested since it would be impossible to take all available information into account. Therefore, the orthogonality condition is toned down to state that the rational forecast error is unpredictable when taking into account the public information subset St. Depending on which information from the public information set the subset St contains, different degrees of rationality can be defined. Hence, St is used as a proxy for the public information set Ψt.
According to Bonham and Dacy (1991, pp. 247 sq.) a forecast is weakly rational if it satisfies the necessary conditions which are outlined in the following. First, the forecast has to be unbiased, meaning on average agents may not systematically over- or underestimate the variable to be forecasted. Second, the forecast error vt+h,t cannot be predicted from past realizations of the variable of interest. Third, the forecast error has to be serially uncorrelated. In the context of informational efficiency this condition is necessary since past forecast error constitute information that can be potentially used to improve the forecast. If, furthermore, the forecast error is uncorrelated with any variable from the public information set it is called “sufficient rational” (Bonham and Dacy (1991, p. 248)). Additionally, forecasts are described as “strongly rational” (Bonham and Dacy (1991, p. 248)) if their accuracy cannot be enhanced by combining them with other forecasts (see for example: Ashley, Granger, and Schmalensee (1980) and Fair and Shiller (1990)).
Following Coibion and Gorodnichenko (2015), we will call empirical tests of the necessary and sufficient conditions “traditional tests”. One variation of such a test was employed by Coibion and Gorodnichenko (2015, p. 2654) before presenting their new approach to test the FIRE hypothesis. The test is employed in the context of inflation forecasting. The authors use data from the Survey of Professional Forecaster (run by the Philadelphia Fed) which consists of 30-40 professional forecasters. The choice to use this data set stems from the consideration that this constitutes a useful bench- mark since evidence for information rigidities among professionals would also indicate information rigidities throughout the (on average less informed) economy (Coibion and Gorodnichenko 2015, p. 2652). Their traditional test is specified as follows:
πt+3,t − Ft(πt+3,t) = c + γFt(πt+3,t) + δzt−1 + errort (5)
Forecast error
where πt+3,t is the average inflation over the year ahead (current quarter plus next three quarters) and Ft(πt+3,t) is the average forecast of it across agents. The idea behind the specification is to regress the average forecast error on past realizations and a subset
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of publicly available information. This subset is embodied by the control variable zt−1 which includes lagged values of inflation, the average rate for three-month US bonds, the quarterly log change of the oil price and the average unemployment rate. These control variables have been selected because they potentially have predictive power (Coibion and Gorodnichenko (2015, pp. 2645 sqq.)). Under the assumption that forecast errors are serially uncorrelated this specification tests sufficient rationality. Formally, the null hypothesis states:
H0: c=0 γ=0 δ=0
In addition to the features implied by the orthogonality condition, the first part of the null hypothesis states that the intercept is equal to zero. This represents the assumption that the forecast error is not generally distorted, i.e. there is no system- atic over- or underestimation which is independent from the information considered by agents.
The results of this test are outlined in table 1 (concluded version of panel A from table 1 in Coibion and Gorodnichenko (2015, p. 2653)):
Table 1: Coefficient estimates from the traditional test
Intercept c Forecast Ft(πt+3,t) Control zt−1
None
−0.181 0.059
Inflation
−0.045 −0.299∗∗ 0.318∗∗
Bond rate
−0.091 0.210∗ −0.125∗
Oil price
−0.181 0.045 1.603∗∗
Unemployment rate
1.449∗∗ 0.095 −0.281∗∗
**: p<0.5,*: p<0.1
Consequently, they reject the null hypothesis of the FIRE hypothesis as all of the control variables can significantly predict the forecast error which implies that agents on average do not form rational expectations. The rejection of the null hypothesis however does not give us any information about the economic significance of the rejection or clues about the reasons for the rejection (Coibion and Gorodnichenko 2015, p. 2645).
Possible reasons for the rejection could be information rigidities that arise if agents do not have full access to all publicly available information at any time. Thus a re- jection of the FIRE hypothesis might not be reasoned in agents forming “irrational expectations” in the sense of interpreting available information wrongly but may be
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due to imperfect access to information. According models which incorporate informa- tion rigidities will be the subject of the next three subsections.
2.2 Sticky Information Models
One of the most important information rigidity models is the sticky information model which was proposed by Mankiw and Reis (2002) in order to present an explanation of nominal rigidities. In their fundamental model, only a fraction (1 − λ) of producers update their information set which enables these producers to obtain full information. This allows them to form forecasts consistent with the FIRE hypothesis. This infor- mation is then used to compute a path of optimal prices according to the price setting formula
p∗t =pt+αyt (6)
where p∗t is the optimal price, pt the overall price level, α the degree of stickiness and yt the output gap. This relationship resembles the short-run Philips curve interconnection according to which higher prices are charged in booms and lower prices are charged in recessions (Mankiw and Reis 2002, pp. 1299 sqq.).
Since not all firms update their price in every period some firms set their price based on outdated information. The price which is charged in period t by a firm whose most recent information set update was j periods ago is defined as
xjt =Et−j(p∗t) (7) which is the optimal price rationally calculated on the basis of a (full) information set
obtained j periods ago. The aggregate price level is then expressed by

pt =(1−λ)λjxjt j=0

=(1−λ)λjEt−j(pt +αyt) (8)
j=0
which is the weighted average of all the prices which are charged (Mankiw and Reis
2002, p. 1300). 1 An intuitive explanation of the reasons why firms do not continuously
1The notation here is slightly altered compared to the original paper by Mankiw and Reis (2002) for the purpose of a consistent notation since Coibion and Gorodnichenko (2015) deviate in the same way.
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update their information is given by Reis (2006) who shows that this behavior is optimal. Under the assumption that gathering and processing information is costly (Mankiw and Reis 2002, p. 2316, Reis 2006, p. 795), producers maximize their expected present discounted profit including these costs. Solving this maximization problem is done by weighing the value of updating information and prices against the incurred costs (Reis 2006, p. 802).
As could be suspected the degree of inattention increases with planning costs and decreases with the volatility of shocks in which case inattention is especially costly (Reis 2006, p. 803). Carroll (2003) supports the assertion that agents are able to form (close-to-)rational expectations when updating their information sets. It is shown that forecasts by professionals – which come closest to being rational – spread slowly through the population where they are adopted by agents (Carroll 2003, pp. 269 sqq.).
Coibion and Gorodnichenko (2015, pp. 2648 sqq.) proceed to generalize this model to make it applicable to other contexts. In the following specification not the overall price level is computed by weighting individual forecasts, but more generally the average time-t-forecast of variable x at time t + h is computed by weighting individual forecasts which are formed at different points in time:

Ft (xt+h ) = (1 − λ) λj Et−j (xt+h )
j=0
= (1 − λ)Etxt+h + λFt−1(xt+h) (9)
In the bottom line of equation above the idea becomes clearer: The average forecast across agents is composed of a share (1−λ) of agents who form truly rational forecasts, i.e. based on all information that is currently available, while the remaining share λ forms forecasts based on older information. Within this remaining share is a fraction (1 − λ) who formed rational forecasts based on all information available at time t − 1 (within the whole population this is a share of (1 − λ)λ) while the remaining fraction formed their forecast based on even older information (overall share of (1−λ)λ2). This argument carries on infinitely up to the agents who formed their expectation of xt+h based on the full information set ∞ periods ago. It is noteworthy that these expectations suffice the FIRE hypothesis at the time they were formed since they incorporate all available information at that time efficiently. However, at the time of the evaluation (period t) these forecasts are most likely deemed irrational since the agents at that time did not and could not consider new information which became available in the
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meantime and would improve the forecast. This again elucidates the difficulty with the term “irrational” since it does not imply that agents formed arbitrary forecasts – contrary to what might be supposed at first sight of the expression.
Ensuing, Coibion and Gorodnichenko (2015, p. 2649) combine equation (9) with the definition of the rational forecast error (equation (4)) to present a relationship between the forecast error and forecast revisions:
xt+h − Ft(xt+h) = λ (Ft(xt+h) − Ft−1(xt+h)) + vt+h,t (10) 1−λ
In the relationship pointed out above the coefficient on forecast revisions solely depends on the degree of information rigidity λ. This finding is compatible with our intuition: If all agents update their information sets on which they form expectations in every period ((1 − λ) = 1 ⇒ λ = 0) no information rigidities would be present. Thus the forecast error would be unpredictable and the average forecast would satisfy the FIRE hypothesis.
2.3 Noisy Information Models
In the class of noisy information models agents are modeled as if they know the un- derlying economic model and continuously update their information sets. In contrast to the classical theory of rational expectations though, agents do not observe the true state of the variable they are interested. Instead they only observe a signal that is afflicted with noise. In order to form expectations procedures to detach this noise have to be performed.
A relatively intuitive and the most famous model of this class is the Lucas islands model. In the model developed by Lucas (1973, pp. 327 sq.) a number of separated competitive markets (“islands”) exist. On each island (denoted by z) is one producer who charges the price pt(z). By assumption, the price level on each island is based on the overall price level and differs only due to market specific shocks zt that are zero on average:
pt(z) = pt + zt (11) Based on the price pt(z) relative to the overall or average price level pt the producer
chooses to produce a specific quantity. This is apparent in the supply curve postu-
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lated by Lucas (1973, p. 327). However, since the overall price level is not observable, producers have to form rational expectations about it and the equation above becomes
pt(z) = E(pt) + zt
=pt −vt +zt (12)
where vt is the rational forecast error (compare equation (4)).
The rational forecasts error can be attributed to an aggregate shock which requires
no response because it only implies nominal changes. On the other hand a market specific shock affects the producer in real terms and thus requires a change in supplied quantity (Blanchard and Fischer 1989, p. 356). The problem the producers face is that they cannot distinguish between a market specific shock and an aggregate (nominal) shock since only the change in price on the respective island, pt(z), is observed. To solve this problem a method is proposed which relies on historical data on vt and zt to generate a forecast that uses past information to estimate the probability of both types of shocks (Lucas 1973, pp. 328 sqq.).
The noisy information model of Woodford (2001) on the other hand makes use of the Kalman filter in extracting the signal we are interested in – such as the overall price level above. In this model the variable of interest, xt, is assumed to follow a first-order autoregressive process:
xt = ρxt + ut (13)
where ρ ∈ (0, 1) is the degree of serial correlation, ut is white noise with mean zero, i.e. unpredictable, and the value of x only depends on its prior realization (Woodford 2001, p. 13, Coibion and Gorodnichenko 2015, p. 2650). Again the agent cannot observe the real state of this variable but a noisy signal of it,
yit = xt + ωt, (14) where ωt is mean-zero white noise (Woodford 2001, p. 14, Coibion and Gorodnichenko
2015, p. 2650). Under the assumption that the agents have knowledge of the underlying
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economic model, expectations of the variable we are interested in can be formed on the individual level using the Kalman filter:2
Fit(xt) = Gyit + (1 − G)Fit−1(xt) (15) respectively, by recalling equation (13):
Fit(xt+h) = ρhFit(xt) (16)
where G is the Kalman gain which is the weight that is placed on new noisy informa- tion. Consequently (1 − G), the weight put on outdated forecasts, is called the degree of information rigidity (Coibion and Gorodnichenko 2015, p. 2650). As done before, Coibion and Gorodnichenko (2015, p. 2650) average forecasts across agents and combine the expression for the average forecast from the noisy information model with the defi- nition of the rational forecast error (equation (4)) in order to point out the relationship between the ex post mean forecast errors and ex ante mean forecast revision:
xt+h − Ft(xt+h) = 1 − G(Ft(xt+h) − Ft−1(xt+h)) + vt+h,t (17) G
Here, the coefficient on forecast revisions only depends on the Kalman gain G, i.e. the weight put on new noisy information. The Kalman gain is determined by factors such as the persistence of the series and the signal-to-noise ratio (Coibion and Gorodnichenko 2015, p. 2651). In the case that agents only rely on the most recent but noisy information (G = 1) the forecast error would be unpredictable. The fact that agents do not only rely on the newest information is intuitively described by Coibion and Gorodnichenko (2015, p. 2651)) who point out that agents do not solely rely on the new noisy information since they do not know if changes only reflect noise. Consequently, new information only enters the expectation formation process gradually.
2.4 Rational Inattention Models
Completing the overview of the three main types of information rigidity theories this subsection will deal with rational inattention models. Since Coibion and Gorodnichenko (2015) treat this class of models rather negligibly as the main implications are similar to
2The Kalman filter is a mathematical procedure to calculate the value of a variable which follows an autoregressive process if only noisy observations of it are known. It requires that the value which is to be estimated is described by a dynamic system of equations (Hamilton 1994, pp. 372 sqq.).
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