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In economics, adaptive expectations is a hypothesized process by which people form their expectations about what will happen in the future based on what has happened in the past. For example, if people want to create an expectation of the inflation rate in the future, they can refer to past inflation rates to infer some consistencies and could derive a more accurate expectation the more years they consider.

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One simple version of adaptive expectations is stated in the following equation, where $p^{e}$ is the next year's rate of inflation that is currently expected; $p_{-1}^{e}$ is this year's rate of inflation that was expected last year; and $p$ is this year's actual rate of inflation:

$p^{e}=p_{-1}^{e}+\lambda (p-p_{-1}^{e})$ where $\lambda$ is between 0 and 1.  This says that current expectations of future inflation reflect past expectations and an "error-adjustment" term, in which current expectations are raised (or lowered) according to the gap between actual inflation and previous expectations. The error-adjustment term, also called partial adjustment, allows for variations in inflation rates over the previous years, especially years that have abnormally high or low rates.

$\lambda (p-p_{-1}^{e})$ The above term is the partial adjustment error term, this term allows for variances that occur between actual values and expected values. The importance of considering the error prevents over and under expecting values of in the above example inflation rates. The adjustment means that the expectation can tend toward the direction of the future expected value that would be closer to the actual value, this allows a prediction to be made and consideration to be added or removed so as to be accurate of the future expectation. This consideration or error term is what allows the predicted value to be adaptable, thus creating an equation that is adaptive of the expectation being inferred.

The theory of adaptive expectations can be applied to all previous periods so that current inflationary expectations equal:

$p^{e}=\lambda \sum _{j=0}^{\infty }((1-\lambda )^{j}p_{j})$ where $p_{j}$ equals actual inflation $j$ years in the past. The adding of a time series portion to the expectation equations accounts for multiple previous years and their respective rates in forecasting like the above example of the future inflation rate. Thus, current expected inflation reflects a weighted average of all past inflation rates, where the weights get smaller and smaller as we move further in to the past. The initial previous year has the highest weighting and the subsequent years take lesser weighting the further back the equation accounts for.

When an agent makes a forecasting error (as in incorrectly recording a value or mistyping), the stochastic shock will cause the agent to incorrectly forecast the price expectation level again even if the price level experiences no further shocks, since the previous expectations only ever incorporates part of their errors. The backward nature of expectation formulation and the resultant systematic errors made by agents (see cobweb model) had become unsatisfactory to economists such as John Muth, who was pivotal in the development of an alternative model of how expectations are formed, called rational expectations. The use of rational expectations have largely replaced adaptive expectations in macroeconomic theory since its assumptions rely on an optimal expectations approach which is consistent with economic theory. However, it must be stressed that confronting adaptive expectations and rational expectations aren't necessarily justified by either use, in other words, there are situations in which following the adaptive scheme is a rational response.

The first use adaptive expectations hypothesis was to describe agent behavior in The Purchasing Power of Money by Irving Fisher (1911), then later used to describe models such as hyperinflation by Philip Cagan (1956).  Adaptive expectations were instrumental in the consumption function (1957) and Phillips curve outlined by Milton Friedman. Friedman suggests that workers form adaptive expectations of the inflation rate, the government can easily surprise them through unexpected monetary policy changes. As agents are trapped by the money illusion, they are unable to correctly perceive price and wage dynamics, so based on Friedman's theory, unemployment can always be reduced through monetary expansions. If the government chooses to fix a low unemployment rate the result is an increasing level of inflation for an extended period of time. However, in this framework, it is clear why and how adaptive expectations are problematic. Agents are arbitrarily supposed to ignore sources of information which, otherwise, would affect their expectations. For example, government announcements are such sources. Agents are expected to modify their expectations and break with the former trends when changes in economic policy necessitate it. This is the reason why the theory of adaptive expectations is often regarded as a deviation from the rational tradition of economics. 

## Related Research Articles In economics, inflation refers to a general progressive increase in prices of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reduction in the purchasing power of money. The opposite of inflation is deflation, a sustained decrease in the general price level of goods and services. The common measure of inflation is the inflation rate, the annualised percentage change in a general price index. In economics, "rational expectations" are model-consistent expectations, in that agents inside the model are assumed to "know the model" and on average take the model's predictions as valid. Rational expectations ensure internal consistency in models involving uncertainty. To obtain consistency within a model, the predictions of future values of economically relevant variables from the model are assumed to be the same as that of the decision-makers in the model, given their information set, the nature of the random processes involved, and model structure. The rational expectations assumption is used especially in many contemporary macroeconomic models. New Keynesian economics is a school of macroeconomics that strives to provide microeconomic foundations for Keynesian economics. It developed partly as a response to criticisms of Keynesian macroeconomics by adherents of new classical macroeconomics. Interest, in finance and economics, is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum, at a particular rate. It is distinct from a fee which the borrower may pay the lender or some third party. It is also distinct from dividend which is paid by a company to its shareholders (owners) from its profit or reserve, but not at a particular rate decided beforehand, rather on a pro rata basis as a share in the reward gained by risk taking entrepreneurs when the revenue earned exceeds the total costs. The Phillips curve is a single-equation economic model, named after William Phillips, hypothesizing an inverse relationship between rates of unemployment and corresponding rates of rises in wages that result within an economy. Stated simply, decreased unemployment, in an economy will correlate with higher rates of wage rises. Phillips did not himself state there was any relationship between employment and inflation; this notion was a trivial deduction from his statistical findings. Paul Samuelson and Robert Solow made the connection explicit and subsequently Milton Friedman and Edmund Phelps put the theoretical structure in place. In so doing, Friedman was to successfully predict the imminent collapse of Phillips' a-theoretic correlation.

In finance, arbitrage pricing theory (APT) is a general theory of asset pricing that holds that the expected return of a financial asset can be modeled as a linear function of various factors or theoretical market indices, where sensitivity to changes in each factor is represented by a factor-specific beta coefficient. The model-derived rate of return will then be used to price the asset correctly—the asset price should equal the expected end of period price discounted at the rate implied by the model. If the price diverges, arbitrage should bring it back into line. The theory was proposed by the economist Stephen Ross in 1976. The linear factor model structure of the APT is used as the basis for many of the commercial risk systems employed by asset managers. A macroeconomic model is an analytical tool designed to describe the operation of the problems of economy of a country or a region. These models are usually designed to examine the comparative statics and dynamics of aggregate quantities such as the total amount of goods and services produced, total income earned, the level of employment of productive resources, and the level of prices. The ΛCDM or Lambda-CDM model is a parameterization of the Big Bang cosmological model in which the universe contains three major components: first, a cosmological constant denoted by Lambda associated with dark energy; second, the postulated cold dark matter ; and third, ordinary matter. It is frequently referred to as the standard model of Big Bang cosmology because it is the simplest model that provides a reasonably good account of the following properties of the cosmos:

The Hodrick–Prescott filter is a mathematical tool used in macroeconomics, especially in real business cycle theory, to remove the cyclical component of a time series from raw data. It is used to obtain a smoothed-curve representation of a time series, one that is more sensitive to long-term than to short-term fluctuations. The adjustment of the sensitivity of the trend to short-term fluctuations is achieved by modifying a multiplier . The filter was popularized in the field of economics in the 1990s by economists Robert J. Hodrick and Nobel Memorial Prize winner Edward C. Prescott. However, it was first proposed much earlier by E. T. Whittaker in 1923.

In economics, money illusion, or price illusion, is the name for the human cognitive bias to think of money in nominal, rather than real, terms. In other words, the face value of money is mistaken for its purchasing power at a previous point in time. Viewing purchasing power as measured by the nominal value is false, as modern fiat currencies have no intrinsic value and their real value depends purely on the price level. The term was coined by Irving Fisher in Stabilizing the Dollar. It was popularized by John Maynard Keynes in the early twentieth century, and Irving Fisher wrote an important book on the subject, The Money Illusion, in 1928. Temporal difference (TD) learning refers to a class of model-free reinforcement learning methods which learn by bootstrapping from the current estimate of the value function. These methods sample from the environment, like Monte Carlo methods, and perform updates based on current estimates, like dynamic programming methods.

Neutrality of money is the idea that a change in the stock of money affects only nominal variables in the economy such as prices, wages, and exchange rates, with no effect on real variables, like employment, real GDP, and real consumption. Neutrality of money is an important idea in classical economics and is related to the classical dichotomy. It implies that the central bank does not affect the real economy by creating money. Instead, any increase in the supply of money would be offset by a proportional rise in prices and wages. This assumption underlies some mainstream macroeconomic models. Others like monetarism view money as being neutral only in the long-run.

The cobweb model or cobweb theory is an economic model that explains why prices might be subject to periodic fluctuations in certain types of markets. It describes cyclical supply and demand in a market where the amount produced must be chosen before prices are observed. Producers' expectations about prices are assumed to be based on observations of previous prices. Nicholas Kaldor analyzed the model in 1934, coining the term "cobweb theorem", citing previous analyses in German by Henry Schultz and Umberto Ricci.

The policy-ineffectiveness proposition (PIP) is a new classical theory proposed in 1975 by Thomas J. Sargent and Neil Wallace based upon the theory of rational expectations, which posits that monetary policy cannot systematically manage the levels of output and employment in the economy.

The Lucas islands model is an economic model of the link between money supply and price and output changes in a simplified economy using rational expectations. It delivered a new classical explanation of the Phillips curve relationship between unemployment and inflation. The model was formulated by Robert Lucas, Jr. in a series of papers in the 1970s.

Quantal response equilibrium (QRE) is a solution concept in game theory. First introduced by Richard McKelvey and Thomas Palfrey, it provides an equilibrium notion with bounded rationality. QRE is not an equilibrium refinement, and it can give significantly different results from Nash equilibrium. QRE is only defined for games with discrete strategies, although there are continuous-strategy analogues.

Dynamic stochastic general equilibrium modeling is a macroeconomic method which is often employed by monetary and fiscal authorities for policy analysis, explaining historical time-series data, as well as future forecasting purposes. DSGE econometric modeling applies general equilibrium theory and microeconomic principles in a tractable manner to postulate economic phenomena, such as economic growth and business cycles, as well as policy effects and market shocks.

The random walk model of consumption was introduced by economist Robert Hall. This model uses the Euler numerical method to model consumption. He created his consumption theory in response to the Lucas critique. Using Euler equations to model the random walk of consumption has become the dominant approach to modeling consumption.

In probability theory and statistics, Campbell's theorem or the Campbell–Hardy theorem is either a particular equation or set of results relating to the expectation of a function summed over a point process to an integral involving the mean measure of the point process, which allows for the calculation of expected value and variance of the random sum. One version of the theorem, also known as Campbell's formula, entails an integral equation for the aforementioned sum over a general point process, and not necessarily a Poisson point process. There also exist equations involving moment measures and factorial moment measures that are considered versions of Campbell's formula. All these results are employed in probability and statistics with a particular importance in the theory of point processes and queueing theory as well as the related fields stochastic geometry, continuum percolation theory, and spatial statistics.

The Taylor contract or staggered contract was first formulated by John B. Taylor in his two articles, in 1979 "Staggered wage setting in a macro model'. and in 1980 "Aggregate Dynamics and Staggered Contracts". In its simplest form, one can think of two equal sized unions who set wages in an industry. Each period, one of the unions sets the nominal wage for two periods. This means that in any one period, only one of the unions can reset its wage and react to events that have just happened. When the union sets its wage, it sets it for a known and fixed period of time. Whilst it will know what is happening in the first period when it sets the new wage, it will have to form expectations about the factors in the second period that determine the optimal wage to set. Although the model was first used to model wage setting, in new Keynesian models that followed it was also used to model price-setting by firms.

1. Evans, G.W.; Honkapohja, S. (2001). "Expectations, Economics of". International Encyclopedia of the Social & Behavioral Sciences. pp. 5060–5067. doi:10.1016/B0-08-043076-7/02245-2. ISBN   978-0-08-043076-8.
2. Mollik, Andrea V. "Adaptive Expectations". Encyclopedia.com. Retrieved 16 April 2021.
3. Galbács, Peter (2015). The Theory of New Classical Macroeconomics. A Positive Critique. Contributions to Economics. Heidelberg/New York/Dordrecht/London: Springer. doi:10.1007/978-3-319-17578-2. ISBN   978-3-319-17578-2.

### General references

• George W. Evans and Seppo Honkapohja (2001), Learning and Expectations in Macroeconomics. Princeton University Press, ISBN 978-0-691-04921-2