In making decisions about the future, economic agents and policymakers have to form expectations about the permanence of current developments. First, when deciding how to allocate an increase in income between consumption and savings, individuals need to evaluate the permanence of this increase. Second, a worker’s decision about whether to accept or reject a poor job offer depends on his perception about the permanence of this condition. Third, a firm’s investment decision following strong demand for its product depends on its perception of the persistence of this state. Fourth, when confronted with a strong economy, monetary policymakers may consider an increase in the policy rate. But if they believe the strength is temporary, they are likely to postpone the increase. Similar considerations apply to contractionary fiscal policies.
Generally, even when individuals possess full information about current and past realisations of relevant variables, individuals remain uncertain about their permanence. In many cases, individuals detect the permanence of changes by observing the persistence of those changes over time. As a consequence, when permanent changes occur they are recognised only gradually. Adaptive expectations capture this sluggishness by making the difference between the current and the previous period’s forecasts a positive function of the forecast error committed in the previous period. Almost 60 years ago, Muth (1960) showed that when a stochastic variable is composed of a random walk and a white noise process, neither of which is ever observed separately, adaptive expectations are rational in the sense that they utilise all available information in an efficient manner.1 In spite of this, individuals are never fully certain – not even ex post – about the permanence of economic developments. We refer to this residual uncertainty as the ‘permanent-transitory confusion’.
In this column, based on a recent paper (Cukierman et al. (2018), we summarise basic features of Muth’s rationally adaptive expectations, consider their implications for standard tests of market efficiency in the treasury bills and foreign exchange markets, and utilise data on inflationary expectations from the Israeli capital market to examine the performance of Muth’s model in tracking those expectations during the turbulent 1985 Israeli stabilisation, as well as during the stable 2003-2018 period.
Salient features of Muth’s (1960) rationally adaptive expectations
Let yt be a stochastic variable composed of a (permanent) random walk and a (transitory) white noise, neither of which is ever observed separately. Muth’s optimal predictor of future values of yt has three notable and convenient features:
- First, it implies that it is optimal to utilise all past observations on in order to forecast the future.
- Second, the optimal predictor is a Koyck lag with geometric weights that decrease the further in the past the observation on yt is.
- Third, the weights sum up to one.
Importantly, the larger the adaptive expectations coefficient, θ, the larger the sum of the weights on the most recent past in comparison to the more distant past. Consequently, the larger θ is, the faster the speed at which individuals detect a permanent change when such a change has occurred, implying that θ characterises the speed of learning. Finally, the learning coefficient, θ, is an increasing function of the ratio between the first difference of the permanent variance and of the transitory variance.
The more general message of the preceding discussion is that although predictors of the future are forward looking, they normally rely on past information since the past contains useful, albeit noisy, information about the future. During the early days of the rational expectations revolution, some economists criticised adaptive expectations on the ground that they are backward rather than forward looking. This criticism is probably based on perfect foresight models, like that of Barro and Gordon (1983), that do not feature stochastic terms. In such models, rational expectations reduce to the (known with certainty) values of relevant variables, as predicted by such models. But once the more realistic existence of uncertainty and the permanent-transitory confusion are incorporated into models, the role of past information in predicting the future becomes essential. Muth’s predictor provided an early, convenient way to capture the main features of the permanent-transitory confusion and to relate it to natural intuition.2
Implications of the permanent-transitory confusion for tests of market efficiency in the treasury bills and foreign exchange markets
Fama (1975) type tests of efficiency in the treasury bill market (as predictors of inflation) proceed by regressing the current realisation of inflation on a lagged capital market variable that embody the preceding period’s expectation of inflation. Relying on Fisher’s theory of interest, this signalling variable is taken to be the lagged value of the nominal interest rate. In tests of efficiency of foreign exchange markets, the signalling variable is taken to be the forward exchange rate leading to formulations in which the current rate of change in the exchange rate is regressed on the rate of change implied by the past forward rate. In either case, the appearance of serial correlation in the residuals of those regressions is considered to be evidence against market efficiency. The intuition supporting this view is that, if markets are efficient, rational individuals should have used it in their predictions leading to the disappearance of serial correlation.
The main analytical proposition reported in this section is that, in the presence of the permanent-transitory confusion, the appearance of serial correlation in finite samples does not necessarily imply that markets are inefficient. In particular, when a relatively large permanent change occurs against an environment characterised by a low speed of learning, it takes a long time for individuals to realise that a large change has occurred.3 During this process, forecast errors appear to be serially correlated, since individuals rationally interpret most of the change to be transitory shocks. This result originates from the fact that it is optimal for individuals to utilise all past information to forecast the future in combination with an environment in which the speed of learning is low because large permanent changes are relatively rare.
Stated in more technical terms, when the signal-to-noise ratio in forecasting permanent shocks is low, it is optimal to interpret most of the observed changes in the signal as being transitory. When, a relatively large permanent change occurs in such an environment, it takes some time for individuals to realise this fact.4 In finite samples that are dominated by such a shock, forecast errors appear to be serially correlated. It is important to note that, in the population, forecast errors are uncorrelated, implying that the appearance of serial correlation is limited to finite samples that are dominated by large permanent shocks. In our paper (Cukierman et al. 2018), we present an example of such a case.
The upshot is that in tests of efficiency of the Treasury bill market, the failure to reject serial correlation can be misleading if applied to samples taken shortly after violent changes in the purchasing power of money. Similarly, the serial correlation test may yield wrong conclusions about the efficiency of the foreign exchange market if applied during, or shortly after, large permanent changes in the exchange rate.
Fitting Muth’s process to expectations formation during the Israeli 1985 ‘cold turkey’ stabilisation and during the tranquil inflation targeting period (2003-2018)
A ‘cold turkey’ or ‘shock’ stabilisation refers to a situation in which high inflation is stabilised very aggressively within a short period of time. Following seven years with yearly rates of inflation of 100% or more, and several failed attempts to stabilise inflation, Israel finally managed to stabilise it in July 1985, bringing the rate of inflation down from about 400% to almost zero within a couple of months. This dramatic drop was achieved through the simultaneous deployment of conventional measures, like restrictive fiscal and monetary policies, as well as less conventional measures such as temporary controls on prices, wages, and the exchange rate.
With the benefit of hindsight, it can be concluded that the 1985 cold turkey stabilisation produced a large permanent drop in the rate of inflation. However, at the time of the stabilisation, there was substantialuncertainty about the extent to which this dramatic drop would persist. The uncertainty was induced by wide gyrations in inflation and several failed attempts to stabilise prior to the 1985 successful stabilisation. It is therefore instructive to examine the behaviour of inflationary expectations before and after the 1985 stabilisation. Although capital market inflationary expectations were not calculated on a systematic basis prior to the mid-nineties, they were occasionally estimated prior to that time.5
Fitting Muth’s process to the Israeli 1985 stabilisation produces a speed of learning coefficient of about one third and a corresponding signal-to-noise ratio of 0.14. The estimate of the learning coefficient implies that, in each period, about one third of the previous period’s forecast is revised in the direction of the forecast error committed in the preceding period. It can be seen from the figure below that the fit of the process, particularly after the stabilisation, is quite good.6
Figure 1 Actual and simulated three-months-ahead inflation expectations, January 1984 to October 1986
Since the mid-1990s the Bank of Israel has been deriving estimates of breakeven expected inflation. Due to the absence of long-term nominal bonds at the start of the period, those estimates were initially limited to forecast horizons of one year. But, as inflation subsided at the beginning of the 21stcentury, the Israeli treasury issued nominal bonds with longer maturities, making it possible to derive longer term inflationary expectations from the bond market, up to a ten-year horizon.
Fitting Muth’s process to those long-term expectations during the tranquil 2% inflation-targeting period that started in 2003 and persists to this day does not yield good results. Instead, the estimation procedure supports the conclusion that, during this period, individuals considered all deviations from the long run 2% inflation target to be transitory. This finding is consistent with the view that, since 2003, long-term inflationary expectations in Israel have been well anchored.
Barro, R J and D B Gordon (1983), “A positive theory of monetary policy in a natural rate model,” Journal of Political Economy 91(4): 589–610.
Cagan, P D (1956), “The monetary dynamics of hyper-inflation,” in M Friedman (ed), Studies in the Quantity Theory of Money, University of Chicago Press.
Cukierman, A (1988), “The end of the high Israeli inflation: An experiment in heterodox stabilization,” in M Bruno, G Di Tella, R Dornbusch and S Fischer (eds), Inflation Stabilization: The Experience of Israel, Argentina, Brazil, Bolivia and Mexico, The MIT Press.
Cukierman, A, T Lustenberger and A H Meltzer (2018), “The permanent-transitory confusion: Implications for tests of market efficiency and for expected inflation during turbulent and tranquil times,” CEPR Discussion Paper 13187.
Fama, E F (1975), “Short-term interest rates as predictors of inflation,” The American Economic Review 65(3): 269–282.
Friedman, M (1957), A Theory of the Consumption Function, Princeton University Press.
Kalman, R E (1960), “A new approach to linear filtering and prediction problems,” Transactions of the ASME - Journal of Basic Engineering 85(Series D): 35–45.
Muth, J F (1960), “Optimal properties of exponentially weighted forecasts,” Journal of the American Statistical Association 55(290): 299–306.
Muth, J F (1961), “Rational expectations and the theory of price movements,” Econometrica 29(3): 315–335.
 This paper should not be confused with the Muth’s much better-known 1961 paper that inspired Lucas’ rational expectations revolution in macroeconomics.
 But it is by no means, the only way to do that. A multi-variables generalisation is provided by the Kalman Filter (Kalman 1960). Two well-known applications of adaptive expectations are Friedman (1957) who utilised them to model individual perceptions about permanent income, and Cagan (1956) who used them to model the formation of inflationary expectations during hyperinflationary episodes.
 Although this proposition is demonstrated analytically within the framework of Muth’s rationally adaptive expectations, it extends to most stochastic processes in which the persistence of changes is not known with certainty even after they are realised.
 Although the probability that such a shock occurs is not large, it is not zero.
 Capital market inflation expectations (also known as breakeven expectations) are derived from the difference in yields between nominal and indexed bonds. Data on such expectations for a three-month forecasting horizon is available for January 1984 to October 1986 and is taken from Table 2.2 of Cukierman (1988).
 Details about the fitting procedure appear in Cukierman et al. (2018).