OXFORD BULLETINof

ECONOMICS and STATISTICS

OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 46, 1 (1984)0305-9049 3.00

TESTS FOR RATIONALITY OF THECARLSON-PARKIN INFLATION

EXPECTATIONS DATA*

George Evans and Riyaz Gulamani

INTRODUCTION

Recent theoretical work has emphasized the role played by inflationaryexpectations in macroeconomic models and demonstrated the im-portance of the method of expectation formation in investigatingquestions such as the persistence of unemployment and the potency ofmonetary and fiscal policy. Rational inflation expectations have beenintroduced not only in equilibrium theories of aggregate supply suchas Lucas (1973) but also in overlapping wage contract models, Taylor(1979), and in some macroeconomic models not involving the naturalrate hypothesis, Henry (1980). Rational expectations of inflation ratesalso have implications for other macroeconomic questions such as theterm structure of interest rates, Modigliani and Shiller (1973).

The Livingston data of forecasts of consumer and wholesale prices(by a group of economists in the United States) have been tested forrationality by Pesando (1975), Carlson (1977), Mullineaux (1978,1980) and Figlewski and Wachtel (1981). Livingston panel forecastsof other data have been investigated by Brown and Maital (1981). Acomparable set of data does not exist for the United Kingdom.However, Canson and Parkin (1975) have used the monthly UK GallopPoll qualitative survey results on the expected direction of price

* We wish to thank Ken Holden, David Peel and Robert Ward for information concerningthe expectations data. Pauline Andrews and Michael Sumner provided helpful comments onan earlier version of this paper, and we have also benefited from discussions with TakeshiAmemiya and Tom MaCurdy. The editors and referees of this journal provided several helpfulsuggestions. The authors, of course, retain responsibility for any remaining errors. We grate-fully acknowledge the computer programming assistance of Mrs Carol Archibald, and the useof the University of Stirling Computer Centre facilities.

Volume 46 February 1984 No. 1

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changes to construct a quantitative measure of the general public'sexpectations of price change.

In the UK Gallup Poll a stratified quota sample of about 1000 peoplewere asked whether they expected prices to go up, go down or staythe same over the next six months (a 'don't know' category was alsopermitted). Carlson and Parkin assumed that each respondent had asubjective distribution for the percentage change in the individual'sprice index. The response chosen by each respondent then depended onwhether the median of that individual's probability density fell above.below or within the boundaries of the 'remain the same' category. Byfurther assuming that the medians of the subjective distribution acrossindividuals were normally distributed, Carlson and Parkin showed howit was possible to estimate the population's mean expected pricechange (and, also, the standard deviation across individuals) for eachmonth.

Difficulties with the Carlson-Parkin data are well known anddiscussed by Foster and Gregory (1977), Defris and Williams (1979)and Batchelor and Sheriff (1980). At this stage we simply remark thatwhile the construction of the data undoubtedly introduces measure-ment error into the expectations series, this does not in itself precludeexamining the series for rationality. At the end of Section 111 we returnto a discussion of how the quality of the data affects the rationalitytests and the interpretation of the results obtained.

lt should te noted that the disagreement among respondents doesnot in itself constitute evidence against rational expectations. Respon-dents are assumed to be forecasting the behaviour of the price indexof their own market basket. Since there is considerable dispersion inrates of inflation across households it would not be unusual for aproportion of individuals to expect their prices to fall when the averagerate of inflation is positive.' Furthermore, the natural interpretationof the tests presented in this paper is that they seek to determinewhether expectations are rational in the 'average' sense originally pro-posed by Muth (1961), i.e. whether the average expectation of inflationacross individuals is the same as the mathematical expectation, con-ditional on available information, of the proportional rate of change ofthe general price level.

In view of the central theoretical role played by inflation expecta-tions and the use made of this data series in empirical investigations of

Indeed, differences in the answers reported by respondents are the information exploitedby Canson and Parkin in constructing their series. The rate of inflation would be indeterminateif 100 per cent of the respondents expected prices to increase. However, this never occurredduring the period. In five months (out of 156) no respondent believed prices would go down.but in each case some respondents chose 'remain the same' or 'don't know'. In these months.Carlson and Parkin's usual method of construction breaks down but they were able to useanother related method by assuming that the standard deviation of inflation expectations wasthe same as for neighbouring observations.

CARLSON-PARKIN INFLATION EXPECTATIONS DATA 3

the determinants of wage-change (see, for example, Parkin et al. (1976)and Sargan (1 980)), it is important that the series be thoroughly testedfor rationality. Canson and Parkin (1975) and Holden and Peel (1977)have carried out empirical investigations attempting to identify theexpectation generating mechanism. Holden and Peel (1977) and, inpassing, Batchelor and Sheriff (1980) have also explicitly tested forrationality. However, their tests of rationality have been incompleteand, in some cases, flawed, as discussed below.

In this paper we focus on the narrow question of testing the expecta-tions series for rationality in the sense of being conditionally unbiasedforecasts. Because we have observations on expectations, we are able totest for rationality without having to make assumptions about the trueeconomic model generating the inflation process. The general procedureused is to regress forecast errors on variables, in particular recent pastinflation, money growth and unemployment rates, which are likely tohave been useful in predicting inflation. A statistically significantcorrelation indicates a deviation from full rationality since forecastscould have been improved by taking account of this correlation. Anestimate of the extent of the possible improvement in forecast accuracyby properly using the available information can also be made.

These are the first tests along these lines for rationality of theCarisonParkin inflation expectations series. In conducting these tests,it was necessary to deal carefully with a technical complication arisingfrom the overlapping nature of the forecast horizons, a problem whichhas been discussed by Brown and Maital (1981). Previous tests forrationality of the CarlsonParkin series, carried out by testing for serialcorrelation of forecast errors, are invalid because this complication wasignored. Statistically correct serial correlation tests for rationality arealso presented.

I. IMPLICATIONS OF RATIONALITY AND TESTABLE HYPOTHESES

Expectations of agents are said to be rational if they are equal to thetrue mathematical expectation conditioned on all relevant informationknown at the time forecasts were made. Such expectations must possessseveral properties. Let Pt = actual rate of inflation during period t and

p be the rate of inflation during period t expected by agents at theend of period t - 1. If expectations are rational then

r_iP =E(pII_1) (1)

where J is the information available at the end of t - 1. It followsthat

= 0 (2)

where Se_1 is any set of information constituting a subset of I. Fordetails see Shiller (1978) and Begg (1982, Ch. 4.1).

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According to (2) forecast errors, Pt - t-iP, are uncorrelated witheach variable in the information set. If forecast errors were correlatedwith available information then forecasts could not be rational in theabove sense since they could be improved by taking account of thecorrelation. Since t-iP is in forecast errors are uncorrelated withforecasts. Assuming also that p_ for i I (and _ip for s <t) are inle_i, so that past forecast errors are part of the information set, itfollows that forecast errors are serially uncorrelated.

The above results suggest the following tests of rationality for anexpected inflation series:

Unbiasedness. From (I) and (2) we have shown that, underrationality,

Pt=t_iP+Ut (3)

where is uncorrelated with t-iP and is serially uncorrelated. Thiscan be tested by estimating

Pt = + irtiP + U (4)

and testing the joint hypothesis (cr, ir) = (0, 1).No serial correlation. Let 1 = Pt t-iP represent measured

forecast error. The hypothesis of zero serial correlation can be testedby regressing on lagged values of itself. That is, we estimate

Kí3iø_i-4-u (5)

i= I

and test the null hypothesis H0: j3 0, i = 1,. .. , K for a range ofchoices of K.

Efficiency. Assuming past price information is readily availableand clearly relevant (i.e. Pt-i S contained in for i 1), an obvioustest of rationality is that can be uncorrelated with lagged rates ofinflation. The equation

K

= (6)i= j

is estimated and we test the null hypothesis H0: f3 = 0, i = 1,. . . , K.Orthogonality. Finally, we can test for lack of correlation between

forecast errors and other variables contained in An importantquestion is what variables are included in the information set. I mustinclude all information which is costless to acquire and process. Feigeand Pearce (1976) have emphasized that the information set is a choicevariable and that agents with economically rational expectations willset the marginal cost equal to the marginal benefit of acquiringinformation. In this paper we concentrate simply on the question of

CARLSONPARKIN INFLATION EXPECTATIONS DATA 5

whether expectations fully incorporate the information available in anumber of variables plausibly related to inflation. The model

=w1+ui (7)

is estimated where w_1 is a vector of information variables dated t -or earlier, 3 is a vector of coefficients, and the hypothesis H0:13 O istested.

In conducting the orthogonality test we chose, as regressors, laggedunemployment rates and money supply growth rates, in addition topast rates of inflation. Unemployment rates are relevant in predictinginflation under some versions of the Phillips curve, while money supplygrowth rates are considered important by many macroeconomists aspredictors of inflation over various horizons. Of course, many othervariables might also be good candidates for regressors,2 but this wasnot explored in order to conserve degrees of freedom.

The tests of rationality just described are the appropriate ones whenexpectations data are available. Typically the assumption of rationalexpectations can only be tested conditional on an assumed model ofthe economy. Because we have observations on expectations, we areable to test rationality without taking as a maintained hypothesis anyspecific economic model. Under full rationality, the agents in theeconomy do know the true process generating inflation (or, at least,make forecasts on average as if they do) but we as econometriciansneed make no assumption about what that process is.

Each of the four tests looks for violations of rationality in a particulardirection. The unbiasedness test looks for systematic over- or under-prediction correlated with the expected rate of inflation. The efficiencytest establishes whether a minimal condition for rationality is met,namely that all past price-change information has been fully utilized inforecasting inflation. The orthogonality test applies the strictercriterion of whether other specified information in addition to pastinflation has been properly used in forecasting price changes. 1f eithertoo much or too little weight (given the actual inflationary process)has been assigned, say, to past money growth in making forecasts, thiswill be detected as a correlation of forecast errors with that variable.Finally, the test of no serial correlation is essentially an omnibus test.Failure to use some unspecified information in making forecasts maywell lead to serial correlation of forecast errors. However, because noparticular alternative is specified, this test may have low power.

2 In particular, some economists may feel that lags of wages and oil prices are particularlyuseful in forecasting the price level. Of course, the relevant question is what variables provideinformatlon about inflation which was not properly exploited by forecastors, and this makesthe best choice of regressors far from obvious. However, in view of our results in Section III,the question is moot. Using the regressors described in the text, we are able to reject rationality,and this conclusion could only be strengthened by the discovery of other variables in the in-formation set correlated with forecast errors.

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Because the extent of each type of deviation from rationality isof interest, it was decided to perform separately each test (except that,for reasons discussed below, the unbiasedness test was not under-taken). However, when more than one test is performed, care must hetaken in assigning an overall significance level to the results. This isdiscussed in Section III below.

II. PROBLEMS OF RATIONALITY TESTS USING THE CARLSON-PARKIN DATA

The forecast horizon of the Carison-Parkin data is six months and thedata is available monthly from January 1961 to December 1973. Usingthis data Canson and Parkin investigated various hypotheses about theformation of expectations in the UK.3 Holden and Peel (1977) cori-ducted further empirical investigations of the Carison-Parkin data. Inaddition, they carried out the first two rationality tests described inSection I, and concluded that expectations exhibited irrationality onboth counts. These tests, however, were flawed, as now discuss.

For the unbiasedness test, Holden and Peel reversed the equation,regressing expected inflation on actual inflation and testing for a zerointercept and a unit slope. That is, they rewrite equation (3) as ==

PtUt and then estimate t-iP = + ypU, testing for t/i = O andy 1. This, however, is incorrect. Since, under rationality, u is un-correlated with it is necessarily the case that the independertvariable, Pt, m their estimated equation is negatively correlated with theerror tenu, uj, so that their estimate of 'y will be biased downwards.Holden and Peel did, in fact, find ' substantially less than 1, and thismay reflect only their incorrect procedure.

Their test of serial correlation suffers from difficulties inherent inthe use of data with overlapping forecast horizons. The quarterlyinflation forecasts they used have six month horizons, and so quarterlydata on forecast errors are in part affected by common innovations. Asdiscussed below (see also Begg (1982, p. 73)) this generates un&rrationality an MA(1), i.e. first-order moving average, form of seria!correlation. Consequently, testing for serial correlation by a simpleregression of forecast error on lagged forecast error is an invalid test cfrationality.4 Such a test would tend to find (as Holden and Peel did

In estimating error-learning mechanisms, Canson and I'arkin appear to have used 12-monthrates of change of prices for the actual price change. lt should be noted that their appendixprovides, at each date, their estimate of expected inflation over the next six months, butactual inflation over the last 12 months. In this paper we always compare expected inflatinwith actual innation over the corresponding six-month period computed directly from RetjilPrice Index data.

4Another complication with Holden and Peel's rationality tests is that forecast errors hasebeen calculated by subtracting inflation forecasts from the actual rate of inflation over a 12-month period. The investigation of the determinants of forecast error described in Holden andPeel (1979) are subject to the saine complications.

CARLSON-PARKIN INFLATION EXPlCTATI0NS DATA 7

find) a pattern of first order serial correlation which would lead investi-gators to reject rationality even when expectations were rational.

Batchelor and Sheriff (1980) carry out similar tests for rationality.While their estimate of (4) does not reveal any bias, they claim thatrationality is violated by the strong serial correlation indicated by thelow value of the DurbinWatson statistic. Again, however, this may bean artifact generated by the use of overlapping data.

As an aside, this is a convenient point to observe that rationalitycannot be casually assessed by examining a plot of actual and expectedinflation over time, such as Figure 1 of Carlson and Parkin (1975), forsystematic prediction errors at peaks or troughs of the inflation rate.As discussed above, forecast errors under rationality will necessarilybe positively correlated with the actual rate of inflation. Also, asdiscussed at the end of this section, because expectations are measuredwith error, observed forecast errors will be correlated with the con-structed expectations series under rationality. Finally, because of themoving average pattern of correlation present under rationality, withthis data, the statistical significance of apparent patterns in forecasterrors cannot be readily evaluated by looking at plots.

We turn, now, to a brief formal discussion of serial correlation withoverlapping forecast horizons. We discuss both why this problemarises and how it can be dealt with, since, if ignored, it would invalidateeach of the four rationality tests. Although the CarlsonParkin seriesis available monthly, because of the unavailability of monthly data onall infonnation variables over the sample period, we use quarterly datain our rationality tests, picking inflation forecasts made every thirdmonth. Because the forecasts concern price change over the next sixmonths, the problem is simply to consider two-period forecasts underrationality.

This problem, which has also been discussed by Brown and Maital(1981), may be analysed as follows. Let y be a vector of macro-economic variables of interest, the first component of which is theone-period rate of inflation, Pt. Assume that y has a well-defined vectorautoregressive representation, so that

Yt = ay_1 + ii (8)i= I

where a1 are constant coefficient matrices and u is a vectOr innova-tion which is serially uncorrelated and which is uncorrelated withlagged Yt We make no specific assumption about the a1. That is, weleave open the question of what particular economic process isgenerating inflation, though under rationality agents are assumed tomake forecasts as if they know the process. At the end of this sectionwe discuss the importance of restrictions implicit in (8).

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We assume that the information set consists of Yt. for s 1 Scthat under rationality the forecast error for y1 is just the innovationu1, i.e. y1-11 = ui.. The forecast for period t+ 1, made in periodt - 1, can be obtained by applying the chain rule of forecasting (seeSargent (1979, p. 268) or Begg (1982, pp. 73-74)) to equation (8).It follows that the forecast error for period t + 1 is given by y1 +1 -t-iYt+i a1u1+ U1.

We are concerned with forecasts of the total rate of inflation over thetwo-period interval covering times t and t + 1. Provided that inflationhas been measured as the difference in the logarithm of the price level,this is just Pt + Pt +, the sum of the one-period rates of inflation,5 andis forecast to be t-iP + j-P+j. Defining F1 as the analogous two-period forecast error for the entire vector y1, we have

Yt+i +y,-1_1y-1_1y1=u11 +(J+a1)u1

where I is the identity matrix. Since the first component of F + 1 is

f1+1 Pt+i + Pt t-iP - t-iP+i, we can write(10)

where u1 is the first component of u and X is the first row of(I+a1).But the u1 are serially uncorrelated and hence

cov(f,f1)=0 forlstI>l (li)

while the correlogram may be non-zero for lag 1. It follows that f1, theforecast error for the rate of inflation over periods t - 1 and t, can becharacterized as a first order moving average process.6

Because a quarterly series of six-month forecast errors is not whitenoise, but has a first-order moving average structure, the standard testsdescribed in Section I are not valid. There are a number of possibleways of dealing with the problem. One would be to choose non-over-lapping data points at six-month intervals. However, this method,which omits half the data, would in our case reduce the sample size to26, which seems unsatisfactorily small.

A second method would be to assume t-iP = t-iP+i so that wecould treat each six-month forecast as a forecast of the rate of inflation(at an annual rate) for the first three month period. We could thencompare the forecast to the actual rate of inflation over these threemonths, yielding a quarterly time series of non-overlapping forecast

1f all rates of inflation are expressed at annual rates we would divide by two, but this doesnot affect the argument.

Another way of looking at this problem is that f1_1 is not in the information set I_ scthat (2) does not imply O correlation with f1. This is the reasoning followed by Brown andMaital. Equation (10) shows how the correlation between f1 and f1_1 depends on the structure(8).

CARLSON-PARKIN INFLATION EXPECTATIONS DATA 9

errors, Ø., which, under rationality, would be white noise. However,= t-iPi holds only in special cases (see Muth (1960)). It is

clearly undesirable to conduct rationality tests which are in fact jointtests of rationality and additional assumptions about the structure.

The strategy followed in this paper is to use the quarterly series ofsix-month forecasts by subtracting them from the actual price changeover the corresponding period to yield a quarterly series of forecasterrors with six-month horizons, f1', but to allow for the first-ordermoving average process which would be generated under rationality.

Consider first the serial correlation test. Under rationality f1' isMA(1), so by Wold's decomposition theorem (see Sargent (1979,pp. 257-8)) there exists a representation

f1'=c1+Oe1'_1 (12)

where I O < 1 and e is white noise. Furthermore is the innovationin predicting f1' by the linear least squares projection of f1' on its ownpast, so that is uncorrelated with all past values off1'.

The serial correlation test thus consists in estimating

K

f1'= 131f1'1+e1+Oe1'_1 t= l,...,T (13)i=J

where f1' is the forecast error covering periods t - 1 and t, made at theend of period t - 2. The null hypothesis of rationality is H0: = O for¡ = J,... , K, where J ( K. This test may be carried out in the followingway. Substituting in recursively for lagged e- in (13), and making theasymptotically valid simplifying assumption of = 0, we find that themodel can be transformed to

Kf1'* ß1f1'i1' + (14)

i=J

where

t-1 t-1f* = (-U)1f1'1 and 1-1*_i = :; (-O)'f1'_1_1 (IS)/=0

Using the method of prediction error decomposition, it is shown inHarvey (1981, pp. 100-105) that if the are normally distributed,maximum likelihood is equivalent to choosing O and ¡3 to minimize thesum of aquares se?. This procedure, which is essentially GLS(generalized least squares) and will be referred to by that name, can heimplemented by choosing a grid of O and running a series of ordinaryleast squares regressions. One can show that the likelihood ratio testof H0 is carried out by computing the statistic T log (S SER /SSE),where SSER and SSE1J are the restricted and unrestricted sum of

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squared residuals. Under the null hypothesis this statistic has a x2distribution with (KJ) degrees of freedom.

We turn now to the efficiency and orthogonality tests. These can becarried out by estimating the model

j; = w12f3 + t = 1,..., T (ls)where i is MA(l) and m is uncorrelated with w_ for s 2. For theefficiency test w5.2 is a vector of past inflation rates known at timet - 2 while for the orthogonality test it may include any relevantvariables known at time t - 2. Under the null hypothesis of rationalitywe have H0: f3 = 0.

In general it is not correct to estimate the model (16) by GLS, asis appropriate in the serial correlation test. This procedure would bevalid if w_2 consisted of variables that were strongly exogenous, inthe sense of Engle, Hendry and Richard (1983), as shown in Harvey(1981, pp. 207-209). However, as Brown and Maital have pointed out,it is not generally valid when w_2 consists of lagged endogenousvariables, as is surely the situation with the efficiency and orthogon-ality tests. In this case the appropriate procedure is to estimate f3 in(16) by OLS (ordinary least squares), which is still consistent, hutto correct the estimated covariance matrix for the serial correlation.

This procedure is - discussed in Hansen (1982) and is the onedescribed and followed by Brown and Maital. Under plausiblieassumptions it can be shown7 that /T(ß-f3) converges in distributionto a normal random vector with O means and covariance matrix Q givenby

Q (1

= plim - w'w) (hm E _W'fW) (pum - w'w)T / T /

where 2 is the covariance matrix of ? and W is the data matrix of theregressors consisting of T rows of w.2. Given that Th is MA(1), &2 canbe estimated consistently by the sample variance and first-order auto-covai-iance of the OLS residuals and used to construct a consistentestimate Q of Q. Restrictions on f3 can then be tested by the usualWald test. In particular, the null hypothesis H0: f3 = O can be tested bythe statistic Tf3Q1/3 which has a limiting x2 distribution with degreesof freedom equal to the number of parameters.

Finally, unbiasedness tests were not undertaken because of thenature of the data. Due to their manner of construction, the Carlson-Parkin expectations series measures true inflation expectations whh

Equation (17) can be derived using the representation of forecast error in terms of gicnby (10). We need to assume that E(ufilwt_2, wt3) =Eu1. SinceE(ut_1Iwt_,, wt..3) = O thiswould follow from strengthening the assumption of zero correlation to independence which,for example, would be guaranteed if the distributions were normal. Q can be estimated byQ= T(W'W)-W'nW(W'W)-'.

(1 7)

CARLSON-PARK1N INFLATION EXPECTATIONS DATA 11

error. Consequently estimates of (4) are likely to lead to downwardlybiased estimates of ir. This difficulty does not affect the other testsin which forecast error is the dependent variable. Provided the measure-ment error is white noise these tests will be valid.

Before turning to the empirical results, we discuss briefly the extentof the restrictions implicit in the model (8) and the subsequent analysis.lt is not strictly necessary to require stationarity of Yt and we couldeven permit the a to depend on calendar time. Provided that expecta-tions are rational in the sense of being conditionally unbiased forecasts,correctly taking account of any such parameter shifts, equation (10)continues to hold. However, in order to conduct valid tests of ration-ality, as set forth above, it is necessary to assume that fi., the forecasterror under rationality specified in (1 0), has a constant variance andfirst-order autocorrelation coefficient over time. Clearly some assump-tion of this sort must be satisfied in order to undertake rationalitytests at all. While under stationarity this assumption is automaticallysatisfied, it is an additional restriction if some model parameters arechanging over time. In any case, since the calculation of conditionallyunbiased forecasts in a nonstationary world would in many circum-stances be an implausibly demanding task, it has been usual foradvocates of the rational expectations hypothesis to assume astationary structure at least over moderate stretches of time. We returnbriefly to the question of model change in Section IV.

III. EMPIRICAL RESULTS

Tables 1, 2 and 3 present the results of the rationality tests using theCarlsonParkin expectations series. In each case the dependent variableis the forecast error constructed by subtracting the expected rate of

TABLE 1Test of No Serial correlation

Generalized least squares estimates

The sample runs from 1963:1 to 1973:4, providing 44 observations. f. forecast errordefined in text. O = first-order moving average parameter. s = estimated standard deviation ofjesidual forecast error, computed as the square-root of (1 + 02) times the estimated standarddeviation of e. The sample standard deviation of actual forecast error over the period was3.295.

f- 2 0.7 0.01 0.920 3.276ft-a,ft-4,ft-s 0.7 10.88 0.012 2.967It- 3ft- 8 0.7 12.59 0.050 3.026

Independent Likelihood ratiovariables o statistic Significance level s

Additional tests:

O coefficients on lagged inflationO coefficients on lagged money growthO coefficients on lagged unemployment

The sample runs from 1961:1 to 1973:4, providing 52 observations. p = first-order auto-correlation of residuals. s = standard error of regression. The sample standard deviation ofactual forecast error over the period was 3.223. See the appendix for precise definition of p.

TABLE 3Orthogonality Tests

Ordinary least squares with correction for covariance matrix

(a) Short sample. 1964 :4 to 1973 :4, 37 observations. Independent variables:3 lags each of inflation rates, money supply growth rates and unemploymentrates

p Chi-square statistic Significance level s

0.17 67.97 below 0.00001 2.309

Chi-square Significancestatistic level

11.62 0.00914.55 0.00211.26 0.010

(b) Long sample: 1961 :1 to 1973:4, 52 observations. Independent variables:3 lags each of inflation rates and unemployment rates

p Chi-square statistic Significance level s

0.15 42.23 below 0.00001 2.455

s = standard error of regression. p = first-order autocorrelation of residuals. The standarddeviation of forecast error was 3.5 15 for the short sample and 3.223 for the tong sample. Seethc appendix for precise definitions of variables.

Independent Ch i-squarevariables p statistic Significance level s

3 lags of Pt 0.41 9.28 0.026 3.0086 lags ofp 0.42 12.18 0.058 3.038

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TABLE 2Efficiency Test

Ordinary least squares estimates with correction for covariance matrixIndependent variables: lags of actual inflation, Pt

CARLSONPARKIN INFLATION EXPECTATIONS DATA 13

inflation from the actual rate of inflation, measured by the RetailPrice Index, over the matching six-month horizon. Data is quarterlyand the MA( I) process generated under rationality is allowed for inone of the two ways discussed in the preceding section. When GLSis used to estimate O under the assumption of full rationality a valueof O = 0.7 is obtained.

Before discussing the test results the issue of information lags mustbe addressed. Independent variables in rationality tests must be in theinformation set at the time forecasts are made, and it is thereforeimportant to allow for publication lags. In the tests reported we haveallowed for a full one-quarter delay in price and unemployment dataand a two-quarter delay in money supply data becoming publiclyavailable when constructing the independent variables used in theregressions.8 To the extent that information was available with a shorterdelay the tests will be biased towards acceptance.

A related issue is that a delay in price information would also meana delay in information about forecast errors so that there is the possi-bility of a second-order moving average process being generated forforecast errors under rationality. In fact, however, the sample second-order autocorrelation for forecast errors is 0.05, so that this does notappear to be important. Furthermore, when forecast error is regressedon a two-quarter lag of forecast error, using GLS to allow for an MA( I)process, the coefficient is not only small but statistically quite in-significant (as may be seen from the first line in Table 1), so that thispossible difficulty can be ignored.

We turn now to the empirical results. We present initially thenominal significance level for each test, not adjusted for the fact thatseveral tests are being conducted. We later consider the overall signifi-cance level of the results.

The tests for serial correlation are presented in Table 1. The GLStests show that the null hypothesis of rationality can be clearly rejectedat the 5 per cent level when forecast error is regressed on its three, fourand five quarter lags.9 The effect is not particularly large in size, with anestimated gain in average forecast error of 9.96 per cent beingobtainable by taking into account lagged forecast errors. One possible

Another problem that we have not dealt with is data revisions. It would be quite difficultand time consuming to determine the exact information available to agents at each time.Furthermore since each revised series constitutes new information which in principle could beincluded as regressors there would probably be a degrees of freedom problem. Mullineaux(1980) found data revisions to be unimportant in rationality tests of the Livingston data.

9A constant was also included in this regression and all others in Tables 1,2 and 3. The teststatistics reported are for the hypothesis that the coefficients arc O for all regressors other thanthe constant. Similar results are obtained if the null hypothesis sets it equal to zero.

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explanation is that agents are not incorporating all the informationavailable to them from other variables which may themselves be seriallycorrelated.

Table 2 reports the results of the efficiency tests. Although the nullhypothesis is rejected at the 5 per cent level for three lags of theinflation rate, there is borderline acceptance for six lags. The size of theeffect can be illustrated by comparing the standard error of the OLSregression with three lags to the sample standard deviation of forecasterror over the period which shows a gain of only 6.67 per cent. Thus,although not all past price change information seems to have been in-corporated in forecasts, it nevertheless appears that not much informa-tion in past inflation data, taken on its own, remained to be exploited.

The orthogonality tests, reported in Table 3, show the clearestresults. Quarterly money supply data, measured as M3, was not regu-larly available until 1963: 1, so results are given both for a short sampleregression with lagged money growth, unemployment and inflationused as independent variables and a long sample regression dropping themoney growth variables. These tests show unequivocably that fullrationality must be rejected, with significance levels of less than one-thousandth of 1 per cent. Additional tests on subsets of coefficientsshow that each of past inflation, money change and unemploymentrate data are not fully incorporated in forecasts. The stronger results forinflation data, compared to the efficiency tests, suggest that they maycontain important information when used in conjunction with othervariables.

The orthogonality results indicate that the gain in forecasting accuracyfrom fully and correctly using money supply, inflation and unemploy-ment information was moderately large. In terms of the standarddeviation of forecast error, the results indicate a potential gain of34.3 1 per cent. The extent of deviation from rationality might be evenlarger since it is possible that other information variables could havebeen used to further reduce the size of forecast errors.

We now consider the overall significance level of these results, giventhat several tests were undertaken. Because the test statistics arecorrelated in an unknown way, it is not possible to compute an exactsignificance level. However, it is sufficient to compute an upper boundusing the Bonferroni inequality. Counting each version of each testseparately, a total of ten tests were performed. Thus, the probability,under the null hypothesis of rationality, that at least one test wouldyield a significance level of less than 0.00001, the level obtained in twoof the orthogonality tests, has an upper bound of 0.0001. In preliminaryversions of this paper, minor variants of some of the tests were con-ducted. Allowing for these in the same conservative way (althoughmany of these test statistics were undoubtedly strongly positivelycorrelated), we still obtain an overall significance level well below

CARLSONPARKIN INFLATION EXPECTATIONS DATA 15

one-tenth of 1 per cent. This clearly indicates a deviation from fullyrational expectations during the period. 10

Finally we consider how these results may be affected by the qualityof the Carlson-Parkin data. In particular we consider the followingthree problems: pure measurement error, the scaling assumption, andthe assumption of a normal distribution.

Clearly the Carlson-Parkin method measured the true mean expectedrate of inflation with error, for example due to sampling fluctuations.As mentioned at the end of Section II, this would result in an 'errorsin the variables' problem in the case of the unbiasedness test, biasingit towards rejection of rationality. Consequently that test was not per-formed. Providing that this measurement error was pure white noise,however, the other tests remain valid since the measurement error isincorporated into the dependent variable. The main problem to beexpected is a reduction in power of the tests. Since the orthogonalitytests reject rationality, this result is not affected. Indeed, it is somewhatstrengthened in the sense that the estimated potential gain in fore-casting accuracy (from using additional information) of 34.31 per centis an underestimate, in proportional terms, to the extent that ourforecast error data include pure measurement errors.

The other two problems involve possible systematic biases in theconstruction of the expectations data. The assumptions made byCanson and Parkin were sufficient only to construct the series up to afactor of proportionality, t5 (the boundary of the 'remain the same'category). ô was then chosen to make the average expected rate ofinflation equal to the average actual rate over the sample period. Sincethis is the assumption most consistent with rational expectations, ourtests may be biased towards acceptance of rationality in the sense thatsome types of systematic error would be undetected. However, sincethe orthogonality test rejects rationality, this result is unaffected.

Finally, we turn to the normality assumption. Foster and Gregory(1977) argue that the population distribution of the median expecta-tions of individuals may be skewed, distorting the estimates of Carlsonand Parkin, and that in particular this may introduce a systematic biasrelated to the level or rate of change of inflation. This raises the possi-bility that our tests falsely indicate irrationality when they are in fact

'°Instead of separate seriar correlation, efficiency and orthogonality tests, an alternativestatistical procedure would have been to conduct one overall test by regressing forecast errorson all relevant information variables, allowing for an MA(1) error, and also to conduct aLagrange Multiplier test for the residual being white noise. Of course, since the number of lagsof each variable to be included cannot be determined a priori, multiple tests would remain acomplication. Still, it is possible that such a procedure, using nested hypotheses, would havebeen more powerful than ours. The procedure actually followed was chosen in part becausethere was interest in the extent of deviation from rationality under each of the alternatives. Inany event, since we are able to reject rationality, the question of power turns out not to havebeen crucial.

16 BULLETIN

detecting a systematic pattern of measurement error correlated withrecent inflation experience. However, while it may be possible toexplain the apparent lack of full efficiency of forecasts in this way, itis difficult to see how the stronger orthogonality test results could beexplained on this basis. This is because, as discussed above, theorthogonality test showed clearly that both money growth and un-employment information could have been used to improve forecastswhen controlling for recent inflation rates.11

In summary, while the manner of construction of the Canson andParkin expectation series should be kept in mind when interpreting thetest results, it seems quite unlikely that the quality of the data can beresponsible for the principal result obtained by the orthogonality test,namely, that there were substantial deviations of inflation forecastsfrom full rationality during the period.

IV. CONCLUSIONS

Taking the widely-used UK inflation expectations series, covering196 1-73, devised by Canson and Parkin, we have tested expectationsfor rationality in the sense of being conditionally unbiased forecastsgiven available information. Allowance has been made for overlappingforecasts which lead forecast errors to follow a first-order movingaverage process under rationality.

The serial correlation and efficiency tests suggest a deviation fromfull rationality, but the size of the effect is not large. In particular, theefficiency test, which tests for rationality by examining whether pastinflation data has been fully utilized in making the forecasts, indicatesthat full exploitation of past inflation information would haveimproved forecast accuracy, in terms of the standard deviation offorecast error, by less than 7 per cent. However, the orthogonality test.which examines whether forecasts made full use not just of inflationdata but also of other publicly available information, indicates both ahighly significant deviation from full rationality and a large effect. Thecorrelation of forecast errors with past price change, money supplychange and unemployment data, taken together, is significant at theone-tenth of 1 per cent level (conservatively adjusting the significancelevel to allow for other tests undertaken) and the possible gain inpredictive accuracy was more than a 30 per cçnt decrease in thestandard deviation of the forecast error.

Foster and Gregory also argue that there is a bias in Carlson and Parkin's special trcatmenfor the five months (out of 156) in which no respondent believed prices would go down. Whilemeasurement error is likely to have been larger in these months, the arguments given by Fostc-and Gregory do not appear to establish a bias (which would depend in turn on a bias in theestimate of the standard deviation). In any case, only one or two of these points are likely tobe in our quaj-terly saniple. (We cannot determine which points they are because Canson aociParkin do not report the categorical data for reasons of confidentiality.)

CARLSONPARKIN INFLATION EXPECTATIONS DATA 17

Several caveats are in order. The extent to which the results mightbe affected by the quality of the CarisonParkin data was discussed atthe end of Section III. Apart from this issue, there are still two qualifi-cations to the conclusion of deviation from full rationality.

First there is the question of economic rationality discussed inFeige and Pierce (1976). We do not know whether improved forecastswould have been cost-effective because we do not know the costsinvolved. Secondly, there is the possibility of changes in the economicstructure over the period. Since structural change can alter thestochastic properties of forecast errors under rationality, it is possiblethat the significance levels of our tests are affected. More fundamen-tally, it raises deeper issues of rationality, in particular the question oflearning discussed by B. Friedman (1979). If, when the economicstructure changes, it takes time for agents to learn the new inflationgenerating process, then forecasts will not be conditionally unbiasedover this period. Thus the requirement that expectations be condition-ally unbiased is implausibly demanding when model changes areoccurring. Indeed, if structural change occurs too rapidly, agentsfollowing sensible learning rules may never converge on the trueprocess.12 A number of deep related issues concerning rational be-haviour, learning, and stability are discussed in Frydman and Phelps(1983).

Thus our results do not show that agents were irrational whenrationality is defined in a broad sense. Our basic conclusions, however,stand. Provided the CarlsonParkin technique is sound, our testsindicate that over the sample period the average forecast of agentscould have been substantially improved using available economicinformation. Expectations were not 'rational' in the sense of con-ditionally unbiased forecasts, and the extent of the deviation was nottrivial. Depending on the macroeconomic model, this could haveimportant implications for policy.

Stanford UniversityDate of Receipt of Final Manuscript: Mai 1983.

By splitting the sample one might be able to say something further about learning orstructural breaks. We decided not to pursue this because of the moderate size of the sample andthe asymptotic justification of the tests.

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APPENDIX: SOURCES OF DATA

Inflation expectations is the pe series from the Carlson and Parkin(1975) appendix. January, April, July and October points chosen asquarterly data (1961:1 to 1973:4). Forecast errors are actual minusexpectation inflation, where the former is measured as the annualizedrate of change of the RPI over the corresponding six-month period. Thusthe first data point for forecast error is the percentage rate of change ofthe RPI (at an annual rate) from January to July 1961 minus theCanson and Parkin pe for January 1961.

Efficiency and orthogonality tests use data constructed from theRetail Price Index for January, April, July and October, taken from theMonthly Digest of Statistics, various issues, the money supply, adjustedend of quarter M3, taken from the Economic Trends Annual Supple-ment 1978, and the unemployment rate, UK percent unemployed,quarterly average, taken from the Economic Trends Annual Supple-ment 1978.

Independent variables for the efficiency and orthogonality tests areconstructed as follows: inflation is the percentage change in the RPIover six months; money supply change is the change in M3 over twoquarters; and the unemployment rate is the average over two quarters.The most recent lag of each independent variable allows for a possibleinformation delay, as discussed in the text. Additional lags are con-structed by further quarterly shifts of these variables.