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Does inflation uncertainty increase with the level of inflation?
European Economic Review 25 (1984) 2 13-221. North-Holland
DOES INFLATION UNCERTAINTY INCREASE WITH THE LEVEL OF INFLATION?
George BULKLEY* Univwsity of Exe,ter; Exeter EX4 4RJ, UK
Receiv;ud February 1983, final version received March 1984
This paper presents evidence that if agents forecast inflation rationally, using an est’“nate of the reduced form equation which generated the data, then the size of their forecast errors is positively correlated with the level of inflation. Forecast errors are measured first as the residuals from a full sample OLS regression, and secondly from one period ahead, outside sample, forecasts using a regression estimated from only data available! at the time of the forecast. Thus, agents who form rational expectations about the variance, as well as the mean, of inflation should form conditional variances dependent on the level of inflation, at the date of the forecast.
1. introduction The question. of whether there 1.) ‘c a correlation between the level of inflation and the degree of uncertainty about future inflation is impsrtanr for several reasons. In measuring the welfare costs of inflation, the greater part of the costs is related to uucertainty about inflation and not its 1eveI per se. Such uncertainty distorts resource allocation and income distribution. If this uncertainty varies systematically with the rate of inflation, then choice of inflation targets must take it into account. It is also relevant to the estimation and interpretation of macroeconomic -4els where the level of inflation and uncertainty about inflation are both candi&&~~.., ?up@natory variables. The incorrect exclusion of a variable measuring Mation un&&z *I!~611lead to biased coefficients on an inflation level term. For example, it is this corr& Li;:Cduhichprovides the basis for the explanation by Friedman (1977) of the upward slops+. -34lips curve, and for the argument of Levi and Makin (1979) that finding the coefficient on inflation is less than unity in an equation explaining nominal interest rates does not refute the Fisher proposition about the constancy of the real interest rate. The objective of this paper is to present some additional evidence for this correlation. Section 2 discusses methods of obtaining a measure of agents’ *I am grateful to J.M. Black, J. Foreman-Pet-k, A.C. Harvey and M. Timbre11 for helpful discussions and two referees for comments 01 an earlier draft. Special thanks are due to J. Rabe and M. Brown who carried out the computations. 0014-2921/84,$3.00 0 19’84,Elsevier Science Publishers B.V. (North-Holland)
G. Bu.Yey, hjhztion uncertdnt,rymd be1 of i@htion
214
subjective xmccrtainty, and sections 3 and 4 present the results of testing for a correlation between the two proposed measures and the level of inflation.
The empiricA support for a correlation between the level of inflation and uncertainty is inter-country and intra-county evidence of a correlation between the average level of inflation over some period and its variability within that period. Variability is measured either as the variance of inflation rates or as the average acceleration in inflation within the period. Crosscountry evidence of this kind has been provided by Okun (1971), Gordon (197I), Lucz~ (1973), M&e and Kleiman (1975), Logue and Willett (19?6), and Foster (1979). i&a-country evidence has been contributed by Gale (1981) who al-so presents a good survey and summary of the cross-country studies. However variability of inflation cannot ntxx~sariiy be identified with unpredictability and uncertainty. Oniy if the expected inflation rate were the f&l period mean rate, in the case of the variance measure, or the previous period’s rate, in the case of the average acceleration measure, could variability be equated with fonzza&ng errors. Even accepting that agents may not forecast using the: true model, it wo&d not seem that either of these expectation S&SZXMS .mentioned abo*reis even being used as a ruie of thu&, judging &J the eAence on the determinants of the Livingston Price cxpectat&ns data. [k for example, Turnovsky (19’70) and Figiewski and Wachte! QZ9%1).-j In this paper it is assumed that agents form rational expcctiirtions, using thaf esti.ma~~%d reduced form equation which best explains inllation using only variabh whose vahzes wzrc: puMicauy rrva&ble iu the previous time per&. If th absollute size of & remking predi&on errors is wmehlted with the leveis &f inflation at t&42fonXast datq then awn& forming rational eqectatitnu; about the variance, as well as the mean, should form condition& foreG& villriaaoesbaby+ on the ~~:r:v&ngIevcl of inflation. i.n the tests which Two di&rent ILItEasures of prediction errors are u fuouow: e-e
1.
Ftdl .mple
rep?ssim2
,
Tiis
assutnes that agents at
date L< T obtain forecasts for it + 1) by su~t~tut~ng kaown values of exogenous variables in&ding the current error, in the case of an autoconeMed error process., into an quatioon estimated aa~r the whole s.~~ple El, TJ.
G. Bulkley, Inflation uncertairlty and level of inJlation
21s
If the true model is constant in the wholle sample [I, TJ, then whilst full
sample residuals are not identical to the prediction errors which would actually have been made, the tiw of any bias is independent of the level of inflation, and hence, if the null hypothesis is rejected using full sample residuals, it may be inferred that actual absolute forecast errors are correlated with the level of inflation. (See the appendix). However if the true model is not constant, the relationship between recursive, outside sample, forecast errors made by agents using a model estimated only from currently available data and full sample residuals cannot be specifkd, and hence: the hypothesis should be tested using Measure II. Section 3 reports the results of testing for the correlation using Measure I. This amounts to testing for heteroscedasticity. Since there is some evidence that there may have been structural change in the model within the sample, the results using Measure II are reported in section 4. 3. Empirical result, when full sample residuals are wed as a rn~e
of
fbreaut errors
Reduced form equations to explain the inflation rate were estimated for the UK using quarterly data 19W-1980. Since the objective is to obtain a forecasting equation, all indepndent variables were lagged at least one quarter. If contemporaneous exogenous variables had been included it would have been necessary to estimate a mechanism for forecasting them, and this two-step procedure wirs considered beyond the scope of this paper. Pretesting with up to six lags of inflation, the rate of increase on the money supply (MI and MT), and the rate of increase of money wage rates (hourly,
UK manufacturing) resulted in the choice of the following model for the rate of growth of the Consumers Price Index, judged as best by R2, parameter significance, and absence of autocorrelation: 7*O.SP,_.‘+0.43P,_~-ot4P,_~, (4.81) ( .- 1.48) (2.11) I6.23)
R2==0.51,
(1,
Durbin h statistic s+ - 1.45.
Using the test of Godfrey (1978) for higher order serial correlation in the de~ndent variable gave a x: = 1.3, xi = 1.34, presence of a so the hypothesis of no serial correlation rn:ky x-; :=: 1.34, xi = 8.9 be accepted at the 0.05 SijJ
explanatory variables, other than the Although the absence history of the variable in es inflation technically exogenous in the sense of ~~~n~~~Sirns causality testmg, it is of course yoscible, and indeed likely, that contemporanl:ous values of other e~~la~~to~ variables woaild contribute Ft R
C’
la~a~,~o~ of’ i~~ati~~. The fact that their lagged
values do not is in keeping with the argument that prices at date t embody all information, including current values of exogenous variables, at th ly for heteroscedasticity the test proposed by is appbed, This test is based on direct use uf uals from 5tisingle rqressiorr On the com#ete test in ,that the precise nature of ver, &.ree the o&xtive here is to establish whether he&m&i&y is present or not, and there is no iuterrtion to re+&imate tie equations if heter0xzdasticity is four& this is adequate fur the purpose of this p&per3.and is co:~lnputation&llycor&derably s&pier than the constructive tests. [See3 for example, Goldfeld and Quandt ( ~9W.3 The Tut involves the calcuhttion of the ratio of the sum of squares of a given subst of residuals to the total residual sum of squares That is b calcufated as e'Ae b=-----
e’e
’
where e is the (n x 1) vector oF regression residuals and A is a selector matrix of order (n x rrj with itl (Utm K nj ones and (n - nrj zeros on its principal diagonal, and zerus c&where. The choice of zeros and on= along the diagonal of A was such that A selected the m residmals where inflation in the previous period was at an annual rate of less than loo//,. Since both the numerator and denomin;itor of b depend on X, the matrix of observations, it I?snot possible to formulate a test criterilon for direct use with Ix However, Harrison and McCabe (1979) show that, in ihe case where there are more observations than independent variables in. both subset!; of the data, 172and (n-m), then b, < h c b,,, where for a given level of significance r, the critical values of & and by may be determined as: (n-m)
be
1 *(n-.m-k)
_
u[
F,*(n--nz,m-k) -m-k F,*(n--m--.k,m) m
1
I, -l
where Fz (. *)deeot?s the tabled c1!critical value of F.
G. Bulkiey, InJation
uncertainty and level of inflation
217
Then, if b as calculated above is less than b,*, the hypothesis that there is no heteroscedasticity would be rejected at that level of significance. A problem in applying the test in the context of eqs. (1) or (2) is that they include lagged values of the dependent variable and the bounds test, like the Durbin-Watson statistic, is only correct in the case of fixed regressors. We respond tcr this problem in two ways. Firstly, instead of conducting tests formally at the 0.05 significance level, all tests are reported for the 0.01 signif&mce level. Secondly an asymtotic LM test using the caiculated value of b is av,ailable.2 Under the mull hypothesis the appropriate LM statistic is approxim.atef
y
Values of this statistic are reported below. Far the UK, selecting the ones and zeros alonf, the diagonal of A_ whenever inflation in the previous quarter is less thar! an annual rate ol IO:‘, gives a value for m of 861,II= f 17 and k = 3 so that b .--3.419. bz = 6.57 at the 0.01 significance level. Selecting 4% inflation as the div”Jing line gave a value of b = 0.11 and bt = 0 19; similarly 50/ inflation gave a value for b of 0.239 and b; -0.253; 7% ie!t‘ration gave a value for b of 0.323 and bZ = 0.45, all *values of bZ being calculatzzd at the 0.01 significance levet. Thus the hypothesis of homoscedasticity disturbances may b_ rejected at this significance level. Calculating the Lagrange multiplier statistic for eq. {I) when the l;alue of b was 0.419, gave a figure of -5.47, so on the basis If this statistic the nul! hypothzsis of homoscedasticity may be rejected as the 0.005 significance level. In order to obtain point estimates of the responsiveness of inflation uncertainty. measured by OLS residuals, rI, to the level of inflation P, _ 1, several functional fonms for the relationship were tried, the following being finally chosen, on the criterion of W2and absence of serial correlation in the error process: r,l -20. t 3.
85 -t O.O84P,2.. 1, (4.13
‘t’ statistics in I~ar~~~~b~sis~R2 =O.P 3,
H,f= 2.:%4.
This implies an elasticity of response of iariance to inflation at an a~ni~~~linf ation rate of IO?;. The hypothesis of Kahe.
who pointed this statistic out to me in private corKspondence.
tiety in the error process of eq. (1) cannot be directly tested using the ‘t’ statistic on Pf.-I in eq. (2) because if the error process in eq. (1) is heteroscedastic, so also will be the error in eq. (2), and so the QLS estimates of the variance of the estimated coefficients are incorrect? However the OLS estimators of the coefficients are consistent, and asymptoticahy unbiased. Thus, there is powdid ~:vi&nce from the UK that the residuals from the full sample regression exhibit heteroscedasticity. As discussed above however, full sample residuals are only an appropriate measure of forecasting errors of the structural model remained constant. There ic evidence to suggest it did not. Dividing the data into two equal size subperiods and performing a CZhowtest for structural stability, an F statistic of 5 was obtained, so the hypothesis of structural constancy can be rejected at the 0.01 significance level. Before turning in the next section to an examination of the properties if the actual forecast errors that would h.ave been made using stepwise reggrescions, the methodology of this section may be compared to that of Engle (1982,1983), who also bases the main body of his analysis on the assumption that agents at da.te t < T work with a model estimated over the full sample period 7’. Engle (1982), using the ARCH (autoregressive conditional heteroscedasticity) model assumes that the variance of the true model is not constant, but instead depends on the values of past squared disturbances. Agents who use this model to forecast inflation will therefore form conditional forlecast variances, obtained by substituting the realized values of past disturbances into the estimated autogression for the variance. Eng#e (1982), using UK data, examines whether such conditional inflation variances are correlated with the level of inflation, and finds they are. However he only examines whether the condition forecasts of variance, a.nd not the actual disturbances, are correlated with the level of inflation. The results of this section suggest a structural explanation for the behaviour of the variance of the error process, which was specified by Engle as an autoregression.
tioaforecast
errors when 0 y current infor~~~~~n is used to estimate
If the hypothesis of structural constancy cannot be accepted, then the OLS residuals can no longer be sonsidered acceptable proxies for the actual forecast errors of agents who are continually making outside sample gredictions. In this section we obtigin a measure of forecast errors which would have actually been made by agents who re-estimated the coefidents of eq. (1) using only currently available data, and examine whether such errors are tolerated with the level of i atjon. Eq. (1) is first re-estimated over the first “Ifi:I the:erlWrin the tfLne
G. Bulkley,
Inflation
uncerrainty
and level C$ inflation
219
40 quarters. Using this estimated model a forecast for quarter 41 is obtained
and the actual forecast error is Irecorded. The data set is then updated by the 41st observation and a forecast for period 42 recorded, and so forth.4 Before these forecast errors can be identified as those made by fully informed agents behaving optimally, it is necessary to check that there is no other way in which these forecasts could, on average, be improved upon. In particular, given the evidence that structural change occurred during the period, it is possible that agents could do better by discarding the earliest observations as new ones become available. The gains in precision from a larger sample size have to be balanced against the probability that the early observations belong to a different model. A priori, the net effect could go eith.er way and so it is necessary to examine empirically whether better forecasts could have been made on average by only using more recent observations. Inflation forecasts, and forecast errors, were recorded for the same period as before, 1960-1980, except that at each date only the previous x observations were used to estimate the forecasting equation, values of x between 22 and 40 being tried. The mean square forecasting error was 1.81% using t!re continuously augmented sample. Limiting the sample size resulted in fluctuations in the mean square forecasting error between 1.85% (sample size held fixed at 22) and 1.78% (sample size held fixed at 32). Wence working with a smaller sample siz,p makes very marginal difference to the size of forecast errors, and the difference may go in either direction, depending on the exact sample size chosen. Thus, we proceed with the interpretation of the forecast errors from the continuously augmented sample as those made by agents forecasting optimally. It may also be of interest to compare the errors made by agents forecasting using these stepwise regressions with the mean of the squared values of the OLS residuals over the same period. The latter was 1.49x, confirming that there is some larger amount of uncertainty to be explained. than the OLS residuals measure. The relationship between the level of inflation, P,, and the size of the forecast error, c,,, was estimated as
e;==o.oooo35 -t- O.l95P,2_ (1.18) R2=0.54,
1,
(3)
(9.8) DIY=2.31,
years 196Q-1980, “?’statistics in parenthesis.
The level of inflation explains over 50% of the variance in the size of squared forecast errors, and the hypothesis that squared forecast errors are positively 4Although structural instability could take the form of different explanatory var;;bles being appropriate in dliff~xent subperiods, the task rf ~~etest~~g to otbtain the eizsr spcification fobr eaoh of the 82 eqluations was beyond the scope oi this paper.
with the’level of inflation may be accepted at a high confidence level. The elasticity of conditional. foreeast variance, the fitted value #, with respect to the level of inflation is 1.55, evaluated at 107; inflatior.
correlated
Whether it is IX& residuals or one step ahead forecatst errors which are us& as rn~ures -of agents r~$stakes in forecasting inflaGon, there is strong evidence “that the size of these mistakes is correlated. wit.h the level of inflation. Thereforee, agents who form rational. expectations about the variance, as well as the mean, of jn&+tion~will have conditional forecast variances which increase with the level of inflation. Although these results are contingent upon the correct spzzification of the forecasting equation, the degree of confidence with which the hypothesis may be accepted suggests that the overall conclusions should be robust with respect to small changes in specification.
TG see this let a2/t be the forecast variance at date t using a recursive regression, $ the variance of the full sample residuals at date t, cr2 the true error variance, Tthe full sample size, and F, the sample size t mean, where Pt is the level of inflation. Consider the autoregression Pt =G + Wt _ 1 + r, .s me&
Whilst it might be expected that c;,,+ I > r2 t+ 1, the possibility remains that Pt could lie closer to P, than does P,, so offsetting the effects of a smaller sample size However what is important is that the extent of the bias depends on the deviation of inflation from the sample mean, and not its absolute lev& 5T&se cm 1~ &r&d easily from, for example, the exprewion for forecast errors in Piadyck and ~~bi~~;~~ (1976_ p. 163).
!X& A~tore~~s~~e conditiod atim: ~~~~~et~i~a
heterosmlasticity 50, 987-1
with estimates cf the variance of
G. &11&y, Injlation uncertainty and level of inflation
221
Engle, R.F., 1983, Estimates of the variance of U.S. inflation ba,sed upon the ARCH Model, Journal of Money, Credit and Banking 15,286301. Foster, E., 1979, The variability of inflation, Review of Economics and Statistics 61, 346-350. Figlewski, 3. a?d Wachtel, 1981. The formation of inflationary expectations, Review of Economic.~ id Statistic:: 63, I-10. Friedman, M., ;977, Nobel lecture: inflation and Jnemployment, Journal of Political Economy 85,451-472. Gale, W.A., 1981, Temporal variability of the United States consumer price index, Journal of Money, Credit r.nd Banking 13,273-297. Godfrey, L.C., 1978, Testing for higher order serial correlation ia regression equations when the regressors include lagged dependent variables, Econometrica 46, 1303-13 10. Goldfeld, S.M. and R.E. Quandt, 1965, Some tests for homoscedasticity, Journal of the American Statistical Association 60, 584-595. Gordon, R.J., 1971, Steady anticipated infl?tinn: Mirage or oasis, Brookings Papers on Economic Activity 2, 499-510. Harrison, M.J. and B. McCabe, 1979, A .est for heterosccdasticity based on ordinary least squares residuals, Journal of the Amerifaan Statistical Association 74,494-499. JaII&e, D. and E. Kleiman, 1975, The welfare implication of uneven inflation, Paper -resented at the I.E.A. conference on inflatio;, theory and anti-inflation policy, Saltsjobaden, %veden. Levi, M.D. and J.H. Makin, 1919, Fisher, Phillips, Friedman and the measured impact of inflation on interest, Journal of Finance 31, 35-52. Logue, D.E. and T.D. Willett, 1976, A note on the relation between the rate and variability of inflation, Economics 43, 151-158. Lucas, R.E., 1972, Econometric testing of the natural rate hypothesis, Conference on the econometrics of price determination (Washington, DC) 50-59. Lucas, R.E., 1973, Some intemationai evidence on output-inflation trade-offs, American Economic Review 63,326334. Piuth, J.F., 1961, Rational expectations and the theory of price movements, Econometrica 29. 315-335. Okun, A., 1971, The mirage of steady inflation, Brookings Papers on Economic Activity 2, 485498. Pesando, J.E., 1975, A note on the rationality of the Livingsiolr mice expectations, Journal of Political Economy 83,849858. Pindyck, R.S. and D.L. Rubinfeld., 1976, Econometric methods and economic forecasts (Mc&aw-Hill, New York). Sargent, T.J., 1972, Rational expecta.tions and the term structure of interest rates, Journal of Money. Credit and Banking 4,74--97. Tumovsky, S.J., 1970, Empirical evidence of the formati*n of price expectations, Journal of rhe American Statistical Association 65, 1441-1454.
DOES INFLATION UNCERTAINTY INCREASE WITH THE LEVEL OF INFLATION?
George BULKLEY* Univwsity of Exe,ter; Exeter EX4 4RJ, UK
Receiv;ud February 1983, final version received March 1984
This paper presents evidence that if agents forecast inflation rationally, using an est’“nate of the reduced form equation which generated the data, then the size of their forecast errors is positively correlated with the level of inflation. Forecast errors are measured first as the residuals from a full sample OLS regression, and secondly from one period ahead, outside sample, forecasts using a regression estimated from only data available! at the time of the forecast. Thus, agents who form rational expectations about the variance, as well as the mean, of inflation should form conditional variances dependent on the level of inflation, at the date of the forecast.
1. introduction The question. of whether there 1.) ‘c a correlation between the level of inflation and the degree of uncertainty about future inflation is impsrtanr for several reasons. In measuring the welfare costs of inflation, the greater part of the costs is related to uucertainty about inflation and not its 1eveI per se. Such uncertainty distorts resource allocation and income distribution. If this uncertainty varies systematically with the rate of inflation, then choice of inflation targets must take it into account. It is also relevant to the estimation and interpretation of macroeconomic -4els where the level of inflation and uncertainty about inflation are both candi&&~~.., ?up@natory variables. The incorrect exclusion of a variable measuring Mation un&&z *I!~611lead to biased coefficients on an inflation level term. For example, it is this corr& Li;:Cduhichprovides the basis for the explanation by Friedman (1977) of the upward slops+. -34lips curve, and for the argument of Levi and Makin (1979) that finding the coefficient on inflation is less than unity in an equation explaining nominal interest rates does not refute the Fisher proposition about the constancy of the real interest rate. The objective of this paper is to present some additional evidence for this correlation. Section 2 discusses methods of obtaining a measure of agents’ *I am grateful to J.M. Black, J. Foreman-Pet-k, A.C. Harvey and M. Timbre11 for helpful discussions and two referees for comments 01 an earlier draft. Special thanks are due to J. Rabe and M. Brown who carried out the computations. 0014-2921/84,$3.00 0 19’84,Elsevier Science Publishers B.V. (North-Holland)
G. Bu.Yey, hjhztion uncertdnt,rymd be1 of i@htion
214
subjective xmccrtainty, and sections 3 and 4 present the results of testing for a correlation between the two proposed measures and the level of inflation.
The empiricA support for a correlation between the level of inflation and uncertainty is inter-country and intra-county evidence of a correlation between the average level of inflation over some period and its variability within that period. Variability is measured either as the variance of inflation rates or as the average acceleration in inflation within the period. Crosscountry evidence of this kind has been provided by Okun (1971), Gordon (197I), Lucz~ (1973), M&e and Kleiman (1975), Logue and Willett (19?6), and Foster (1979). i&a-country evidence has been contributed by Gale (1981) who al-so presents a good survey and summary of the cross-country studies. However variability of inflation cannot ntxx~sariiy be identified with unpredictability and uncertainty. Oniy if the expected inflation rate were the f&l period mean rate, in the case of the variance measure, or the previous period’s rate, in the case of the average acceleration measure, could variability be equated with fonzza&ng errors. Even accepting that agents may not forecast using the: true model, it wo&d not seem that either of these expectation S&SZXMS .mentioned abo*reis even being used as a ruie of thu&, judging &J the eAence on the determinants of the Livingston Price cxpectat&ns data. [k for example, Turnovsky (19’70) and Figiewski and Wachte! QZ9%1).-j In this paper it is assumed that agents form rational expcctiirtions, using thaf esti.ma~~%d reduced form equation which best explains inllation using only variabh whose vahzes wzrc: puMicauy rrva&ble iu the previous time per&. If th absollute size of & remking predi&on errors is wmehlted with the leveis &f inflation at t&42fonXast datq then awn& forming rational eqectatitnu; about the variance, as well as the mean, should form condition& foreG& villriaaoesbaby+ on the ~~:r:v&ngIevcl of inflation. i.n the tests which Two di&rent ILItEasures of prediction errors are u fuouow: e-e
1.
Ftdl .mple
rep?ssim2
,
Tiis
assutnes that agents at
date L< T obtain forecasts for it + 1) by su~t~tut~ng kaown values of exogenous variables in&ding the current error, in the case of an autoconeMed error process., into an quatioon estimated aa~r the whole s.~~ple El, TJ.
G. Bulkley, Inflation uncertairlty and level of inJlation
21s
If the true model is constant in the wholle sample [I, TJ, then whilst full
sample residuals are not identical to the prediction errors which would actually have been made, the tiw of any bias is independent of the level of inflation, and hence, if the null hypothesis is rejected using full sample residuals, it may be inferred that actual absolute forecast errors are correlated with the level of inflation. (See the appendix). However if the true model is not constant, the relationship between recursive, outside sample, forecast errors made by agents using a model estimated only from currently available data and full sample residuals cannot be specifkd, and hence: the hypothesis should be tested using Measure II. Section 3 reports the results of testing for the correlation using Measure I. This amounts to testing for heteroscedasticity. Since there is some evidence that there may have been structural change in the model within the sample, the results using Measure II are reported in section 4. 3. Empirical result, when full sample residuals are wed as a rn~e
of
fbreaut errors
Reduced form equations to explain the inflation rate were estimated for the UK using quarterly data 19W-1980. Since the objective is to obtain a forecasting equation, all indepndent variables were lagged at least one quarter. If contemporaneous exogenous variables had been included it would have been necessary to estimate a mechanism for forecasting them, and this two-step procedure wirs considered beyond the scope of this paper. Pretesting with up to six lags of inflation, the rate of increase on the money supply (MI and MT), and the rate of increase of money wage rates (hourly,
UK manufacturing) resulted in the choice of the following model for the rate of growth of the Consumers Price Index, judged as best by R2, parameter significance, and absence of autocorrelation: 7*O.SP,_.‘+0.43P,_~-ot4P,_~, (4.81) ( .- 1.48) (2.11) I6.23)
R2==0.51,
(1,
Durbin h statistic s+ - 1.45.
Using the test of Godfrey (1978) for higher order serial correlation in the de~ndent variable gave a x: = 1.3, xi = 1.34, presence of a so the hypothesis of no serial correlation rn:ky x-; :=: 1.34, xi = 8.9 be accepted at the 0.05 SijJ
explanatory variables, other than the Although the absence history of the variable in es inflation technically exogenous in the sense of ~~~n~~~Sirns causality testmg, it is of course yoscible, and indeed likely, that contemporanl:ous values of other e~~la~~to~ variables woaild contribute Ft R
C’
la~a~,~o~ of’ i~~ati~~. The fact that their lagged
values do not is in keeping with the argument that prices at date t embody all information, including current values of exogenous variables, at th ly for heteroscedasticity the test proposed by is appbed, This test is based on direct use uf uals from 5tisingle rqressiorr On the com#ete test in ,that the precise nature of ver, &.ree the o&xtive here is to establish whether he&m&i&y is present or not, and there is no iuterrtion to re+&imate tie equations if heter0xzdasticity is four& this is adequate fur the purpose of this p&per3.and is co:~lnputation&llycor&derably s&pier than the constructive tests. [See3 for example, Goldfeld and Quandt ( ~9W.3 The Tut involves the calcuhttion of the ratio of the sum of squares of a given subst of residuals to the total residual sum of squares That is b calcufated as e'Ae b=-----
e’e
’
where e is the (n x 1) vector oF regression residuals and A is a selector matrix of order (n x rrj with itl (Utm K nj ones and (n - nrj zeros on its principal diagonal, and zerus c&where. The choice of zeros and on= along the diagonal of A was such that A selected the m residmals where inflation in the previous period was at an annual rate of less than loo//,. Since both the numerator and denomin;itor of b depend on X, the matrix of observations, it I?snot possible to formulate a test criterilon for direct use with Ix However, Harrison and McCabe (1979) show that, in ihe case where there are more observations than independent variables in. both subset!; of the data, 172and (n-m), then b, < h c b,,, where for a given level of significance r, the critical values of & and by may be determined as: (n-m)
be
1 *(n-.m-k)
_
u[
F,*(n--nz,m-k) -m-k F,*(n--m--.k,m) m
1
I, -l
where Fz (. *)deeot?s the tabled c1!critical value of F.
G. Bulkiey, InJation
uncertainty and level of inflation
217
Then, if b as calculated above is less than b,*, the hypothesis that there is no heteroscedasticity would be rejected at that level of significance. A problem in applying the test in the context of eqs. (1) or (2) is that they include lagged values of the dependent variable and the bounds test, like the Durbin-Watson statistic, is only correct in the case of fixed regressors. We respond tcr this problem in two ways. Firstly, instead of conducting tests formally at the 0.05 significance level, all tests are reported for the 0.01 signif&mce level. Secondly an asymtotic LM test using the caiculated value of b is av,ailable.2 Under the mull hypothesis the appropriate LM statistic is approxim.atef
y
Values of this statistic are reported below. Far the UK, selecting the ones and zeros alonf, the diagonal of A_ whenever inflation in the previous quarter is less thar! an annual rate ol IO:‘, gives a value for m of 861,II= f 17 and k = 3 so that b .--3.419. bz = 6.57 at the 0.01 significance level. Selecting 4% inflation as the div”Jing line gave a value of b = 0.11 and bt = 0 19; similarly 50/ inflation gave a value for b of 0.239 and b; -0.253; 7% ie!t‘ration gave a value for b of 0.323 and bZ = 0.45, all *values of bZ being calculatzzd at the 0.01 significance levet. Thus the hypothesis of homoscedasticity disturbances may b_ rejected at this significance level. Calculating the Lagrange multiplier statistic for eq. {I) when the l;alue of b was 0.419, gave a figure of -5.47, so on the basis If this statistic the nul! hypothzsis of homoscedasticity may be rejected as the 0.005 significance level. In order to obtain point estimates of the responsiveness of inflation uncertainty. measured by OLS residuals, rI, to the level of inflation P, _ 1, several functional fonms for the relationship were tried, the following being finally chosen, on the criterion of W2and absence of serial correlation in the error process: r,l -20. t 3.
85 -t O.O84P,2.. 1, (4.13
‘t’ statistics in I~ar~~~~b~sis~R2 =O.P 3,
H,f= 2.:%4.
This implies an elasticity of response of iariance to inflation at an a~ni~~~linf ation rate of IO?;. The hypothesis of Kahe.
who pointed this statistic out to me in private corKspondence.
tiety in the error process of eq. (1) cannot be directly tested using the ‘t’ statistic on Pf.-I in eq. (2) because if the error process in eq. (1) is heteroscedastic, so also will be the error in eq. (2), and so the QLS estimates of the variance of the estimated coefficients are incorrect? However the OLS estimators of the coefficients are consistent, and asymptoticahy unbiased. Thus, there is powdid ~:vi&nce from the UK that the residuals from the full sample regression exhibit heteroscedasticity. As discussed above however, full sample residuals are only an appropriate measure of forecasting errors of the structural model remained constant. There ic evidence to suggest it did not. Dividing the data into two equal size subperiods and performing a CZhowtest for structural stability, an F statistic of 5 was obtained, so the hypothesis of structural constancy can be rejected at the 0.01 significance level. Before turning in the next section to an examination of the properties if the actual forecast errors that would h.ave been made using stepwise reggrescions, the methodology of this section may be compared to that of Engle (1982,1983), who also bases the main body of his analysis on the assumption that agents at da.te t < T work with a model estimated over the full sample period 7’. Engle (1982), using the ARCH (autoregressive conditional heteroscedasticity) model assumes that the variance of the true model is not constant, but instead depends on the values of past squared disturbances. Agents who use this model to forecast inflation will therefore form conditional forlecast variances, obtained by substituting the realized values of past disturbances into the estimated autogression for the variance. Eng#e (1982), using UK data, examines whether such conditional inflation variances are correlated with the level of inflation, and finds they are. However he only examines whether the condition forecasts of variance, a.nd not the actual disturbances, are correlated with the level of inflation. The results of this section suggest a structural explanation for the behaviour of the variance of the error process, which was specified by Engle as an autoregression.
tioaforecast
errors when 0 y current infor~~~~~n is used to estimate
If the hypothesis of structural constancy cannot be accepted, then the OLS residuals can no longer be sonsidered acceptable proxies for the actual forecast errors of agents who are continually making outside sample gredictions. In this section we obtigin a measure of forecast errors which would have actually been made by agents who re-estimated the coefidents of eq. (1) using only currently available data, and examine whether such errors are tolerated with the level of i atjon. Eq. (1) is first re-estimated over the first “Ifi:I the:erlWrin the tfLne
G. Bulkley,
Inflation
uncerrainty
and level C$ inflation
219
40 quarters. Using this estimated model a forecast for quarter 41 is obtained
and the actual forecast error is Irecorded. The data set is then updated by the 41st observation and a forecast for period 42 recorded, and so forth.4 Before these forecast errors can be identified as those made by fully informed agents behaving optimally, it is necessary to check that there is no other way in which these forecasts could, on average, be improved upon. In particular, given the evidence that structural change occurred during the period, it is possible that agents could do better by discarding the earliest observations as new ones become available. The gains in precision from a larger sample size have to be balanced against the probability that the early observations belong to a different model. A priori, the net effect could go eith.er way and so it is necessary to examine empirically whether better forecasts could have been made on average by only using more recent observations. Inflation forecasts, and forecast errors, were recorded for the same period as before, 1960-1980, except that at each date only the previous x observations were used to estimate the forecasting equation, values of x between 22 and 40 being tried. The mean square forecasting error was 1.81% using t!re continuously augmented sample. Limiting the sample size resulted in fluctuations in the mean square forecasting error between 1.85% (sample size held fixed at 22) and 1.78% (sample size held fixed at 32). Wence working with a smaller sample siz,p makes very marginal difference to the size of forecast errors, and the difference may go in either direction, depending on the exact sample size chosen. Thus, we proceed with the interpretation of the forecast errors from the continuously augmented sample as those made by agents forecasting optimally. It may also be of interest to compare the errors made by agents forecasting using these stepwise regressions with the mean of the squared values of the OLS residuals over the same period. The latter was 1.49x, confirming that there is some larger amount of uncertainty to be explained. than the OLS residuals measure. The relationship between the level of inflation, P,, and the size of the forecast error, c,,, was estimated as
e;==o.oooo35 -t- O.l95P,2_ (1.18) R2=0.54,
1,
(3)
(9.8) DIY=2.31,
years 196Q-1980, “?’statistics in parenthesis.
The level of inflation explains over 50% of the variance in the size of squared forecast errors, and the hypothesis that squared forecast errors are positively 4Although structural instability could take the form of different explanatory var;;bles being appropriate in dliff~xent subperiods, the task rf ~~etest~~g to otbtain the eizsr spcification fobr eaoh of the 82 eqluations was beyond the scope oi this paper.
with the’level of inflation may be accepted at a high confidence level. The elasticity of conditional. foreeast variance, the fitted value #, with respect to the level of inflation is 1.55, evaluated at 107; inflatior.
correlated
Whether it is IX& residuals or one step ahead forecatst errors which are us& as rn~ures -of agents r~$stakes in forecasting inflaGon, there is strong evidence “that the size of these mistakes is correlated. wit.h the level of inflation. Thereforee, agents who form rational. expectations about the variance, as well as the mean, of jn&+tion~will have conditional forecast variances which increase with the level of inflation. Although these results are contingent upon the correct spzzification of the forecasting equation, the degree of confidence with which the hypothesis may be accepted suggests that the overall conclusions should be robust with respect to small changes in specification.
TG see this let a2/t be the forecast variance at date t using a recursive regression, $ the variance of the full sample residuals at date t, cr2 the true error variance, Tthe full sample size, and F, the sample size t mean, where Pt is the level of inflation. Consider the autoregression Pt =G + Wt _ 1 + r, .s me&
Whilst it might be expected that c;,,+ I > r2 t+ 1, the possibility remains that Pt could lie closer to P, than does P,, so offsetting the effects of a smaller sample size However what is important is that the extent of the bias depends on the deviation of inflation from the sample mean, and not its absolute lev& 5T&se cm 1~ &r&d easily from, for example, the exprewion for forecast errors in Piadyck and ~~bi~~;~~ (1976_ p. 163).
!X& A~tore~~s~~e conditiod atim: ~~~~~et~i~a
heterosmlasticity 50, 987-1
with estimates cf the variance of
G. &11&y, Injlation uncertainty and level of inflation
221
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