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International Journal of Forecasting 14 (1998) 457–468

The comparative forecast performance of univariate andmultivariate models: an application to real interest rate forecasting

*Prasad V. BidarkotaSchool of Business, La Trobe University, Bundoora, VIC 3083, Australia

Abstract

Does the use of information on the past history of the nominal interest rates and inflation entail improvement in forecastsof the ex ante real interest rate over its forecasts obtained from using just the past history of the realized real interest rates?To answer this question we set up a univariate unobserved components model for the realized real interest rates and abivariate model for the nominal rate and inflation which imposes cointegration restrictions between them. The two modelsare estimated under normality with the Kalman filter. It is found that the error-correction model provides more accurateone-period ahead forecasts of the real rate within the estimation sample whereas the unobserved components model yieldsforecasts with smaller forecast variances. In the post-sample period, the forecasts from the bivariate model are not only moreaccurate but also have tighter confidence bounds than the forecasts from the unobserved components model. 1998Elsevier Science B.V. All rights reserved.

Keywords: Unobserved components models; ARIMA models; Cointegration; Kalman filtering; Forecast evaluation

1. Introduction and then forming the forecast for the real rate fromthe component forecasts.

Our interest is in forecasting the ex ante real The nominal interest rate is usually taken as theinterest rate. This variable is a purely theoretical default-free rate of return on Treasury bills. Theconstruct with no obvious counterpart in the actual expected inflation rate is unobserved, but relative toeconomy. In the literature, we use as proxy the any information set, the observed realized inflationdifference between a nominal interest rate and a rate can be written as the sum of the expectedmeasure of the expected inflation rate in the inflation rate, given that information set, and a zeroeconomy. What is the optimal way for economet- mean random disturbance term. Under rational ex-ricians to forecast the real interest rate? The question pectations, the expectational error is orthogonal toasked here is whether a better forecast of the real the information set on which the expectation isinterest rate, in terms of some specified criterion, is based. Thus, expected inflation rate can be estimatedobtained by forecasting the real rate directly, or by from observations on observed inflation rate (thisforecasting its components in a multivariate setting simply involves application of signal extraction

methods). The realized inflation rate is measured by* the rate of change in some measure of an aggregateCorresponding author. Tel.: 161-3-9479-3079; fax: 161-3-

9479-1654; e-mail: [email protected] price index, such as the CPI or the GDP deflator.

0169-2070/98/$ – see front matter 1998 Elsevier Science B.V. All rights reserved.PI I : S0169-2070( 98 )00036-3

458 P.V. Bidarkota / International Journal of Forecasting 14 (1998) 457 –468

Empirical evidence clearly fails to reject a unit and M0. They compare the root mean squaredroot in the nominal interest rate but there seems to be forecast errors (RMSEs) obtained by aggregatingsome controversy regarding stochastic nonstationari- forecasts of individual components with the directty of the inflation rate. Economic theory, such as the forecast of the aggregate. They find that forecastingconsumption-based asset pricing theory, would tell the aggregate directly is more accurate for M3 butus that it might be reasonable to think of the real aggregating individual forecasts results in a smallerinterest rate as being stationary in any real world RMSE for M0.economy. This would suggest a cointegrating rela- Our problem differs from the one considered bytionship between the nominal interest rate and the Lutkepohl (1984b), (1985), (1986) in the followinginflation rate, with the real rate being the cointegrat- respects. Firstly, our problem involves forecasting aing combination. variable which is unobserved but which, however,

Forecasting the ex ante real interest rate directly can be estimated using signal extraction methodsinvolves fitting a univariate model for the realized from observed variables. Secondly, the observedreal rate and then projecting into the future. Instead, variables are integrated rather than stationary, withhowever, if we were to estimate a multivariate an apparent cointegrating relationship and our objec-process for the nominal rate and inflation, we can use tive is to forecast the cointegrating combination.forecasts for these to construct forecasts of the real This paper is organized as follows. In Section 2rate directly. we describe the data sources and the results of unit

Lutkepohl (1984a), Lutkepohl (1986) considers root tests. In Section 3 we formally state the prob-forecasting contemporaneous aggregates of station- lem, and then introduce the univariate unobservedary variables. He shows that in the event the true components model and present its estimation results.data generation process is known, forecasting the The multivariate time series model is formulated anddisaggregated series, and then forming aggregates of estimated in Section 4. In Section 5 we investigatethe component forecasts generally outperforms fore- the comparative forecast performance of the twocasting the aggregate quantity directly in terms of the models. In the last section we summarize our mainmean squared errors (MSEs) of the forecasts. This findings and suggest some useful extensions.result follows from the fact that the disaggregate datacontain at least as much information as the aggregatedata and the general principle that one should not do 2. Description of the dataworse with more information.

However, it is possible that the two methods yield We use monthly data on the CPI-U until Juneidentical MSEs, for instance, when the disaggregate 1967, and then switch to the CPI-X from July 1967variables are uncorrelated and have identical stochas- onwards, when it first became available. The basetic structures. Moreover, difficulty in modeling the year for the CPI-U/CPI-X is 1967. The CPI series isdisaggregate series and formally aggregating the taken from ‘‘The Consumer Price Index—Detailedforecasts, and measurement error in the disaggregate Report’’, published by the Bureau of Labor Statistics.series can lead to violation of the general principle. The inflation rates are constructed as the first differ-This can also happen when the data generation ences of the natural logarithms of the CPI series.process is unknown and has to be specified and Data on nominal interest rates are taken from theestimated from the sample data, and restrictions on Federal Reserve Bulletin, Table 1.35, available onthe stochastic processes of the disaggregate series the Internet via anonymous ftp to town.hall.org /such as those above are not taken into account during other / fed /h15. These are the three-month U.S.the estimation procedure. Treasury bill rates in the auction market quoted on a

In an empirical illustration of the issue of forecast- discount basis and correspond to the rates in theing contemporaneous aggregates of time series Mills secondary market published in the newspapers. Weand Stephenson (1985) construct forecasts for two convert the nominal interest rates to continuouslydifferent measures of money in the UK, namely M3 compounded rates because continuous compounding

P.V. Bidarkota / International Journal of Forecasting 14 (1998) 457 –468 459

enables us to use the linear form of the Fisher stationarity of the real rate implies a cointegratingequation. relationship between the nominal rate and inflation,

We use quarterly data, with observations recorded with the real rate being the cointegrating combina-in the first month of each quarter, for analysis. tion.Earlier studies (e.g., Fama (1975); Mishkin (1981))report small differences in coefficient estimates withquarterly versus monthly data. As Mishkin notes on 3. The univariate unobserved components modelFama’s findings, ‘‘little additional information iscontained in the more noisy monthly series, since the Let R denote the one-period nominal interest ratet

standard errors of the coefficients using the monthly from period t to t 1 1, and p denote the realizedt11

data are only slightly smaller that those found with inflation rate during that period. Then, the observedpthe quarterly data.’’ Our data spans the period 1954.I ex post real interest rate, r , for that period ist

to 1993.III. We estimate the models using the data up defined as:to 1991.III and hold out the last eight observations, pr 5 R 2 p . (1)t t t111991.IV to 1993.III, for postsample evaluation of the

emodel. We adjust the two series, the nominal rate and If p denotes the inflation rate during the period t totinflation, by a one-sided, three period moving aver- t11, as expected by the agents in the market at timeage filter. The seasonally adjusted ex post real rate is t, then the ex ante real interest rate, r , is defined as:tobtained from the seasonally adjusted nominal rate

er 5 R 2 p . (2)t t tand inflation.The augmented Dickey–Fuller test failed to reject

eAssuming rational expectations, p is, then, thea unit root in the nominal interest rate at the 10 t

mathematical expectation of p conditional on allpercent significance level. This outcome was robust t11

the relevant information available to the agents atto the number of lag differenced terms and to thetime, t. This information set is not available to thepresence of a time trend in the right hand side of theeconometricians in its entirety, who observe only aregression. For the real rate we rejected a unit root atsubset of this complete information set. Let I be thisthe 10 percent level when the number of lag differ- t

subset, available to the econometricians at time t,enced terms on the right hand side was smaller thancomprising the complete history of R and p , up to9, whether or not a time trend was included. The t t

and including, the values at time t.results of the unit root test for inflation wereWe can assume that I summarizes all the relevantsomewhat ambiguous. The test failed to reject a unit t

information available to the agents at the timeroot in inflation at the 10 percent level when theinflation expectations are formed, in which case weminimum Akaike Information Criterion (AIC) washave:used to select the lag length and a constant and a

elinear time trend were included in the regression. p 5 E (p ) (3)t t t11Overfitting of the AIC being only too well-known we

where E is the mathematical expectation conditionalrepeated the test with lags ranging from 0 to 11 and t

on I . This implies:were unable to reject a unit root only when the lag t

elength exceeded 5. When the test was repeated p 5 p 1 u (4)t11 t t11without including a time trend we were unable towhere u is the zero mean random disturbancereject a unit root only when the lag exceeded 8. t11

term (inflation forecast error). By construction, uFrom above we conclude that there seems to be t11

is uncorrelated with I . Eqs. (2) and (3) imply that:strong evidence of a unit root in the nominal rate and t

strong evidence against a unit root in the real rate. r 5 R 2 E (p ). (5)t t t t11But, there appears to be mixed evidence regardingthe nonstationarity of the inflation rate. If the infla- Alternatively, for the purposes of our exposition, wetion rate is in fact integrated of order one, I(1), then could define the ex ante real interest rate by (5).


In either case, we have, from (2) and (4) above: we find that the ARMA(1, 0) model is preferable tothe ARMA(2, 0) and ARMA(2, 1) models in termsR 2 p 5 r 1 n , (6)t t11 t t11 of both criteria. Also, ARMA(3, 0) model is prefer-

where v 52u . In this section our objective is able to ARMA(3, 1) model in terms of both criteria.t11 t11

to obtain forecasts of r , given just the past history It is well known that the AIC tends to favor biggerT 1lpof r , where T is the sample size and l is the models. It is perhaps not surprising then that thet

forecast horizon. ARMA(3, 2) model ranks first in terms of thisIn order to implement the above we set up an criterion. We initially chose the ARMA(1, 0) model

unobserved components model: since it has the minimum SBC value and is also thep most parsimonious of all the models. However, onr 5 r 1 n (7a)t t t11 estimation, this model failed the diagnostic test for

no serial correlation in the estimated residuals.f(L)r 5u(L)w (7b)t t Hence, we settled on the ARMA(3, 0) model. Withwhere f(.) and u(.) are scalar polynomials in the lag this specification for r , the unobserved componentst

operator, L, and w is a zero mean white noise error model becomes:t

term. A plot of the sample autocorrelations and the p 2r 5 r 1 n , n | NID(0, s ), (8a)t t t11 nsample partial autocorrelations of r (not shown)t

revealed significant correlations up to order 3 at ther 5 f r 1 f r 1 f r 1 w ,95 percent level. t 1 t21 2 t22 3 t23 t

2Identifiability of (7) requires that the AR lag, p, w | NID(0, s ), Cov(n, w) 5 0. (8b)wbe greater than or equal to one plus the MA lag, q,i.e. p$(q11); see Harvey (1992) (p. 206). There- The above system implies a reduced form ARMA(3,fore, we estimate all identifiable models for which r 3) model for the ex post real rate. Hamilton (1988),t

follows an ARMA( p, q) process, with max( p)53. following Sargent (1979) and Hansen and SargentEstimation is carried out via the Kalman filter under (1980), uses an AR(4) model for short-term realthe assumption of mutually and serially independent interest rates as a benchmark model. We thereforeGaussian errors for the disturbances. While the estimated an AR(4) plus white noise and found theassumption of normality is made here primarily for AR(4) coefficient to be 20.028 with a standard errorease of estimation it may not hold under strict of 0.083, and therefore decided against adopting thescrutiny. However, the estimation of such models AR(4) plus white noise model in favor of (8).becomes analytically intractable under most nonnor- Misspecification of our model, if any, should bemal error distributions. The numerical implementa- evident in terms of residual serial correlation whention of the filtering recursions under nonnormality is we conduct diagnostic tests.an alternative that is only feasible for small state The results of estimating (8) are tabulated in Tabledimensions (see Kitagawa, 1987). The values of the 2. The standard errors are computed using the

21 21 21Akaike Information Criterion (AIC) and the Schwarz diagonal elements of the matrix, T (I I I )2 op 2

Bayesian Criterion (SBC), computed for each of where I is the Hessian, I is the outer product of2 op

these models, are reported in Table 1. From the table the gradient and T is the sample size. These standard

Table 1Model selection for the unobserved components model

pr 5r 1nt t t11

f(L)r 5u(L)wt t

ARMA order (1, 0) (2, 0) (2, 1) (3, 0) (3, 1) (3, 2)

AIC 608.20 608.80 608.50 604.24 606.11 601.24SBC 611.22 614.84 617.55 613.29 618.18 616.33

AIC522 ln(likelihood)12k, SBC522 ln(likelihood)1k ln(T ) where k is the number of parameters and T is the sample size.


Table 2Estimation of ARMA(3, 0) model for the real rate

p 2r 5r 1n , n |NID(0, s )t t t11 v2r 5f r 1f r 1f r 1w , w|NID(0, s ), Cov(v, w)50t 1 t21 2 t22 3 t23 t w

2 2s s f f f Box–Ljung Statisticv w 1 2 3

0.004 3.027 0.106 0.217 0.262 0.940(0.015) (0.432) (0.091) (0.085) (0.091) (0.332)

1. The numbers in parentheses are robust standard errors for the hyperparameter estimates.]2 Œ2. The Box–Ljung statistic is distributed asymptotically as a x , where P5 T, T is the sample size and n is the number ofP2(n21)

hyperparameters. The number in parentheses gives the area under the right tail.

errors are robust to the assumption of normality (see across rival models. The coefficient of determination,2White, 1982; Hamilton, 1994). The estimated param- R , measures the goodness of fit of the modelD

eters yield an AR real root at 1.26 and two complex against a random walk model with drift. This is aconjugate roots, 1.0471.39i, with a modulus of 1.74. useful yardstick for time series models as it is a very

pThe reduced form ARMA(3, 3) model for r has simple model but nonetheless has been found to fitt

one real MA root at 220.01 and two complex most economic time series surprisingly well. For our2conjugate roots at 8.85716.70i. Thus none of the AR ARMA(3, 0) model R is positive and moderatelyD

or MA roots lie near the unit circle, and they do not high, suggesting that our model performs substantial-have any common roots either. The Box–Ljung ly better than the naive random walk model withstatistic is designed to test for no serial correlation in drift.the estimated residuals from the model. From the We obtain forecasts of r from (7), usingt

table we find that this statistic is not significant at the E (w )50. It can be shown that forecasts of rT T 1l t]Œ10 percent level. The above test based on T lags, from its univariate process are unbiased, both un-where T is the sample size, is unlikely to be very conditionally and conditional on the history of rt

powerful. The results of this test, however, remain alone. However, conditional on I , these suffer fromt

qualitatively unchanged when the test is repeated aggregation bias; see Rose (1977).with lags ranging from 6 through 12. Thus the AR(3) To evaluate the out of sample performance of theplus white noise model seems adequate for modeling model we obtain the one-step ahead prediction errorsthe ex post real rate. out of sample. The postsample predictive test statis-

To evaluate the goodness of fit of the model we tic, reported in Table 4, is found to be statisticallytabulate some useful statistics in Table 3. The insignificant, suggesting that the prediction errors areprediction error variance and the prediction error jointly insignificant. The mean squared error and themean deviation are primarily useful for comparison mean absolute error of the forecasts are also reported

in Table 4. We shall use these measures later forcomparison with forecasts from the multivariate

Table 3 model.Goodness of fit statistics To evaluate the multi-step ahead predictive per-Prediction error Prediction error Coefficient of formance of the model we compute the forecastvariance mean deviation determination function which gives forecasts of the future values of

p3.015 1.347 0.439 r, based on the history of r up to T,tp p

¯ E(r ur , . . . ,r ). The absence of a constant term1. Prediction error variance, f, is defined as the limit of the T 1j 1 T

variance of the prediction errors, f , as t→`. in (8b) implies that the multi-step ahead forecasts oft¯ ˜2. Prediction error mean deviation is defined as: ( f /T ) o uy u,t t the real rate rapidly approach its unconditional mean

˜where y is the standardised one-step ahead prediction error and T of zero. The extrapolative sum of squares, defined asis the sample size. l 2o y , where upsilon is the forecast error, andj51 T 1j uT3. Coefficient of determination512SSE/SSRW, where SSE is the

the extrapolative sum of absolute errors, defined assum of squared errors from our model and SSRW is the sum oflp psquared errors from: r 5r 1b 1h . o uy u, provide a useful basis for comparisonst t21 t j51 T 1j uT


Table 4Postsample evaluation

Mean squared Mean absolute Postsample Extrapolative Extrapolativeerror error predictive test sum of squares sum of

statistic absolute errors

1.641 1.119 0.535 20.848 11.324(0.829)

l 2 T 2˜ ˜ ˜1. Postsample predictive test statistic5[o y /l] / [o y /T ], where y is the standardised one-step ahead prediction error, l is thej51 T 1j t51 t

forecast horizon and T is the sample size. This is distributed as an F(l, T ). The number in parentheses gives the area under the right tail.l 22. Extrapolative sum of squares5o y .j51 T 1j uTl3. Extrapolative sum of squares5o uy u.j51 T 1j uT

between rival models based on multi-step ahead ARMA (VARIMA) system for the nominal rate andpredictions. These are also reported in Table 4. inflation, y 5[R p ]9. Assuming for a moment thatt t t

there is no cointegration between R and p , y hast t t

the following representation:4. The multivariate time series model

F(L)Dy 5 Q(L)U , (9)t t

One can see that the unobserved components where, F(.) and Q(.) are 2-by-2 matrices of polyno-approach involves fitting a pure time series model to mials in the lag operator, D is the first-differencethe ex ante real rate. It seems plausible that when operator, (12L), F(0) and Q(0) are 2-by-2 identityforming expectations rational economic agents may matrices, and U is a two-dimensional serially un-t

draw on an information set larger than just the past correlated vector white noise process with zerohistory of the variable being expected. Nelson (1975) mean.examines the conditions under which a larger in- Empirical evidence seems to indicate unit rootformation set will be relevant for the formation of nonstationarity in R and p , whereas, on the basis oft t

rational expectations and demonstrates that when the economic theory, r is best deemed stationary. Incor-t

economic system is subject to multiple shocks the porating this knowledge suggests reexpressing (9) inrational expectation of a variable can no longer be error-correction form:expressed as its univariate extrapolation.

F *(L)Dy 5 a(L) R 2 p 1 Q(L)U , (10)h jt t22 t21 tIn this section we focus on a bigger informationset that is deemed to consist of the past histories of where F *(.) is a 2-by-2 matrix polynomial and a(.)the nominal rate and inflation (the history of the is a 2-by-1 matrix polynomial. Both Eqs. (9) andrealized real rate can be constructed from these two). (10) correctly specify the model but when theAs noted in the introduction, within the context of variables are cointegrated, (9) omits the cross-equa-forecasting aggregates of variables, it has been tion cointegration restrictions.shown that it is not always optimal to forecast While the evidence on the stochastic nonstationari-disaggregate variables and construct forecasts of the ty of the nominal rate is pretty convincing such isaggregate from the disaggregate forecasts. This unfortunately not the case with inflation. A numberhappens especially when the true data generation of authors have therefore modeled inflation lately asprocess is unknown and has to be estimated from the a process with a changing mean that is represented assample data. However, the models of the disaggre- a Markov process; see Kim (1993). While thegate variables introduced in this section incorporate shifting mean process may be viewed as an alter-cointegration restrictions imposed by economic native to the linear Gaussian ARIMA representation,theory. Therefore, it would be interesting to see how Bidarkota and McCulloch (in press) propose linearthese models fare in comparison to the unobserved state-space models with thick-tailed error distur-components model. bances as continuous alternatives to the discrete

Let us begin by considering a vector integrated Markov switching processes. Specifically, they


model inflation as a local level process with heteros- local level model for inflation, with an ARIMA(0, 1,kedastic symmetric stable shocks, with the scales of 1) reduced form structure, yielded an MA coefficientthese shocks dependent on the trend level of infla- of 20.084 which implies an MA root of 11.92, welltion. The motivation for heteroskedasticity comes away from unity. A near compensating MA root offrom the literature on the links between the level of unity is evidence of over-differencing and arguesinflation and its uncertainty which has sparked a against modeling inflation as an integrated process.number of other papers that have attempted to model Failure to reject a unit root indicates permanentthe heteroskedasticity in inflation (for example, see shocks to inflation. These two somewhat conflictingEngle, 1983; Cosimano and Jansen, 1988, for ARCH pieces of evidence have been used to motivate long-processes, Brunner and Hess, 1993, for state-depen- memory ARFIMA process for inflation; see Baillie etdent models of conditional moments, and Evans and al. (1996). We, however, abstract from fractionalWachtel, 1993, for Markov switching volatilities). differencing here and simply assume (10) toOur simpler error-correction representation for the adequately represent the bivariate process. Based onnominal rate and inflation has precedents in Mishkin our univariate models we choose first order polyno-(1992) who tests for the long run Fisher effect using mials for both F *(.) and Q(.). For the sake ofthe cointegration approach between nominal rates parsimony, we restrict a(.) to be of order zero.and inflation. While the Markov switching and other Thus the model to be estimated, after rearrangingmodels for the univariate inflation process may have terms, can be written as:some merit our objective in this section is to design a

R 1 2 a Rt 1 t21bivariate process for the nominal rates and inflation.5S D F GS Dp 0 (1 2 a ) pIt is convenient to use the error-correction model t 2 t21

here because the theory for comparing forecasts from a 0 R u1 t22 1t1 1the aggregated and disaggregated linear processes F GS D S Da 0 p u2 t22 2thas already been worked out by Lutkepohl (1984a),

u u u11 12 1t21(1986). 1 , (11)F GS Du u uIn order to estimate the error-correction model, 21 22 2t21

(10), we need to specify the orders of the polyno-where E[u u ]5[0 0]9 and the variance–co-1t 2tmials, F *(.), Q(.) and a(.). We fit univariate 2

s rs su u u1 1 2ARIMA( p, 1, q) processes to R and p , with ( p1t t variance matrix of (u , u ) 5 .1t 2t 2F Grs s sq)#2. The AIC and SBC values computed for these u u u1 2 2

models are presented in Table 5. The sample auto- This equation, as it is written, imposes the cross-correlations suggest that a random walk model seems equation cointegration restriction between the nomi-adequate for R whereas an ARIMA(0, 1, 1) or an nal rate and inflation, and also imposes orders on thet

ARIMA(0, 1, 2) model is most suitable for p . Most lag polynomials as discussed above.t

univariate models that have been fitted for inflation Although it is fairly straightforward to write downin the past have invariably been ARIMA(0, 1, 1). A the log-likelihood function for model (11) we find it

Table 5Model selection for the Bivariate model

f(L)Dx 5u(L)u , x 5R , pt t t t t

ARIMA order (0, 1, 1) (1, 1, 0) (1, 1, 1) (2, 1, 0) (2, 1, 1) (2, 1, 2)

R AIC 407.36 407.64 403.08 407.77 399.98 405.13SBC 410.37 410.65 409.10 413.79 409.01 417.17

p AIC 606.87 616.78 604.62 597.38 598.77 595.87SBC 609.98 619.79 610.64 603.41 607.80 607.92

AIC522 ln (likelihood)12k.SBC522 ln (likelihood)1k ln(T ).where k is the number of parameters and T is the sample size.


Table 6Estimation of the Bivariate model

R 1 2 a R a 0 R u u u ut 1 t21 1 t22 1t 11 12 1t215 1 1 1S D F GS D F GS D S D F GS Dp 0 (1 2 a ) p a 0 p u u u ut 2 t21 2 t22 2t 21 22 2t21

2u 0 s rs s1t u u u1 1 2E 5 , var 2 cov(u , u ) 5 .1t 2tS D S D F 2 Gu 0 rs s s2t u u u1 2 2

2 2a a u u u u r s s1 2 11 12 21 22 u u1 2

20.217 20.047 20.203 20.166 0.525 20.759 0.358 0.830 2.949(0.106) (0.092) (0.110) (0.132) (0.179) (0.097) (0.094) (0.171) (0.457)

1. The numbers in parentheses are robust standard errors for the hyperparameter estimates.

convenient to work with the prediction error de- the quasi-maximum likelihood estimators of the ARcomposition form of the likelihood function. While and MA operators in ARMA models are unaffectednone of the standard software packages, to the due to the block diagonality of the informationauthor’s knowledge, are designed for multivariate matrix (i.e., the Hessian). Thus, inferences regardingprocesses the state-space representation provides a these remain valid, although these estimators sufferconvenient formulation which is easy to implement from loss of efficiency.efficiently, especially once the unobserved compo- Plots of the standardized residuals from the twonents model has been estimated by the Kalman filter. equations in (11) suggest no apparent deviation ofOnce again this model has a large state dimension the residuals from zero mean processes. Also there(56) which renders numerical implementation under appears to be no obvious trend in the residuals. Thenonnormality infeasible. Therefore, under the assample autocorrelations of the estimated residuals,sumption of normality for the disturbance vector in along with their standard errors, reveal marginally(11), we estimate the model using the Kalman filter. significant serial correlation in the residuals of theInitialization of the filter is done by the diffuse prior nominal rate at the third and fifth lag. However, themethod. The starting values for the maximization of autocorrelations are only slightly greater than theirthe likelihood are those obtained by fitting univariate 95 percent critical values. The estimated residuals formodels to each of the two variables. The results of inflation show nonnegligible sample autocorrelationsthe estimation of the bivariate model are reported in at the third lag (possibly arising from seasonalTable 6. The robust standard errors reported are once adjustment by the three period moving average filter)again those computed using the diagonal elements of and marginally significant autocorrelations at the first

21 21 21the matrix, T (I I I ) where I is the Hessian, and fourth lags. To evaluate the goodness of fit of2 op 2 2

I is the outer product of the gradient and T is the the model we tabulate the prediction error varianceop

sample size. and the prediction error mean deviation for the realWe find that all the coefficients, except a and u , interest rate in Table 7.2 12

are significant at the 10 percent level. The lack of To obtain forecasts of r we note that (6) implies:t

significance of a means that the error-correction2 r 5 R 2 p 2 n . (12)T 1f T 1f T 1f 11 T 1f 11term has no explanatory power for the inflation rate,once the other terms are included in the model. The But, E (n )50. Therefore,T T 1f 11significance of the cross-equation MA coefficient, u21

E (r ) 5 E (R 2 p )and of the correlation coefficient between the two T T 1f T T 1f T 1f 11

disturbance terms, however, indicates gains to be 5 E [1 0]y 2 [0 1]y . (13)h jT T 1f T 1f 11made from the bivariate representation. As Harvey(1992) (p. 220) points out, even when the normality Given optimal forecasts of y , one can then obtaint

assumption is invalid, the asymptotic distributions of forecasts of r from (13). Rose (1977) demonstratest


Table 7 5. Comparison of forecast performanceGoodness of fit statistics

Prediction error Prediction error Coefficient of In Fig. 1 we plot the realized real interest rates andvariance mean deviation determination the one-step ahead forecasts obtained from the3.980 1.011 0.221 unobserved components model and the bivariate

model. The figures indicate that the bivariate model¯1. Prediction error variance, f, is defined as the limit of theis better able to capture the short run fluctuations invariance of the prediction errors, f , as t→`.t

¯ ˜2. Prediction error mean deviation is defined as: ( f /T ) o u y u, the interest rates. We now turn to more formalt t

˜where y is the standardised one-step ahead prediction error and T evaluation of the forecast performance of the twois the sample size. models.3. Coefficient of determination512SSE/SSRW, where SSE is the

There are a number of dimensions along whichsum of squared errors from our model and SSRW is the sum ofp p model comparisons can be made. These comparisonssquared errors from: r 5r 1b 1h .t t21 t

can be made both within the sample in whichestimation of the unknown parameters of the modelwas carried out and outside the estimation sample

the optimality of (13) as a forecast of r , given (post sample comparison). Within the estimationt

optimal forecasts of y . sample the prediction error mean deviation measurest

To evaluate the out of sample performance of the the accuracy of forecasts in terms of how close themodel we obtain the one-step ahead prediction errors forecasts are to the true realized values. Comparingout of sample. The postsample predictive test statis- the prediction error mean deviations tabulated intics, based on the postsample one-step-ahead predic- Tables 3 and 7 we find that the forecasts from thetion errors, are reported in Table 8. The test statistics error-correction model are closer, on average, to theare found to be statistically insignificant, suggesting realized interest rates than those obtained from thethat the prediction errors are jointly insignificant. univariate model. In the postsample period the meanThe mean squared error of the forecasts and the absolute error of the forecasts from the error-correc-mean absolute error of the forecasts of the real rate tion model is again smaller than that of the forecastsare also reported in Table 8. from the unobserved components model, as is evi-

To evaluate the multi-step ahead predictive per- dent from Tables 4 and 8. Thus in this sense theformance of the model we compute the forecast bivariate model gives more accurate forecasts of thefunction which gives forecasts of the future values of ex ante rate than the unobserved components modelr, based on the history of R and p up to T, both within and outside the estimation sample.t t

E(r uI ). The extrapolative sum of squares and the One can identify at least four sources of uncertain-T 1j T

extrapolative sum of absolute errors, based on multi- ty in model-based forecasts. These arise from thestep ahead predictions, are reported in Table 8. inherently stochastic nature of the true data generat-

ing process, the uncertainty in model specification,uncertainty in parameter estimation and measurement

Table 8 error in the data. Forecasts of variables which arePostsample evaluation integrated of order one, as the nominal rate andj (l) j (l) MSE MAE ESS ESAE1 2 inflation appear to be, can only be made with

increasing forecast error variance as the forecast0.263 0.342 1.478 1.011 22.203 12.133(0.967) (0.934) horizon increases. However, their stationary cointeg-

l 2 T rating combinations (the real interest rate in our case)˜1. Postsample predictive test statistic, j(l)5[o y /l] / [oj51 T 1j t512˜ ˜y /T ], where y is the standardised one-step ahead prediction can be forecast with finite limiting forecast errort

error, l is the forecast horizon and T is the sample size. This is variance.distributed as an F(l, T2d), d is the order of differencing. The Given our assumptions on the information setnumber in parentheses gives the area under the right tail.

available to the econometricians, or alternatively,Subscripts 1 and 2 correspond to u and u respectively.1 2given our definition of the ex ante rate, forecasts of2. ESS is the extrapolative sum of squares.

3. ESAE is the extrapolative sum of absolute errors. the ex ante real rate obtained from the bivariate


Fig. 1. (a) One-step ahead forecasts Unobserved components model. (b) One-step ahead forecasts Bivariate model.


model are rational in the sense that the forecast The extrapolative sum of squares and the ex-errors are uncorrelated with all available past in- trapolative sum of absolute errors summarize theformation. The rational forecast of a variable must be multi-step ahead forecast performance. From Tablesmore efficient than its best extrapolative predictor, in 4 and 8 one can see that these measures are quitesome well-defined sense, since the conditioning close for the two models, although the forecasts frominformation set of the rational expectation subsumes the unobserved components model seem slightlythe past history of the variable being forecast. better in this respect. However, the forecast functions

The gain in efficiency from using a rational from the two models (not shown) indicated that theforecast over a univariate extrapolation in a par- multi-step ahead predictions from the two models areticular situation will depend on the stochastic prop- both very poor.erties of the processes generating the exogenousinputs to the economic system as well as on theparticular economic structure at hand. For instance, 6. Conclusions and extensionsfor a system subject to two exogenous shocks, therelative gain from using a bigger information set than In this paper we addressed the question of whetherjust the past history of the variable being forecast is an improved forecast of the real interest rate, insmaller the bigger the discrepancy in the variances of terms of some well-specified criteria, is obtained bythe two shocks. However, for long horizon predic- taking into account a bigger information set than justtions, the limits of the forecast error variances in the past history of the variable being forecast. Forboth cases must approach the unconditional variance the purposes of this study, the bigger information setof the variable being forecast. comprised of the history of the nominal interest rate

A number of measures of forecast uncertainty are and inflation. The performance of the alternatecurrently in use. Within the estimation sample the models was compared, both within the estimationprediction error variance serves as a useful measure sample and outside, under a set of criteria based onof forecast uncertainty. From Tables 3 and 7 we find both one-step ahead and multi-step ahead forecasts.that the prediction error variance from the univariate The main conclusions that emerged from ourmodel is smaller than that from the error-correction investigations can be summarized as follows. Withinmodel. However, for the postsample forecasts, the the estimation sample the bivariate model providesmean squared error is higher for the forecasts from more accurate forecasts of the real interest rate inthe univariate model, as is evident from Tables 4 and terms of the prediction error mean deviation. On the8. Thus the univariate model provides more precise contrary the forecasts derived from the unobservedestimates of the ex ante real rate within the estima- components model are more precise in that they havetion sample whereas outside the estimation sample a smaller prediction error variance. In the postsamplethe error-correction model provides tighter bounds period the bivariate model outperforms the univariateon the forecasts. model both in terms of the accuracy of the forecasts

As a measure of forecast comparison, the mini- and their precision.mum mean squared error criterion (also termed the The above results are based on one-period aheadmean squared error dominance criterion) is some- forecasts. As regards the multi-step ahead forecaststimes justified on the grounds that conditional ex- we find that both models yield very poor forecasts ofpectations result in the minimum mean squared the real interest rate. This suggests that the long runerrors when sufficient moments exist. But, as pointed dynamics of the true generating process are not beingout by Clements and Hendry (1993), the model with accurately captured in our specified models. Amongthe minimum mean squared error is not necessarily the two models the unobserved components modelthe conditional expectation given the combined seems to perform marginally better in this respect.information set of all the models. Such a situation The models that we have set up can certainly bearises, for instance, when each of the models uses extended and improved upon. Although the growthonly a subset of the combined information set of all rates of the nominal rate do not seem to displaythe models. much clustering conditional heteroskedasticity is


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