University of Toronto Department of Economics

May 16, 2008

By Alonso Gomez, John M Maheu and Alex Maynard

Improving Forecasts of Inflation using the Term Structure ofInterest Rates

Working Paper 319

Improving Forecasts of Inflation using theTerm Structure of Interest Rates∗

Alonso Gomez†, John M. Maheu‡and Alex Maynard§

May 2008

Abstract

Many pricing models imply that nominal interest rates contain in-formation on inflation expectations. This has lead to a large empiricalliterature that investigates the use of interest rates as predictors of fu-ture inflation. Most of these focus on the Fisher hypothesis in whichthe interest rate maturity matches the inflation horizon. In generalforecast improvements have been modest and often fail to improve onautoregressive benchmarks. Rather than use only monthly interestrates that match the maturity of inflation, this paper advocates usingthe whole term structure of daily interest rates and their lagged val-ues to forecast monthly inflation. Principle component methods areemployed to combine information from interest rates across both theterm structure and time series dimensions. We find robust forecastingimprovements in general as compared to both an augmented Fisherequation and autoregressive benchmarks.

JEL Classification: E31,E37,C53,C32Keywords: inflation, inflation forecast, Fisher equation, term struc-ture, principal components

∗Maheu is grateful for financial support from SSHRCC.†Department of Economics, University of Toronto.‡Department of Economics, University of Toronto and RCEA.§Department of Economics, University of Guelph

1 Introduction

There has been a growing empirical literature that finds a statistical re-lationship between interest rates and future inflation. Papers by Mishkin1990a, 1990b, 1991, Jorion and Mishkin (1991), Estrella and Mishkin (1997),and Kozicki (1997) document the predictive power of interest rates for fu-ture inflation. In general, these papers investigate the statistical relationshipbetween a yield spread that matches an inflation spread. This is motivatedby the Fisher equation which states that the nominal interest rate is com-posed of the real interest rate and expected inflation. The stability of thisrelationship is investigated in Estrella, Rodrigues, and Schich (2003). Forthe U.S. they find evidence of a structural break in 1979 and 1982 for shorthorizon regressions. Estrella (2005) discusses the reasons why the yield curvecontains predictive content for both real and nominal variables, and arguesthat monetary policy plays a large role in this relationship.

Nevertheless, the practical forecast improvements that come from employ-ing the Fisher equation sometimes appear modest relative its importance asa theoretical relationship. The literature finds poor predictive power at theshort end of the yield curve but improved prediction around 12 months andthe best performance from 3 to 5 years out. Stock and Watson (2003) alsonote that the Fisher equation often loses its predictive content after control-ling for lagged inflation.

The failure of the Fisher equation to consistently beat autoregressivebenchmarks has two possible implications for inflation forecasting. One pos-sible implication, as argued by Stock and Watson (2003), is that, despitetheir theoretical appeal, interest rates may have limited practical value forinflation forecasting. An alternative interpretation is that the performanceof the Fisher equation is hindered by its rigid restrictions, in which onlythe information in a single interest rate spread with maturities matching theinflation spread is employed in the forecast. While theoretically appealing,using only one source of information goes against some of the more recenttrends in forecasting, in which improvements are found from combing infor-mation from multiple predictors or forecasts (Bernanke, Boivin, and Eliasz(2005), Stock and Watson (2002), Favero, Marcellino, and Neglia (2005)).

Recent research on macro-finance models, including Ang and Piazzesi(2003), Chernov and Biokbov (2006), Dewachter, Lyrio, and Maes (2006),and Diebold, Rudebusch, and Aruoba (2006), demonstrates the importantrole that macro aggregates and monetary policy play in determining the

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shape of the term structure and its dynamics.In this paper we contribute to the literature on inflation forecasting in

several ways. First, we investigate whether, for a given inflation spread,additional forecasting information can be extracted from the entire interestrate term structure. We also investigate if information in daily lags of theterm structure are useful for prediction. In contrast to previous work, wefocus exclusively on the out-of-sample predictive performance relative to au-toregressive type models, and we work with readily available interest rateson government T-bill and bonds.1 Our focus is on the empirical forecastingcontent of interest rates.

One reason why multiple interest rates may improve inflation forecastsis that we face a signal extraction problem. For example, suppose that theFisher equation holds as in ikt = rk

t +Etπkt+k, where ikt is the k period nominal

interest rate, rkt is the matching real interest rate, Etπ

kt+k is the expected

inflation over this period, and πkt+k = log(pt+k/pt), where pt denotes the price

level. If real interest rates are unobserved and stochastic then ikt provides anoisy measure of Etπ

kt . However, the interest rate ilt, l > k, also contains

information on Etπkt+k, since we can decompose the longer inflation rate as

πlt+l = πk

t+k + πl−kt+l to get ilt = rl

t + Etπkt+k + Etπ

l−kt+l . Therefore, ikt and ilt have

Etπkt+k as a common factor. This suggests additional information on Etπ

kt+k

may be available in other interest rates and the term structure as a whole.If inflation expectations are persistent then similar reasoning suggest thatthere may be benefits to using lags of the term structure.

By employing multiple interest rates we may also allow for a more robustpredictive relationship with inflation. For instance, the information contentof the term structure may be constant, but the location of inflation expec-tations may be easier to extract at different maturities, depending on thestance and expectation of monetary policy.

The use of the whole term structure and lags of the term structure resultsin a dimensionality problem. To extract the predictive content but retain par-simony we follow the diffusion index method of Stock and Watson (2002). Weextract principle components from both current term structures and laggedterm structures. We select the principle components using a variance criteriaand a correlation criteria. The latter is based on the correlation between theprinciple components and regression residuals from an AR model of infla-

1The majority of research in this area uses zero coupon yields. This requires estimationof the yield curve at each point in time.

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tion. Therefore, they pick up remaining structure not captured by the ARmodel. We investigate several variable selection methods and report resultson a model average over different principle component specifications.

To gauge the success of our methods we include, as benchmarks, an au-toregressive type model, a Fisher equation model augmented with lags ofinflation and a Phillips curve. To allow for changes in the model relationshipwe use a rolling window for estimation.

In general, our results show that an AR type benchmark is better than anaugmented Fisher model. Our principle component methods improve uponthese results. The forecasting improvements come from two dimensions. Theterm structure (current term structure) dimension and the time (lagged termstructures) dimension. For the 12 month ahead predictive regressions, onlythe term structure dimension improves forecasts, while for the 36 monthregressions both the term structure and time dimension are useful. Therelative performance of the variance and correlation criteria used to selectprinciple components criteria is mixed, and depends on the subsample andthe forecast horizon. Our recommended approach is a model average overspecifications that have between 3 and 5 principle components from both theterm structure and time dimensions. This approach provides good resultsfor all time periods and forecast horizons.

Our results are supportive of the broad implication of the Fisher equationthat interest rates provide a useful tool in inflation forecasting. At the sametime, we find that employing the whole term structure and its recent lagsprovides clear improvements over the restrictions of the Fisher equation.This is consistent with recent developments in forecasting which emphasizethe value of combining information and forecasts from multiple sources.

While this paper focuses on predictive content of the term structure, itis not our intention to suggest that interest rates be used in isolation toforecast inflation. There is a large literature on inflation forecasting andmany useful techniques have been developed using information other thaninterest rates that could potentially be combined with term and lagged termstructure variables considered here. Examples, include methods based onstructural relationships (Fisher, Liu, and Zhou 2002, Orphanides and vanNorden 2005, Stock and Watson 1999), the use of asset prices, (Ang, Bekaert,and Wei 2005, Stock and Watson 2003), and factor models (Banerjee andMarcellino 2006, Camba-Mendez and Kapetanios 2005, Stock and Watson

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2002).2.This paper is organized as follows. Section 2 reviews the data, Section 3

discusses the benchmark models, while the new principle components modelsare introduced in Section 4. Results are found in Section 5 and conclusionsin Section 6.

2 Data

Our data set comprise daily interest rates and monthly data on consumerprice indexes, and the total civilian unemployment rate. The daily interestrates included in our panel are U.S. Treasury yields with maturities of 3-months, 6-months, 1-year, 3-years, 5-years, and 10-years. Inflation was com-puted as the log difference of the seasonally adjusted consumer price indexfor all urban items. Interest rates and consumer price indices were obtainedfrom the Federal Reserve of St. Louis FRED data base. The unemploymentrate was obtained from the Citibase data base. The sample period spansfrom March 1962 to December 2004. This yields a sample size of 514 for themonthly inflation rate data and a sample size of 11,175 for the daily interestrate data. Out-of-sample forecasts are computed from September 1974 toDecember 2004, giving a total of 364 monthly out-of-sample forecasts, withthe proceeding 150 months reserved for the training sample.

The top panel of Figure 1 presents the annual inflation rate for the wholesample. Two major episodes are observed during the post-war U.S. history.The first one is characterized by a period of high inflation from the early1960s through the mid 1980s, and a second period of low inflation runs fromthe mid 1980s to the present.

We define the annualized j-th term inflation rate observed at time t as:

πjt = (1200/j) ln(pt/pt−j).

Following Stock and Watson (1999), we will focus on forecasting annualizedinflation percentage spreads of the form:

πkt+k − π1

t+1 = (1200/k) ln(pt+k/pt) − 1200 ln(pt+1/pt)

2Recent work on time series models of inflation emphasize the changing inflation dy-namics (Cogley and Sargent 2002, Dossche and Everaert 2005, Pivetta and Reis 2006, Stockand Watson 2006).

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where k = {12, 36}.

Finally, note that in the following we use interest rates for T-bills andbonds which may have coupon payments. This has two attractions for ourforecasting exercises. First, it avoids the need to estimate a zero-couponterm structure and the associated estimation errors that can arise form this.Second, the interest rates we use are readily available, for example from theSt. Louis FRED data base. Throughout the paper we use the terminologyFisher equation even though our rates may not be derived from a zero-couponinstrument.

3 Benchmark Models

In order to obtain the benchmark with the best performance in our sample,a set of competing models were estimated.

1. Pure autoregression-type benchmark (AR-type) Our base-linebenchmark model for inflation forecasting is the simple autoregressive-type specification given by:

πkt+k − π1

t+1 = γ +P−1∑p=0

φpπ1t−p + εt+k, (1)

where εt+k is an error term. We also consider two other standardbenchmark models in which this AR-type specification is augmentedby other predictors suggested by economic models.

2. Phillips Curve (PhCu)

The unemployment-based Phillips curve specification used in this paperis

πkt+k − π1

t+1 = γ +P−1∑p=0

φpπ1t−p +

Q−1∑q=0

βqwt−q + εt+k, (2)

where wt is total civilian unemployment rate (LHUR) in month t.

3. Augmented-Fisher Equation

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Finally, a benchmark that appears as the natural counterpart to themodel that includes daily spreads as predictors, is a model that incorpo-rates the Fisher hypothesis. In this case, we extend the autoregressivemodel by including as an additional predictor the monthly yield spreadwith the same maturity as the target inflation forecast. The empiricalspecification of the augmented Fisher equation hypothesis is given by

πkt+k − π1

t+1 = γ +P−1∑p=0

φpπ1t−p + β(ikt − i1t ) + εt+k, (3)

where ikt is the observed treasury yield with maturity k at time t.

All three benchmark models are estimated by OLS, with the number oflagged inflation rates (1 ≤ P ≤ 12) and unemployment rates (1 ≤ Q ≤ 12,PhCu model only) determined recursively using the Bayesian informationcriterion (BIC). For the Phillips curve, P is selected first and then condi-tional on that value Q is selected. This process of lag selection is repeatedrecursively as new information arrives.

The benchmark models above were estimated under the assumption ofan I(1) price index. We also compared these to the equivalent benchmarkmodels that hold under the assumption that prices are I(2), in which casethe terms involving π1

t−p are replaced by ∆π1t−p. As in Stock and Watson

(1999, 2002) we find little difference between the I(1) and I(2) specificationswith respect to the accuracy of the inflation forecasts. Thus we present onlythe results that hold when prices are modelled as I(1) variables. Generallyspeaking, we found that the forecasts using the pure AR-type model withprices modelled as I(1) provided the best benchmark forecasts.

4 Principal Components Models

Models that use interest rate spreads as predictors of future inflation arebased on a version of the Fisher’s hypothesis. The empirical specification ofFisher’s hypothesis includes only monthly spreads with the same maturityas the target inflation rate. Thus, Fisher’s specification discards any otherpotential information contained in the term structure of interest rates or itsdaily lags. In this paper, we propose forecasting inflation by incorporatinginformation contained in both dimensions of the interest rate data: the term

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structure dimension and the time dimension. The time series dimension isincorporated via the use of daily interest rate observations.

The transition from Fisher’s specification that includes just one spreadof interest rates as predictors to the multivariate case that incorporates dif-ferent terms for different dates raises the important problem of parsimony.In order to mitigate overfitting and poor forecast performance we proceedby constructing a small number of factors from the large set of daily spreadsusing principal components. The methodology is based on the premise thatthe most useful information for forecasting purposes can be summarized bythe first few principal components.

4.1 Incorporating Term Structure and Time Series In-formation

We denote by Xt the full vector of interest rates used as inflation predictors.In our main forecast results the interest rates included in Xt span both thematurity and time series dimensions. However, in order to better understandthe source of the forecast improvements, we first consider the two dimensionsseparately.

Let spk,jt = ikt − ijt denote the spread at time t between yields with a

maturity of k and j respectively. To simplify notation, the length of everymonth is assumed to be of 22 business days. We use integer subindicesto denote monthly frequencies, whereas for daily data subindices we usethe conventional notation in which t + h/22 for h = 0, ..., 22 − 1 refers tobusiness day h in month t. Time t refers to the first business day of montht. Inflation rates are assumed to be observed the first business day of everymonth. Following the above notation, and to avoid the use of informationnot available to the econometrician, the information set available at time tis given by inflation rates observed up to time t, and interest rate spreadsobserved up to time t − 1/22.

4.1.1 Term Structure Dimension

In this case, we extend the augmented Fisher equation in the term structuredimension by including the full set of interest interest rate spreads as pre-dictors for future inflation. The matrix of predictors, Xt, is therefore given

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by

X ′t =

[sp12,3

t−1/22, sp36,3t−1/22, sp

36,12t−1/22, sp

60,3t−1/22, sp

60,12t−1/22, sp

60,36t−1/22, sp

120,3t−1/22,

sp120,12t−1/22, sp

120,36t−1/22, sp

120,60t−1/22

]such that only the interest rate spreads observed on the last business day ofmonth t − 1 are included in the information set It.

4.1.2 Time Series Dimension

This case extends the augmented Fisher specification in the time series di-mension by including m daily lags of the interest rate spread with the samematurity as the target inflation spread as predictors. In other words, for thetarget inflation forecast period πk

t+k − πjt+j, k > j, Xt is given by,

X ′t =

[spk,j

t−1/22, spk,jt−2/22, ..., sp

k,jt−m/22

].

4.1.3 Combined term structure and time series information

In this case both dimensions, the term structure and time dimension, arepooled together to construct predictors to forecast future inflation. Thepredictors’ matrix, Xt, is formed as

X ′t = [x1t

′, x′2t, . . . , x

′mt]

xht =[sp12,3

t−h/22, sp36,3t−h/22, sp

36,12t−h/22, sp

60,3t−h/22, sp

60,12t−h/22,

sp60,36t−h/22, sp

120,3t−h/22, sp

120,12t−h/22, sp

120,36t−h/22, sp

120,60t−h/22

], h ∈ {1, ...,m}.

Unfortunately, there is little guidance as to the number of days, m, thatshould be included in Xt. To measure the relative information content indaily data, we constructed three matrices of predictors by setting differentvalues for m. The first matrix sets m equal to 20, including four weeks ofdaily data. In the second matrix m is set equal to 10, and finally, the lastcase includes only a week of daily data setting m equal to 5. In the followinglet n denote the number of columns in Xt.

From the above definitions the information set available for forecastingat time t, It, is given by lag values of inflation rates and Xt,

It = {πt, πt−1, πt−2, ..., Xt} .

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4.2 Dimension Reduction via Principal Components

Following Stock and Watson (2002), forecasting is performed in a two-stepprocedure. First, the principal components of the set of interest spreads arecomputed; second, the estimated factors are used as predictors to forecastfuture inflation.

As discussed above, we denote by Xt the vector of n daily interest ratepredictors, potentially varying across both the time series and maturity di-mensions, that are observed up until the last day of time t − 1. Because thedimension of Xt is large, it is impractical to use all n predictors. Instead, weemploy the principle component decomposition

Xt = AZt + AcZct = AZt + vt, vt ≡ AcZc

t (4)

where Z ′t =

(zt,(1), zt,(2), . . . , zt,(F )

)denotes the first F < n (sorted) principal

components, and Zct contains the remaining n − F components. The factor

loading matrices for Zt and Zct are denoted by A and Ac respectively. This

allows us to achieve dimension reduction by extracting only the first F factorsZt for inclusion as regressors in our forecasting equation for inflation, togetherwith the lagged values of inflation. Thus our forecasting model becomes

πkt+k − π1

t+1 = γ +P−1∑p=0

φpπ1t−p +

F∑f=1

αfzt,(f) + εt+k, (5)

All of the regression coefficients in (5), including the coefficients αf on thefirst F principal components are estimated by OLS. Since the lagged valuesof inflation already enter (5) parsimoniously, we do not include them in theprincipal component analysis. As in the benchmark models, the numberof lagged inflation rates is chosen using the Bayesian Information Criteria(BIC), with a maximum of 12 (1 ≤ P ≤ 12). We discuss the choice of F inSection 4.3.2 below.

The advantage of this approach is that we have reduced the number ofterm structure regressors from n down to F , while allowing the factor decom-position to pick out what we expect to be the most important components inZt for forecasting inflation. This potentially allows us to incorporate substan-tially more information on inflation than in the augmented Fisher regressions,while still maintaining parsimony.

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4.3 Selection of Principal Components

In order to determine which principal components to employ in a principalcomponents model two choices are required. First, one needs a rule forordering the components from first to last. This involves taking a stand onwhich components will be most useful for forecasting inflation. Secondly, oneneeds to decide how many components to use. We discuss these questionsseparately in the two subsections below.

4.3.1 Ordering the principal components

Two different criteria are employed in the selection of factors. The first cri-terion, Variance Sort , consists in selecting the principal components thatexplain the highest percentage of the second moment of the predictors matrix{Xt}. Regardless of this being the most common practice in the selection ofprincipal components, there is no guarantee that this methodology will resultin the set of factors that maximize forecasting power for inflation. Hence, wealso propose and compare to a second criterion for selecting factors, Cor-relation Sort , in which we sort principal components according to theirin-sample correlation with the residual from the AR-type benchmark modelin (1). The idea consists in selecting the principal components that have thehighest in-sample correlation with the residual term from the AR model tocapture the component of inflation not explained by the AR model.

4.3.2 Selecting the number of principal components

Once we have decided on a rule for ordering the components, we must nextchoose the number of components (F) to include. Here we compare threeapproaches. The first is simply to fix F, the second is to select F by applyinga model selection criterion to the forecasting equation, and the third is tomodel average across forecasts using a range of different choices for F.

Because our primary interest is in forecasting inflation, we apply BICto the forecasting equation in (5), while treating the principle componentszt(f) as regressors. Thus we measure the (penalized) fit in terms of theexplanatory power of the principle components of zt(f) for inflation πk

t+k −π1

t+1. This is similar to the approach of Stock and Watson (1999). Analternative approach, not pursued here, is to measure the penalized fit basedon the explanatory power of the principle components zt(f) for Xt itself in(4). This is the approach pursued by Bai and Ng (2002) in a more general

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context, where the primary focus is often on the modeling of Xt itself, ratherthan on the use of the components of Xt in the forecast of another variable,say yt. Bai and Ng (2002) provide consistent model selection procedures forthe choice of F in (4).

For the model averaging approach, we let fpct+k,F denote the out-of-sample

forecast of the principal component model using F factors. The model aver-age forecasts are then given by

fa,t+k =u∑

F=l

ωF,t+kfpcF,t+k, (6)

where l and u are the lower and upper bounds for the number of principalcomponents used to forecast inflation. We compare two simple approachesto choosing the weights {ωF,t}. In the first approach, we simply weight allforecasts equally, setting {ωF,t} ≡ 1/(u − l + 1), so that fa,t+k is just thesimple average of all date t + k forecasts. In our second approach, we chooseweights using an in-sample regression of the realized inflation rate on theinflation predictions from each model. In other words, the weights are givenby the estimated coefficients in the following regression:

πkt+k − π1

t+1 =u∑

F=l

ωF,t+kfpcF,t+k + εt+k. (7)

5 Results

The ability of the different models, benchmarks and factor models, to fore-cast inflation is summarized by the mean-squared-error (MSE) and mean-absolute-error (MAE) of their forecasts relative to the MSE and MAE offorecasts based on the autoregressive type model (AR-type) respectively.Table 1 summarizes the results from different tables while results for thecompeting benchmark models are found in Table 2.

Tables 3-7 present results for the different specifications of the factor mod-els; Term Structure Dimension, Time Series Dimension, and All Information(using both the term structure and time series dimension). For the latter twomodels results are shown for 5, 10, and 20 days of lagged daily interest ratespreads. As mentioned above, principal components were selected using twodifferent methods: 1) the variance sort criteria and 2) the correlation sortcriteria. We also considered three competing approaches to chose the number

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of principal components: a) the specification of parsimonious models withfixed numbers of principal components, b) selection by BIC, and c) modelaveraging across forecasts with different numbers of principal components.Results for the different models are summarized in Tables 3-7 following theorganization described in Table 1.

5.1 Benchmark

Table 2 summarizes results for the different benchmark models: the autoregressive-type model (AR-type), the augmented Fisher model, and the Phillips Curve(PhCu) specification. The columns labelled I(1) and I(2) refer to cases whereprice indices are modelled as I(1) and I(2) respectively. Panel A shows re-sults when the parameters are estimated using recursive least squares3 andin Panel B rolling windows estimates are presented. In order to make com-parisons between the different estimates, the MSE and MAE are computedrelative to the recursive least squares AR type model. Results are shownfor the 12-month and 36-month inflation forecasts, and are divided in twoforecast sub-samples: 1974-1983, 1984-2004, and for the whole period.

There are important differences in the forecasting performance betweenrecursive least squares estimates and rolling windows estimates. Both, MSEand MAE deteriorate dramatically when parameters are estimated usingrolling windows, especially the 36-month inflation rate forecasts. The MSEfor the 12-month inflation forecasts over the period 1974-2004 using the ARmodel when prices are modelled as I(1) increases by 14 percent when es-timated by rolling windows (1.144, panel B.1, AR) compared to the leastsquare estimation (1.000, panel A.1). For the 36-month inflation forecasts,least squares estimation outperforms rolling windows forecasts by 68 per-cent measured by MSE (1.684, panel B.2, AR) and by 20 percent in termsof the MAE (1.206, panel B.2, AR). The augmented Fisher representationand Phillips Curve benchmarks exhibit the same pattern. In the case ofthe Fisher’s representation, the MSE for the 12-month inflation rate goesfrom 0.847 (panel A.2, Fisher) in the least square case to 1.480 (panel B.2,Fisher) in the rolling windows case. Therefore, in the results that follow weconcentrate exclusively on recursive least squares estimates.

The AR type model and the augmented Fisher specification outperform

3In other words the model is re-estimated and a forecast computed as each new obser-vation arrives.

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the rest of the benchmarks. In the case of the 12-month inflation period,the AR type model with an I(1) specification performs the best in both sub-samples.4 For the 36-months inflation rate, forecasts using Fisher’s modeloutperform the rest of the benchmarks with a relative MSE of 0.847 (panelA.2, Fisher) compared to 1.000 and 1.203 for the AR model and PhillipsCurve model respectively. Modelling price indexes as I(1) result in bet-ter forecasts than I(2) specifications in the 1974-1983 sub-sample for boththe 12 and 36-month inflation terms. However, for the 36-period inflation,the AR-type and Fisher’s representation with price indices modelled as I(2)outperform the I(1) specification in the 1984-2004 sub-sample. Despite thisresult, benchmarks using the I(1) specification have a smaller MSE and MAEthan the I(2) specification for the whole sample.

Although no-single benchmark model dominated in all cases, we con-cluded that the AR-type model provided the best overall benchmark model.The comparisons in the proceeding tables (Tables 3-7) all employ this bench-mark when presenting the relative MSEs and MAEs. Since the Fisher modelprovided the starting point for the term structures models proposed here,Tables 3-6 also include a column with the results from the augmented Fishermodel as an additional point of comparison.

5.2 Fixed Number of Principal Components

Tables 3 and 6 present the results for different specifications of the factormodel using a fixed number of principal components (from one to five) forthe variance and correlation sort criteria respectively. In Panel A the resultsfor the 12-period inflation period are shown and in Panel B results for the 36-period inflation are presented. The columns labelled “Term Structure” referto the factor model that includes as predictors the current term structure.The columns labelled “Time” present results for the model that incorporateslags of daily spreads with the same maturity as the target inflation. Finally,both are combined in the columns labelled “5 days”, “10 days” and “20 days”for a factor model that respectively include 5, 10 and 20 lags of the daily termstructure as predictors.

Several findings come out from these tables. All factor models, exceptthe time dimension model, outperform the AR type benchmark uniformly

4We compared I(1) and I(2) specifications in which the left-hand side variable is iden-tical annualized inflation differences, but the right hand side inflation variables where I(1)(π1

t−p) or I(2) (first differences of π1t−p).

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in the 1974-1983 sub-sample. Important differences in forecasting perfor-mance between models are observed depending on the target inflation rate.Forecast errors are smaller when factors are selected using the correlationapproach than the variance approach in the 1974-1983 sub-sample. How-ever, in the 1984-2004 sub-sample, the variance sort produce smaller MSEfor the 12-month inflation period. For example, the relative MSE for theTerm Structure model using three principal components is 0.820 (Table 3,panel A.3, Term Structure) in the case of the variance sort and 0.805 (Ta-ble 4, panel A.3, Term Structure) in the correlation sort for the 1974-1983subsample, however for the 1984-2004 subsample the results for the varianceand correlation sort are 0.933 and 1.300 respectively.

The term structure model outperforms the rest of the factor models whenforecasting the 12-month inflation rate for the whole sample. However, thereis an improvement in forecasts during the 1984-2004 sub-sample when in-cluding the time dimension into the model. Forecasts that use 5 or 10 lagsof the daily term structure of interest rates with three principal componentsperform the best, with a relative MSE of 0.930 and 0.899 (Table 3, panel A.3,5 and 10 days) respectively. In general, models with three or more principalcomponents improve upon the more parsimonious representations with onlyone or two principal components.

For the 36-month inflation rate, the correlation sort criteria producessuperior forecasts than the variance sort. In contrast with the 12-monthinflation, we observe that the time dimension contains information on futureinflation beyond the AR-type specification, and uniformly outperforms theAR-type model in all sub-samples. Forecasts with one principal componentoutperform higher dimension models that include more than one principalcomponent.

Different numbers of lags of daily spreads, m, in the predictors matrix,X for the All Information model are tested. In general, forecasts using 5daily lags of interest rate spreads perform better than when 10 or 20 lags areincluded.

5.3 Bayesian Information Criteria

Table 5 shows the results when the number of principal components is se-lected recursively using BIC. The results are qualitatively similar to thosepresented in Tables 3 and 6. Overall, the term structure dimension appearsmore informative than the time dimension, specially for the 12-month in-

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flation rate. The time dimension of interest rates adds information relativeto the AR-type model for the 36-month inflation, but not for the 12-monthinflation. In contrast to the results obtained in Tables 2 and 3, we obtainmixed results regarding the forecast performance between the correlation andvariance sorts. The specification that performs the best for the 12-month in-flation includes only the term structure dimension and has a relative MSEof 0.908 (panel A, Term Structure) and 0.907 (panel B, Term Structure) forthe correlation and variance sort respectively.

5.4 Model Averaging

Finally, Tables 6 and 7 present results for model averaging. Important im-provements in forecasting performance are observed from model averaging.Two different methods are employed to compute forecast averages. The firstmethod is a simple average across models with different number of princi-pal components. The second method is a weighted average across modelswhere weights are obtained by regressing in-sample inflation realizations onforecasts with different numbers of principal components. OLS averagingoutperform simple averages. Consistent with the results observed in Tables3 and 6, averages of models with 3 to 5 principal components perform thebest when forecasting the 12-period inflation; and for the 36-period inflation,averages from 1 to 3, and from 1 to 5 principal components are the best.The variance criteria outperforms the correlation criteria for the 12-monthinflation rate. The opposite is true for the 36-period inflation rate.

5.5 Significance of Forecast Improvements

Tables 8 and 9 show the results of the Diebold and Mariano (1995) tests usedto assess the significance of the out-of-sample improvements of the forecastsusing the regression based model average relative to the autoregressive bench-mark. In order to generate a sufficient sample size for reliable out-of-sampletesting, the tests are based on the full sample period.

Since the forecast horizons, (k = 12 and k = 36 months) exceed themonthly sampling period, the errors from even an unbiased forecast wouldfollow a moving average process of order k − 1 and it would therefore beunrealistic to expect uncorrelated errors in either forecast model. Thus toimplement the Diebold and Mariano (1995) test we require a kernel estimatorfor the long-run variance of the difference in the squared forecast errors.

16

To this end, we employ the Newey-West (Bartlett) kernel, with a baselinebandwidth choice of k−1. While k−1 seems a natural choice in the contextof a k-period forecast, our relatively large values of k may result in a noisyvariance estimator. For this reason, and to assess the robustness of ourfindings, we also provide test results using several smaller bandwidth values.

The results from the Diebold and Mariano (1995) tests confirm the statis-tical significance of the forecast improvements relative to our baseline model.In fact, all the t-statistics shown in Tables 8 and 9 exceed the standard crit-ical values by a substantial margin. This strong significance appears robustacross the bandwidth choice, forecast horizon (k =12,36), the collection ofprincipal component models included in the average (1 to 3, 3 to 5, 1 to5), and the choice of term structure and lag information used for the princi-ple components (columns 3-7). On the other hand, the tests do not adjustfor the effects of parameter estimation (West 1996), model encompassing(McCracken 2007), or potential data snooping concerns (White 2000). Thiswould be of particular concern in the case of marginally significant results.However, any resulting distortions would have to be quite large in order tooverturn the very highly significant results shown in Tables 8 and 9.

5.6 Summary of results

In summary, the AR type model outperforms all benchmarks for the 12-month inflation rate, and the augmented Fisher model outperforms all bench-marks when forecasting the 36-period inflation. Forecasts improvements overthe benchmark model for the 12-month inflation rate come from the termstructure dimension, with better results obtained when using the variancesort criteria than the correlation sort.

For the 36-period inflation both dimensions, the term structure and timedimension improve upon the benchmark model, with the correlation crite-ria performing better than the variance criteria. However, model averagingprovide robust forecasting improvements as compared with both the Fishermodel and autoregressive type models.

6 Conclusion

This paper proposes a new approach to forecasting inflation using daily in-terest rate data. We consider a large number of potential interest rates

17

predictors and organize them along a term structure and time series dimen-sion. Principal component methods are used to extract useful predictors.For 12 month predictive regressions, only the term structure dimension im-proves forecasts, while for the 36 month regressions both the term structureand time dimension are useful. We find robust forecasting improvements ingeneral as compared to the augmented Fisher equation and autoregressivebenchmarks.

The performance of variance and correlation criteria used to select prin-ciple components criteria is mixed, and depends on the subsample and theforecast horizon. Our recommended approach is the model average acrossmodels using between 3 to 5 principle components from both term struc-ture and time dimensions. This approach provides good results for all timeperiods and forecast horizons.

18

Table 1: Location of Forecast Summary Results for Various Models

Number ofPrincipal Components

Selection of Principal ComponentsVariance Sort Correlation Sort

Fixed from 2 to 6 Table 3 Table 6BIC Table 5 Table 5Model Averaging Tables 6 and 7 Tables 6 and 7

19

Tab

le2:

Bench

mark

Models

:R

elat

ive

MSE

and

MA

Efo

rdiff

eren

tben

chm

arks

com

par

edto

the

au-

tore

gres

sive

type

model

(AR

-type)

esti

mat

edby

recu

rsiv

ele

ast

squar

es.

The

colu

mns

label

led

“PhC

u”

and

“Fis

her

”re

fer

toth

eP

hillips

Curv

em

odel

and

toth

eau

gmen

ted

Fis

her

spec

ifica

tion

.Lab

els

I(1)

and

I(2)

refe

rto

the

case

sw

her

epri

cein

dic

esar

em

odel

led

asei

ther

I(1)

orI(

2).

InPan

elA

resu

lts

usi

ng

Rec

urs

ive

Lea

stSquar

esar

esh

own.

Pan

elB

pre

sent

resu

lts

usi

ng

rollin

ges

tim

ates

wit

ha

win

dow

of10

year

s.

Pan

elA

.R

ecurs

ive

Lea

stSquar

esSam

ple

Rel

ative

MSE

MA

EA

RFis

her

PhC

uA

RFis

her

PhC

uI(

1)

I(2)

I(1)

I(2)

I(1)

I(2)

I(1)

I(2)

I(1)

I(2)

I(1)

I(2)

1.

π12

t+12−

π1 t+

1

1974-1

983

1.0

00

1.0

56

1.0

36

1.0

28

1.0

01

1.0

84

1.0

00

1.0

09

1.0

19

0.9

97

1.0

13

1.0

29

1984-2

004

1.0

00

1.0

38

1.0

75

1.2

08

1.2

12

1.2

13

1.0

00

0.9

35

1.0

28

1.0

14

1.1

12

1.0

37

1974-2

004

1.0

00

1.0

50

1.0

49

1.0

87

1.0

71

1.1

27

1.0

00

0.9

73

1.0

23

1.0

05

1.0

61

1.0

33

2.

π36

t+36−

π1 t+

1

1974-1

983

1.0

00

1.9

99

0.8

27

1.9

31

0.8

25

1.8

86

1.0

00

1.3

95

0.9

03

1.3

05

0.8

72

1.3

53

1984-2

004

1.0

00

0.6

90

0.8

82

0.7

37

1.8

48

1.1

75

1.0

00

0.6

72

0.9

36

0.7

02

1.3

62

0.9

27

1974-2

004

1.0

00

1.5

16

0.8

47

1.4

90

1.2

03

1.6

24

1.0

00

1.0

27

0.9

20

0.9

98

1.1

22

1.1

36

Pan

elB

.R

olling

Win

dow

sSam

ple

Rel

ative

MSE

MA

EA

RFis

her

PhC

uA

RFis

her

PhC

uI(

1)

I(2)

I(1)

I(2)

I(1)

I(2)

I(1)

I(2)

I(1)

I(2)

I(1)

I(2)

1.

π12

t+12−

π1 t+

1

1974-1

983

1.0

78

1.1

59

1.0

60

1.1

11

1.1

19

1.1

01

1.0

30

1.0

53

1.0

37

1.0

24

1.0

31

1.0

31

1984-2

004

1.2

79

1.0

07

1.3

95

1.0

49

1.5

00

1.3

91

1.1

37

0.9

46

1.1

51

0.9

71

1.2

47

1.0

70

1974-2

004

1.1

44

1.1

08

1.1

70

1.0

91

1.2

45

1.1

97

1.0

82

1.0

01

1.0

93

0.9

98

1.1

36

1.0

50

2.

π36

t+36−

π1 t+

1

1974-1

983

1.1

67

2.0

41

0.9

18

2.0

59

1.0

56

1.6

58

1.0

94

1.4

07

0.9

32

1.2

74

1.0

10

1.2

58

1984-2

004

2.5

67

0.7

62

2.4

40

0.7

64

3.7

56

2.3

22

1.3

14

0.7

19

1.4

03

0.7

00

1.7

41

1.2

85

1974-2

004

1.6

84

1.5

69

1.4

80

1.5

81

2.0

53

1.9

03

1.2

06

1.0

56

1.1

72

0.9

81

1.3

83

1.2

72

20

Tab

le3:

Out-

of-Sam

ple

Fore

cast

ing

for

Com

peti

ng

Models

,V

ari

ance

Sort

.Rel

ativ

eM

SE

and

MA

E

for

diffe

rent

spec

ifica

tion

sar

epr

esen

ted.

All

resu

lts

are

rela

tive

toth

eA

R-t

ype

benc

hmar

km

odel

.T

heco

lum

nsla

belle

d“F

ishe

r”pr

esen

tre

sult

sfo

rth

em

odel

that

inco

rpor

ates

aspr

edic

tors

the

mon

thly

yiel

dsp

read

wit

hth

esa

me

mat

urity

asth

efo

reca

sted

infla

tion

diffe

renc

e.T

heco

lum

nsla

belle

d“T

erm

Stru

ctur

e”pr

esen

tre

sult

sfo

rth

em

odel

that

inco

rpor

ates

aspr

edic

tors

the

prin

cipa

lco

mpo

nent

sof

the

mon

thly

term

stru

ctur

esp

read

s.In

the

colu

mns

labe

lled

“Tim

e,”

resu

lts

for

the

mod

elth

atus

esda

ilysp

read

sw

ith

the

sam

em

atur

ity

asin

flati

onar

epr

esen

ted.

The

last

thre

eco

lum

nssh

owth

ere

sult

spr

inci

pal

com

pone

ntm

odel

inco

rpor

atin

gbo

thth

eti

me

and

term

stru

ctur

edi

men

sion

s,w

ith

eith

er5,

10,or

20da

ysof

daily

inte

rest

rate

lags

.

Num

.Sam

ple

Augm

ente

dTer

mStr

uct

ure

Tim

eB

oth

Dim

ensi

ons

(Tim

e&

Ter

mStr

uct

ure

)of

Per

iod

Fis

her

Dim

ensi

on

Dim

ensi

on

5D

ays

10

Days

20

Days

PC

sM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EPanel

A.π

12

t+12−

π1 t+

1

11974-1

983

1.0

36

1.0

19

0.9

16

0.8

41

1.0

34

1.0

20

0.9

21

0.8

43

0.9

60

0.8

55

0.9

59

0.8

65

1984-2

004

1.0

75

1.0

28

1.1

74

1.0

58

1.0

93

1.0

40

1.1

56

1.0

54

1.1

29

1.0

44

1.1

29

1.0

46

1974-2

004

1.0

49

1.0

23

1.0

01

0.9

46

1.0

53

1.0

30

0.9

99

0.9

46

1.0

16

0.9

46

1.0

15

0.9

52

21974-1

983

0.9

54

0.8

83

1.0

51

1.0

18

0.9

64

0.8

89

1.0

02

0.9

03

1.0

32

0.9

28

1984-2

004

1.1

36

1.0

68

1.0

72

1.0

36

1.1

24

1.0

64

1.1

10

1.0

58

1.0

91

1.0

57

1974-2

004

1.0

14

0.9

73

1.0

58

1.0

26

1.0

17

0.9

74

1.0

38

0.9

78

1.0

51

0.9

91

31974-1

983

0.8

20

0.7

98

1.0

60

1.0

19

0.8

27

0.8

01

0.8

60

0.8

20

0.9

60

0.9

01

1984-2

004

0.9

33

0.9

60

1.0

51

1.0

22

0.9

30

0.9

66

0.8

99

0.9

41

1.0

89

1.0

45

1974-2

004

0.8

57

0.8

77

1.0

57

1.0

20

0.8

61

0.8

81

0.8

73

0.8

79

1.0

03

0.9

71

41974-1

983

0.8

10

0.8

10

1.0

72

1.0

26

0.8

33

0.8

08

0.8

57

0.8

17

0.9

00

0.8

51

1984-2

004

1.1

02

1.0

28

1.0

53

1.0

22

1.1

87

1.0

57

1.0

53

1.0

19

0.9

18

0.9

58

1974-2

004

0.9

06

0.9

16

1.0

66

1.0

24

0.9

50

0.9

29

0.9

22

0.9

15

0.9

06

0.9

03

51974-1

983

0.7

73

0.7

90

1.0

96

1.0

27

0.8

44

0.8

08

0.8

86

0.8

29

0.8

91

0.8

29

1984-2

004

1.0

81

1.0

17

1.0

55

1.0

23

1.1

34

1.0

41

1.1

36

1.0

43

1.2

57

1.0

76

1974-2

004

0.8

75

0.9

00

1.0

83

1.0

25

0.9

40

0.9

21

0.9

69

0.9

33

1.0

12

0.9

49

Panel

B.π

36

t+36−

π1 t+

1

11974-1

983

0.8

27

0.9

03

0.8

31

0.9

04

0.8

69

0.9

22

0.8

47

0.8

43

0.9

60

0.8

55

0.9

21

0.8

43

1984-2

004

0.8

82

0.9

36

0.7

50

0.8

45

0.8

96

0.9

43

0.7

59

1.0

54

1.1

29

1.0

44

1.1

56

1.0

54

1974-2

004

0.8

47

0.9

20

0.8

01

0.8

74

0.8

79

0.9

33

0.8

15

0.9

46

1.0

16

0.9

46

0.9

99

0.9

46

21974-1

983

1.0

61

1.0

00

0.8

42

0.9

03

1.0

81

0.8

89

1.0

02

0.9

03

0.9

64

0.8

89

1984-2

004

0.8

50

0.8

43

0.8

93

0.9

41

0.8

70

1.0

64

1.1

10

1.0

58

1.1

24

1.0

64

1974-2

004

0.9

83

0.9

20

0.8

61

0.9

23

1.0

03

0.9

74

1.0

38

0.9

78

1.0

17

0.9

74

31974-1

983

0.7

23

0.7

93

0.8

80

0.9

24

0.7

63

0.8

01

0.8

60

0.8

20

0.8

27

0.8

01

1984-2

004

1.1

76

1.0

49

0.8

95

0.9

42

1.1

94

0.9

66

0.8

99

0.9

41

0.9

30

0.9

66

1974-2

004

0.8

90

0.9

24

0.8

86

0.9

33

0.9

22

0.8

81

0.8

73

0.8

79

0.8

61

0.8

81

41974-1

983

0.8

22

0.8

53

0.8

98

0.9

32

0.8

87

0.8

08

0.8

57

0.8

17

0.8

33

0.8

08

1984-2

004

0.8

48

0.8

78

0.8

88

0.9

38

0.8

48

1.0

57

1.0

53

1.0

19

1.1

87

1.0

57

1974-2

004

0.8

32

0.8

65

0.8

95

0.9

35

0.8

72

0.9

29

0.9

22

0.9

15

0.9

50

0.9

29

51974-1

983

0.7

97

0.8

44

0.8

89

0.9

21

0.8

65

0.8

08

0.8

86

0.8

29

0.8

44

0.8

08

1984-2

004

0.9

32

0.9

19

0.9

04

0.9

43

0.8

56

1.0

41

1.1

36

1.0

43

1.1

34

1.0

41

1974-2

004

0.8

47

0.8

82

0.8

95

0.9

33

0.8

62

0.9

21

0.9

69

0.9

33

0.9

40

0.9

21

21

Tab

le4:

Out-

of-Sam

ple

Fore

cast

ing

for

Com

peti

ng

Models

,C

orr

ela

tion

Sort

.Res

ults

for

the

MS

E

and

MA

Efo

rdi

ffere

ntsp

ecifi

cati

ons

are

pres

ente

d.A

llre

sult

sar

ere

lati

veto

the

AR

benc

hmar

km

odel

.T

heco

lum

nsla

belle

d“F

ishe

r”pr

esen

tre

sult

sfo

rth

em

odel

that

inco

rpor

ates

aspr

edic

tor

the

mon

thly

yiel

dsp

read

wit

hth

esa

me

mat

urity

than

the

fore

cast

edin

flati

ondi

ffere

nce.

The

colu

mns

labe

lled

“Ter

mSt

ruct

ure”

pres

ent

resu

ltsfo

rth

em

odel

that

inco

rpor

ates

aspr

edic

tors

the

prin

cipa

lcom

pone

ntsof

the

mon

thly

term

stru

ctur

esp

read

s.In

the

colu

mns

labe

lled

“Tim

e”,re

sult

sfo

rth

em

odel

that

uses

daily

spre

ads

wit

hth

esa

me

mat

urity

asin

flati

onar

epr

esen

ted.

The

last

thre

eco

lum

nssh

owth

ere

sult

spr

inci

palc

ompo

nent

mod

elin

corp

orat

ing

both

the

tim

ean

dte

rmst

ruct

ure

dim

ensi

ons,

wit

hei

ther

5,10

,or

20da

ysof

daily

inte

rest

rate

lags

.

Num

.Sam

ple

Augm

ente

dTer

mStr

uct

ure

Tim

eB

oth

Dim

ensi

ons

(Tim

e&

Ter

mStr

uct

ure

)of

Per

iod

Fis

her

Dim

ensi

on

Dim

ensi

on

5D

ays

10

Days

20

Days

PC

sM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EPanel

A.π

12

t+12−

π1 t+

1

11974-1

983

1.0

36

1.0

19

0.9

16

0.8

41

1.0

70

1.0

19

0.9

21

0.8

43

0.9

60

0.8

55

0.9

59

0.8

65

1984-2

004

1.0

75

1.0

28

1.1

74

1.0

58

1.0

93

1.0

40

1.1

56

1.0

54

1.1

29

1.0

44

1.1

29

1.0

46

1974-2

004

1.0

49

1.0

23

1.0

01

0.9

46

1.0

78

1.0

29

0.9

99

0.9

46

1.0

16

0.9

46

1.0

15

0.9

52

21974-1

983

0.8

77

0.8

14

1.0

78

1.0

28

0.8

87

0.8

20

0.9

24

0.8

27

0.9

35

0.8

47

1984-2

004

1.5

72

1.1

97

1.0

79

1.0

36

1.6

82

1.2

45

1.5

63

1.2

11

1.6

53

1.2

01

1974-2

004

1.1

07

1.0

00

1.0

78

1.0

32

1.1

49

1.0

27

1.1

35

1.0

13

1.1

72

1.0

19

31974-1

983

0.8

05

0.7

81

1.0

91

1.0

28

0.8

14

0.7

82

0.8

31

0.7

95

0.8

83

0.8

28

1984-2

004

1.3

00

1.0

88

1.0

78

1.0

33

1.3

70

1.1

15

1.3

31

1.1

01

1.4

44

1.1

26

1974-2

004

0.9

69

0.9

30

1.0

87

1.0

30

0.9

98

0.9

44

0.9

96

0.9

43

1.0

68

0.9

72

41974-1

983

0.7

94

0.7

81

1.0

98

1.0

42

0.8

20

0.7

90

0.8

13

0.7

97

0.8

43

0.7

97

1984-2

004

1.1

02

1.0

28

1.0

64

1.0

34

1.2

16

1.0

61

1.0

71

1.0

31

1.2

45

1.0

78

1974-2

004

0.8

96

0.9

01

1.0

87

1.0

38

0.9

50

0.9

22

0.8

98

0.9

11

0.9

76

0.9

33

51974-1

983

0.7

73

0.7

90

1.1

38

1.0

53

0.8

07

0.7

83

0.8

08

0.8

08

0.8

06

0.7

83

1984-2

004

1.0

81

1.0

17

1.0

45

1.0

26

1.1

71

1.0

26

1.1

13

1.0

22

1.1

99

1.0

46

1974-2

004

0.8

75

0.9

00

1.1

07

1.0

40

0.9

27

0.9

01

0.9

09

0.9

12

0.9

36

0.9

11

Panel

B.π

36

t+36−

π1 t+

1

11974-1

983

0.8

27

0.9

03

0.6

37

0.7

68

0.8

70

0.9

21

0.6

92

0.8

15

0.6

77

0.7

91

0.6

56

0.7

70

1984-2

004

0.8

82

0.9

36

0.6

77

0.8

03

0.9

29

0.9

59

0.6

87

0.7

98

0.7

73

0.8

49

0.7

23

0.8

15

1974-2

004

0.8

47

0.9

20

0.6

52

0.7

86

0.8

92

0.9

40

0.6

90

0.8

06

0.7

12

0.8

21

0.6

81

0.7

93

21974-1

983

0.6

62

0.7

56

0.9

36

0.9

60

0.6

84

0.7

94

0.7

25

0.7

92

0.6

28

0.7

43

1984-2

004

1.3

25

1.0

81

0.9

32

0.9

56

1.2

61

1.0

29

0.8

30

0.8

51

1.2

85

1.0

24

1974-2

004

0.9

07

0.9

21

0.9

35

0.9

58

0.8

97

0.9

14

0.7

64

0.8

22

0.8

71

0.8

86

31974-1

983

0.6

91

0.7

65

0.9

43

0.9

81

0.8

04

0.8

31

0.8

54

0.8

83

0.7

32

0.7

65

1984-2

004

0.9

11

0.8

86

0.9

52

0.9

70

0.9

15

0.8

80

0.6

09

0.7

49

1.3

08

1.0

57

1974-2

004

0.7

72

0.8

27

0.9

47

0.9

75

0.8

45

0.8

56

0.7

63

0.8

15

0.9

44

0.9

14

41974-1

983

0.7

33

0.7

96

0.9

09

0.9

33

0.7

59

0.8

09

0.8

80

0.8

83

0.8

03

0.7

98

1984-2

004

0.9

96

0.9

31

0.9

42

0.9

55

0.9

46

0.9

00

0.8

43

0.8

68

0.8

88

0.8

69

1974-2

004

0.8

30

0.8

65

0.9

21

0.9

44

0.8

28

0.8

55

0.8

67

0.8

76

0.8

34

0.8

34

51974-1

983

0.7

97

0.8

44

0.9

56

0.9

56

0.8

01

0.8

30

0.7

69

0.7

99

0.7

58

0.7

76

1984-2

004

0.9

32

0.9

19

0.9

40

0.9

48

0.9

64

0.9

08

0.7

70

0.8

15

0.9

24

0.8

82

1974-2

004

0.8

47

0.8

82

0.9

50

0.9

52

0.8

61

0.8

70

0.7

69

0.8

07

0.8

19

0.8

30

22

Tab

le5:

Out-

of-Sam

ple

Fore

cast

ing

usi

ng

BIC

todete

rmin

eth

enum

ber

of

pri

nci

palco

mpo-

nents

:π

12

t+12−

π1 t+

1.R

elat

ive

MS

Ean

dM

AE

for

the

diffe

rent

spec

ifica

tion

sar

epr

esen

ted.

MS

Ean

dM

AE

for

each

spec

ifica

tion

isca

lcul

ated

rela

tive

toth

eA

R-t

ype

mod

el.

The

colu

mns

labe

lled

“Ter

mSt

ruct

ure”

pres

ent

re-

sult

sfo

rth

em

odel

that

inco

rpor

ates

aspr

edic

tors

the

prin

cipa

lcom

pone

nts

ofth

em

onth

lyte

rmst

ruct

ure

spre

ads.

Inth

eco

lum

nsla

belle

d“T

ime”

,res

ults

for

the

spec

ifica

tion

that

inco

rpor

ates

aspr

edic

tors

prin

cipa

lcom

pone

nts

ofda

ilym

atch

edsp

read

sar

esh

own.

The

last

thre

eco

lum

nssh

owth

ere

sult

spr

inci

palc

ompo

nent

mod

elin

corp

orat

ing

both

the

tim

ean

dte

rmst

ruct

ure

dim

ensi

ons,

wit

hei

ther

5,10

,15

,or

20da

ysof

daily

inte

rest

rate

lags

.Pan

elA

pres

ents

the

resu

lts

for

the

case

whe

reth

epr

inci

palc

ompo

nent

sw

ith

the

larg

est

in-s

ampl

eco

rrel

atio

nsbe

twee

nth

eA

Rre

sidu

als

and

the

esti

mat

edpr

inci

palc

ompo

nent

sar

ese

lect

ed.

InPan

elB

show

sth

ere

sult

sw

here

the

prin

cipa

lco

mpo

nent

sw

ith

the

high

est

vari

ance

are

sele

cted

are

pres

ente

d.

Sam

ple

Ter

mStr

uct

ure

Tim

eB

oth

Dim

ensi

ons

(Tim

e&

Ter

mStr

uct

ure

)Per

iod

Dim

ensi

on

Dim

ensi

on

5D

ays

10

Days

15

Days

20

Days

MSE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EPanelA

.C

orrela

tion

Sort

π12

t+12−

π1 t+

1

1974-1

983

0.7

99

0.7

86

1.0

47

1.0

00

0.8

34

0.8

09

0.8

27

0.7

99

0.9

19

0.8

46

0.8

17

0.7

93

1984-1

990

1.4

94

1.1

97

1.1

56

1.0

45

1.6

55

1.2

43

1.5

07

1.2

20

1.8

20

1.3

51

1.9

79

1.3

41

1991-2

004

0.8

39

0.9

37

1.1

64

1.0

95

0.8

10

0.9

29

0.8

11

0.9

24

0.8

49

0.9

48

0.8

54

0.9

51

1974-2

004

0.9

08

0.9

09

1.0

83

1.0

35

0.9

50

0.9

28

0.9

24

0.9

17

1.0

39

0.9

74

0.9

93

0.9

45

π36

t+12−

π1 t+

1

1974-1

983

0.7

07

0.7

68

0.8

72

0.9

12

0.7

42

0.8

04

0.8

33

0.8

46

0.7

85

0.8

21

0.7

22

0.7

61

1984-1

990

0.7

81

0.8

57

0.9

67

1.0

08

0.7

70

0.8

34

0.6

43

0.8

03

0.5

47

0.7

10

0.4

54

0.6

32

1991-2

004

1.0

60

0.9

80

0.9

60

0.9

61

1.0

69

0.9

66

1.1

52

0.9

94

1.1

72

1.0

11

1.1

24

0.9

92

1974-2

004

0.7

97

0.8

51

0.9

05

0.9

45

0.8

20

0.8

60

0.8

79

0.8

84

0.8

39

0.8

60

0.7

76

0.8

09

PanelB

.V

aria

nce

Sort

π12

t+12−

π1 t+

1

1974-1

983

0.7

98

0.7

85

1.0

18

1.0

04

0.8

51

0.8

28

0.8

80

0.8

40

0.9

34

0.8

78

0.8

82

0.8

31

1984-1

990

1.4

94

1.1

97

1.1

53

1.0

35

1.7

92

1.3

03

1.5

00

1.2

25

1.9

09

1.3

49

1.9

50

1.3

44

1991-2

004

0.8

39

0.9

37

1.1

55

1.0

96

0.8

43

0.9

52

0.8

31

0.9

39

0.8

57

0.9

63

0.8

49

0.9

50

1974-2

004

0.9

07

0.9

09

1.0

61

1.0

36

0.9

88

0.9

56

0.9

62

0.9

43

1.0

63

0.9

95

1.0

33

0.9

65

π36

t+12−

π1 t+

1

1974-1

983

0.7

25

0.7

85

0.8

80

0.9

17

0.8

11

0.8

25

0.7

74

0.8

03

0.7

39

0.7

66

0.7

55

0.7

69

1984-1

990

0.7

86

0.8

59

0.8

98

0.9

76

0.7

71

0.8

38

1.0

57

0.9

72

0.7

88

0.8

53

0.7

45

0.8

27

1991-2

004

0.9

47

0.9

27

0.9

60

0.9

61

0.9

63

0.9

33

0.9

64

0.9

38

0.9

62

0.9

39

0.9

45

0.9

31

1974-2

004

0.7

84

0.8

43

0.9

00

0.9

42

0.8

40

0.8

62

0.8

55

0.8

77

0.7

96

0.8

37

0.7

97

0.8

30

23

Tab

le6:

ModelA

vera

gin

g:

π12

t+12−

π1 t+

1.

See

table

not

eon

the

nex

tpag

e.

Num

.Sam

ple

Ter

mStr

uct

ure

Tim

eB

oth

Dim

ensi

ons

(Tim

e&

Ter

mStr

uct

ure

)of

Per

iod

Dim

ensi

on

Dim

ensi

on

5D

ays

10

Days

20

Days

PC

sM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EPanel

A.C

orr

elation

Sort

Sim

ple

Aver

age

1to

31974-1

983

0.8

44

0.7

95

1.0

59

1.0

16

0.8

57

0.8

04

0.8

83

0.8

11

0.8

99

0.8

24

1984-1

990

1.9

05

1.3

49

1.1

16

1.0

22

2.0

10

1.3

84

1.8

68

1.3

47

2.0

81

1.4

18

1991-2

004

0.8

61

0.9

42

1.1

53

1.0

87

0.9

01

0.9

69

0.8

95

0.9

56

0.8

61

0.9

30

1974-2

004

1.0

02

0.9

45

1.0

83

1.0

37

1.0

33

0.9

65

1.0

29

0.9

57

1.0

65

0.9

70

3to

51974-1

983

0.7

81

0.7

80

1.0

97

1.0

36

0.8

13

0.7

92

0.8

09

0.7

97

0.8

34

0.7

98

1984-1

990

1.6

52

1.2

68

1.0

68

1.0

05

1.8

11

1.3

00

1.6

01

1.2

45

1.8

89

1.3

20

1991-2

004

0.7

97

0.9

22

1.1

45

1.0

86

0.8

39

0.9

38

0.8

42

0.9

41

0.8

57

0.9

31

1974-2

004

0.9

11

0.9

16

1.1

01

1.0

44

0.9

64

0.9

33

0.9

31

0.9

26

0.9

92

0.9

38

1to

51974-1

983

0.8

08

0.7

81

1.0

68

1.0

20

0.8

28

0.7

85

0.8

42

0.8

00

0.8

57

0.8

05

1984-1

990

1.7

32

1.2

93

1.0

85

1.0

10

1.8

62

1.3

26

1.6

96

1.2

81

1.9

23

1.3

46

1991-2

004

0.8

22

0.9

26

1.1

42

1.0

84

0.8

44

0.9

41

0.8

44

0.9

36

0.8

32

0.9

12

1974-2

004

0.9

46

0.9

23

1.0

83

1.0

36

0.9

82

0.9

35

0.9

67

0.9

33

1.0

09

0.9

41

Reg

ress

ion

base

d1

to3

1974-1

983

0.7

71

0.7

77

1.0

37

1.0

01

0.8

27

0.8

11

0.7

89

0.7

92

0.8

87

0.8

31

1984-1

990

1.5

22

1.1

72

1.1

89

1.0

27

1.3

32

1.1

20

1.0

54

1.1

07

1.3

73

1.1

20

1991-2

004

0.8

54

0.9

58

1.1

70

1.0

68

1.0

27

1.0

16

1.1

76

1.1

32

1.0

73

1.0

68

1974-2

004

0.8

96

0.9

06

1.0

82

1.0

25

0.9

35

0.9

29

0.8

94

0.8

97

0.9

90

0.9

54

3to

51974-1

983

0.7

44

0.7

57

1.0

60

1.0

11

0.7

70

0.7

89

0.8

20

0.7

80

0.7

99

0.8

12

1984-1

990

1.3

75

1.1

22

1.1

82

1.0

26

1.4

63

1.1

43

1.3

22

1.2

34

1.3

29

1.1

30

1991-2

004

0.9

36

0.9

67

1.1

96

1.0

75

0.8

19

0.9

27

0.8

62

0.8

63

0.9

87

0.9

98

1974-2

004

0.8

69

0.8

88

1.1

02

1.0

32

0.8

80

0.8

97

0.9

01

0.8

61

0.9

09

0.9

27

1to

51974-1

983

0.8

02

0.7

95

1.0

78

1.0

13

0.8

72

0.8

47

0.8

75

0.8

21

0.8

98

0.8

61

1984-1

990

1.2

87

1.0

85

1.1

87

1.0

27

1.2

87

1.0

87

0.9

21

1.1

71

1.1

41

1.0

34

1991-2

004

0.7

68

0.9

08

1.1

78

1.0

68

0.8

29

0.9

32

0.8

87

1.1

38

0.9

56

1.0

07

1974-2

004

0.8

67

0.8

84

1.1

11

1.0

32

0.9

25

0.9

18

0.8

84

0.9

27

0.9

44

0.9

36

Panel

B.Vari

ance

Sort

Sim

ple

Aver

age

1to

31974-1

983

0.8

79

0.8

28

1.0

41

1.0

16

0.8

90

0.8

32

0.9

23

0.8

46

0.9

71

0.8

94

1984-1

990

1.3

82

1.1

48

1.1

10

1.0

18

1.3

55

1.1

39

1.2

84

1.1

03

1.4

21

1.1

74

1991-2

004

0.8

16

0.9

30

1.1

42

1.0

86

0.8

12

0.9

24

0.8

25

0.9

34

0.8

86

0.9

86

1974-2

004

0.9

42

0.9

19

1.0

69

1.0

36

0.9

45

0.9

18

0.9

59

0.9

21

1.0

22

0.9

75

3to

51974-1

983

0.7

93

0.7

96

1.0

69

1.0

22

0.8

26

0.7

99

0.8

57

0.8

16

0.8

91

0.8

37

1984-1

990

1.3

74

1.1

62

1.0

96

1.0

04

1.5

17

1.2

22

1.3

16

1.1

45

1.3

60

1.1

60

1991-2

004

0.7

97

0.9

22

1.1

24

1.0

81

0.8

28

0.9

43

0.8

30

0.9

42

0.8

35

0.9

47

1974-2

004

0.8

79

0.9

04

1.0

82

1.0

35

0.9

28

0.9

23

0.9

20

0.9

16

0.9

50

0.9

32

1to

51974-1

983

0.8

26

0.7

95

1.0

51

1.0

19

0.8

52

0.8

00

0.8

85

0.8

16

0.9

18

0.8

55

1984-1

990

1.4

09

1.1

68

1.1

04

1.0

11

1.4

73

1.1

98

1.3

42

1.1

46

1.3

74

1.1

57

1991-2

004

0.7

62

0.8

97

1.1

34

1.0

84

0.7

77

0.9

05

0.7

83

0.9

07

0.8

15

0.9

35

1974-2

004

0.9

00

0.8

97

1.0

73

1.0

36

0.9

30

0.9

08

0.9

34

0.9

07

0.9

67

0.9

37

Reg

ress

ion

base

d1

to3

1974-1

983

0.7

39

0.7

63

1.0

28

1.0

00

0.7

42

0.7

51

0.7

92

0.7

79

0.8

45

0.8

45

1984-1

990

1.1

65

1.0

37

1.2

22

1.0

42

1.2

02

1.0

94

1.1

07

1.0

14

1.6

70

1.2

48

1991-2

004

1.0

20

1.0

44

1.1

65

1.0

70

1.1

37

1.1

00

1.1

32

1.0

90

1.0

56

1.0

37

1974-2

004

0.8

50

0.8

96

1.0

80

1.0

28

0.8

77

0.9

17

0.8

97

0.9

13

1.0

02

0.9

78

3to

51974-1

983

0.7

10

0.7

30

1.0

27

0.9

85

0.7

50

0.7

59

0.7

80

0.7

84

0.8

05

0.7

91

1984-1

990

1.4

25

1.1

20

1.1

77

1.0

20

1.2

69

1.0

71

1.2

34

1.0

27

1.2

73

1.0

68

1991-2

004

0.9

39

0.9

68

1.1

43

1.0

60

0.8

52

0.9

50

0.8

63

0.9

52

0.9

09

0.9

81

1974-2

004

0.8

54

0.8

74

1.0

69

1.0

13

0.8

43

0.8

74

0.8

61

0.8

79

0.8

92

0.8

99

1to

51974-1

983

0.7

69

0.7

70

1.0

76

1.0

14

0.7

66

0.7

60

0.8

21

0.8

07

0.8

04

0.8

16

1984-1

990

1.2

10

1.0

52

1.2

16

1.0

34

1.2

77

1.1

16

1.1

71

1.0

42

1.4

51

1.1

48

1991-2

004

1.0

91

1.0

57

1.1

48

1.0

59

1.1

72

1.1

13

1.1

38

1.0

95

1.1

24

1.0

79

1974-2

004

0.8

89

0.9

06

1.1

09

1.0

30

0.9

10

0.9

30

0.9

27

0.9

34

0.9

54

0.9

55

24

Model Averaging: π12t+12−π1

t+1.Relative MSE and MAE for equally weightedforecasts using 1 to 3 principal components, 3 to 5 principal components, and 1 to5 principal components. All results are relative to the AR benchmark model. Thecolumns labelled “Term Structure” present results for the model that incorporatesas predictors the principal components of the monthly term structure spreads. Inthe columns labelled “Time”, results for the model that uses daily spreads withthe same maturity as inflation are presented. The last three columns show the re-sults principal component model incorporating both the time and term structuredimensions, with either 5, 10, or 20 days of daily interest rate lags.

25

Tab

le7:

ModelA

vera

gin

g:

π36

t+36−

π1 t+

1.

See

table

not

eon

the

nex

tpag

e.

Num

.Sam

ple

Ter

mStr

uct

ure

Tim

eB

oth

Dim

ensi

ons

(Tim

e&

Ter

mStr

uct

ure

)of

Per

iod

Dim

ensi

on

Dim

ensi

on

5D

ays

10

Days

20

Days

PC

sM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EM

SE

MA

EPanel

A.C

orr

elation

Sort

Sim

ple

Aver

age

1to

31974-1

983

0.6

23

0.7

39

0.9

16

0.9

47

0.7

79

0.8

28

0.7

08

0.7

97

0.6

35

0.7

34

1984-1

990

0.5

22

0.7

45

0.9

96

1.0

29

0.4

98

0.7

15

0.6

76

0.7

88

0.5

37

0.7

26

1991-2

004

1.1

82

1.0

41

0.9

59

0.9

58

1.0

49

0.9

74

0.7

15

0.8

13

1.4

19

1.1

27

1974-2

004

0.7

35

0.8

35

0.9

37

0.9

66

0.8

01

0.8

53

0.7

05

0.8

01

0.7

98

0.8

56

3to

51974-1

983

0.7

40

0.7

98

0.9

38

0.9

48

0.7

66

0.8

06

0.8

10

0.8

34

0.7

51

0.7

67

1984-1

990

0.6

67

0.7

84

1.0

14

1.0

33

0.6

40

0.7

53

0.5

51

0.7

56

0.5

79

0.7

28

1991-2

004

1.1

46

1.0

09

0.9

60

0.9

47

1.1

00

0.9

71

0.8

38

0.8

63

1.3

11

1.0

73

1974-2

004

0.8

22

0.8

62

0.9

53

0.9

64

0.8

24

0.8

48

0.7

81

0.8

28

0.8

54

0.8

56

1to

51974-1

983

0.6

65

0.7

65

0.9

17

0.9

39

0.7

39

0.8

06

0.7

20

0.8

00

0.6

67

0.7

42

1984-1

990

0.5

78

0.7

70

0.9

95

1.0

26

0.5

29

0.7

23

0.5

76

0.7

56

0.4

87

0.6

80

1991-2

004

1.1

49

1.0

25

0.9

56

0.9

52

1.0

39

0.9

61

0.8

11

0.8

61

1.2

93

1.0

77

1974-2

004

0.7

62

0.8

48

0.9

37

0.9

59

0.7

78

0.8

39

0.7

21

0.8

11

0.7

84

0.8

36

Reg

ress

ion

base

d1

to3

1974-1

983

0.5

84

0.7

08

0.9

06

0.9

26

0.8

16

0.7

79

0.6

56

0.5

69

0.5

92

0.6

92

1984-1

990

0.4

75

0.6

89

1.0

50

1.0

42

0.3

94

0.6

05

0.7

71

1.5

68

0.5

47

0.7

28

1991-2

004

0.7

85

0.8

57

0.9

17

0.9

33

0.9

76

0.9

47

0.9

13

1.0

54

1.2

21

1.0

52

1974-2

004

0.6

14

0.7

51

0.9

28

0.9

50

0.7

94

0.7

99

0.7

29

0.8

15

0.7

28

0.8

12

3to

51974-1

983

0.6

56

0.7

23

0.9

49

0.9

34

0.7

89

0.7

89

0.8

14

0.7

75

0.7

38

0.7

38

1984-1

990

0.7

70

0.8

60

1.0

97

1.0

55

0.6

76

0.7

55

0.5

02

1.1

54

0.7

79

0.8

78

1991-2

004

1.2

50

1.0

19

0.9

08

0.9

12

1.0

91

0.9

60

0.6

40

0.9

80

1.2

14

1.0

19

1974-2

004

0.8

06

0.8

42

0.9

60

0.9

50

0.8

42

0.8

36

0.7

32

0.8

73

0.8

51

0.8

53

1to

51974-1

983

0.5

64

0.6

83

0.8

89

0.9

04

0.7

84

0.7

42

0.6

21

0.5

48

0.6

19

0.7

03

1984-1

990

0.4

48

0.6

62

1.0

97

1.0

44

0.4

03

0.5

96

0.5

60

0.9

92

0.4

92

0.6

85

1991-2

004

0.8

04

0.8

61

0.9

17

0.9

23

0.9

41

0.9

31

0.8

31

1.1

43

1.0

54

0.9

84

1974-2

004

0.6

02

0.7

35

0.9

24

0.9

36

0.7

67

0.7

74

0.6

60

0.7

43

0.7

00

0.7

88

Panel

A.Vari

ance

Sort

Sim

ple

Aver

age

1to

31974-1

983

0.8

29

0.8

80

0.8

69

0.9

11

0.8

66

0.9

03

0.8

67

0.8

97

0.9

62

0.9

41

1984-1

990

0.9

71

0.9

73

0.8

88

0.9

76

1.0

28

0.9

91

1.1

23

1.0

21

1.0

92

0.9

64

1991-2

004

0.8

28

0.8

74

0.9

60

0.9

59

0.8

29

0.8

76

0.8

24

0.8

74

0.7

60

0.8

10

1974-2

004

0.8

48

0.8

96

0.8

92

0.9

38

0.8

80

0.9

11

0.8

92

0.9

13

0.9

34

0.9

04

3to

51974-1

983

0.7

87

0.8

25

0.8

97

0.9

22

0.8

30

0.8

38

0.7

82

0.8

11

0.7

89

0.8

07

1984-1

990

0.9

68

0.9

40

0.8

87

0.9

79

0.9

70

0.9

37

1.1

87

1.0

38

1.1

90

1.0

40

1991-2

004

1.0

20

0.9

68

0.9

62

0.9

56

0.9

79

0.9

49

0.9

62

0.9

45

0.8

31

0.8

83

1974-2

004

0.8

64

0.8

92

0.9

10

0.9

43

0.8

82

0.8

92

0.8

78

0.8

96

0.8

54

0.8

75

1to

51974-1

983

0.8

12

0.8

61

0.8

78

0.9

14

0.8

40

0.8

72

0.8

07

0.8

52

0.8

21

0.8

49

1984-1

990

0.8

32

0.8

98

0.8

86

0.9

78

0.8

59

0.9

02

0.9

91

0.9

57

1.0

52

0.9

75

1991-2

004

0.8

67

0.9

03

0.9

60

0.9

57

0.8

48

0.8

92

0.8

43

0.8

90

0.7

94

0.8

60

1974-2

004

0.8

27

0.8

81

0.8

97

0.9

40

0.8

44

0.8

84

0.8

40

0.8

84

0.8

47

0.8

76

Reg

ress

ion

base

d1

to3

1974-1

983

0.5

53

0.6

47

0.8

79

0.9

11

0.5

55

0.6

67

0.5

69

0.6

72

0.8

22

0.8

45

1984-1

990

1.3

14

1.1

69

0.9

72

1.0

00

1.4

87

1.2

32

1.5

68

1.2

83

0.7

64

0.8

79

1991-2

004

1.1

16

1.0

18

0.9

30

0.9

36

1.1

05

1.0

14

1.0

54

0.9

98

0.8

63

0.8

88

1974-2

004

0.7

84

0.8

62

0.9

04

0.9

36

0.8

06

0.8

83

0.8

15

0.8

90

0.8

23

0.8

65

3to

51974-1

983

0.6

18

0.7

09

0.9

45

0.9

20

0.7

87

0.7

98

0.7

75

0.7

74

0.7

79

0.7

35

1984-1

990

1.7

36

1.1

92

0.9

39

1.0

01

1.3

01

1.0

12

1.1

54

0.9

75

0.7

36

0.7

77

1991-2

004

1.1

26

1.0

00

0.9

52

0.9

48

0.9

21

0.9

10

0.9

80

0.9

22

0.8

91

0.8

90

1974-2

004

0.8

85

0.8

91

0.9

46

0.9

44

0.8

87

0.8

73

0.8

73

0.8

58

0.7

98

0.7

91

1to

51974-1

983

0.4

92

0.6

21

0.9

15

0.9

09

0.5

43

0.6

42

0.5

48

0.6

67

0.4

88

0.6

09

1984-1

990

1.4

26

1.2

03

0.9

97

1.0

09

1.2

56

1.1

09

0.9

92

0.9

73

1.0

28

0.9

64

1991-2

004

1.2

21

1.0

41

0.9

46

0.9

42

1.1

44

1.0

00

1.1

43

0.9

88

1.1

39

1.0

08

1974-2

004

0.7

84

0.8

63

0.9

33

0.9

38

0.7

76

0.8

43

0.7

43

0.8

25

0.7

09

0.8

01

26

Model Averaging: π36t+36 −π1

t+1. Relative MSE and MAE for equally weightedforecasts using 1 to 3 principal components, 3 to 5 principal components, and 1 to 5principal components. All results are relative to the AR benchmark model. The columnslabelled “Term Structure” present results for the model that incorporates as predictorsthe principal components of the monthly term structure spreads. In the columns labelled“Time”, results for the model that uses daily spreads with the same maturity as inflationare presented. The last three columns show the results principal component model incor-porating both the time and term structure dimensions, with either 5, 10, or 20 days ofdaily interest rate lags.

27

Tab

le8:

Die

bold

Mari

ano

Test

son

Regre

ssio

nbase

dM

odelA

vera

ges.

Res

ults

for

the

Die

bold

Mar

iano

test

stat

isti

cfo

rre

gres

sion

base

dav

erag

efo

reca

sts

usin

g1

to3

prin

cipa

lco

mpo

nent

s,3

to5

prin

cipa

lco

mpo

nent

s,an

d1

to5

prin

cipa

lco

mpo

nent

sar

epr

esen

ted.

All

resu

lts

are

rela

tive

toth

eA

Rbe

nchm

ark

mod

el.

The

colu

mns

labe

lled

“Ter

mSt

ruct

ure”

pres

entre

sult

sfo

rth

em

odel

that

inco

rpor

ates

aspr

edic

tors

the

prin

cipa

lcom

pone

ntsof

the

mon

thly

term

stru

ctur

esp

read

s.In

the

colu

mns

labe

lled

“Tim

e”,r

esul

tsfo

rth

em

odel

that

uses

daily

spre

ads

wit

hth

esa

me

mat

urity

asin

flati

onar

epr

esen

ted.

Fin

ally

,th

eco

lum

nsla

belle

d“5

days

”,“1

0da

ys”

and

“20

days

”pr

esen

tre

sult

sfo

rth

em

odel

that

inco

rpor

ates

prin

cipa

lco

mpo

nent

sof

pred

icto

rsth

atin

corp

orat

ebo

thdi

men

sion

s:Ter

m-S

truc

ture

dim

ensi

onan

dT

ime

dim

ensi

on.

Lags

Num

.Ter

mT

ime

5D

ays

10

Days

20

Days

NW

est

ofP

CStr

uct

ure

Panel

A.π

12−

π1

Corr

elation

Sort

31

to3

7.0

87.3

27.4

46.7

17.2

13

to5

7.2

37.4

37.3

37.2

47.6

01

to5

7.1

87.3

47.2

86.4

17.3

96

1to

35.9

26.4

16.3

45.7

06.0

73

to5

6.1

76.4

76.2

96.1

36.4

01

to5

6.0

96.4

16.2

55.4

86.2

111

1to

35.0

95.7

45.5

54.9

95.2

63

to5

5.3

35.7

35.5

25.3

15.4

91

to5

5.2

55.7

25.5

04.8

15.3

5Vari

ance

Sort

31

to3

6.6

77.2

46.8

16.6

67.4

83

to5

7.2

77.2

46.9

56.7

36.5

81

to5

6.9

37.1

96.7

36.3

56.8

46

1to

35.1

55.9

15.2

65.1

35.9

33

to5

5.6

75.8

85.3

55.1

04.9

81

to5

5.4

45.8

45.2

34.8

85.2

311

1to

34.3

95.2

24.4

84.3

55.0

43

to5

4.8

55.1

84.5

84.3

54.2

51

to5

4.6

55.1

64.4

54.1

34.4

3

28

Table 9: Diebold Mariano Tests on Regression based Model Aver-ages. Results for the Diebold Mariano test statistic for regression based average forecastsusing 1 to 3 principal components, 3 to 5 principal components, and 1 to 5 principal com-ponents are presented. All results are relative to the AR benchmark model. The columnslabelled “Term Structure” present results for the model that incorporates as predictorsthe principal components of the monthly term structure spreads. In the columns labelled“Time”, results for the model that uses daily spreads with the same maturity as inflationare presented. Finally, the columns labelled “5 days”, “10 days” and “20 days” presentresults for the model that incorporates principal components of predictors that incorporateboth dimensions: Term-Structure dimension and Time dimension.

Lags Num. Term Time 5 Days 10 Days 20 DaysNWest of PC StructurePanel A. π36 − π1

Correlation Sort3 1 to 3 8.27 7.93 8.14 8.55 9.10

3 to 5 8.50 8.30 7.77 7.71 8.181 to 5 8.27 8.14 7.15 7.65 8.79

9 1 to 3 5.79 5.55 5.85 5.91 6.343 to 5 6.10 5.82 5.65 5.47 5.691 to 5 5.88 5.68 5.20 5.44 6.05

15 1 to 3 4.91 4.72 4.94 4.97 5.343 to 5 5.26 4.93 4.86 4.58 4.801 to 5 5.02 4.81 4.42 4.57 5.05

20 1 to 3 4.49 4.36 4.52 4.53 4.873 to 5 4.86 4.52 4.46 4.13 4.411 to 5 4.60 4.43 4.04 4.11 4.59

30 1 to 3 4.05 4.03 4.07 4.06 4.413 to 5 4.44 4.12 4.00 3.64 4.011 to 5 4.13 4.05 3.60 3.61 4.11

35 1 to 3 3.91 3.93 3.91 3.90 4.263 to 5 4.27 3.99 3.83 3.47 3.871 to 5 3.96 3.94 3.45 3.45 3.96

Variance Sort3 1 to 3 6.86 8.21 6.92 6.99 8.08

3 to 5 7.82 8.12 8.22 7.27 6.881 to 5 7.04 8.23 6.77 6.74 6.63

9 1 to 3 4.78 5.77 4.79 4.84 5.623 to 5 5.64 5.69 5.76 5.08 4.791 to 5 4.96 5.77 4.70 4.71 4.61

15 1 to 3 4.03 4.91 4.05 4.08 4.773 to 5 4.86 4.84 4.85 4.33 4.061 to 5 4.20 4.91 3.98 4.01 3.90

20 1 to 3 3.67 4.52 3.71 3.72 4.413 to 5 4.46 4.46 4.44 3.97 3.721 to 5 3.82 4.53 3.64 3.69 3.57

30 1 to 3 3.28 4.16 3.36 3.35 4.043 to 5 4.01 4.09 3.99 3.59 3.381 to 5 3.41 4.16 3.29 3.36 3.24

35 1 to 3 3.16 4.05 3.25 3.24 3.913 to 5 3.84 3.97 3.84 3.48 3.281 to 5 3.26 4.05 3.19 3.26 3.15

29

Figure 1: Inflation

Apr61 Feb66 Dec70 Oct75 Aug80 Jun85 Apr90 Feb95 Dec99 Oct040

5

10

15Annualized 12−month Inflation

Apr61 Jun65 Aug69 Oct73 Dec77 Feb82 Apr86 Jun90 Aug94 Oct98 Dec020

5

10

15Annualized 36−month Inflation

30

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