Short-term inflation forecasting: The M.E.T.A. approach

International Journal of Forecasting 33 (2017) 1065–1081

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International Journal of Forecasting journal homepage: www.elsevier.com/locate/ijforecast

Short-term inflation forecasting: The M.E.T.A. approach Giacomo Sbrana a , Andrea Silvestrini b, *, Fabrizio Venditti b a b

NEOMA Business School, 59 Rue Pierre Taittinger, 51100 Reims, France Bank of Italy, Directorate General for Economics, Statistics and Research, Via Nazionale 91, 00184, Rome, Italy

article Keywords: Inflation Forecasting Aggregation State space models

info

a b s t r a c t Forecasting inflation is an important and challenging task. This paper assumes that the core inflation components evolve as a multivariate local level process. While this model is theoretically attractive for modelling inflation dynamics, its usage thus far has been limited, owing to computational complications with the conventional multivariate maximum likelihood estimator, especially when the system is large. We propose the use of a method called ‘‘moments estimation through aggregation’’ (M.E.T.A.), which reduces the computational costs significantly and delivers fast and accurate parameter estimates, as we show in a Monte Carlo exercise. In an application to euro-area inflation, we find that our forecasts compare well with those generated by alternative univariate and multivariate models, as well as with those elicited from professional forecasters. © 2017 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

1. Introduction Traditionally, the interest in inflation forecasting has been motivated by the existence of nominal contracts whose real values are determined by changes in the purchasing power of money, among other factors. Thus, the forecasting of inflation is crucial for nominal obligations, including those of governments. The importance of this issue has increased since the adoption by a large number of Central Banks of explicit or implicit inflation targets that have a forward-looking flavour, like the European Central Bank’s (ECB) stated medium-term target of below but close to 2%, or the long-run 2% target adopted by the Fed in 2012. Two issues arise when one is building models for predicting inflation. First, it is hard to find models that outperform naïve inflation forecasts; see for instance Marcellino, Stock, and Watson (2003), D’Agostino, Giannone, and Surico (2006), and Bańbura and Mirza (2013). Second, models that work well at very short horizons tend to perform poorly at longer horizons. Similarly, models that track changes in the inflation rate over the medium term author. * Corresponding E-mail address: [email protected] (A. Silvestrini).

reasonably well often have difficulty in getting the starting point right, so that they can end up missing the level of inflation quite markedly. The tension between short- and medium-term forecasting models is at the heart of the comprehensive review by Faust and Wright (2013). These authors stress that two crucial ingredients are necessary in order to obtain accurate inflation forecasts at different horizons: first, the starting point must be predicted accurately; second, longer-horizon forecasts must be anchored somewhat to the inflation target adopted by monetary policy. In practice, they find that it is hard to outperform a strategy in which the starting point is elicited from professional forecasters and the subsequent inflation path is obtained as a smooth transition to the inflation target. These findings have far-reaching implications for the research on inflation forecasting, since they reduce the emphasis on long-term and trend inflation forecasting, which are the focus of a large and growing body of literature (Chan, Koop, and Potter, 2013; Clark and Doh, 2014; Cogley, 2002; Garnier, Mertens, and Nelson, 2013, among others), and instead emphasize shorter horizons, for which the literature is relatively scant. Motivated by this observation, our paper contributes to the debate on inflation forecasting

http://dx.doi.org/10.1016/j.ijforecast.2017.06.007 0169-2070/© 2017 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

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by focusing on the short end of the inflation forecast curve. Specifically, we propose a modelling framework that provides accurate one-step-ahead inflation predictions, and therefore can be seen as a useful starting platform for longer-horizon forecasts. Central to our forecasting framework is a multivariate local level model (MLL henceforth) that extracts the permanent components from a panel of elementary inflation series.1 The model represents a multivariate extension of the approach originally proposed by Muth (1960) and subsequently employed for forecasting U.S. inflation (see for example Barsky, 1987 and Nelson and Schwert, 1977). In its univariate version, the local level model has attracted the attention of the recent literature due mainly to its ease of computation. For example, Stock and Watson (2007) use it for forecasting inflation in the U.S. during the Great Moderation period, allowing for changes in the signal to noise ratio over time. On the other hand, the multivariate version of the local level model has attracted less attention due to the computational issues that arise with maximum likelihood estimation, even for low-dimensional systems.2 Indeed, when the dimension of the local level model becomes relatively large, the maximum likelihood estimator may not perform well and can be computationally intensive to obtain. In this case, there are various other strategies that have been suggested in the literature, such as the univariate treatment of the Kalman filter as per Koopman and Durbin (2000) (see also Anderson and Moore, 1979 and Fahrmeir and Tutz, 1994). However, this paper makes use of a new estimation method for the MLL model that was proposed recently by Poloni and Sbrana (2015), which allows us to circumvent these computational problems and makes it possible to evaluate the forecasting performances of relatively large systems. The method, defined as ‘‘moments estimation through aggregation’’ (M.E.T.A. henceforth), consists of breaking down the complex problem of estimating a (potentially large) multivariate system into the more manageable problem of estimating the parameters of many univariate processes. The latter are used to estimate the moments of the system, and eventually to derive the parameters of the multivariate model through a closed-form relationship between the moments and the model parameters. Our contribution is twofold. First, an extensive Monte Carlo exercise is performed to show that the M.E.T.A. is considerably faster and more accurate (especially for large dimensional systems) than the traditional multivariate maximum likelihood estimator, and indeed that it is the only viable method beyond a certain model size,3 which 1 Our application focuses mainly on a measure of the core inflation, i.e., the percentage change in the overall index net of food and energy, which is watched closely by the ECB. The elementary series that compose this inflation measure are strongly persistent (unlike food and energy prices), and therefore lend themselves best to being modelled using the MLL. As a robustness check, we also consider forecasting the overall inflation; see Section 4 for a detailed discussion. 2 Notable exceptions are the studies by Proietti (2007), in the context of U.S. monthly core inflation rates, and Stella and Stock (2012). 3 Refer to Kascha (2012) for an overview and a comparison of the estimation algorithms proposed in the literature for the general class of VARMA models.

opens up the use of the MLL model to a wide set of applications. Our second contribution is to employ this model in an empirical application to the forecasting of euro-area inflation at short horizons. We find that the predictions derived from the MLL model estimated using the M.E.T.A. approach compare well with those generated by vector autoregressions (VAR) or alternative univariate constant and time-varying parameter models. Moreover, they are as accurate as those obtained on the basis of factor models that use large datasets. Furthermore, by making the estimation feasible regardless of the model dimension, the M.E.T.A. approach allows us to assess the relative benefits, in terms of the predictive accuracy, of different levels of aggregation of the elementary price indices. In this respect, we find that a preliminary aggregation of the price indices improves the forecast accuracy, which is a result that could not have been obtained on the basis of the traditional multivariate maximum likelihood estimator. The rest of the paper is structured as follows. Section 2 describes the model, points out the problematic aspects of estimation and presents the M.E.T.A. methodology in detail. It shows the results as per Poloni and Sbrana (2015), as well as using a new closed-form expression. Through a Monte Carlo exercise, Section 3 then illustrates the computational and accuracy gains attained by the M.E.T.A. approach. Section 4 discusses the empirical application, and Section 5 concludes. 2. The multivariate local level model Our paper uses a multivariate local level model to construct the inflation forecasts. This model posits that all of the series in the system are driven by series-specific random walks (Harvey, 1989). Its state space representation, which is also known as its structural form, is yt = µt + ϵt µt = µt −1 + ηt .

(1)

The vector yt , of dimension d, collects the percentage change over the previous period of the elementary items that constitute the core index, and t = 1, 2, . . . , T is the number of observations. Thus, Eq. (1) decomposes the multivariate time series yt into a stochastic trend µt that evolves as a multivariate random walk and a vector white noise (ηt ). It is assumed that the noises are i.i.d., with zero mean and the following covariances:

( ) ( ϵt Σϵ cov ηt = 0

0 Ση

)

,

(2)

where Σϵ and Ση are (d × d) error covariance matrices (also defined as structural parameters). It is also assumed that Σϵ and Ση are both positive definite (i.e., all of their eigenvalues are non-negative). The positive definiteness assumption can be relaxed by allowing Ση to have a reduced rank and eigenvalues that are equal to zero. This is the case when cointegration arises such that the model is driven by a smaller number of random walks. This is a special case that will be the object of a separate paper.

G. Sbrana et al. / International Journal of Forecasting 33 (2017) 1065–1081

The empirical application will experiment with two different levels of aggregation of the core inflation index, so that d in Eq. (1) will take values between 3 and around 40. 2.1. The mapping between the structural and reduced form parameters Taking first differences in Eq. (1), we obtain the stationary representation of the multivariate local level model (see Harvey, 1989): zt = yt − yt −1 = ηt + ϵt − ϵt −1 = ξ t + Θ ξ t −1 ,

Γ0 = E(zt ztT ) = Ση + 2Σϵ = Ω + ΘΩΘ T Γ1 = E(zt ztT−1 ) = −Σϵ = ΘΩ

(4)

Thus, the stationary vector process zt is a moving average process of order one and yt can be represented as an integrated vector moving average process of order one, i.e., an integrated VMA(1).4 Note that the process zt can also be rewritten as a vector autoregression (VAR) with an infinite number of lags, under the assumption of invertibility. Furthermore, when common trends arise, Harvey (2006) shows that the process zt has a vector error correction model (VECM) representation. Based on Eq. (4), the structural parameters Σϵ and Ση can be recovered easily using the autocovariances of the stationary representation zt , i.e., Γ0 and Γ1 : Σϵ = − Γ 1 Ση = Γ0 + 2Γ1 ,

Θ =

(

2

(

−1

Γ0 Γ1

−1

Ω = 2 Γ0 Γ1

(

−1

−1

+ Γ0 Γ1 Γ0 Γ1 − 4Id

(

−1

−1

+ Γ0 Γ1 Γ0 Γ1 − 4Id

) 12

) 21

)

)−1

Γ1 .

(6)

The proof of Eq. (6) is given by Poloni and Sbrana (2015). It can be observed from Eq. (6) that Σϵ has to be non-singular in order to be inverted in the expression of Θ . That is, all of the eigenvalues of Σϵ must be strictly positive in order for its inverse to exist. A simpler, equivalent expression that does not require a non-symmetric matrix such as Γ0 Γ1−1 to be made diagonal as in Eq. (6) is as follows. Consider the symmetric matrix −1

1

the matrix G = Σϵ2 V . Then, an equivalent expression of Eq. (6) is: Θ =

1 2

−1

Q = Σϵ 2 Ση Σϵ 2 = V ΛV T , where V are the eigenvectors of Q (such as VV T = Id ) and Λ is a diagonal matrix 4 See e.g. Brockwell and Davis (2002, Proposition 2.1.1, p. 50).

[

(

) 12

2

G −Λ − 2Id + Λ + 4Λ

]

G −1 .

(7)

This expression, which was not provided by Poloni and Sbrana (2015), can be implemented easily by software such as Eviews that only diagonalise symmetric matrices.5 Eq. (7) also allows it to be observed that the eigenvalues of Θ are always included between −1 and 0. First, note that −1

the symmetric matrix Q = Σϵ 2 Ση Σϵ 2 is positive semidefinite, implying that its eigenvalues are non-negative. Moreover, if one or more eigenvalues of Q are equal to zero (that is, one or more eigenvalues of Ση are zero), Θ has one or more eigenvalues equal to −1 (this can be seen easily by setting one or more of the elements of Λ equal to zero in Eq. (7)).6 In addition, when the eigenvalues of Q are strictly positive, Θ has eigenvalues that are strictly greater than −1 and less than or equal to 0. Finally, note that Θ has one or more eigenvalues that are equal to zero when one or more eigenvalues of Q go to infinity (which happens when one or more eigenvalues of Σϵ get close to zero, for example). An important aspect of this model is that both Γ0 and Γ1 are symmetric matrices, as is determined by the properties of the noises in Eq. (2). The symmetry of Γ1 in particular turns out to be a necessary condition for the implementation of the M.E.T.A. approach, as will become clear in the next subsection.7 The mapping between Γ0 , Γ1 and Θ is particularly important in our context, since Θ in Eq. (3) is crucial for the forecasting of the yt vector. Indeed, once an estimate of Θ is available, the following recursion can be employed to derive the forecasting error (Harvey, 1989; Muth, 1960): 8

(5)

and the reduced form parameters Θ and Ω can also be recovered using the autocovariances of zt . Indeed, as was shown by Poloni and Sbrana (2015), there exists a unique mapping between Γ0 and Γ1 and the reduced form parameters Θ and Ω . That is: 1

containing the corresponding eigenvalues. Now consider

−1

(3)

where Θ is a (d × d) matrix and ξ t is a vector process such that E(ξ t ξ Tt ) = Ω and E(ξ t ξ Tt+j ) = 0 (for j = ±1, ±2, . . .). The autocovariances of zt are:

Γn = E(zt ztT−n ) = 0 ∀n ≥ 2.

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ξˆ t +1 =

∞ ∑

ˆ )j zt +1−j . (− Θ

(8)

j=0

This recursion is an exponentially weighted moving average of current and past observations, which assigns a higher weight to more recent data points. As long as the eigenvalues of Θ are smaller than one in absolute value, the weights attached to past data will die out exponentially.9 The estimator for Θ as in Eq. (6) (or Eq. (7)) is simple and quick to compute. However, its accuracy relies 5 The authors are grateful to Federico Poloni for suggesting this expression. 6 This paper only considers the case of an invertible integrated VMA(1). However, as was noted above, this assumption may be relaxed by allowing for common trends (cointegration). This is left for future research. 7 The symmetry of Γ does not hold for more general (non-orthogonal) 1

specifications of Eq. (2), where for example cov (εt ηTt ) ̸ = 0. 8 For a proof that (I − Θ L)−1 = ∑∞ (Θ L)j , see Abadir and Magnus j=0 (2005, p. 249). 9 Eq. (8) will be used in our empirical analysis when forecasting the underlying items of the core industrial goods and services inflation aggregates. Then, a forecast of the core industrial goods (or services) inflation is obtained by aggregating predictions for the disaggregate items: wtT+1 zt +1 , where wt +1 is a vector of known core industrial goods/services weights.

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upon the estimates of the autocovariances of zt , and unfortunately, the use of the autocovariances Γˆ 0 = ∑N ∑Nsample 1 1 T T ˆ z z and Γ = z z 1 t =1 t t t =1 t t −1 might not be accurate N N enough, especially in small samples. Poloni and Sbrana (2015) overcome this problem by developing a simple estimation method defined as ‘‘moment estimation through aggregation’’ (i.e., M.E.T.A.), and establish the asymptotic properties (consistency and asymptotic normality) of the proposed estimator.10 Since this approach is novel to the forecasting of economic time series, we describe it in the next subsection, which draws heavily on the work of Poloni and Sbrana (2015).

with the moving average parameter δ(i+j) and innovation variance E(a2(i+j),t ) = σ(i2+j) .

2.2. The estimation problem: The M.E.T.A. solution

and for the pairwise aggregate

Let us explain the motivation for the M.E.T.A. estimation method by making two observations: 1. A linear transformation of a vector MA(1) process itself possesses an MA(1) representation. Indeed, the moving average (MA) class of models (of generic q order) is closed with respect to linear transformations (see Lütkepohl, Proposition 11.1, p. 435, 2007). As a consequence, if we take two generic elements of the vector zt = yt − yt −1 in Eq. (3), their sum is itself an MA(1). 2. Maximum likelihood is faster and more precise when it is applied to the estimation of the parameters of univariate processes rather than vector processes. The M.E.T.A. approach exploits these two facts in order to obtain estimates of Θ and Ω via Γ0 and Γ1 following Eq. (6). A simple bivariate example illustrating the building of Γ0 and Γ1 is presented in Appendix A. In the general multivariate case, in accordance with the notation used in Appendix A, let us define a pairwise aggregate process as z(i+j),t := zit + zjt (i, j = 1, 2, . . . , d, i ̸ = j), for building aggregating pairs of individual components. (i+j) be its autocovariances, with k = 0, 1. Then, the Let γk (i, j)th entry of Γk is given by:

⎧ (i) ⎨γk ) (Γk )i,j = 1 ( (i+j) ⎩ γ − γk(i) − γk(j) k 2

i = j, i ̸ = j.

(9)

In particular, the d diagonal entries are just the auto(i) covariances of the individual components γk , while the off-diagonal entries are determined uniquely based on the d(d−1) autocovariances of the 2 pairwise aggregate processes. That is, for each individual component we have zit = vit + ψi vi,t −1 ,

E [vit2 ] = σi2 ,

and for the pairwise aggregate we have z(i+j),t = zit + zjt = a(i+j),t + δ(i+j) a(i+j),t −1 , 10 Poloni and Sbrana (2015) show that these asymptotic properties hold when ϵt and ηt are i.i.d., meaning that the noise should not necessarily be Gaussian. This also explains why we do not assume normality in Eq. (1).

As was seen in the bivariate case, it is possible to establish a mapping between the MA(1) parameters of the individual components ψi and σi2 and the autocovariances γ0(i) and γ1(i) . Of course, this also holds for the pairwise aggregate processes, so that one can use the parameters δ(i+j) and σ(i2+j) to recover γ0(i+j) and γ1(i+j) . More specifically, for the individual components we have

γ0(i) = (1 + ψi2 )σi2 γ1(i) = ψi σi2 ,

(10)

γ0(i+j) = (1 + δ(i2+j) )σ(i2+j) ,

(11)

γ1(i+j) = δ(i+j) σ(i2+j) .

This suggests an estimation procedure that can be summarised in four steps. Given T observations of the multivariate process as in Eq. (3), the M.E.T.A. can be implemented as follows: 1. Estimate an MA(1) model for each individual component, say zit = yit − yi,t −1 , and each pairwise aggregate process, defined as z(i+j),t := zit + zjt , ˆ i , σˆ i2 ) and (δˆ(i+j) , σˆ (i2+j) ) respectively. obtaining (ψ (i)

(i+j)

2. Construct γˆk and γˆk for k = 0, 1, using Eqs. (10) and (11). 3. Recover estimates of the autocovariance matrices Γ0 and Γ1 using Eq. (9). 4. Recover estimates of Θ (and Ω ) using the closedform relationship in Eqs. (6) or (7) (since these are equivalent). The estimated Θ is then used to produce forecasts based on Eq. (8). Remark 1. The M.E.T.A. approach does not ensure that the error covariance matrices Ση and Σϵ (see Eq. (5)) will be positive definite. A simple fix for this issue is to enforce positivity by adding a suitable multiple of the identity matrix, such as, for ˆ η = Σˆ η + cId , and similarly for Σϵ (see Poloni and instance, Σ Sbrana, 2016). This procedure, which is implemented in this paper, is inspired by the so-called Tikhonov regularization and shrinkage estimation. It delivers regularized estimates of Ση and Σϵ (such that the resulting Θ has all eigenvalues strictly between −1 and 0, and Ω is positive definite). Fig. 1 provides a graphical representation of the M.E.T.A. approach in order to assist in visualising the four steps that make up the estimation procedure. A multivariate maximum likelihood estimator of the local level model in Eqs. (1) and (2) would directly follow the ‘‘missing arrow’’ between the observed vector zt and the reduced form ˆ and Ωˆ on the diagram in Fig. 1. parameters Θ The main advantage of the procedure described in steps 1–4 is that it exploits the univariate maximum likelihood estimation for MA(1) processes, yielding remarkably accurate values for both the autocovariances and the parameters, and has a low computational cost.11 Most 11 In fact, it reduces the problem from one d-dimensional maximum d(d+1) likelihood estimation to 2 univariate maximum likelihood problems.

G. Sbrana et al. / International Journal of Forecasting 33 (2017) 1065–1081

1069

Fig. 1. The M.E.T.A. estimation approach.

importantly, this estimation method is much faster than a multivariate maximum likelihood estimator and has clear computational advantages. Indeed, a likelihood routine estimating the MLL model as in Eq. (1) directly is affected by numerical convergence issues and bad complexity, growing with the dimensionality of the model (see, e.g., Kascha, 2012). The next section provides convincing evidence on this point by means of a Monte Carlo exercise.

Model 1: {−0.815, −0.412, −0.168}

3. M.E.T.A. vs. multivariate maximum likelihood: Comparing speed and accuracy

Model 5: {−0.868, −0.753, −0.582, −0.565, −0.524, − 0.485, −0.466, −0.434, −0.407, −0.329, −0.316, − 0.262, −0.248, −0.222, −0.138}

This section presents the results of a Monte Carlo experiment that is designed to compare the performances of the M.E.T.A. approach and the standard multivariate maximum likelihood estimator (MLE). We generate the multivariate local level model as in Eq. (1) using six different dimensions. Models 1 to 6 refer to models with dimensions d = 3, 6, 9, 12, 15, 18, respectively. While we considered dimensions higher than 18, we decided not to pursue this direction, given the difficulty of getting the multivariate MLE to converge. On the other hand, the M.E.T.A. always converges, regardless of the dimension, since the likelihood of each univariate process achieves the maximum easily. As was mentioned in Remark 1, the only issue with the M.E.T.A. estimator is the lack of positivity of the error covariance matrices, ensuring that Θ has eigenvalues between −1 and 0. When the lack of positive definiteness arises in the simulation study, we regularize the error ˆ η = Σˆ η + cId , where c makes the smallest covariance as Σ ˆ η greater than 0.0001 (the same procedure eigenvalue of Σ ˆ ϵ ). applies to Σ Given the results of Eq. (6), it should be noted that the matrix of parameters Θ is only a function of the covariances of the noises in Eq. (2). In addition, the eigenvalues of Θ are always between −1 and 0. More specifically, it can be observed that the higher the norm of Σϵ compared with that of Ση (i.e., the noisier the observed data), the closer the eigenvalues of Θ are to −1. On the other hand, the smaller the norm of Σϵ with respect to Ση , the closer the eigenvalues of Θ are to zero. Our first Monte Carlo simulation deals with the most general setting, in which we consider six models with Θ having eigenvalues between −0.9 and −0.1. More specifically, the eigenvalues of Θ spanned by the covariances of the noises in Eq. (2) for the first set of six models are as follows.

Model 6: {−0.848, −0.699, −0.636, −0.616, −0.586, − 0.581, −0.574, −0.533, −0.466, −0.433, −0.406, − 0.399, −0.378, −0.354, −0.329, −0.176, −0.162, − 0.148}

Model 2: {−0.857, −0.415, −0.392, −0.382, −0.293, − 0.089} Model 3: {−0.869, −0.441, −0.369, −0.353, −0.348, − 0.310, −0.281, −0.218, −0.164} Model 4: {−0.853, −0.661, −0.595, −0.540, −0.436, − 0.411, −0.372, −0.364, −0.356, −0.314, −0.109, − 0.093}

Given that the Θ matrix for the euro-area inflation tends to have roots between −1 and −0.5, we perform a second Monte Carlo simulation using six additional models with Θ having eigenvalues between −0.95 and −0.5. The eigenvalues of Θ for this second set of six models are reported below. Model 1: {−0.909, −0.778, −0.684} Model 2: {−0.870, −0.831, −0.776, −0.734, −0.647, − 0.519} Model 3: {−0.925, −0.919, −0.879, −0.788, −0.772, − 0.703, −0.659, −0.545, −0.506} Model 4: {−0.929, −0.863, −0.853, −0.843, −0.822, − 0.809, −0.804, −0.778, −0.763, −0.737, −0.679, − 0.469} Model 5: {−0.923, −0.889, −0.879, −0.846, −0.829, − 0.819, −0.801, −0.765, −0.726, −0.673, −0.639, − 0.595, −0.556, −0.505, −0.429} Model 6: {−0.912, −0.892, −0.848, −0.837, −0.830, − 0.822, −0.801, −0.801, −0.791, −0.782, −0.778, − 0.767, −0.749, −0.739, −0.700, −0.655, −0.642, − 0.593} It should be noted that the eigenvalues considered mimic those estimated in the empirical application using the euro-area inflation sub-indexes. For example, since the prices of core industrial goods are noisier, they tend to have

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roots closer to −1, while those of services tend to have roots closer to −0.5.12 Note that Stock and Watson (2007, Table 3, p. 13) also find MA coefficients of between −0.5 and −1 for US inflation in the 1984:I–2004:IV sub-sample. We compare the performances of the two methods using the ratio of the Frobenius norm (root mean squared error of the matrix entries) for the estimates of Θ and Ω . That is: RATIO ∥ΘM .E .T .A. − Θ ∥F RMSE(ΘM .E .T .A. ) = Θ = ∥ΘMLE − Θ ∥F RMSE(ΘMLE ) (12) RATIO ∥ΩM .E .T .A. − Ω ∥F RMSE(ΩM .E .T .A. ) = , Ω = ∥ΩMLE − Ω ∥F RMSE(ΩMLE ) where ∥X ∥F :=

2 1/2 . j=1 Xij

(∑m ∑n i=1

)

Table 1 reports the results of the simulation experiments. The results using Θ and Ω refer to the first set of models, in which the eigenvalues of Θ are between −0.9 and −0.1. Those using Θ and Ω refer to the second set of models, in which the eigenvalues of Θ are between −0.95 and −0.5. The simulation study is carried out for two sets of sample sizes: Models 1 to 3 are estimated on a sample size of T = 200, 400 and 800 observations, while Models 4 to 6 consider T = 400, 800 and 1200. This has been done because the multivariate MLE faces big convergence difficulties as the system dimension grows, especially when there are not enough observations. Note that we run 1000 replications of each experiment. The table of results is structured as follows. The first four columns, comparing M.E.T.A. and the multivariate MLE, report the average ratios of root mean squared errors of the estimated Θ and Ω , respectively, multiplied by 1000. Numbers smaller than one indicate that the root mean squared error of the multivariate MLE is greater than that of M.E.T.A. The last four columns report the average numbers of seconds required for estimating specific models using M.E.T.A. and multivariate MLE. As per Poloni and Sbrana (2015), all simulations are carried out using the TimeSeries 1.4.1 package in Mathematica 8 by Wolfram.13 Unlike the simulations reported by Poloni and Sbrana (2015), both the M.E.T.A. and multivariate MLE estimators here rely on the conditional maximum likelihood approach. This has been done because the exact MLE employed by Poloni and Sbrana (2015) is time-consuming, especially for multivariate processes. When combined with the fact that the conditional MLE delivers estimates similar to those from the exact MLE, this is what motivated our decision to employ this alternative estimation technique. This choice, together with the use of a faster processor, is what explains the gaps (in terms of seconds) between the results here and those of Poloni and Sbrana (2015). Note that the conditional MLE for the univariate MA(1) processes uses the parameters provided by autoregressive processes of order 10 as initial values. Similarly, the conditional multivariate MLE for the vector MA(1) process uses 12 Two plots of the eigenvalues for core industrial goods and services are available from the authors upon request. 13 See the webpage http://media.wolfram.com/documents/TimeSeries Documentation.pdf.

the parameters provided by a VAR model of order 10 as initial values (see the documentation of the TimeSeries 1.4.1 package). Overall, the results are clearly in favor of the M.E.T.A. estimator, which seems to outperform its rival estimator all the time in terms of accuracy. Indeed, for Θ , the RMSE of the M.E.T.A. estimator is below that of the multivariate MLE in all cases (this is generally more evident when considering the set of models with eigenvalues between −0.95 and −0.5). As for Ω , the M.E.T.A. tends to outperform the multivariate MLE, but this is less clear-cut, with the ratios being close to one. These results are in line with those obtained by Poloni and Sbrana (2015, see their Table 1). Moreover, the columns comparing the average numbers of seconds reveal the computational difficulties of the multivariate MLE for large medium-dimensional systems. This is evident from the results for Models 5 and 6 in particular, where the M.E.T.A. tends to be faster than the multivariate MLE. More specifically, when the dimension increases and the sample size is not large enough, the multivariate MLE faces convergence issues, which impacts on its performance, as is shown by the RMSE ratios. Such is not the case for Models 1 to 3, where the multivariate MLE seems to be faster than the M.E.T.A., although less accurate. Summing up, these results show that the M.E.T.A. estimator tends to be more accurate than the multivariate MLE, especially for large dimensional systems. In addition, the multivariate MLE faces evident difficulties when the dimension increases and the number of observations is not large enough. Based on this extensive Monte Carlo evaluation, we can fairly claim that the M.E.T.A. is probably the only feasible method for estimating a system with more than 20 equations, leading to very significant time savings. These nice features are clearly due to the fact that we have replaced the traditional multivariate MLE with a procedure that requires the estimation of univariate processes only. This makes the estimation quick and accurate, surmounting the computational difficulties that are faced when maximizing the multivariate likelihood. The benefits will become clearer in the empirical application, where we will also be able to assess the relative gains of different disaggregation levels of the core price sub-indexes for inflation forecasting, an appraisal that becomes infeasible as the dimension increases when using the multivariate MLE. 4. Empirical application Our empirical investigation consists of a pseudo real time out-of-sample forecasting exercise of euro-area consumer price inflation, measured on the basis of the Harmonized Index of Consumer Prices (HICP). The HICP consists of 88 elementary series, which are grouped conventionally into four categories: core goods (non-energy industrial goods, or NEIG), services, food and energy prices. The bulk of our empirical analysis focuses on the first two of these four categories (core goods and services prices) and on their aggregation, the HICP net of food and energy, which is typically used to measure the core, or underlying, inflation. Our motivation for looking more closely at this subset of prices is threefold. The first reason is their policy relevance. In fact, it is mostly through these prices

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Table 1 Ratio(s) of the mean RMSEs of the estimates of Θ and Ω , with computation times. Sample

RATIO

RATIO

RATIO

RATIO

Time

Time

Time

Time

size

Θ

Θ

Ω

Ω

M .E .T .A.

MLE

M .E .T .A.

MLE

Model 1

200 400 800

0.83 0.83 0.82

0.86 0.92 0.88

0.99 0.99 0.99

0.96 0.98 1.00

0.17 0.30 0.56

0.07 0.12 0.22

0.20 0.33 0.66

0.11 0.14 0.24

Model 2

200 400 800

0.78 0.84 0.84

0.75 0.84 0.84

0.97 0.98 0.99

0.99 0.98 0.99

0.57 1.17 1.99

0.31 0.37 0.56

0.63 1.12 2.71

0.64 0.45 1.25

Model 3

200 400 800

0.71 0.76 0.76

0.59 0.78 0.89

0.89 0.98 0.99

0.96 0.99 0.98

1.13 2.21 4.11

1.36 1.13 1.70

1.52 2.94 4.81

2.26 3.9 3.14

Model 4

400 800 1200

0.77 0.78 0.77

0.66 0.74 0.81

0.98 0.99 0.98

0.98 0.98 0.99

3.86 7.35 10.9

3.83 4.23 6.01

4.82 8.38 11.9

9.87 9.37 8.05

Model 5

400 800 1200

0.72 0.80 0.80

0.66 0.81 0.88

0.97 0.98 0.99

0.97 0.90 0.99

6.31 11.8 17.9

14.7 10.7 14.4

6.59 11.9 18.2

20.01 22.33 18.48

Model 6

400 800 1200

0.74 0.76 0.77

0.47 0.71 0.80

0.95 0.97 0.99

0.99 0.98 0.99

8.20 15.2 23.2

23.2 14.1 18.1

9.81 17.3 25.9

32.68 53.44 27.63

Notes: Models 1 to 6 refer to systems with 3, 6, 9, 12, 15 and 18 equations, respectively. Θ and Ω refer to the first set of models, in which the eigenvalues of Θ are between −0.9 and −0.1. Θ and Ω refer to the second set of models, in which the eigenvalues of Θ are between −0.95 and −0.5. The first two columns report the average root mean squared errors of the estimated ΘM .E .T .A. relative to ΘMLE for the first and second sets of models, respectively, multiplied by 1000 (see Eq. (12)). The third and fourth columns report the average root mean squared errors of the estimated ΩM .E .T .A. relative to ΩMLE for the first and second sets of models, respectively, multiplied by 1000 (again see Eq. (12)). The last four columns report the average numbers of seconds required to estimate Θ and Ω using M.E.T.A. and the multivariate MLE for the first and second sets of models.

that monetary policy affects consumer price inflation, since food and energy prices are determined largely by factors over which monetary policy has little control, such as commodity prices and seasonal factors. Second, these prices are highly persistent, a feature which is confirmed by a wealth of micro-data based studies that have found that the median frequency at which they are reset is considerably lower than for the remaining categories (see Bils and Klenow, 2004, for the U.S.; and Altissimo, Ehrmann, and Smets, 2006, for the euro area). This implies that their time series properties match the assumptions of the integrated VMA(1) model best. The third reason, which is related to the first two, is that, given its persistence, core inflation is seen as the main driver of the medium-term headline (or overall) inflation, and is therefore related strongly to the ECB medium-run inflation target. Indeed, the behaviour of core inflation was quoted frequently by President Draghi in 2014 in order to de-emphasize prospective deflation risks in the euro area.14 The rest of this section is structured as follows. Section 4.1 discusses core inflation forecasts and also presents a broad description of both the forecasting experiment and the competing models (the multivariate local level model and VAR models on disaggregated data, ARIMA models and the random walk model on aggregate data). Section 4.2 assesses the forecasting abilities of factor models built using a large dataset of 154 real and nominal macroeconomic variables for forecasting core inflation. Next, Section 4.3 focuses on headline inflation forecasting. A key advantage 14 See for example https://www.ecb.europa.eu/press/pressconf/2013/ html/is131205.en.html. The strength of the empirical link between shortterm core inflation developments and medium-term headline inflation has been challenged by several studies (see for instance Bullard, 2011).

Table 2 HICP sub-index weights. Year

NEIG

Services

Food

Energy

Others

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

0.31 0.31 0.30 0.30 0.30 0.29 0.29 0.28 0.27 0.27

0.41 0.41 0.41 0.41 0.41 0.42 0.41 0.41 0.42 0.43

0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

0.09 0.09 0.10 0.10 0.10 0.10 0.10 0.11 0.11 0.11

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

Notes: The source is Eurostat. The data are for the euro area (changing composition), are neither seasonally nor working-day adjusted, and represent annual averages of monthly observations.

of using headline (instead of core) inflation as the target variable is that its predictions can be compared directly with those of the professional forecasters surveyed each month by Bloomberg. Additional robustness checks are discussed in Section 4.4. 4.1. Forecasting core inflation The core inflation index is obtained by aggregating the core goods and services prices, accounting for around 70% of the HICP basket. Table 2 reports a snapshot of the weights of these components in the overall basket in order to show that the relative importance of each of these items in the consumption basket has remained relatively stable over time. In this context, since the integrated VMA(1) process is particularly suited to model persistent processes, we start our empirical analysis by applying it to core goods and

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services inflation separately, and derive a prediction for the year-on-year core inflation (our target variable) through a weighted aggregation of these two sub-indexes. Given that the M.E.T.A. approach makes estimation feasible for virtually any model size, we explore the performance of the integrated VMA(1) using different levels of aggregation of the underlying core price series. By comparing the results obtained from models of different scales, we can then determine the appropriate level of granularity for forecasting the aggregate inflation, an assessment that would not be feasible with the traditional multivariate maximum likelihood method. We compare the performance of the integrated VMA(1) with those of other popular multivariate and univariate benchmarks. The multivariate models that we consider include VARs, which are the workhorse of economic forecasting. Of our univariate benchmarks, we are particularly interested in the performance of the IMA(1,1) model, which has dynamic properties similar to those of the integrated VMA(1) but lies at the opposite end of the aggregation spectrum, being estimated on the aggregate series directly rather than on the disaggregated components. Thus, a comparison of the forecast accuracies of the integrated VMA(1) and IMA(1,1) is an indirect assessment of the information value of cross-sectional price data for forecasting aggregate inflation. As an additional benchmark, we also consider the IMA(1,1) model with time-varying parameters that was developed by Stock and Watson (2007) in the context of inflation forecasting. We estimate all of the models using data as of January 1996 and run the forecast exercise using an out-of-sample period from January 2005 to April 2014 (112 observations). We initially use an expanding window estimation scheme, then employ a fixed rolling window scheme (5-year, 7-year and 9-year) as a robustness check.15 Unless otherwise specified, all models are estimated on seasonally adjusted monthly price changes.16 The full battery of models employed in the forecasting exercise is as follows: 1. ‘‘Integrated VMA(1)-large’’ uses the integrated VMA(1) model estimated using the M.E.T.A. approach on the disaggregated core goods and services inflation series, exploiting a relatively high level of granularity of the core sub-indexes (around 25 elementary consumer price series for core goods and around 40 for services; see Tables A.1 and A.2). As has been noted, if the estimated error covariance matrices are not positive definite, we apply the regularization method as described in Remark 1 and Section 3. This delivers regularized estimates of the reduced form integrated VMA(1) parameters. The same procedure is also implemented for the ‘‘integrated VMA(1)-small’’ model described below. 15 Note that the model in Eq. (1) is specified for variables in first differences, implying that the corresponding price levels are I(2); standard ADF unit root tests performed on the series confirm that this assumption is satisfied overall. Detailed test results are available from the authors upon request. 16 The seasonal adjustment is carried out by regressing the monthly price changes on a constant and eleven time dummies. The forecasts for the seasonal and deterministic components are then added to those of the seasonally adjusted month-on-month inflation series.

2. ‘‘Integrated VMA(1)-small’’ uses the integrated VMA(1) model estimated with the M.E.T.A. approach on the disaggregated core goods and services inflation series, using a relatively high aggregation level (five sub-indexes for core goods and three subindexes for services; see Tables A.3 and A.4). The choice of this level of aggregation is based on the results of Sbrana and Silvestrini (2013). Using the multivariate local level model, these authors derive conditions on the parameters under which the variance of the contemporaneously aggregated process (core inflation in this case) achieves its minimum (p. 188). These conditions are met here using five sub-indexes for goods and three sub-indexes for services.17 3. ‘‘VAR(1)’’, ‘‘VAR(2)’’, ‘‘VAR(3)’’ and ‘‘VAR(p)’’ employ the same type of information as ‘‘integrated VMA(1)large’’, but use vector autoregressions to model the behaviour of the core goods and services inflation series, taken in first differences. In the ‘‘VAR(p)’’ model, a standard lag length selection criterion such as the BIC (Schwarz’ Bayesian information criterion) is used to choose the optimal lag length (assuming a maximum lag length of three). 4. ‘‘AR(p)’’ fits univariate AR(p) models to the aggregate core goods and services inflation series, taken in first differences. The same information criterion as above is used to select the optimal order for the AR model. 5. ‘‘IMA(1,1)’’ employs the univariate integrated moving average process with constant parameters to model the two core goods and services aggregate inflation series. 6. ‘‘T.V.P. IMA(1,1)’’ uses the univariate integrated moving average process with time-varying parameters that was popularised by Stock and Watson (2007) to model the two core goods and services aggregate inflation series.18 7. Finally, we compute a direct random walk (RW) forecast of aggregate core inflation that constitutes a fairly challenging benchmark, given the high persistence of the year-on-year underlying inflation rate. In order to enable a better understanding of the design of the forecasting exercise, Fig. 2 presents a flowchart of its main steps. The results of the out-of-sample forecasting exercise are presented in Table 3, which shows the root mean squared forecast errors (RMSFE) of the models described above. These results refer to an expanding window estimation scheme (the estimation is carried out over the period January 1996–December 2004, and then recursive forecasts 17 We preserve the out-of-sample nature of the forecasting exercise by estimating the parameters of the multivariate local level model once using information from January 1996 up to December 2004, to choose the preferred aggregation levels for core goods and services, then keeping these values fixed over the whole out-of-sample period. 18 We follow the UC-SV specification as per Stock and Watson (2007) and fix the variances of the stochastic volatilities shocks to γ = 0.22 . The model is estimated by Markov chain Monte Carlo (MCMC) methods. The estimation results are based on 5000 draws and a burn-in period of 100 draws. The results are robust to changes in these parameters.

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Fig. 2. The forecasting exercise for core (goods and services) inflation. Notes: The flowchart shows the forecasting procedures for the univariate (j)

ARIMA(p, 1, 0), the IMA(1,1)/T.V.P. IMA(1,1) models, the integrated VMA(1) model, and the VARs. We let pi,t (i = 1, 2, . . . , d(j) ) be the elementary prices for each sub-index (j = goods, ser v ices). Consistent with the previous notation, (j)

(j)

(j)

(j) yi,t

are the corresponding month-on-month percentage changes, whereas (j)

(j)

the first differences are zi,t = yi,t − yi,t −1 . We denote the aggregate core (goods/services) inflation sub-index by yt ; similarly, zt is the aggregate core (goods/services) inflation sub-index in first differences, on which univariate models such as the AR(p), the MA(1) and the T.V.P. MA(1) are estimated directly.

are generated by adding a month at a time, up to April 2014). The left column displays results for the whole outof-sample period (January 2005–April 2014), while the center and right columns show the results for two equal sub-samples (January 2005–September 2009 and October 2009–April 2014). A number of interesting findings emerge. First, on average, the point forecasts produced using the integrated VMA(1) models, both small and large, are more accurate than those obtained by other multivariate models. This indicates that the integrated VMA(1) model is indeed a competitive benchmark for other multivariate setups (VARs), the popularity of which is related partly to the fact that they are very easy to estimate. Since the use of the M.E.T.A. approach makes the integrated VMA(1) equally appealing from a computational point of view, we consider this result encouraging for researchers who wish to apply the local level model in a multivariate context. Second, the integrated VMA(1)-small model improves slightly on both the IMA(1,1) with constant parameters and the T.V.P. IMA(1,1). Moreover, the univariate IMA(1,1) and T.V.P. IMA(1,1) models have comparable performances, which suggests that stochastic volatility has not been a relevant feature of euro-area (core) inflation over the last decade. Note that these three models are closely related.

Table 3 Forecast accuracy (RMSFE): core inflation, expanding window estimation scheme.

VMA(1)-small VMA(1)-large VAR(1) VAR(2) VAR(3) VAR(p) Random walk AR(p) IMA(1,1) T.V.P. IMA(1,1)

2005–2014

2005–2009

2009–2014

0.155 0.156 0.174 0.182 0.183 0.174 0.158 0.156 0.161 0.163

0.125 0.126 0.152 0.161 0.159 0.153 0.136 0.124 0.129 0.133

0.181 0.181 0.193 0.200 0.204 0.193 0.177 0.182 0.188 0.189

Notes: The target variable is the underlying core inflation rate. The RMSFEs in the 2005–2014 column are based on 112 recursive (i.e., expanding window) forecasts between January 2005 and April 2014. Those in the 2005–2009 column are based on 56 recursive forecasts between January 2005 and August 2009. Those in the 2009–2014 column are based on 56 recursive forecasts between September 2009 and April 2014. VMA(1)small, VMA(1)-large, VAR(1), VAR(2), VAR(3) and VAR(p) are estimated on the disaggregated inflation series taken in first differences. Similarly, the AR(p) model is estimated on the aggregate inflation series in first differences. Random walk refers to a random walk forecast of the aggregate inflation rate.

Specifically, the model generated after contemporaneous aggregation of the integrated VMA(1) is itself an integrated

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MA(1): as has been explained, the vector moving average of order one is in fact closed with respect to linear transformations. In turn, the univariate IMA(1,1) with constant parameters is a restricted version of the T.V.P. IMA(1,1). In this particular application, we find that, on average, a prediction built by aggregating the forecasts of the whole integrated VMA(1) process is more accurate than a prediction built by forecasting the aggregate univariate inflation series directly using an IMA(1,1) model, with both constant and time-varying parameters.19 This outcome indicates that cross-sectional price data contain useful information for predicting aggregate core inflation, which further underscores the importance of having an efficient estimation method available for multivariate versions of the local level model. Third, in terms of the optimal level of granularity, we find that an intermediate level of aggregation of sectorial inflation rates yields a slightly better performance than the highest level of disaggregation of the index, as the forecasts of the integrated VMA(1)-small are marginally more accurate than those of the integrated VMA(1)-large, in a mean squared error sense. Finally, the AR(p) model performs slightly worse than the integrated VMA(1)-small model and similarly to the integrated VMA(1)-large model both over the whole sample and over the sub-samples. The RW forecast also performs well, which is not surprising given the high persistence of the core inflation rate; however, the sub-sample analysis shows that the results over the two sub-periods are mixed. It should be mentioned that we use the weighted average of forecasts of the disaggregate sub-indices to get the aggregate core inflation forecasts from the multivariate models (the integrated VMA(1) and VARs on the inflation series in first differences). With this aim, we employ the actual sub-component weights from the HICP basket in Table 2, though in reality, the future time-varying weights are not available beforehand. Therefore, as a validation check, our out-of-sample forecasting exercise compares the VMA and VAR predictions obtained by using (onemonth) lagged and current inflation weights. The results are no different in terms of root mean squared forecast errors.20 We gauge whether the differences in forecast accuracy highlighted in Table 3 are statistically significant by running equal forecast accuracy and forecast encompassing tests. The hypothesis of equal forecast accuracy is analysed through the standard test of Diebold and Mariano (1995), in which the null hypothesis is that the differences in either the squared or absolute prediction errors of two competing models are not significantly different from each other. 19 This result has been proven formally by Lütkepohl (1987) under the assumptions of a known data generation process and no estimation uncertainty. 20 The expenditure weights are reviewed and updated by Eurostat only once a year, not every month. Indeed, the quality requirements for inflation weights call for a minimum of review and adjustment to ensure sufficient quality. This means that the weights in a given year are constant, not time-varying. Furthermore, as can be seen in Table 2, even the differences across years are relatively small. Consequently, the impact of considering current or lagged expenditure weights turns out to be negligible, and no significant differences in terms of forecast accuracy can be observed.

The null hypothesis of forecast encompassing tests, on the other hand, is that, given two competing forecasts for the same target variable, there is no significant gain from (linearly) combining the predictions of the two models. In this latter case, we use the test by Harvey, Leybourne, and Newbold (1998). We take our preferred model, i.e., the integrated VMA(1)-small model, as the benchmark, and run both forecast accuracy and encompassing tests against this benchmark. The results are reported in Table 4. The Diebold-Mariano test results show that the null hypothesis of equal forecasting accuracy can be rejected at the 5% level in all pairwise comparisons except when dealing with the IMA(1,1) model, the integrated VMA(1)large, the AR(p) model and the RW. This is true whether we consider mean squared errors or absolute errors. Turning to forecast encompassing tests, the integrated VMA(1)small model encompasses all of the competitors (except the integrated VMA(1)-large and the AR(p)), but is not encompassed by any of them (except the RW) at the 5% level. So far, we have employed a recursive estimation procedure, implicitly assuming that there are no structural breaks in the sample under examination. However, our full sample (January 1996–April 2014) corresponds to an almost 20-year span which includes the global financial crisis and the sovereign debt crisis, with widespread instabilities. Thus, the remainder of this subsection tests the influence of structural breaks and sub-sample model instability by considering forecasts obtained using rolling windows of the most recent observations. Rolling window parameter estimates are based on a partial sample of a fixed size that moves forward as the forecast point progresses, making this a useful technique for handling model instability. The resulting forecasts enable the researcher to evaluate the forecasting ability in the presence of breaks, but they also allow us to assess the extent to which the predictive content of time series predictors/models is stable over time (Rossi, 2013). We consider rolling windows of three different lengths, in order to check the robustness of the empirical results to the choice of the window size. Given that our dataset starts in January 1996 and the out-of-sample forecasting period is from January 2005 to April 2014, the maximum size of the rolling window is 108 monthly observations, i.e., nine years of data. Table 5 therefore reports the results of fiveyear, seven-year and nine-year rolling forecasts. Overall, our empirical findings are consistent with the evidence obtained from the recursive estimation scheme, which requires the use of all available data when estimating the parameters. The results suggest that, on average, the most accurate forecasts are those produced by the integrated VMA(1)-small model, followed by the univariate models (IMA(1,1), T.V.P. IMA(1,1) and AR(p)), and finally the integrated VMA(1)-large. The random walk provides rather good predictions as well. However, the VAR models estimated on disaggregated inflation series in first differences do not perform satisfactorily. Of the VARs, the VAR(1) yields the forecasts with the smallest out-of-sample RMSFEs. The rolling window scheme generally delivers lower RMSFE statistics than the recursive scheme. For instance,

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Table 4 Equal forecast accuracy and forecast encompassing tests (with the benchmark model as VMA(1)-small): core inflation, expanding window estimation scheme. Forecast accuracy tests

VMA(1)-large VAR(1) VAR(2) VAR(3) VAR(p) Random walk AR(p) IMA(1,1) T.V.P. IMA(1,1)

Forecast encompassing tests

Mean squared errors

Absolute errors

VMA(1)-small encompasses

VMA(1)-small is encompassed

Stat

p-val

Stat

p-val

Stat

p-val

Stat

p-val

−0.165 −2.203 −2.559 −2.549 −2.237 −0.191 −0.057 −1.820 −2.965

0.869 0.028 0.011 0.011 0.025 0.848 0.955 0.069 0.003

0.461 −2.111 −2.158 −2.271 −2.149 0.348 0.139 −0.625 −1.663

0.645 0.035 0.031 0.023 0.032 0.728 0.889 0.532 0.096

1.630 4.509 4.592 4.697 4.609 6.209 1.474 2.764 4.054

0.103 0.000 0.000 0.000 0.000 0.000 0.140 0.006 0.000

1.123 0.996 1.090 0.888 1.056 6.394 1.232 0.892 1.858

0.262 0.319 0.276 0.375 0.291 0.000 0.218 0.373 0.063

Notes: The forecasting sample is January 2005–April 2014. The table shows the results of the tests of equal forecast accuracy and forecast encompassing between the model VMA(1)-small and the models indicated in the rows. The forecast accuracy test is the Diebold-Mariano test based on squared and absolute errors, respectively. The forecast encompassing test is the Harvey-Leybourne-Newbold test. VMA(1)-small, VMA(1)-large, VAR(1), VAR(2), VAR(3) and VAR(p) are estimated on the disaggregated inflation series taken in first differences. Similarly, the AR(p) model is estimated on the aggregate inflation series in first differences. Random walk refers to a random walk forecast of the aggregate inflation rate. Table 5 Forecast accuracy (RMSFE): core inflation, rolling window estimation scheme.

VMA(1)-small VMA(1)-large VAR(1) VAR(2) VAR(3) VAR(p) Random walk AR(p) IMA(1,1) T.V.P. IMA(1,1)

5-year window

7-year window

9-year window

0.133 0.140 0.162 0.232 – 0.162 0.158 0.139 0.137 0.137

0.141 0.148 0.164 0.192 – 0.164 0.158 0.143 0.141 0.143

0.147 0.155 0.173 0.174 0.184 0.173 0.158 0.146 0.144 0.147

Notes: The table summarises the results for rolling forecasts. The target variable is the underlying’ core inflation rate. The RMSFEs are based on 112 forecasts between January 2005 and April 2014. The models are estimated based on rolling 5-year, 7-year and 9-year increasing windows (with the corresponding RMSFEs presented in the second, third and fourth columns, respectively). No RMSFEs are reported for the VAR(3) model with 5- and 7-year rolling windows because there are not sufficient observations to allow these models to be estimated. VMA(1)-small, VMA(1)-large, VAR(1), VAR(2), VAR(3) and VAR(p) are estimated on the disaggregated inflation series taken in first differences. Similarly, the AR(p) model is estimated on the aggregate inflation series in first differences. Random walk refers to a random walk forecast of the aggregate inflation rate.

it does for all of the univariate models, the integrated VMA(1)-small model and the VAR(1). When considering the VAR(2), though, a seven-year rolling window forecast has a RMSFE of 0.192, compared to a RMSFE of 0.182 for the recursive scheme. Furthermore, when the VAR(2) is estimated on a five-year rolling window, the variance of the parameter estimates is so high as to give a RMSFE of 0.232. At the same time, no RMSFEs are reported for the VAR(3) with five- or seven-year rolling windows because there are not sufficient observations to estimate these highly parameterised models. In general, the lowest RMSFEs are achieved by employing a fixed rolling window of five years. If we increase the size of the rolling window, e.g., move from a five-year to a seven-year window, the corresponding out-of-sample RMSFEs approach those produced by the recursive scheme. Taken together, these results, based on a rolling window estimation scheme, provide strong empirical evidence of model instability and structural change in the sample considered. Our takeaway message from the empirical analysis that we have conducted so far is that, when considering persistent processes like core inflation, the multivariate local

level model is a serious benchmark. This is the case for both the recursive and rolling window estimation schemes. 4.2. A comparison with large dynamic factor models The previous subsection compared the forecast accuracy of our method with those of different time series models using a similar information set (inflation data at different levels of aggregation). However, the literature on inflation forecasting has shown that exploiting the information contained in other macroeconomic variables can improve the forecast accuracy. In particular, there is an extensive body of literature on forecasting using many predictors which suggests that a small number of linear combinations of macroeconomic indicators (factors) contains useful predictive content for inflation and business cycle variables. In the context of inflation forecasting, Liu and Jansen (2011) and Stock and Watson (1999) document the good performance of Phillips curve models where cyclical developments are captured in a single index that aggregates a large number of real economic indicators. Since these models are very popular, they represent a natural benchmark against which to gauge the performance of our method.

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A number of choices need to be made when setting up a factor-based forecast model, relating to both the estimation method used for recovering the factor space and the specification of the forecasting equation. In theory, when the number of time series (N) is large, consistent estimation of the factor space can be obtained by either static principal components (see Bai and Ng, 2002 and Stock and Watson, 2002) or dynamic principal components (see Forni and Lippi, 2001 and Forni, Hallin, Lippi, and Reichlin, 2004). In practice, static principal components estimation is considerably easier to implement, since it requires fewer parameters to be specified (only r factors).21 When the data are generated by a factor model, each individual time series is also driven by an idiosyncratic component. Hence, a prediction for each series can be obtained either by simply using the information contained in the common factors or by combining this information with a forecast of the idiosyncratic term. These two issues, estimation and forecasting, are analysed accurately by Boivin and Ng (2005), who compare the forecasting performances of static and dynamic principal components and evaluate the forecast accuracy gains obtained by modelling not only the common but also the idiosyncratic terms. Based on a Monte Carlo experiment and a pseudo real time forecasting exercise, they conclude that forecasts based on (1) static principal components and (2) a forecasting equation that only considers the information contained in the lagged dependent variable or the common factors are generally superior. Building on their results, we work with a forecasting equation that is specified as

ˆ πt +1|t = ˆ α+

p ∑

ˆ βj πt −j+1 +

m ∑

j=1

ˆ λTj Ft −j+1 ,

(13)

j=1

where πt is the month-on-month inflation rate, λ is an r-dimensional vector of estimated coefficients, and Ft are r common factors extracted from a large panel of macroeconomic series. We work with two different specifications, one in which the factors Ft are computed on a large dataset including both real and nominal variables, and the other in which they are estimated on a subset of real economic indicators. The coefficients in Eq. (13) are obtained via OLS estimation using the following equation:

πt = αh +

p ∑ j=1

βhj πt −j−h+1 +

m ∑

λThj Ft −j−h+1 + ϵt ,

(14)

j=1

where h is the prediction horizon. 4.2.1. The dataset for the factor models The large dataset used for the factor model application includes 154 monthly macroeconomic series, referring to both the euro area as a whole and its major economies. The dataset reflects the information set on which the business cycle indicator Eurocoin, developed by Altissimo, Cristadoro, Forni, Lippi, and Veronese (2010), is estimated. The 21 Moreover, if the data generating process does not contain lagged (i.e., dynamic) factors, the estimation of the spectra performed in the dynamic method could induce efficiency losses.

Table 6 Forecast accuracy (RMSFE): core inflation, expanding window estimation scheme, comparison with dynamic factor models.

VMA(1)-small VMA(1)-large DFM (5 factors) DFM (6 factors) DFM (7 factors) DFM (8 factors) Real-DFM (5 factors) Real-DFM (6 factors) Real-DFM (7 factors) Real-DFM (8 factors)

2005–2014

2005–2009

2009–2014

0.155 0.156 0.160 0.160 0.161 0.160 0.155 0.156 0.158 0.159

0.125 0.126 0.130 0.132 0.132 0.132 0.124 0.126 0.127 0.130

0.181 0.181 0.185 0.184 0.184 0.184 0.182 0.182 0.183 0.184

Notes: The RMSFEs in the 2005–2014 column are based on 112 recursive forecasts between January 2005 and April 2014. Those in the 2005–2009 column are based on 56 recursive forecasts between January 2005 and August 2009. Those in the 2009–2014 column are based on 56 recursive forecasts between September 2009 and April 2014. VMA(1)-small and VMA(1)-large are estimated on disaggregated inflation series taken in first differences.

database is organised into homogeneous blocks, namely industrial production indexes (30 series), prices (26), money aggregates (8), interest rates (7), spreads and stock prices (17), demand indicators (11), surveys (32), trade variables (8), leading indicators (6), exchange rates (3), employment indicators (4) and wage indicators (2). All series are transformed to remove seasonal factors and non-stationarity; see Altissimo et al. (2010) for a detailed description of the data treatment. The factor model that considers only real indicators uses 85 indicators, namely industrial production indexes (30 series), demand indicators (11), surveys (32), trade variables (8) and employment indicators (4). 4.2.2. Factor model specification: Choosing the number of factors We determine the number of factors by relying on both informal screening and formal tests. An informal rule of thumb requires us to keep at least enough factors to explain around 50–60% of the overall dataset variance (see Boivin and Ng (2005)). Using this criterion, we should keep no fewer than four or five factors; see Fig. 3 for the dynamic factor model (DFM) based on the whole dataset and Fig. 4 for the DFM based only on real indicators. Formal tests such as those proposed by Bai and Ng (2002) favour richer models, with r being between five and eight factors. We therefore experiment with different specifications, letting r vary between five and eight. In each exercise, we determine the lag order p in Eq. (14) at each forecasting step (i.e., in pseudo real time) via the Akaike criterion, and fix m at 1. 4.2.3. Forecasting results Table 6 shows the one-step-ahead RMSFEs attained by the integrated VMA models (small and large) and by the different dynamic factor models on the whole sample and different sub-samples. The analysis confirms that, on average, factor models computed on real indicators perform better than those based on a larger set of variables, regardless of the sub-sample considered. The performance of the integrated VMA(1)-small model is broadly in line with that of the best factor model (Real-DFM, five factors), while the

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Fig. 3. Scree plot, full dataset. Notes: The figure shows the variance explained by each common factor (top panel) and the cumulative variance (bottom panel) for factor models that use all of the indicators in the large dataset.

Fig. 4. Scree plot, real indicators. Notes: The figure shows the variance explained by each common factor (top panel) and the cumulative variance (bottom panel) for factor models that use only the real indicators.

use of a lower level of aggregation, as in the integrated VMA(1)-large model, worsens the forecast accuracy. We test whether the differences in forecast accuracy are statistically meaningful by performing Diebold-Mariano and encompassing tests, taking the integrated VMA(1)small specification as the benchmark. The results, reported in Table 7, indicate that the differences across these models are not statistically significant: none of the DieboldMariano tests reject the null of equal forecast accuracy, while all of the encompassing tests suggest that a model

combination of the integrated VMA(1)-small and a factor model would achieve some gains in predictive accuracy. Overall, we take the results of this analysis to be quite supportive of our method for the purposes of inflation forecasting, since it enables us to recover the information contained in a large panel of macroeconomic series from a small number of disaggregated prices. As a further robustness check, we also estimate these forecasting models using a (five-year) rolling window rather than an expanding window. While the results are

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Table 7 Equal forecast accuracy and forecast encompassing tests (benchmark model is VMA(1)-small): core inflation, expanding window estimation scheme, comparison with dynamic factor models. Forecast accuracy tests Mean squared errors

DFM (5 factors) DFM (6 factors) DFM (7 factors) DFM (8 factors) Real-DFM (5 factors) Real-DFM (6 factors) Real-DFM (7 factors) Real-DFM (8 factors)

Forecast encompassing tests Absolute errors

VMA(1)-small encompasses

VMA(1)-small is encompassed

Stat

p-val

Stat

p-val

Stat

p-val

Stat

p-val

−0.864 −0.856 −1.004 −0.906

0.388 0.392 0.316 0.365 0.999 0.864 0.656 0.521

−0.868 −1.011 −1.020 −1.105 −0.087 −0.288 −0.545 −0.954

0.386 0.312 0.308 0.269 0.931 0.773 0.586 0.340

6.669 6.845 6.852 6.845 6.342 6.379 6.420 6.637

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

6.301 6.359 6.353 6.346 6.204 6.213 6.227 6.292

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.001

−0.172 −0.446 −0.641

Notes: The forecasting sample is January 2005–April 2014. The table shows the results of the tests of equal forecast accuracy and forecast encompassing between the model VMA(1)-small and the models indicated in the rows. The forecast accuracy test is the Diebold-Mariano test based on squared and absolute errors, respectively. The forecast encompassing test is the Harvey-Leybourne-Newbold test. VMA(1)-small is estimated on the disaggregated inflation series taken in first differences.

not reported here for the sake of brevity, they show that the forecast accuracy improves for all of the models, with the integrated VMA(1)-small achieving the lowest RMSFE (0.133, compared to 0.138 attained by the best performing factor model). This confirms that factor models do not provide forecasts that are superior to those of the integrated VMA(1) model, nor are they more robust to structural change. 4.3. Forecasting headline inflation This subsection contrasts our model forecasts with those elicited from professional forecasters, motivated by Faust and Wright’s (2013) finding that the short-term predictions of professional forecasters, which also include elements of judgment, outperform those obtained by formal models. However, one complication arises, owing to the fact that the only measure for which a reasonably long time series of forecasts polled by Bloomberg is available is euro-area headline (rather than core) inflation, defined as the year-on-year growth of the overall HICP. Therefore, we perform this comparison by aggregating the core inflation predictions obtained using the integrated VMA(1) models with forecasts for the volatile components (energy and food prices) derived from auxiliary models.22 The results in Table 8 show that the predictions produced using the integrated VMA(1) models improve over the Bloomberg forecast in terms of forecast accuracies, with a RMSFE reduction of almost 15%. However, taking the integrated VMA(1)-small as a benchmark again, the differences in accuracy relative to the Bloomberg forecast turn out not to be statistically significant, whether one considers mean squared errors or absolute errors (Table 9). Furthermore, the results of 22 For energy prices, we use the information on fuel prices that was available from the European Commission, as well as models that closely follow those of ECB (2010). For food prices, we rely on Ferrucci, JimenezRodriguez, and Onorante (2012) and Porqueddu and Venditti (2013). We refer the interested reader to Appendix B.1 for details of the models used for energy and food prices. Another complication is that we need to replicate the information set that was available to professional forecasters when their forecasts were elicited as closely as possible. Appendix B.2 explains how we address this issue.

Table 8 Forecast accuracy (RMSFE): headline inflation, expanding window estimation scheme.

VMA(1)-small VMA(1)-large VAR(1) VAR(2) VAR(3) VAR(p) Bloomberg Random walk AR(p) IMA(1,1) T.V.P. IMA(1,1)

2005–2014

2005–2009

2009–2014

0.128 0.131 0.143 0.150 0.143 0.143 0.147 0.271 0.122 0.134 0.134

0.113 0.117 0.131 0.145 0.134 0.130 0.134 0.304 0.106 0.117 0.118

0.142 0.144 0.155 0.155 0.152 0.155 0.160 0.233 0.136 0.149 0.147

Notes: The target variable is the headline HICP inflation rate. The RMSFEs in the 2005–2014 column are based on 112 recursive forecasts between January 2005 and April 2014. Those in the 2005–2009 column are based on 56 recursive forecasts between January 2005 and August 2009. Those in the 2009–2014 column are based on 56 recursive forecasts between September 2009 and April 2014. VMA(1)-small, VMA(1)-large, VAR(1), VAR(2), VAR(3) and VAR(p) are estimated on disaggregate inflation series taken in first differences. Similarly, the AR(p) model is estimated on the aggregate inflation series in first differences. Random walk refers to a random walk forecast of the aggregate inflation rate.

the encompassing tests show that the integrated VMA(1)small model both encompasses and is encompassed by the Bloomberg forecasts, indicating that a combination of model and judgmental forecast could lead to significant improvements in precision. 4.4. Robustness checks We continue assessing the robustness of our proposed approach by conducting two further exercises. The former is an extension to the three largest euro-area countries (Germany, France and Italy). This is necessary to dispel the possibility that there could be something specific about the euro-area data that favours our approach. The latter consists of considering two-step-ahead forecasts. In both cases, the focus is on the overall (headline) inflation rate. Note that this is not the ideal setting for our model as it has only one lag memory, since it is an integrated VMA(1) in reduced form. This implies that its forecast for multi-period predictions reverts to the unconditional mean.

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Table 9 Equal forecast accuracy and forecast encompassing tests (where the benchmark model is VMA(1)-small): headline inflation, expanding window estimation scheme. Forecast accuracy tests

VMA(1)-large VAR(1) VAR(2) VAR(3) VAR(p) Bloomberg Random walk AR(p) IMA(1,1) T.V.P. IMA(1,1)

Forecast encompassing tests

Mean squared errors

Absolute errors

Stat

p-val

Stat

p-val

Stat

p-val

Stat

p-val

−1.257 −2.515 −2.689 −1.744 −2.441 −1.370 −4.314

0.209 0.012 0.007 0.081 0.015 0.171 0.000 0.089 0.008 0.005

−0.336 −2.321 −2.952 −1.702 −2.188 −0.522 −5.236

0.737 0.020 0.003 0.089 0.029 0.601 0.000 0.048 0.207 0.212

2.419 4.572 4.261 3.726 4.595 4.866 5.141 0.579 3.489 3.679

0.016 0.000 0.000 0.000 0.000 0.000 0.000 0.563 0.001 0.000

0.129 0.210 0.058 0.934 0.361 3.308 0.709 2.648 1.852 1.896

0.898 0.834 0.953 0.350 0.718 0.001 0.478 0.008 0.064 0.058

1.700

−2.669 −2.792

1.976

−1.262 −1.248

VMA(1)-small encompasses

VMA(1)-small is encompassed

Notes: The forecasting sample is January 2005–April 2014. The table shows the results of the tests of equal forecast accuracy and forecast encompassing between the model VMA(1)-small and the models indicated in the rows. The forecast accuracy test is the Diebold-Mariano test based on squared and absolute errors, respectively. The forecast encompassing test is the Harvey-Leybourne-Newbold test. VMA(1)-small, VMA(1)-large, VAR(1), VAR(2), VAR(3) and VAR(p) are estimated on the disaggregated inflation series taken in first differences. Similarly, the AR(p) model is estimated on the aggregate inflation series in first differences. Random walk refers to a random walk forecast of the aggregate inflation rate. It should be stressed that the same models for the food and energy components are employed when constructing forecasts using the small and large VMA systems, as well as the VARs and IMA(1,1) models. As such, the differences in forecast accuracy across these models depend on the accuracy of the core inflation forecasts. Table 10 Forecast accuracy, headline inflation: expanding window estimation scheme, country results. VMA(1)-small

VAR(1)

Random walk

Germany

0.214

0.363

0.338

France

0.187

0.202

0.300

Italy

0.258

0.331

0.312

Notes: The table shows the one-step-ahead RMSFEs of the models indicated in the columns. The target variable is the headline HICP inflation rate. The RMSFEs are based on 108 recursive forecasts between January 2005 and December 2013. The VMA(1)-small and VAR(1) models are estimated on disaggregated inflation series taken in first differences. We have underlined the RMSFEs for which the Diebold-Mariano test rejects the null hypothesis of equal forecast accuracy between the forecasts produced with the VAR(1)/RW and those of the benchmark model (VMA(1)-small).

We shorten the discussion by considering fewer models in both analyses, namely the integrated VMA(1)-small model, the VAR(1) model (estimated on disaggregate inflation series in first differences) and the RW. The results obtained for the individual countries are shown in Table 10.23 Two key results emerge. First, on the full out-of-sample period, the RMSFE of the integrated VMA(1) model is lower than those of the remaining models for Germany, France and Italy. Second, for Germany and Italy, the improvement on the VAR(1) model is statistically significant at the 10% confidence level according to the Diebold-Mariano test. Furthermore, the improvement on the RW is statistically significant at the 10% level for all three countries. Taken together, the results are quite reassuring about the robustness of the integrated VMA(1) estimated with the M.E.T.A. approach for one-step-ahead inflation predictions. 23 The forecast exercise for the three largest euro-area countries assumes the same information flow as for the euro area, and therefore combines one-step-ahead forecasts for the core components with nowcasts of energy prices based on fuel price data for the current month.

Table 11 Forecast accuracy (RMSFE), headline inflation: expanding window estimation scheme, two-step-ahead forecasts. 2005–2014 VMA(1)-small VAR(1)

0.241 0.236

Notes: The RMSFEs in the 2005–2014 column are based on 110 recursive forecasts between January 2005 and April 2014. The VMA(1)-small and VAR(1) models are estimated on disaggregate inflation series taken in first differences.

The results of the two-step-ahead forecast exercise are reported in Table 11. In this case, the performances of the integrated VMA(1)small model and the VAR(1) are similar, with the competitor being slightly superior. Overall, it appears that the differences in favour of the integrated VMA(1) model that are observed when considering one-step-ahead forecasts vanish at longer horizons. As was mentioned above, this is not surprising, given the lag memory of the MA(1) model. Finally, we discuss how our model forecasts compare with those produced by alternative methods that have appeared recently in the literature, namely the MIDAS model of Monteforte and Moretti (2013) and the factor model of Modugno (2011, 2013). The former analyses the information content of financial variables for inflation forecasting, while the latter uses a mixed-frequency dynamic factor model to forecast inflation in a large dataset environment. Both have a forecasting application to euro-area inflation. For the sake of brevity, we do not discuss the relative merits of the three approaches here, as our main concern is in gauging whether the performance of our method is comparable to those of other state-of-the-art methodologies. We do this in the only feasible (albeit rather coarse) way: that is, by collecting forecasts over the sample periods considered by those studies and comparing the results. Between June 2002 and September 2007, the integrated VMA(1)-small model yields a RMSFE of 0.096, as opposed

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to the 0.148 that is achieved by aggregating the daily onestep-ahead forecasts obtained using the best MIDAS specification of Monteforte and Moretti (2013, Table 1). Between January 2002 and December 2009, the integrated VMA(1)small model achieves a RMSFE of 0.10, whereas the factor model of Modugno (2011, Table 1), obtains a RMSFE of 0.156.24 Needless to say, we do not take these results as conclusive evidence in favour of our approach; however, we do consider them to indicate that our methodology constitutes a serious benchmark for short-term inflation forecasting. 5. Conclusions This paper proposes the forecasting of euro-area inflation under the assumption that the underlying components of the core inflation index evolve as a multivariate local level model. We circumvent the estimation problems faced by the multivariate maximum likelihood estimator by means of the ‘‘moments estimation through aggregation’’ (M.E.T.A.) approach that was proposed recently by Poloni and Sbrana (2015). We illustrate the advantages of this estimation method by means of a Monte Carlo experiment and an application to the forecasting of euro-area inflation at short horizons. The Monte Carlo simulations show that the M.E.T.A. approach outperforms the standard multivariate maximum likelihood estimator in both speed and accuracy. The empirical application demonstrates that the multivariate local level model estimated with the M.E.T.A. provides predictions that compare well with those generated by alternative univariate constant and time-varying parameter models, as well as with those of professional forecasters and vector autoregressions. Furthermore, the performance of our benchmark model is broadly in line with that of the best factor model, and it is not outperformed by the other factor specifications. These results are quite supportive of our approach for the purposes of inflation forecasting, since it enables the information contained in a large panel of macroeconomic series to be recovered from a small number of disaggregated prices. In terms of future research, further work may refer to the recent literature on the forecasting of inflation using univariate unobserved components models with stochastic volatility, along the lines of the IMA(1,1) process with timevarying parameters that was proposed by Stock and Watson (2007). A challenging extension would be to consider a multivariate generalization of this model, developing an estimation procedure that allows us to overcome the computational difficulties that are known to emerge even in the univariate framework. Acknowledgments While assuming the scientific responsibility for any errors in this paper, the authors would like to thank Prof. Rob Hyndman (the Editor), Prof. George Kapetanios 24 The latter figure is obtained by multiplying the ratio to the random walk RMSFE (0.56) presented in Table 1 of Modugno (2011) by the random walk RMSFE over the same period, 0.277.

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