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Econometrics 2018, 6(2), 23; doi:10.3390/econometrics6020023

Article
Forecasting Inflation Uncertainty in the G7 Countries
1
Department of Economics (CQE), Westfälische Wilhelms-Universität Münster, Am Stadtgraben 9, 48143 Münster, Germany
2
Department of Accounting and Finance, Athens University of Economics and Business, Trias 2, GR 11362 Athens, Greece
3
Department of Economics, European University Institute, Via delle Fontanelle 18, I-50014 Florence, Italy
*
Author to whom correspondence should be addressed.
Received: 16 February 2018 / Accepted: 16 April 2018 / Published: 27 April 2018

Abstract

:
There is substantial evidence that inflation rates are characterized by long memory and nonlinearities. In this paper, we introduce a long-memory Smooth Transition AutoRegressive Fractionally Integrated Moving Average-Markov Switching Multifractal specification [ STARFIMA ( p , d , q ) - MSM ( k ) ] for modeling and forecasting inflation uncertainty. We first provide the statistical properties of the process and investigate the finite sample properties of the maximum likelihood estimators through simulation. Second, we evaluate the out-of-sample forecast performance of the model in forecasting inflation uncertainty in the G7 countries. Our empirical analysis demonstrates the superiority of the new model over the alternative STARFIMA ( p , d , q ) - GARCH -type models in forecasting inflation uncertainty.
Keywords:
inflation uncertainty; smooth transition; multifractal processes; GARCH processes
JEL Classification:
C22; E31

1. Introduction

The financial crisis of 2007–2009 and the long-lasting economic recovery has renewed the interest in studying and measuring inflation uncertainty. Studies by Baker et al. (2015) and Jurado et al. (2015), for example, discuss new approaches to defining and measuring inflation and, more generally, macroeconomic uncertainty. Theoretical and empirical studies indicating that uncertainty negatively affects economic growth are well-documented in the literature (see Bernanke 1983; Bloom 2009, 2014; Stock and Watson 2012; Henzel and Rengel 2017). In this context, Stock and Watson (2012) find that liquidity risk and uncertainty shocks account for about two-thirds of the US GDP decline during the Great Recession. Bloom (2014) and Henzel and Rengel (2017) provide evidence for a countercyclical behavior of uncertainty. Gurkaynak and Wright (2012) and Wright (2011) argue that inflation uncertainty may explain the behavior of bond risk premia and thus plays a major role in understanding the different effects of monetary policy on short- and long-term interest rates. As stressed in Goodhart (1999) and Greenspan (2003), effective monetary policy purposes prevail as reliable, easy-to-update, and accurate measures of inflation uncertainty.
In spite of being inherently unobservable, inflation uncertainty can be estimated from econometric models. One of the most frequently used approaches to measuring inflation uncertainty consists of applying Engle’s (1982) AutoRegressive Conditional Heteroscedasticity (ARCH) processes and their generalized variants. These models are motivated by stylized facts on inflation uncertainty, in particular volatility clustering, high persistence, and asymmetry (see, among others, Baillie et al. 1996; Fountas et al. 2004; Karanasos and Schurer 2008; Caporale et al. 2012; Clements 2014; Makarova 2018). The popularity of GARCH-type models stems from their formal simplicity, flexibility, low computational costs, and their capacity to reproduce clustering effects. However, thorough investigations reveal that alternative inflation uncertainty measures (distinct absolute powers of inflation rates) typically exhibit structural dynamics and persistence patterns that GARCH-type models cannot reproduce. This leads to the question as to which econometric models may be appropriate for modeling (and producing accurate measures of) inflation uncertainty.
In this paper, we consider a new modeling approach by combining long-memory Smooth Transition AutoRegressive Fractionally Integrated Moving Average (STARFIMA) specifications with Markov Switching Multifractal (MSM) models, as recently developed by Calvet and Fisher (2004). MSM processes represent an alternative tool for modeling and forecasting volatility in financial and commodities markets, which regularly outperform GARCH-type models in out-of-sample forecasting evaluations (see Lux et al. 2016; Wang et al. 2016; Segnon et al. 2017). Owing to its formal construction, MSM models properly reproduce the structural dynamics observed in different absolute powers of inflation rates.1
The rest of the paper is organized as follows. Section 2 introduces the STARFIMA-MSM model. The statistical properties of the model are established in Section 3. Section 4 briefly outlines maximum likelihood estimation and optimal forecasting. Section 5 presents the data analysis for the G7 countries, forecasting methodologies, and the empirical results. Section 6 concludes.

2. The STARFIMA ( p , d , q ) -MSM ( k ) Model

We define the STARFIMA ( p , d , q ) -MSM ( k ) model to be a discrete-time stochastic process { x t } satisfying the equation
Φ s t ; η ( L ) ( 1 L ) d x t = Θ ( L ) ϵ t
where ϵ t | Ω t 1 N ( 0 , h t ) and
h t = σ 2 j = 1 k M t ( j ) .
In Equations (1) and (2), L denotes the lag operator and Ω t 1 is the σ -field generated by the information set { ϵ t 1 , ϵ t 2 , } . The lag polynomials are defined as Φ s t ; η ( L ) = 1 ϕ 1 ( s t ; η 1 ) L ϕ p ( s t ; η p ) L p , where the p autoregressive coefficients ϕ i s t , η i = ϕ i 0 + ϕ i 1 G s t ; τ , c are nonlinear functions of the state variable s t . η i = ϕ i 0 , ϕ i 1 , τ , c is a vector of parameters, and Θ ( L ) = 1 + θ 1 L + + θ q L q . d ( 0.5 , 0.5 ) is a real number, and ( 1 L ) d is the fractional differencing operator given by
( 1 L ) d = j = 0 Γ ( j d ) L j Γ ( d ) Γ ( j + 1 )
where Γ ( · ) denotes the gamma function.
In Equation (2), M t ( 1 ) , M t ( 2 ) , , M t ( k ) denote the random volatility components (called multipliers). At date t, each multiplier M t ( j ) is drawn from the base distribution M (to be specified) with positive support. Depending on its rank within the hierarchy of multipliers, M t ( j ) changes from one period to the next with probability γ j and remains unchanged with probability 1 γ j . We specify these transition probabilities as
γ j = 2 j k , j = 1 , , k
so that the transition matrix related to the jth multiplier is given by
P j = 1 0.5 γ j 0.5 γ j 0.5 γ j 1 0.5 γ j .
In this paper, we draw each multiplier M t ( j ) (in case of a change) from a binomial distribution with support { m 0 , 2 m 0 } , 1 < m 0 < 2 , and (binomial) probability 0.5 , implying the unconditional expectation E ( M t ( j ) ) = 1 . If we assume stochastic independence among the multipliers, the transition matrix of the vector M t ( M t ( 1 ) , , M t ( k ) ) becomes the 2 k × 2 k matrix P = P 1 P 2 P k , where ⊗ denotes the Kronecker product. Using the binomial base distribution for the single multipliers implies the finite support Γ m 0 , 2 m 0 k for M t .
Remark 1.
The stochastic process in Equation (1) can be viewed as a special case of the model proposed by Hillebrand and Medeiros (2016) with constant conditional variance and multiple regimes. The process reduces to the linear AutoRegressive Fractionally Integrated Moving Average (ARFIMA) model, when setting ϕ i s t , η i = ϕ i , i = 1 , , p . In this paper, we consider only two regimes, since this turns out to be sufficient in our empirical application below. We allow the conditional variance in Equation (2), which we model as the product of the time-varying multipliers and the positive scaling factor σ 2 , to vary over time (see Calvet and Fisher 2004). As the transition function, we specify the first-order logistic function, G ( s t ; τ , c ) = 1 + exp { τ ( s t c ) } 1 , τ > 0 , which is arbitrarily often differentiable and satisfies lim s t G ( s t ; τ , c ) 0 and lim s t + G ( s t ; τ , c ) 1 . For τ + the function G ( s t ; τ , c ) approaches the indicator function 1 ( s t > c ) . The parameter τ regulates the smoothness of the transition from one regime to another (cf. van Dijk et al. 2002).
Remark 2.
The transition probabilities defined in Equation (4) have been proposed by Lux (2008). This specification reduces the number of parameters to be estimated and enables us to obtain some statistical properties of the model. In Calvet and Fisher (2001), the k transition probabilities are specified as γ j = 1 ( 1 γ 1 ) ( b j 1 ) with γ 1 ( 0 , 1 ) and b > 1 , which guarantees the convergence of the discrete-time MSM model to the Poisson multifractal process in the continuous-time limit. Calvet and Fisher (2004) assume binomial and log-normal base distributions for the multipliers. Liu et al. (2007) find that assuming other base distributions, such as lognormal and gamma, makes little difference in empirical applications.
Our Markov-Switching Multifractal (MSM) volatility process as specified in Equations (2) and (4) could alternatively be specified as a GARCH-type process. In our out-of-sample forecasting analysis below, we compare the performance of our STARFIMA ( p , d , q ) -MSM ( k ) model to that of a number of STARFIMA ( p , d , q ) -GARCH-type processes. He and Terasvirta (1999) propose a general class of GARCH ( 1 , 1 ) models of the form
h t δ = g ( u t 1 ) + c ( u t 1 ) h t 1 δ
with Pr ( h t δ > 0 ) = 1 , δ > 0 , and where { u t } is a sequence of i.i.d. standard normal random variables, and g ( x ) , c ( x ) are nonnegative functions. This class of GARCH-type models includes, among others, the specifications of Bollerslev (1986) (standard GARCH), Glosten et al. (1993) (GJR-GARCH), Nelson (1991) (EGARCH), Sentana (1995) (QGARCH), and Ding et al. (1993) (APGARCH).

3. Statistical Properties

In this section, we consider statistical properties of the STARFIMA ( p , d , q ) -MSM ( k ) and the general STARFIMA ( p , d , q ) -GARCH-type processes, as defined in Section 2.
Assumption .
The roots of the characteristic polynomials Φ s t ; η ( L ) and Θ ( L ) lie outside the unit circle and the logistic transition function G ( s t ; τ , c ) is well-defined.
Assumption 2.
The volatility components M t ( 1 ) , M t ( 2 ) , , M ( t ) k with E ( M t ( 1 ) ) = E ( M t ( k ) ) = 1 are nonnegative and independent of each other for all t, and for the transition probabilities, we have γ 1 , , γ k ( 0 , 1 ) .
Proposition 1.
Under Assumptions 1 and 2, the STARFIMA ( p , d , q ) -MSM ( k ) model defined in Equations (1)–(4) has a unique, second-order stationary solution. It follows that { x t } , { ϵ t } , a n d { h t } are strictly stationary, ergodic, and invertible.
Proof. 
Under Assumption 2, the conditions of Theorem 1 in (Shiryaev 1995, p. 118) are satisfied. It follows that the Markov chain underlying the dynamics of the multipliers M t ( 1 ) , , M t ( k ) is geometrically ergodic. The probabilities of the ergodic distribution are given by π l = 1 / 2 k , l = 1 , , 2 k . Under Assumptions 1 and 2, { x t } , { ϵ t } , a n d { h t } are strictly stationary, ergodic, and invertible. ☐
Proposition 2.
Under Assumption 1, the STARFIMA(p,d,q)-GARCH model specified in Equations (1), (3), and (5) has a unique, α δ -order stationary solution (α a positive integer). It follows that { x t } , { ϵ t } , a n d { h t } are strictly stationary, ergodic, and invertible.
Proof. 
The proof follows from Assumption 1 and the conditions in Theorem 2.1 of Ling and McAleer (2002a), where we replace the constant mean process with our stationary STARFIMA ( p , d , q ) process from Equations (1) and (3). ☐
Proposition 3.
Under Proposition 1 and with m denoting a positive integer, it follows that the 2 m -th moments of { x t } a n d { ϵ t } are finite.
Proof. 
The proof follows from Proposition 1 and the conditions in Theorem 1 in Shiryaev (1995, p. 118). ☐
Proposition 4.
Under Proposition 2, it follows that the m δ -th moments of { x t } , { ϵ t } exist.
Proof. 
The proof follows from Proposition 2 and Theorem 2.2 in Ling and McAleer (2002a). ☐
Remark 3.
Second moments and autocovariances of the MSM ( k ) process for binomial and lognormal base distributions of the multipliers are given in Lux (2008). As argued in Ling and McAleer (2002a), Proposition 4 cannot easily be extended to higher-order generalized GARCH processes, as specified in Equation (5). However, Ling (1999) provides a sufficient condition for the existence of 2 m -th moments for the standard GARCH ( p , q ) process. Ling and McAleer (2002b) establish necessary and sufficient higher-order moment conditions for standard GARCH ( p , q ) and APARCH ( p , q ) processes.
Next, we present results for (i) the autocorrelation function of the process { x t } from Equation (1), which we denote by ρ ( n ) = Cov ( x t , x t n ) / Var ( x t ) , and (ii) the q-order autocorrelation function of the process ϵ t denoted by ρ q ( n ) = Cov ( | ϵ t | q , | ϵ t n | q ) / Var ( | ϵ t | q ) , for every moment q and every integer n. For this purpose, we consider the two arbitrary numbers κ 1 , κ 2 ( 0 , 1 ) , κ 1 < κ 2 , which we use to define the following set of integers (as before, k denotes the number of volatility multipliers in Equation (2)): S k = n : κ 1 k log 2 ( n ) κ 2 k . It is easy to check that S k contains a wide range of intermediate lags.
Proposition 5.
Under Assumption 1, we have ρ ( n ) c | n | 2 d 1 as n , where c is a real constant.
Proof. 
The proof follows from Proposition 2 and Theorem 2.4 in Hosking (1981). ☐
Proposition 6.
Under Assumption 2, it follows that ln ρ q ( n ) ψ ( q ) ln n as k , where ψ ( q ) = log 2 E ( M q ) [ E ( M q / 2 ) ] 2 . (M is a random variable distributed as the base distribution of the multipliers M t ( 1 ) , , M t ( k ) ).
Proof. 
The proof follows from Proposition 2 and the proof of Proposition 1 in Calvet and Fisher (2004). ☐
Remark 4.
MSM processes only exhibit apparent long memory with asymptotic hyperbolic decay in the autocorrelation of absolute powers over a finite horizon. This does not coincide with the traditional definition of long memory with asymptotic power-law behavior of the autocorrelation function in the limit or divergence of the spectral density (see Beran 1994).

4. Maximum Likelihood Estimation and Optimal Forecasting

4.1. Maximum Likelihood Estimation

Hillebrand and Medeiros (2016) suggest using Nonlinear Least Squares (NLS) for parameter estimation of the STARFIMA model. We collect all parameters of the STARFIMA specification in the vector χ and denote (i) an appropriately defined subset of the parameter space by Ξ and (ii) the true parameter vector by χ 0 . Then, for a sample of T observations, the NLS estimator is given by
χ ^ = arg min χ Ξ t = 1 T ϵ t 2 .
In the case of normally distributed innovations ϵ t , NLS is equivalent to Maximum Likelihood Estimation (MLE), whereas for non-normal innovations NLS can be interpreted as Quasi MLE (QMLE). Wooldridge (1994), Pötscher and Prucha (1997), and Hillebrand and Medeiros (2016) show that the NLS estimator is consistent and asymptotically normal under appropriate regularity conditions. Li and McLeod (1986) derive asymptotic properties of the MLE for the ARFIMA processes, and a portmanteau test for checking model adequacy.
Proposition 7.
Let χ ^ be the solution to the minimization problem (6). Under Assumption 1, it follows that χ ^ is (i) a consistent estimator of χ 0 and (ii) asymptotically normal.
Proof. 
Under Assumption 1, the conditions of Theorems 1 and 2 in Hillebrand and Medeiros (2016) are satisfied, yielding the proof. ☐
Using a binomial base distribution for the k multipliers, Calvet and Fisher (2004) derive a closed-form solution for the log-likelihood and exact ML estimators of the parameters in the MSM(k) model. In fact, discrete base distributions with positive support for the multipliers imply a finite number of states for the hidden Markov process in the MSM model. This allows us to derive the exact likelihood function via Bayesian updating. For pre-specified k, it is known that the MLE is consistent and asymptotically efficient.
Since the off-diagonal blocks in the information matrix of a STARFIMA ( p , d , q ) - MSM ( k ) model are zero, the parameters in the STARFIMA ( p , d , q ) and MSM ( k ) specifications can be estimated separately, without asymptotic efficiency loss (see Lundbergh and Terasvirta 1999). Therefore, in a first stage, we estimate the conditional mean via NLS, thus providing consistent estimates of the ϵ t ’s, which we use in the second stage to estimate the parameters of the conditional variance from the specification
ϵ ^ t = u t h t .
Denoting the parameter vector by ξ = ( m 0 , σ ) (defined on a compact subset of the parameter space), we obtain the parameters in the second stage by maximizing the log-likelihood
ξ ^ = arg max ξ i = 1 T ln ω ( ϵ ^ t ; ξ ) π t 1 P .
In Equation (8), ω ( ϵ ^ t ; ξ ) is a 1 × 2 k vector containing the conditional densities of any observation ϵ ^ t given by
f ( ϵ ^ t | M t = m j ) = 1 h ( m j ) ϕ ϵ ^ t h ( m j )
where ϕ ( · ) denotes the standard normal density and h ( m j ) = σ i = 1 k m j ( i ) with m j ( i ) being the i-th element of vector m j . The transition matrix P has the components p i , j = Pr ( M t + 1 = m j | M t = m i ) . M t is latent, but we can recursively compute the conditional probabilities π t i = Pr ( M t = m i | ϵ ^ t , , ϵ ^ 1 ) through Bayesian updating as
π t = ω ( ϵ ^ t ; ξ ) · ( π t 1 P ) ω ( ϵ ^ t ; ξ ) · ( π t 1 P ) .
Proposition 8.
Let ϑ = ( χ , ξ ) denote the complete parameter vector of the STARFIMA( p , d , q )-MSM(k) model. Under Assumptions 1 and 2 and Propositions 3 and 4, there exists an MLE ϑ ^ that is consistent and asymptotically efficient.
Proof. 
Under the given assumptions, the conditions of Theorem 1 in Hillebrand and Medeiros (2016) are met, yielding the proof. ☐
Remark 5.
The shortcoming of the exact MLE is that it becomes computationally demanding for a large number of multipliers ( k > 10 ) . Furthermore, a continuous base distribution with positive support for the multipliers implies an infinite state space of the hidden Markov chain, so that the MLE is not applicable. To circumvent these issues, Lux (2008) proposes a generalized method-of-moments estimator with linear forecasting. Recently, Žikeš et al. (2017) established the Whittle estimation approach.
In Section 4.3, we show that numerical optimization of the MSM(k) log-likelihood function produces satisfactory results for a moderate number of volatility components.

4.2. Optimal Forecasting

Using the maximum likelihood estimation approach, we easily obtain volatility forecasts in the MSM(k) model via Bayesian updating of the conditional probabilities. The h-step-ahead volatility forecasts of the MSM ( k ) model are given by
E ( h t + h | Ω t ) = σ ^ 2 j = 1 k E ( M t + h ( j ) | Ω t ) .
In fact, to produce volatility forecasts over arbitrary, long-term horizons as given in Equation (11), we need the conditional probabilities of future multipliers. These conditional state probabilities can be iterated forward via the transition matrix P as follows:
π ^ t , t + h = π t P h .
For GARCH-type models the formula for the h-step-ahead volatility forecasts are available in the literature (see, for example, Lux et al. 2016, Appendix A).

4.3. Monte Carlo Simulation

We assess the robustness of the MLE in small samples via Monte Carlo simulations. We choose the number of volatility components as k = 8 , which turns out to be optimal in our empirical application below.2 As the base distribution, we consider a binomial distribution taking on the values m 0 and 2 − m 0 each with probability 0.5 . Along with the switching probabilities from Equation (4), our simulation of the MSM model only requires two parameters: the binomial parameter m 0 and the scale factor (unconditional standard deviation) σ , which we normalize to unity. We simulate 500 independent sample paths of our restricted MSM model for (i) the three different binomial parameters m 0 { 1.1 , 1.2 , 1.3 } and (ii) the three different sample sizes T { 250 , 500 , 1000 } .
Table 1 reports the Monte Carlo maximum likelihood estimation results for small sample sizes. The first two rows provide the average bias and the mean squared error (MSE) of the parameter estimates, relative to the true parameters. The results of the ML estimation appear reasonable and exhibit a decrease in the MSEs with increasing sample size T. From T 1 = 250 to T 2 = 500 , the MSEs decrease roughly with a factor of about 2. Overall, our Monte Carlo simulation demonstrates that ML estimation produces reliable results.
Next, we analyze the capacity of the MSM model for reproducing the statistics of empirical data. We first estimate the binomial parameter m 0 and the scaling factor σ 2 for each G7 country and then use the parameter estimates to simulate 500 independent sample paths with country-specific sample sizes corresponding to those from the empirical data. The country-specific averaged means, standard deviations, skewness and kurtosis values, and the Hurst exponents are reported in Table 2. Overall, the results indicate that the MSM model reproduces the inflation-rate characteristics accurately. We note, however, that the MSM model is not able to capture the asymmetric properties observed in the data.

5. Empirical Application

5.1. Data

Our data set consists of seasonally adjusted consumer-price-index (CPI)-based inflation rates for the G-7 countries (USA, UK, Germany, France, Italy, Canada and Japan). The monthly data were compiled from the International Financial Statistics (IFS). Our data cover the following country-specific time spans: (i) January 1985–December 2015 for the USA, France, and Italy; (ii) January 1985–November 2015 for Canada and Japan; (iii) January 1989–December 2015 for UK; (iv) January 1992–December 2015 for Germany.
The descriptive statistics of the inflation rates are reported in Table 3. The inflation-rate time series exhibit positive skewness and excess kurtosis (greater than 3) for all G7 countries. This indicates a deviation from the normal distribution that is confirmed by the Jarque-Bera test. To test for stationarity, we apply the Phillips-Perron unit-root test, which does not reject the null hypothesis of a unit root at the 1% level for any of G7 countries (see Table 4). We also apply the KPSS test for the stationarity, the results of which are also reported in Table 4. Here, the null hypothesis of stationarity is rejected for all G7 countries at any conventional significance level. In order to analyze the decay in the tails of the unconditional distributions, we also disclose the country-specific tail indices in Table 3, which range between 2 and 13. For the USA, UK, France, Germany, Italy, and Canada, the tail indices are substantially larger than 2, indicating convergence under time-aggregation towards the normal distribution. For Japan, the tail index is close to 2, indicating that the unconditional distribution exhibits tail behavior like the normal distribution. The results of the ARCH tests in Table 3 suggest the presence of heteroscedasticity in the G7 inflation-rate time series. Figure 1 displays the inflation-rate series.

5.2. Forecasting Methodology

To analyze the predictive ability of our proposed model in forecasting inflation uncertainty, we adopt a rolling forecasting scheme that keeps fixed the estimation sample size over the out-of-sample period and adds new (and removes old) observations on a monthly basis. We define the following in-sample (out-of-sample) periods: (i) January 1958–October 2009 (November 2009–November 2015) for the USA, Canada, and Japan; (ii) January 1989–November 2009 (December 2009–December 2015) for the UK; (iii) January 1958–November 2009 (December 2009–December 2015) for France and Italy; (iv) January 1992–November 2009 (December 2009–December 2015) for Germany. For each country and model specification, we consider inflation uncertainty forecasts for the horizons h = 1 , 2 , 3 , 4 , 5 , 6 months. We consider the end of the global financial crisis 2007–2009 as the splitting point in our forecasting analysis.
In a first step, we first evaluate the forecasting performance of our specifications on the basis of two loss functions, (i) the mean squared error (MSE) and (ii) the mean absolute error (MAE), given by
MSE = T 1 i = 1 T h f , t σ a , t 2 2
MAE = T 1 i = 1 T h f , t σ a , t 2
with h f , t denoting the volatility forecast obtained from the binomial MSM or GARCH-type models, and σ a , t 2 the monthly actual inflation uncertainty proxy obtained from the monthly squared residuals from suitably selected STARFIMA model specifications. (Here, T is the number of out-of-sample observations.)
Next, we use of the predictive ability tests of Hansen (2005) and Diebold and Mariano (1995) to test the relative forecasting performance of our proposed specification against competitor models. The Equal Predictive Ability (EPA) test of Diebold and Mariano (1995) enables us to directly compare the forecasting accuracy of two competing models (say, M 1 and M 2 ) under a predefined loss function. The null hypothesis of no difference in the forecasting accuracy between the competing models is stated as
H 0 : E d t = 0 for   all   t
where d t = L ( ε t , M 1 ) L ( ε t , M 2 ) , with ε t , M 1 = h f , t , M 1 σ a , t 2 and ε t , M 2 = h f , t , M 2 σ a , t 2 denoting the forecast errors obtained from the models M 1 and M 2 , respectively. The loss function L ( · ) is either the squared error loss L ( ε t , M i ) = ε t , M i 2 or the absolute error loss L ( ε t , M i ) = | ε t , M i | . The Diebold-Mariano test statistic is given by
EPA = d ¯ V ( d ¯ ) 1 / 2
where d ¯ = T 1 t = 1 T d t , and V ( d ¯ ) = T 1 j = N N γ ^ j is the heteroscedasticity and autocorrelation consistent (HAC) variance estimator. ( γ ^ j is the estimate of the autocovariance function at lag j, N is the nearest integer larger than T 1 / 3 .) Under the null hypothesis, the E P A test statistic in Equation (16) is asymptotically standard normally distributed.
Based on the framework of the Reality Check (RC) proposed by White (2000), the Superior Predictive Ability (SPA) test of Hansen (2005) enables us to compare a benchmark forecast model, M 0 , with K alternative competing models, M 1 , , M K , under predefined loss functions. The null hypothesis, stating that the benchmark model is not outperformed by any of the K competing models, is formalized as
H 0 : max E ( d t , M 1 ) , , E ( d t , M K ) 0 for   all   t
where d t , M i = L ( ε t , M 0 ) L ( ε t , M i ) for i = 1 , , K and L ( · ) denotes either the squared-error or the absolute-error loss function, as defined above. To formally state the test statistic, we consider (i) the sample mean of the ith loss differential, d ¯ M i = 1 / T t = 1 T d t , M i , and (ii) the estimated variance V a r ^ ( T · d ¯ M i ) for i = 1 , , K . We refer the reader to Hansen (2005) for the technical details on how to estimate this latter variance by bootstrapping. To test the null hypothesis in Equation (17), we use the test statistic
SPA = max T d ¯ M 1 V a r ^ T · d ¯ M 1 , , T d ¯ M K V a r ^ T · d ¯ M K ,
the p-values of which can be obtained via a stationary bootstrap procedure.

5.3. Forecasting Results

The G7 country-specific root mean squared error (RMSE) and mean absolute error (MAE) values for alternative STARFIMA-MSM and STARFIMA-GARCH-type specifications at the forecasting horizons h = 1 , 2 , 3 , 4 , 5 , 6 months are reported in Table 5, Table 6, Table 7 and Table 8. Instead of considering the general STARFIMA( p , d , q ) class in modeling our mean process, we restrict attention to two special cases, namely, the STARFI model (by setting q = 0 ) and the ARFIMA model (by setting ϕ i ( s t , η i ) = ϕ i for i = 1 , , p in the lag polynomial on the left side of Equation (1)).
Based on the RMSEs and MAEs in Table 5 and Table 6, the ARFIMA-MSM specification appears to fit best the US and UK inflation rates. For France, Germany, Italy, Canada, and Japan, the ARFIMA-GARCH model yields relatively similar RMSEs and MAEs, that are superior to those of the ARFIMA-MSM model. In order to test whether the observed RMSE- and MAE-differences between the ARFIMA-MSM and -GARCH-type models are statistically significant, we apply the SPA test of Hansen (2005). The p-values obtained from 5000 bootstrap samples using both, the squared and absolute error loss functions, are reported in Table 9 and Table 10. While the null hypothesis (that the ARFIMA-MSM model is not outperformed by any of the ARFIMA-GARCH specifications) cannot be rejected for the US, UK, and France at the 10% level, we find rejection of the null hypothesis for Germany, Italy, Canada, and Japan. We also apply the EPA test in order to compare the ARFIMA-MSM specification with each of the ARFIMA-GARCH-type models (see Table 11 and Table 12). The null hypothesis (no difference in forecast accuracy) can only be rejected for the US, UK, and France (in most cases) at the 10% level. For Germany, Italy, Canada, and Japan, the ARFIMA-MSM and -GARCH models appear to exhibit similar forecasting performance.
When modeling the inflation-rate mean process by the STARFI specification, we obtain substantial gains in forecast accuracy. The RMSEs and MAEs in Table 7 and Table 8 as well as the SPA and EPA tests in Table 13, Table 14, Table 15 and Table 16 indicate that the STARFI-MSM specification systematically outperforms the respective STARFI-GARCH specifications for all G7 countries, except for Japan. Our results suggest that the STARFI-MSM model fits the G7 inflation rates considerably well, thus producing accurate inflation uncertainty forecasts. For Japan, all models perform well, but it appears impossible to find a specific model systematically dominating the others.

6. Conclusions

This paper proposes the ARFIMA- and STAR-MSM models for forecasting inflation uncertainty in the G7 countries. The specifications are found to model the dynamics of inflation uncertainty appropriately, since they are able to capture (i) dual long memory, (ii) clustering effects, (iii) non-linearities, and (iv) asymmetries observed in inflation rates. Our out-of-sample forecasting analysis confirms these capacities and the robustness of our models, which yield accurate forecasts of inflation uncertainty. In particular, the performance of the STARFI-MSM is interesting and should have major implications for monetary policy, which merit careful investigation in future research.

Author Contributions

All authors contributed equally to the paper.

Conflicts of Interest

The authors declare no conflict of interest.

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  • 1See Lux and Segnon (2018) for details on the genesis and alternative applications of multifractal processes in finance.
  • 2Technical details on the determination of the optimal number of multipliers are available upon request.
Figure 1. Consumer-price-index (CPI)-based inflation rates for the G7 countries.
Figure 1. Consumer-price-index (CPI)-based inflation rates for the G7 countries.
Econometrics 06 00023 g001
Table 1. Monte-Carlo maximum-likelihood-estimation results for small sample sizes.
Table 1. Monte-Carlo maximum-likelihood-estimation results for small sample sizes.
m 0 = 1.1 m 0 = 1.2 m 0 = 1.3
T 1 = 250 T 2 = 500 T 3 = 1000 T 1 = 250 T 2 = 500 T 3 = 1000 T 1 = 250 T 2 = 500 T 3 = 1000
Binomial parameter
Bias−0.036−0.028−0.010−0.025−0.023−0.036−0.022−0.027−0.018
MSE0.0050.0030.0010.0040.0020.0020.0030.0020.001
Scaling factor
Bias−0.108−0.072−0.061−0.214−0.150−0.105−0.315−0.232−0.187
MSE0.0130.0060.0040.0480.0240.0120.1010.0550.036
Note: The table reports average biases and mean-squared errors (MSEs) of parameter estimates from 500 MSM(k) simulation paths for k = 8 .
Table 2. Simulated moments and Hurst exponents via the binomial MSM(k) model.
Table 2. Simulated moments and Hurst exponents via the binomial MSM(k) model.
G7 Countries
USUKFranceGermanyItalyCanadaJapan
Mean1.659 × 10 4 9.152 × 10 4 −5.509 × 10 4 4.574 × 10 4 0.001−1.472× 10 4 −6.959 × 10 4
Standard deviation0.2830.2600.3170.2240.2810.3650.737
Skewness−6.932 × 10 4 0.032−0.0030.0210.0060.006−0.029
Kurtosis4.3697.1305.6275.7758.2024.6868.380
Hurst Exponent for the G7 Countries
ϵ t 0.4990.4910.5080.5040.5150.4990.508
ϵ t 2 0.6730.8140.7800.7720.8670.6940.901
| ϵ t | 0.6300.7730.7660.7210.8130.6730.848
Note: For each country, we estimate the parameters in the binomial MSM ( 8 ) specification and use the estimates to simulate data with country-specific sample sizes corresponding to those of the estimated residuals from the STARFIMA model. The moments and Hurst exponents are averaged over the number of replications.
Table 3. Descriptive statistics of the G7 inflation-rate time series.
Table 3. Descriptive statistics of the G7 inflation-rate time series.
USUKFranceGermanyItalyCanadaJapan
No. of Obs696324696288696695695
Mean3.7782.6434.5821.7716.0043.8153.157
Standard deviation2.8641.8023.9801.1715.7393.0824.264
Skewness1.5361.3951.2381.4821.4241.2482.206
Kurtosis5.4284.6243.7285.9934.1053.68710.080
Tail index7.68811.71912.2516.72010.92213.0502.09
Q ( 8 ) 4.825 × 10 3 2.056 × 10 3 4.769 × 10 3 1.359 × 10 3 5.123 E× 10 3 5.030 × 10 3 4.712 × 10 3
ARCH(1)685.983309.386684.018271.658684.284684.306660.613
Jarque-Bera444.528140.674193.272213.002207.722194.0392.095 × 10 3
Note: Q ( 8 ) denotes the Ljung-Box test for serial correlation out to lag 8. ARCH(1) denotes the Engle test for ARCH effects at lag 1.
Table 4. Unit root tests for inflation time series.
Table 4. Unit root tests for inflation time series.
H 0 : I ( 1 )   H 0 : I ( 0 )
CountryPPPP * STST *
US−2.044−1.876 3.6346.001
UK−1.851−1.669 2.1013.963
France−2.225−2.243 2.75313.571
Germany−3.357−3.363 1.1483.495
Italy−1.728−1.324 4.5498.382
Canada−2.052−1.798 4.3568.033
Japan−3.119−2.345 1.70413.913
Note: PP and PP* represent the Phillips-Perron adjusted t-statistics of the lagged dependent variable in a regression with (i) intercept and time trend and (ii) intercept only. The critical values at the 1% level are 3.975 and 3.441 . ST and ST * denote the KPSS test statistics using residuals from regressions with (i) intercept and time trend and (ii) intercept only. The critical values at the 1% level are 0.216 and 0.739 .
Table 5. Root mean squared errors (RMSEs), mean process: AutoRegressive Fractionally Integrated Moving Average (ARFIMA).
Table 5. Root mean squared errors (RMSEs), mean process: AutoRegressive Fractionally Integrated Moving Average (ARFIMA).
Forecasting Horizons1M 2M 3M 4M 5M 6M
CountriesGARCH
US0.179 0.184 0.187 0.183 0.178 0.173
UK0.110 0.115 0.120 0.123 0.124 0.126
France0.065 0.065 0.066 0.067 0.068 0.069
Germany0.078 0.077 0.076 0.069 0.068 0.066
Italy0.076 0.078 0.078 0.078 0.078 0.078
Canada0.213 0.210 0.210 0.211 0.209 0.209
Japan0.512 0.510 0.509 0.508 0.505 0.506
GJR
US0.196 0.201 0.206 0.204 0.198 0.192
UK0.117 0.119 0.121 0.125 0.128 0.130
France0.063 0.064 0.065 0.067 0.067 0.069
Germany0.082 0.081 0.079 0.073 0.072 0.071
Italy0.076 0.077 0.077 0.078 0.078 0.078
Canada0.213 0.210 0.210 0.211 0.210 0.209
Japan0.513 0.511 0.509 0.507 0.504 0.504
EGARCH
US0.169 0.171 0.171 0.169 0.162 0.156
UK0.117 0.115 0.116 0.116 0.116 0.117
France0.078 0.083 0.079 0.082 0.080 0.081
Germany0.085 0.084 0.082 0.077 0.075 0.073
Italy0.078 0.078 0.079 0.080 0.080 0.080
Canada0.215 0.210 0.210 0.210 0.211 0.211
Japan0.505 0.503 0.502 0.500 0.498 0.497
QGARCH
US0.180 0.183 0.185 0.183 0.177 0.171
UK0.103 0.104 0.106 0.107 0.108 0.110
France0.061 0.063 0.064 0.065 0.067 0.066
Germany0.083 0.082 0.080 0.072 0.071 0.069
Italy0.077 0.078 0.078 0.079 0.079 0.078
Canada0.213 0.210 0.210 0.211 0.209 0.208
Japan0.509 0.508 0.507 0.505 0.503 0.503
APGARCH
US0.203 0.209 0.216 0.212 0.206 0.201
UK0.120 0.123 0.123 0.128 0.128 0.125
France0.057 0.057 0.058 0.058 0.058 0.059
Germany0.086 0.086 0.086 0.078 0.076 0.075
Italy0.075 0.077 0.078 0.079 0.078 0.078
Canada0.214 0.212 0.213 0.214 0.212 0.211
Japan0.509 0.506 0.505 0.504 0.410 0.500
MSM
US0.153 0.157 0.156 0.154 0.151 0.150
UK0.104 0.104 0.103 0.105 0.106 0.106
France0.060 0.061 0.061 0.062 0.062 0.062
Germany0.082 0.081 0.080 0.075 0.074 0.073
Italy0.082 0.086 0.089 0.091 0.092 0.093
Canada0.219 0.215 0.215 0.216 0.214 0.214
Japan0.518 0.514 0.513 0.511 0.508 0.511
Note: RMSEs are computed for the following out-of-sample periods: November 2009–November 2015 for Canada and Japan; December 2009–December 2015 for the US, UK, France, Germany, and Italy. The lag orders in the ARFIMA specification (not displayed here) were obtained from the Bayesian Information Criterion (BIC).
Table 6. Mean absolute errors (MAEs), mean process: ARFIMA.
Table 6. Mean absolute errors (MAEs), mean process: ARFIMA.
Forecasting Horizons1M 2M 3M 4M 5M 6M
CountriesGARCH
US0.140 0.142 0.144 0.142 0.139 0.137
UK0.083 0.088 0.091 0.094 0.093 0.096
France0.059 0.060 0.061 0.062 0.063 0.064
Germany0.063 0.064 0.064 0.060 0.060 0.060
Italy0.049 0.049 0.050 0.050 0.049 0.050
Canada0.151 0.148 0.150 0.151 0.150 0.149
Japan0.199 0.199 0.200 0.199 0.198 0.199
GJR
US0.150 0.151 0.152 0.152 0.148 0.145
UK0.091 0.093 0.092 0.093 0.096 0.098
France0.056 0.058 0.059 0.061 0.061 0.063
Germany0.066 0.066 0.066 0.063 0.062 0.061
Italy0.048 0.047 0.049 0.049 0.048 0.048
Canada0.151 0.150 0.150 0.152 0.151 0.149
Japan0.214 0.215 0.214 0.214 0.213 0.215
EGARCH
US0.133 0.133 0.132 0.132 0.128 0.125
UK0.092 0.091 0.089 0.088 0.088 0.089
France0.072 0.078 0.073 0.077 0.074 0.076
Germany0.070 0.070 0.069 0.066 0.065 0.064
Italy0.050 0.050 0.051 0.051 0.051 0.050
Canada0.144 0.137 0.139 0.140 0.141 0.142
Japan0.191 0.192 0.190 0.189 0.190 0.192
QGARCH
US0.140 0.140 0.140 0.140 0.136 0.133
UK0.077 0.078 0.079 0.079 0.080 0.080
France0.056 0.058 0.060 0.060 0.062 0.061
Germany0.067 0.067 0.066 0.063 0.061 0.060
Italy0.049 0.049 0.050 0.051 0.050 0.050
Canada0.151 0.149 0.149 0.151 0.150 0.148
Japan0.204 0.205 0.207 0.207 0.207 0.210
APGARCH
US0.154 0.155 0.158 0.156 0.152 0.150
UK0.089 0.090 0.087 0.090 0.091 0.090
France0.052 0.052 0.053 0.054 0.054 0.055
Germany0.071 0.072 0.072 0.068 0.067 0.067
Italy0.048 0.048 0.050 0.050 0.048 0.049
Canada0.155 0.154 0.155 0.157 0.154 0.154
Japan0.186 0.188 0.186 0.187 0.188 0.190
MSM
US0.123 0.126 0.127 0.126 0.123 0.122
UK0.080 0.082 0.081 0.082 0.084 0.086
France0.051 0.052 0.053 0.054 0.054 0.055
Germany0.067 0.068 0.068 0.066 0.065 0.064
Italy0.063 0.066 0.071 0.073 0.074 0.077
Canada0.162 0.159 0.161 0.163 0.161 0.161
Japan0.216 0.219 0.227 0.231 0.234 0.240
Note: MAEs are computed for the following out-of-sample periods: November 2009–November 2015 for Canada and Japan; December 2009–December 2015 for the US, UK, France, Germany, and Italy. The lag orders in the ARFIMA specification (not displayed here) were obtained from the Bayesian Information Criterion (BIC).
Table 7. Root mean squared errors (RMSEs), mean process: Smooth Transition AutoRegressive Fractionally Integrated (STARFI).
Table 7. Root mean squared errors (RMSEs), mean process: Smooth Transition AutoRegressive Fractionally Integrated (STARFI).
Forecasting Horizons1M 2M 3M 4M 5M 6M
CountriesGARCH
US0.176 0.181 0.184 0.179 0.173 0.169
UK0.108 0.113 0.112 0.114 0.118 0.120
France0.067 0.067 0.067 0.068 0.068 0.069
Germany0.080 0.080 0.079 0.072 0.071 0.070
Italy0.075 0.076 0.076 0.076 0.076 0.076
Canada0.206 0.203 0.203 0.204 0.203 0.203
Japan0.516 0.514 0.513 0.511 0.509 0.509
GJR
US0.192 0.197 0.203 0.202 0.195 0.190
UK0.121 0.115 0.117 0.122 0.125 0.129
France0.066 0.066 0.067 0.068 0.068 0.069
Germany0.082 0.083 0.081 0.076 0.075 0.074
Italy0.074 0.075 0.075 0.075 0.075 0.075
Canada0.205 0.203 0.202 0.205 0.203 0.204
Japan0.517 0.515 0.513 0.511 0.507 0.508
EGARCH
US0.169 0.171 0.172 0.170 0.163 0.157
UK0.117 0.117 0.118 0.119 0.119 0.122
France0.078 0.084 0.079 0.081 0.080 0.081
Germany0.086 0.085 0.083 0.077 0.075 0.078
Italy0.077 0.077 0.077 0.078 0.078 0.078
Canada0.209 0.203 0.203 0.204 0.204 0.205
Japan0.508 0.507 0.506 0.503 0.501 0.503
QGARCH
US0.176 0.179 0.181 0.180 0.173 0.167
UK0.105 0.105 0.107 0.109 0.110 0.112
France0.061 0.060 0.059 0.060 0.060 0.060
Germany0.085 0.085 0.082 0.076 0.075 0.074
Italy0.076 0.076 0.077 0.077 0.077 0.077
Canada0.206 0.203 0.202 0.204 0.202 0.202
Japan0.513 0.512 0.510 0.508 0.506 0.508
APGARCH
US0.193 0.202 0.210 0.204 0.198 0.194
UK0.111 0.118 0.121 0.126 0.130 0.128
France0.060 0.060 0.061 0.061 0.062 0.062
Germany0.086 0.086 0.092 0.084 0.082 0.082
Italy0.073 0.076 0.076 0.076 0.075 0.074
Canada0.205 0.204 0.206 0.206 0.204 0.204
Japan0.512 0.509 0.507 0.505 0.503 0.504
MSM
US0.155 0.158 0.158 0.155 0.152 0.151
UK0.103 0.103 0.102 0.104 0.104 0.105
France0.061 0.061 0.061 0.062 0.062 0.062
Germany0.083 0.084 0.083 0.078 0.077 0.076
Italy0.082 0.086 0.088 0.089 0.089 0.091
Canada0.213 0.208 0.208 0.210 0.208 0.208
Japan0.522 0.518 0.517 0.515 0.511 0.515
Note: RMSEs are computed for the following out-of-sample periods: November 2009–November 2015 for Canada and Japan; December 2009–December 2015 for the US, UK, France, Germany, and Italy. The lag orders in the STARFI specification (not displayed here) were obtained from the Bayesian Information Criterion (BIC).
Table 8. Mean absolute errors (MAEs), mean process: STARFI.
Table 8. Mean absolute errors (MAEs), mean process: STARFI.
Forecasting Horizons1M 2M 3M 4M 5M 6M
CountriesGARCH
US0.136 0.138 0.138 0.136 0.133 0.131
UK0.081 0.087 0.086 0.088 0.091 0.093
France0.062 0.062 0.062 0.063 0.063 0.064
Germany0.064 0.064 0.064 0.061 0.061 0.060
Italy0.049 0.049 0.050 0.049 0.048 0.050
Canada0.152 0.149 0.150 0.151 0.150 0.149
Japan0.198 0.197 0.200 0.199 0.198 0.200
GJR
US0.142 0.144 0.146 0.145 0.142 0.141
UK0.095 0.091 0.090 0.094 0.097 0.097
France0.060 0.060 0.061 0.062 0.063 0.064
Germany0.065 0.066 0.066 0.063 0.062 0.061
Italy0.046 0.047 0.047 0.047 0.047 0.047
Canada0.150 0.149 0.149 0.152 0.152 0.151
Japan0.216 0.216 0.217 0.217 0.216 0.218
EGARCH
US0.128 0.129 0.129 0.128 0.123 0.121
UK0.093 0.093 0.092 0.091 0.092 0.093
France0.072 0.078 0.074 0.075 0.075 0.075
Germany0.071 0.070 0.068 0.067 0.066 0.067
Italy0.050 0.050 0.051 0.052 0.051 0.052
Canada0.147 0.140 0.142 0.143 0.144 0.145
Japan0.192 0.193 0.194 0.194 0.195 0.198
QGARCH
US0.132 0.132 0.134 0.134 0.130 0.127
UK0.081 0.080 0.083 0.083 0.084 0.084
France0.055 0.055 0.055 0.056 0.056 0.055
Germany0.067 0.068 0.066 0.064 0.063 0.061
Italy0.049 0.049 0.050 0.050 0.050 0.051
Canada0.151 0.149 0.149 0.152 0.150 0.149
Japan0.204 0.206 0.208 0.209 0.209 0.213
APGARCH
US0.144 0.149 0.151 0.146 0.143 0.143
UK0.085 0.091 0.088 0.091 0.094 0.093
France0.055 0.055 0.056 0.057 0.057 0.057
Germany0.072 0.073 0.075 0.070 0.070 0.069
Italy0.046 0.048 0.049 0.047 0.046 0.047
Canada0.153 0.151 0.154 0.156 0.152 0.152
Japan0.185 0.187 0.186 0.188 0.189 0.192
MSM
US0.123 0.126 0.126 0.124 0.122 0.122
UK0.081 0.082 0.081 0.082 0.084 0.085
France0.052 0.052 0.053 0.054 0.054 0.055
Germany0.067 0.068 0.068 0.066 0.065 0.064
Italy0.062 0.066 0.070 0.072 0.073 0.076
Canada0.163 0.159 0.161 0.162 0.159 0.160
Japan0.214 0.219 0.226 0.230 0.233 0.239
Note: MAEs are computed for the following out-of-sample periods: November 2009–November 2015 for Canada and Japan; December 2009–December 2015 for the US, UK, France, Germany, and Italy. The lag orders in the STARFI specification (not displayed here) were obtained from the Bayesian Information Criterion (BIC).
Table 9. Superior Predictive Ability (SPA) test, squared error loss, mean process: ARFIMA.
Table 9. Superior Predictive Ability (SPA) test, squared error loss, mean process: ARFIMA.
Forecasting Horizons1M 2M 3M 4M 5M 6M
Benchmark ModelsUS
GARCH0.088 0.066 0.047 0.070 0.049 0.045
GJR0.034 0.032 0.026 0.024 0.021 0.039
EGARCH0.055 0.088 0.081 0.072 0.143 0.284
QGARCH0.041 0.047 0.041 0.031 0.031 0.032
APGARCH0.038 0.038 0.031 0.031 0.033 0.043
MSM1.000 1.000 1.000 1.000 0.857 0.716
UK
GARCH0.110 0.255 0.094 0.031 0.164 0.109
GJR0.017 0.028 0.036 0.034 0.033 0.031
EGARCH0.046 0.122 0.086 0.134 0.161 0.100
QGARCH0.621 0.748 0.259 0.328 0.287 0.208
APGARCH0.070 0.051 0.058 0.061 0.057 0.109
MSM0.379 0.735 0.741 0.672 0.713 0.792
France
GARCH0.006 0.002 0.001 0.001 0.000 0.000
GJR0.053 0.026 0.020 0.010 0.002 0.001
EGARCH0.000 0.000 0.000 0.000 0.000 0.000
QGARCH0.080 0.006 0.004 0.001 0.000 0.000
APGARCH0.780 0.856 0.821 0.859 0.839 1.000
MSM0.274 0.172 0.224 0.179 0.161 0.145
Germany
GARCH1.000 1.000 1.000 1.000 1.000 1.000
GJR0.020 0.049 0.076 0.016 0.034 0.037
EGARCH0.009 0.018 0.022 0.005 0.008 0.007
QGARCH0.012 0.021 0.064 0.037 0.075 0.091
APGARCH0.013 0.006 0.002 0.005 0.005 0.004
MSM0.055 0.049 0.101 0.010 0.011 0.016
Italy
GARCH0.240 0.367 0.232 0.474 0.553 0.476
GJR0.245 0.717 0.898 0.926 0.599 0.559
EGARCH0.092 0.202 0.144 0.151 0.061 0.109
QGARCH0.099 0.180 0.141 0.160 0.053 0.134
APGARCH0.781 0.770 0.312 0.387 0.765 0.785
MSM0.018 0.012 0.003 0.001 0.000 0.000
Canada
GARCH0.953 0.869 0.582 0.840 0.803 0.791
GJR0.727 0.521 0.630 0.450 0.431 0.416
EGARCH0.383 0.558 0.549 0.598 0.437 0.347
QGARCH0.802 0.838 0.951 0.870 0.993 0.998
APGARCH0.468 0.116 0.048 0.139 0.154 0.125
MSM0.004 0.059 0.025 0.041 0.033 0.005
Japan
GARCH0.210 0.267 0.217 0.238 0.114 0.172
GJR0.130 0.188 0.267 0.268 0.379 0.148
EGARCH0.749 0.733 0.993 0.993 0.853 0.764
QGARCH0.047 0.037 0.031 0.017 0.018 0.015
APGARCH0.373 0.492 0.451 0.400 0.541 0.350
MSM0.063 0.060 0.052 0.014 0.029 0.013
Note: The displayed numbers are the p-values of the SPA test of Hansen (2005) using the squared error loss. We test the null hypothesis that the benchmark model is not outperformed by any of the other candidate models. The p-values are obtained for the following out-of-sample periods: November 2009–November 2015 for Canada and Japan; December 2009–December 2015 for the US, UK, France, Germany, and Italy. The inflation-rate mean process is ARFIMA.
Table 10. Superior Predictive Ability (SPA) test, absolute error loss, mean process: ARFIMA.
Table 10. Superior Predictive Ability (SPA) test, absolute error loss, mean process: ARFIMA.
Forecasting Horizons1M 2M 3M 4M 5M 6M
Benchmark ModelsUS
GARCH0.075 0.032 0.014 0.026 0.025 0.018
GJR0.024 0.014 0.0150 0.017 0.017 0.028
EGARCH0.089 0.156 0.180 0.168 0.236 0.346
QGARCH0.053 0.076 0.036 0.052 0.078 0.051
APGARCH0.023 0.020 0.009 0.016 0.023 0.030
MSM1.000 0.844 0.820 0.832 0.764 0.678
UK
GARCH0.136 0.130 0.057 0.016 0.098 0.058
GJR0.007 0.004 0.014 0.012 0.003 0.002
EGARCH0.0122 0.020 0.051 0.073 0.050 0.036
QGARCH0.814 1.000 0.742 0.802 1.000 1.000
APGARCH0.058 0.043 0.159 0.058 0.049 0.091
MSM0.258 0.145 0.375 0.265 0.175 0.155
France
GARCH0.001 0.000 0.000 0.000 0.000 0.000
GJR0.008 0.003 0.001 0.000 0.000 0.000
EGARCH0.000 0.000 0.000 0.000 0.000 0.000
QGARCH0.098 0.004 0.001 0.002 0.000 0.001
APGARCH0.359 0.537 0.470 0.544 0.542 0.578
MSM0.641 0.463 0.530 0.456 0.458 0.422
Germany
GARCH1.000 1.000 0.847 1.000 0.756 0.691
GJR0.161 0.154 0.204 0.123 0.188 0.229
EGARCH0.006 0.007 0.014 0.004 0.004 0.003
QGARCH0.086 0.105 0.250 0.220 0.393 0.484
APGARCH0.033 0.018 0.006 0.012 0.008 0.001
MSM0.009 0.010 0.020 0.010 0.016 0.031
Italy
GARCH0.272 0.204 0.249 0.402 0.271 0.200
GJR0.611 1.000 1.000 0.913 0.822 0.715
EGARCH0.062 0.065 0.063 0.082 0.052 0.133
QGARCH0.045 0.024 0.030 0.043 0.012 0.060
APGARCH0.630 0.281 0.063 0.374 0.559 0.462
MSM0.000 0.000 0.000 0.000 0.000 0.000
Canada
GARCH0.365 0.084 0.066 0.094 0.139 0.261
GJR0.268 0.078 0.087 0.096 0.112 0.200
EGARCH0.880 1.000 1.000 1.000 1.000 1.000
QGARCH0.388 0.091 0.092 0.102 0.155 0.337
APGARCH0.050 0.012 0.010 0.006 0.036 0.009
MSM0.000 0.000 0.000 0.000 0.000 0.000
Japan
GARCH0.121 0.162 0.092 0.140 0.179 0.175
GJR0.007 0.007 0.003 0.003 0.004 0.003
EGARCH0.433 0.469 0.271 0.373 0.377 0.542
QGARCH0.045 0.039 0.013 0.007 0.010 0.005
APGARCH0.731 0.704 0.729 0.627 0.623 0.651
MSM0.006 0.001 0.000 0.000 0.000 0.000
Note: The displayed numbers are the p-values of the SPA test of Hansen (2005) using the absolute error loss. We test the null hypothesis that the benchmark model is not outperformed by any of the other candidate models. The p-values are obtained for the following out-of-sample periods: November 2009–November 2015 for Canada and Japan; December 2009–December 2015 for the US, UK, France, Germany, and Italy. The inflation-rate mean process is ARFIMA.
Table 11. Equal Predictive Ability (EPA) test, squared error loss, mean process: ARFIMA.
Table 11. Equal Predictive Ability (EPA) test, squared error loss, mean process: ARFIMA.
Forecasting Horizons
Model 1Model 21M2M3M4M5M6M
US
GARCHMSM0.0260.0870.0870.0680.1360.192
GJR 0.0150.0670.0740.0550.1060.155
EGARCH 0.0450.1400.1420.0660.1690.308
QGARCH 0.0220.0840.0860.0500.1180.190
APGARCH 0.0120.0640.0660.0610.1130.154
UK
GARCHMSM0.0570.1630.0340.0210.1200.067
GJR 0.0180.0590.0920.1350.1480.162
EGARCH 0.0540.0890.0790.1580.1940.167
QGARCH 0.6330.5490.3170.3970.3820.334
APGARCH 0.0370.0730.1070.1440.1680.204
France
GARCHMSM0.0110.0200.0100.0230.0020.002
GJR 0.0090.0200.0120.0040.0000.000
EGARCH 0.0000.0000.0000.0000.0000.000
QGARCH 0.3950.3490.2780.3220.1920.251
APGARCH 0.7770.8000.7280.7450.7180.706
Germany
GARCHMSM0.9650.9690.9300.9580.9400.919
GJR 0.4640.6330.6700.8060.8250.791
EGARCH 0.0400.1360.2250.3600.4030.402
QGARCH 0.2840.3820.5580.7600.7760.751
APGARCH 0.1270.1260.0830.3430.3800.363
Italy
GARCHMSM0.9470.9680.9790.9850.9870.9871
GJR 0.8900.9660.9800.9820.9810.981
EGARCH 0.7830.9290.9560.9630.9590.966
QGARCH 0.8460.9500.9710.9750.9700.976
APGARCH 0.9690.9670.9680.9710.9810.984
Canada
GARCHMSM0.9950.9720.9770.9780.9740.996
GJR 0.9970.9510.9590.9270.9260.972
EGARCH 0.7380.7470.7910.7970.6890.677
QGARCH 0.9950.9660.9780.9670.9750.996
APGARCH 0.9910.8140.7430.8530.8460.981
Japan
GARCHMSM0.9370.8310.7720.7360.6570.809
GJR 0.7290.6580.6750.7010.6910.941
EGARCH 0.9880.9850.9680.9760.9170.996
QGARCH 0.9660.9460.8930.9220.8230.973
APGARCH 0.9960.9870.9310.8790.8770.986
Note: The displayed number are p-values of the EPA test of Diebold and Mariano (1995) using the squared error loss. We test the null hypothesis that the forecasts at horizon h of Model 1 are equal to those of Model 2 against the one-sided alternative that forecasts of Model 1 are inferior to those of Model 2. The p-values are obtained for the following out-of-sample periods: November 2009–November 2015 for Canada and Japan; December 2009–December 2015 for the US, UK, France, Germany, and Italy. The inflation-rate mean process is ARFIMA.
Table 12. Equal Predictive Ability (EPA) test, absolute error loss, mean process: ARFIMA.
Table 12. Equal Predictive Ability (EPA) test, absolute error loss, mean process: ARFIMA.
Forecasting Horizons
Model 1Model 21M2M3M4M5M6M
US
GARCHMSM0.0150.0850.0990.1280.1850.225
GJR 0.0100.0640.0890.1080.1580.206
EGARCH 0.0570.2160.2560.2540.3300.416
QGARCH 0.0210.1260.1590.1730.2370.293
APGARCH 0.0070.0560.0630.0930.1510.189
UK
GARCHMSM0.2460.1490.0540.0450.1880.102
GJR 0.0240.0860.1420.1910.1990.200
EGARCH 0.0360.1270.1660.2710.3510.371
QGARCH 0.8340.8350.6960.7240.7750.764
APGARCH 0.0860.1700.3010.2660.3280.384
France
GARCHMSM0.0000.0000.0000.0010.0000.000
GJR 0.0000.0000.0000.0000.0000.000
EGARCH 0.0000.0000.0000.0000.0000.000
QGARCH 0.0460.0490.0590.1220.0600.099
APGARCH 0.3240.5150.4640.5150.5140.532
Germany
GARCHMSM0.9990.9930.9630.9570.9300.891
GJR 0.8330.9170.9170.9350.9560.939
EGARCH 0.0550.1670.3150.4350.4800.467
QGARCH 0.4940.6410.8040.8440.8550.826
APGARCH 0.0860.1370.1560.2890.3340.307
Italy
GARCHMSM1.0001.0001.0001.0001.0001.000
GJR 1.0001.0001.0001.0001.0001.000
EGARCH 1.0001.0001.0001.0001.0001.000
QGARCH 1.0001.0001.0001.0001.0001.000
APGARCH 1.0001.0001.0001.0001.0001.000
Canada
GARCHMSM1.0001.0001.0001.0001.0001.000
GJR 1.0000.9990.9990.9970.9910.997
EGARCH 0.9950.9970.9990.9990.9970.991
QGARCH 1.0001.0001.0001.0001.0001.000
APGARCH 0.9960.9580.9760.9860.9980.999
Japan
GARCHMSM1.0001.0001.0001.0001.0001.000
GJR 0.5630.6340.7770.8360.8600.899
EGARCH 0.9970.9910.9940.9960.9950.999
QGARCH 0.9830.9690.9820.9920.9930.997
APGARCH 1.0001.0001.0001.0001.0001.000
Note: The displayed number are p-values of the EPA test of Diebold and Mariano (1995) using the absolute error loss. We test the null hypothesis that the forecasts at horizon h of Model 1 are equal to those of Model 2 against the one-sided alternative that forecasts of Model 1 are inferior to those of Model 2. The p-values are obtained for the following out-of-sample periods: November 2009–November 2015 for Canada and Japan; December 2009–December 2015 for the US, UK, France, Germany, and Italy. The inflation-rate mean process is ARFIMA.
Table 13. Superior Predictive Ability (SPA) test, squared error loss, mean process: STARFI.
Table 13. Superior Predictive Ability (SPA) test, squared error loss, mean process: STARFI.
Forecasting Horizons1M 2M 3M 4M 5M 6M
Benchmark ModelsUS
GARCH0.108 0.068 0.051 0.099 0.058 0.038
GJR0.041 0.036 0.027 0.029 0.024 0.041
EGARCH0.081 0.111 0.094 0.081 0.142 0.292
QGARCH0.049 0.053 0.037 0.027 0.026 0.052
APGARCH0.058 0.047 0.035 0.052 0.045 0.055
MSM1.000 1.000 0.906 1.000 0.858 0.737
UK
GARCH0.197 0.099 0.088 0.044 0.080 0.063
GJR0.004 0.045 0.077 0.048 0.040 0.034
EGARCH0.026 0.046 0.066 0.080 0.092 0.062
QGARCH0.542 0.280 0.101 0.145 0.142 0.120
APGARCH0.217 0.080 0.050 0.068 0.055 0.101
MSM0.715 0.720 1.000 1.000 1.000 1.000
France
GARCH0.001 0.001 0.000 0.000 0.000 0.000
GJR0.029 0.012 0.006 0.003 0.001 0.000
EGARCH0.000 0.000 0.000 0.000 0.000 0.000
QGARCH0.623 0.817 0.719 0.776 0.758 0.833
APGARCH0.798 0.685 0.058 0.145 0.028 0.022
MSM0.422 0.360 0.318 0.254 0.283 0.195
Germany
GARCH1.000 0.901 1.000 1.000 1.000 1.000
GJR0.094 0.050 0.090 0.034 0.049 0.034
EGARCH0.023 0.056 0.095 0.040 0.090 0.005
QGARCH0.036 0.030 0.068 0.038 0.044 0.026
APGARCH0.150 0.171 0.007 0.007 0.001 0.004
MSM0.092 0.036 0.030 0.008 0.007 0.010
Italy
GARCH0.052 0.108 0.103 0.318 0.341 0.188
GJR0.263 1.000 1.000 0.942 0.542 0.397
EGARCH0.004 0.021 0.013 0.005 0.006 0.022
QGARCH0.044 0.023 0.019 0.013 0.004 0.019
APGARCH0.801 0.271 0.177 0.368 0.849 0.775
MSM0.007 0.006 0.003 0.000 0.001 0.000
Canada
GARCH0.638 0.801 0.350 0.914 0.875 0.722
GJR0.743 0.796 0.653 0.657 0.517 0.352
EGARCH0.356 0.612 0.510 0.622 0.447 0.381
QGARCH0.824 0.978 0.918 0.939 0.988 0.999
APGARCH0.795 0.453 0.108 0.529 0.419 0.512
MSM0.006 0.077 0.033 0.076 0.028 0.002
Japan
GARCH0.217 0.266 0.219 0.224 0.1456 0.198
GJR0.109 0.158 0.200 0.199 0.327 0.225
EGARCH0.712 0.680 0.985 0.987 0.907 0.995
QGARCH0.054 0.031 0.024 0.010 0.008 0.018
APGARCH0.430 0.561 0.543 0.514 0.459 0.388
MSM0.052 0.035 0.059 0.011 0.010 0.030
Note: The displayed numbers are the p-values of the SPA test of Hansen (2005) using the squared error loss. We test the null hypothesis that the benchmark model is not outperformed by any of the other candidate models. The p-values are obtained for the following out-of-sample periods: November 2009–November 2015 for Canada and Japan; December 2009–December 2015 for the US, UK, France, Germany, and Italy. The inflation-rate mean process is STARFI.
Table 14. Superior Predictive Ability (SPA) test, absolute error loss, mean process: STARFI.
Table 14. Superior Predictive Ability (SPA) test, absolute error loss, mean process: STARFI.
Forecasting Horizons1M 2M 3M 4M 5M 6M
ModelsUS
GARCH0.000 0.000 0.000 0.000 0.000 0.000
GJR0.000 0.000 0.000 0.000 0.000 0.000
EGARCH0.000 0.000 0.000 0.000 0.000 0.000
QGARCH0.000 0.000 0.000 0.000 0.000 0.000
APGARCH0.000 0.000 0.000 0.000 0.000 0.000
MSM1.000 1.000 1.000 1.000 1.000 1.000
UK
GARCH0.000 0.000 0.000 0.000 0.000 0.000
GJR0.000 0.000 0.000 0.000 0.000 0.000
EGARCH0.000 0.000 0.000 0.000 0.000 0.000
QGARCH0.000 0.000 0.000 0.000 0.000 0.000
APGARCH0.000 0.000 0.000 0.000 0.000 0.000
MSM1.000 1.000 1.000 1.000 1.000 1.000
France
GARCH0.000 0.000 0.000 0.000 0.000 0.000
GJR0.000 0.000 0.000 0.000 0.000 0.000
EGARCH0.000 0.000 0.000 0.000 0.000 0.000
QGARCH0.000 0.000 0.000 0.000 0.000 0.000
APGARCH0.000 0.000 0.000 0.000 0.000 0.000
MSM1.000 1.000 1.000 1.000 1.000 1.000
Germany
GARCH0.000 0.000 0.000 0.000 0.000 0.000
GJR0.000 0.000 0.000 0.000 0.000 0.000
EGARCH0.000 0.000 0.000 0.000 0.000 0.000
QGARCH0.000 0.000 0.000 0.000 0.000 0.000
APGARCH0.000 0.000 0.000 0.000 0.000 0.000
MSM1.000 1.000 1.000 1.000 1.000 1.000
Italy
GARCH0.000 0.000 0.000 0.000 0.000 0.000
GJR0.000 0.000 0.000 0.000 0.000 0.000
EGARCH0.000 0.000 0.000 0.000 0.000 0.000
QGARCH0.000 0.000 0.000 0.000 0.000 0.000
APGARCH0.000 0.000 0.000 0.000 0.000 0.000
MSM1.000 1.000 1.000 1.000 1.000 1.000
Canada
GARCH0.000 0.000 0.000 0.000 0.000 0.000
GJR0.000 0.000 0.000 0.000 0.000 0.000
EGARCH0.000 0.000 0.000 0.000 0.000 0.000
QGARCH0.000 0.000 0.000 0.000 0.000 0.000
APGARCH0.000 0.000 0.000 0.000 0.000 0.000
MSM1.000 1.000 1.000 1.000 1.000 1.000
Japan
GARCH0.115 0.174 0.101 0.148 0.179 0.217
GJR0.006 0.007 0.004 0.004 0.006 0.008
EGARCH0.395 0.437 0.390 0.443 0.450 0.441
QGARCH0.041 0.046 0.022 0.016 0.020 0.014
APGARCH0.931 0.935 0.936 0.935 0.934 0.931
MSM0.229 0.243 0.239 0.244 0.252 0.257
Note: The displayed numbers are the p-values of the SPA test of Hansen (2005) using the absolute error loss. We test the null hypothesis that the benchmark model is not outperformed by any of the other candidate models. The p-values are obtained for the following out-of-sample periods: November 2009–November 2015 for Canada and Japan; December 2009–December 2015 for the US, UK, France, Germany, and Italy. The inflation-rate mean process is STARFI.
Table 15. Equal Predictive Ability (EPA) test, squared error loss, mean process: STARFI.
Table 15. Equal Predictive Ability (EPA) test, squared error loss, mean process: STARFI.
Forecasting Horizons
Model 1Model 21M2M3M4M5M6M
US
GARCHMSM0.0300.0940.0960.0810.1490.210
GJR 0.0230.0810.0860.0700.1190.163
EGARCH 0.0680.1670.1580.0960.1860.319
QGARCH 0.0400.1120.1050.0640.1340.215
APGARCH 0.0240.0810.0760.0780.1340.173
UK
GARCHMSM0.0930.0560.0450.0690.0730.086
GJR 0.0010.0720.1180.1360.1520.161
EGARCH 0.0150.0570.0880.1470.1830.170
QGARCH 0.2770.3060.1530.2330.2640.262
APGARCH 0.0990.0770.0810.1300.1550.187
France
GARCHMSM0.0030.0060.0080.0180.0040.003
GJR 0.0050.0050.0030.0040.0000.000
EGARCH 0.0000.0000.0000.0000.0000.000
QGARCH 0.5530.6100.6340.6740.6500.690
APGARCH 0.5870.6000.4930.5870.4930.554
Germany
GARCHMSM0.9700.9810.9630.9680.9530.934
GJR 0.6510.7000.7710.8660.8640.823
EGARCH 0.2120.3710.5000.5730.6740.389
QGARCH 0.2450.3570.5770.7270.7400.706
APGARCH 0.2650.3010.0380.0980.1150.044
Italy
GARCHMSM0.9760.9830.9840.9890.9870.988
GJR 0.9600.9810.9810.9800.9760.980
EGARCH 0.8990.9640.9650.9570.9550.968
QGARCH 0.9200.9670.9650.9600.9530.970
APGARCH 0.9880.9730.9660.9730.9770.985
Canada
GARCHMSM0.9940.9660.9710.9700.9690.995
GJR 0.9940.9350.9510.8780.8760.902
EGARCH 0.7750.7830.7790.7920.6930.714
QGARCH 0.9950.9580.9740.9470.9740.996
APGARCH 0.9930.8860.7030.9070.8550.983
Japan
GARCHMSM0.9310.8330.7660.7410.6770.813
GJR 0.6960.6370.6490.6840.7160.935
EGARCH 0.9800.9860.9720.9890.9590.988
QGARCH 0.9540.9490.9050.9450.8810.944
APGARCH 0.9970.9940.9640.9470.8970.969
Note: The displayed number are p-values of the EPA test of Diebold and Mariano (1995) using the squared error loss. We test the null hypothesis that the forecasts at horizon h of Model 1 are equal to those of Model 2 against the one-sided alternative that forecasts of Model 1 are inferior to those of Model 2. The p-values are obtained for the following out-of-sample periods: November 2009–November 2015 for Canada and Japan; December 2009–December 2015 for the US, UK, France, Germany, and Italy. The inflation-rate mean process is STARFI.
Table 16. Equal Predictive Ability (EPA) test, absolute error loss, mean-process: STARFI.
Table 16. Equal Predictive Ability (EPA) test, absolute error loss, mean-process: STARFI.
Forecasting Horizons
Model 1Model 21M2M3M4M5M6M
US
GARCHMSM0.0000.0000.0000.0000.0000.000
GJR 0.0000.0000.0000.0000.0000.000
EGARCH 0.0000.0000.0000.0000.0000.000
QGARCH 0.0000.0000.0000.0000.0000.000
APGARCH 0.0000.0000.0000.0000.0000.001
UK
GARCHMSM0.0000.0000.0000.0000.0000.000
GJR 0.0000.0000.0000.0000.0000.000
EGARCH 0.0000.0000.0000.0000.0000.000
QGARCH 0.0000.0000.0000.0000.0000.000
APGARCH 0.0000.0000.0000.0000.0000.000
France
GARCHMSM0.0000.0000.0000.0000.0000.000
GJR 0.0000.0000.0000.0000.0000.000
EGARCH 0.0000.0000.0000.0000.0000.000
QGARCH 0.0000.0000.0000.0000.0000.000
APGARCH 0.0000.0000.0000.0000.0000.000
Germany
GARCHMSM0.0000.0000.0000.0000.0000.000
GJR 0.0000.0000.0000.0000.0000.000
EGARCH 0.0000.0000.0000.0000.0000.000
QGARCH 0.0000.0000.0000.0000.0000.000
APGARCH 0.0000.0000.0000.0000.0000.000
Italy
GARCHMSM0.0000.0000.0000.0000.0000.000
GJR 0.0000.0000.0000.0000.0000.000
EGARCH 0.0000.0000.0000.0000.0000.000
QGARCH 0.0000.0000.0000.0000.0000.000
APGARCH 0.0000.0000.0000.0000.0000.000
Canada
GARCHMSM0.0000.0000.0000.0000.0000.000
GJR 0.0000.0000.0000.0000.0000.000
EGARCH 0.0000.0000.0000.0000.0000.000
QGARCH 0.0000.0000.0000.0000.0000.000
APGARCH 0.0000.0000.0000.0000.0000.000
Japan
GARCHMSM0.7380.7380.7360.7420.7460.759
GJR 0.6840.6790.6800.6840.6920.711
EGARCH 0.7540.7520.7610.7690.7790.795
QGARCH 0.7190.7140.7110.7130.7170.726
APGARCH 0.7750.7690.7790.7810.7870.794
Note: The displayed number are p-values of the EPA test of Diebold and Mariano (1995) using the absolute error loss. We test the null hypothesis that the forecasts at horizon h of Model 1 are equal to those of Model 2 against the one-sided alternative that forecasts of Model 1 are inferior to those of Model 2. The p-values are obtained for the following out-of-sample periods: November 2009–November 2015 for Canada and Japan; December 2009–December 2015 for the US, UK, France, Germany, and Italy. The inflation-rate mean process is STARFI.

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