Signal smoothing for score-driven models: a linear approach

Abstract In the present article, a linear approach of signal smoothing for nonlinear score-driven models is suggested, by using results from the literature on minimum mean squared error signals. Score-driven location, trend, and seasonality models with constant and score-driven scale parameters are used, for which the parameters are estimated by using the maximum likelihood method. The smoothing procedure is computationally fast, and it uses closed-form formulas for smoothed signals. Applications for monthly data of the seasonally adjusted and the not seasonally adjusted the United States inflation rate variables for the period of 1948–2020 are presented.


Introduction
In this article, we apply results from the literature on signal extraction, and we present a new approach of minimum mean squared error (MSE) signal extraction for score-driven state space models of location, trend, seasonality, and scale for macroeconomic time series variables.Signal extraction from economic variables is important for statistical analyses that support the decisions of economic agents (Erceg and Levin 2003), or policymakers (Ghysels 1987;Bryan and Pike 1991;Cristadoro et al. 2005).From the literature, some relevant works are Bell (1984), Bell and Hillmer (1988), McElroy (2008), and McElroy and Maravall (2014), in which minimum MSE signal smoothing formulas are presented for signal plus noise models.Those works use a variety of models, for which signal or noise or both is non-stationary, and the error terms in the signal and noise equations are heteroskedastic, non-Gaussian, and correlated.As an empirical contribution we apply the results of McElroy and Maravall (2014) to signal smoothing for score-driven models using the United States (US) inflation rate data, and as a theoretical contribution we show that the assumptions of those results are satisfied for the score-driven models.
The class of score-driven model is the observation-driven (Cox 1981) state space models (Harvey 1989;Durbin and Koopman 2012), introduced in the works of Creal, Koopman, and Lucas (2008, 2011, 2013), Harvey and Chakravarty (2008), and Harvey (2013).The filters of score-driven models are updated by using the scaled conditional score of the log-likelihood (LL) with respect to a time-varying parameter (hereinafter, the updating term of score-driven filters is named score function), and score-driven models are estimated by using the maximum likelihood (ML) method (e.g.Harvey 2013;Creal, Koopman, and Lucas 2013;Blasques et al. 2018;Blasques, van Brummelen, et al. 2021).Score-driven models are robust to outliers and missing observations, and the filters that update score-driven models are generalizations of the updating mechanisms of classical time series models (Harvey 2013).We also refer to the information-theoretically optimal updating mechanisms of score-driven filters (Blasques, Koopman, and Lucas 2015).The aforementioned statistical advantages of score-driven models may motivate their use for macroeconomic time series (e.g.Blazsek, Escribano, and Licht 2021a, 2021b, 2021c), the need for straightforward smoothing procedures for score-driven models.
In the work of Harvey (2013), the filtered signal Eðs t jy 1 , :::, y tÀ1 Þ for t ¼ 1, :::, T and the smoothed signal Eðs t jy 1 , :::, y T Þ for t ¼ 1, :::, T are estimated for score-driven models, and an application of a state space smoothing recursions procedure (Koopman and Harvey 2003) to first-order score-driven location models is presented.According to Harvey (2013, p. 87), the generalization of the state space smoothing recursions procedure applied to score-driven models with several lags is not straightforward.The smoothing procedure suggested in the present article can be applied to score-driven models with several lags in a straightforward way.Another article from the literature that is relevant to our article is the work of Buccheri et al. (2019), in which a state space smoothing recursions procedure is suggested for the score-driven location, scale, or duration model.Those authors note that, as score-driven models are observation-driven models, the time-varying parameters are one-step-ahead predictable, hence, the estimate of the filtered signal is improved in score-driven models, by using information from contemporaneous and future observations in the smoothed signal.This is the main motivation for the development of the smoothing method for score-driven models in our article.
In the signal extraction procedure of this article, the ML estimates of parameters of the scoredriven models are obtained in the first step by using numerical maximization of the LL function, and the minimum MSE signals are estimated in the second step by using a closed-form signal smoothing formula.We suggest a smoothing procedure for score-driven models, which are more complex than the score-driven models of Harvey (2013, p. 87) and Buccheri et al. (2019).In the score-driven models of the present article, score-driven location, trend, and seasonality filters with constant and score-driven scale parameters are included.The smoothing procedure is easy to apply in practice, because the minimum MSE signal extraction formula for score-driven models is available in closed form.
In the score-driven signal plus noise models, the updating terms of the signal and noise are correlated time series.As a theoretical contribution, we present that, asymptotically at the true values of parameters, the assumptions of the minimum MSE signal extraction filter of McElroy and Maravall (2014) are satisfied for the following models: (i) score-driven location with constant scale; (ii) score-driven location with score-driven scale; (iii) score-driven trend plus score-driven seasonality with constant scale; (iv) score-driven trend plus score-driven seasonality with scoredriven scale.The statistical properties of the score functions of model (i) are presented in the work of Harvey (2013, p. 61), and the same properties for models (ii) to (iv) are shown in the present article.Models (iii) and (iv) include new score-driven seasonality specifications, which are applied to signal smoothing.
By using the asymptotic properties of the score functions at the true values of parameters, we show that the assumptions of the linear signal smoothing results for correlated signal and noise of McElroy and Maravall (2014) are satisfied for all score-driven models of the present article.Therefore, the linear signal smoothing formulas can be used for the non-linear score-driven models, greatly simplifying the statistical estimation of the smoothed signals.This result is our main contribution.
For the empirical application, we use US inflation rate data for the period of January 1948 to May 2020, for which signal smoothing can be motivated by the following points: First, the series derived from the inflation smoothing procedure could allow the private sector to infer the present and future stance of monetary policy in a better way than the official inflation series (which would be understood to contain more noise and less signal than the smoothed inflation series).In relation to the use of smoothed inflation in the private sector, we refer to the work of Erceg and Levin (2003).Second, in central banks, it is common to monitor one or more measures of core inflation (Rich and Steindler 2005).The core inflation measure corrects the excessive volatility that the official inflation usually shows, since the latter is affected by some relative price changes that are not of direct interest to the monetary policy.There is a significant body of literature on alternative measures of core inflation, from which we refer to the works of Bryan and Pike, Bryan and Pike (1991) and Cristadoro et al. (2005).In the present article, a novel procedure to generate a measure of core inflation is presented.
In the remainder of this article: Section 2 reviews the literature.Section 3 presents the minimum MSE signal for correlated signal and noise.Section 4 presents the method of signal smoothing for score-driven location models for constant and score-driven scales.Section 5 presents the method of signal smoothing score-driven trend plus score-driven seasonality models for constant and score-driven scales.Section 6 presents the empirical application for the US inflation rate.Section 7 concludes.

Signal extraction
In the works of Bell (1984), and Bell and Hillmer (1988), minimum MSE signal extraction filters are presented, for which either signal or noise or both is non-stationary, and the signal and noise components are uncorrelated.In those works, signal extraction formulas are presented by using the transformation approach (Ansley and Kohn 1985), which eliminates the effects of the non-stationary initial conditions.In relation to this, we also refer to the works of Bell and Hillmer (1984) and Bell (2004).In the work of Bell (1984, p. 662), it is shown that minimum MSE signal extraction filters can be applied to non-Gaussian observations.In the work of McElroy (2008), minimum MSE signal extraction filters are presented for finitely-sampled non-stationary ARIMA (autoregressive integrated moving average) processes.Signal smoothing formulas are provided, for which the updating terms of signal and noise, fu t g T t¼1 and fv t g T t¼1 , respectively, are uncorrelated.It is also shown that the minimum MSE signal extraction filter can be used for non-Gaussian observations (McElroy 2008, p. 991).
Several works perform signal smoothing for correlated signal and noise; for example, Beveridge and Nelson (1981), Snyder (1985), Ghysels (1987), Ord, Koehler, and Snyder (1997), Hyndman et al. (2002), Proietti (2006), andMcElroy andMaravall (2014).From these works, McElroy and Maravall (2014) is an extension of the aforementioned works of Bell (1984), Bell andHillmer (1988), andMcElroy (2008).In the work of McElroy and Maravall (2014), minimum MSE signal extraction filters for correlated updating terms of signal and noise are presented, for which either signal or noise or both is non-stationary, with heteroskedastic and non-Gaussian error terms.The work of McElroy and Maravall (2014) is relevant to the present article, because signal and noise are correlated, and the updating terms are non-Gaussian for the score-driven models.For all score-driven models of this article, we show that u t and v t are contemporaneously uncorrelated, but for some lags or leads u t and v t are correlated.

Score-driven time series models
Score-driven models are named generalized autoregressive score (GAS) models (Creal, Koopman, and Lucas 2008, 2011, 2013), or dynamic conditional score (DCS) models (Harvey and Chakravarty 2008;Harvey 2013).Score-driven models are observation-driven time series models (Cox 1981), which are alternatives to several classical observation-driven time series models.For several score-driven models, the sufficient conditions of consistency and asymptotic normality of the ML estimates are known (Blasques et al. 2018;Blasques, van Brummelen, et al. 2021).Scoredriven models are applied to I(0), co-integrated I(1), and fractionally integrated variables, and an advantage of score-driven models is that they are more robust to outliers and missing observations than classical time series models (Harvey 2013).
An example of score-driven models is the quasi-AR (QAR) location model (Harvey 2013), which is an alternative to the classical ARMA model (Box and Jenkins 1970).Another example of score-driven models is the Beta-t-EGARCH (exponential generalized autoregressive conditional heteroskedasticity) model (Harvey and Chakravarty 2008;Harvey 2013), which is an alternative to the classical GARCH (Engle 1982;Bollerslev 1986), and EGARCH (Nelson 1991) models.
Univariate score-driven models, such as Beta-t-EGARCH, implement an optimal filtering mechanism, according to the Kullback-Leibler divergence in favor of the true data-generating process.In the work of Blasques, Koopman, and Lucas (2015), it is shown that, asymptotically, a score-driven update of the time series model reduces the distance between the true conditional density and the conditional density implied by the score-driven model, in expectation and at every step, even for misspecified score-driven models.Those authors show that only score-driven updates have this property, by providing an information-theoretic support for the use of scoredriven models.Moreover, the work of Blasques, Lucas, et al. (2021) supports the information-theoretic effective filtering mechanism of score-driven volatility models for finite samples in practically significant cases.

Signal extraction for correlated signal and noise
In this section, we summarize the results of McElroy and Maravall (2014) for minimum MSE signal extraction filters for correlated signal and noise, which we use for score-driven models.The signal plus noise model is y t ¼ s t þ n t for t ¼ 1:::, T, for which the updating terms are defined as follows: where the vectors ða 1 , :::, (3) where U is a ðT À pÞ Â 1 vector, D S is a ðT À pÞ Â T matrix, and S is a T Â 1 vector (Bell and Hillmer 1988).Moreover, in matrix notation (Bell and Hillmer 1988) where V is a ðT À mÞ Â 1 vector, D N is a ðT À mÞ Â T matrix, and N is a T Â 1 vector.
In the following, matrix D S with dimensions ðT À dÞ Â ðT À mÞ is also used, where d ¼ p þ m, which is defined by the first ðT À dÞ rows and the first ðT À mÞ columns of D S .Matrix D N with dimensions ðT À dÞ Â ðT À pÞ is also used, which is defined by the first ðT À dÞ rows and the first ðT À pÞ columns of D N .Moreover, matrix D ¼ D S D N is also defined.The covariance matrices of U and V are C U with dimensions ðT À pÞ Â ðT À pÞ, and C V with dimensions ðT À mÞ Â ðT À mÞ, respectively.The covariance matrix of the elements of U and V, with dimensions ðT À pÞ Â ðT À mÞ is C U, V , and C V, U ¼ C 0 U, V : Moreover, the following matrix is included in the minimum MSE signal extraction formula: , and C U, V are available in closed form for all score-driven models of this article.
Second, in the work of McElroy and Maravall (2014, Theorem 1) it is assumed that: (A1) Y Ã ðy 1 , :::, y d Þ 0 are uncorrelated with u t and v t , for t > d. (A2) a p ðÁÞ and b m ðÁÞ are relatively prime polynomials (i.e.polynomials a p ðÁÞ and b m ðÁÞ share no common zeros).(A3) fu t g and fv t g have mean zero, and are covariance stationary, correlated, and purely nondeterministic.(A4) C U , C V , and C W are invertible.In Sections 4 and 5, we show that, asymptotically at the true values of parameters, these assumptions are satisfied for the score-driven location, trend, seasonality, and scale models.Under assumptions (A1)-(A4), the minimum MSE signal is Ŝ ¼ FY, where the T Â T matrix F is: All score-driven models of this article can be written according to the representation of Equations ( 1) and (2).Therefore, if (A1) to (A4) hold, then the signal smoothing results of McElroy and Maravall (2014) can be applied to the score-driven models.In the following Sections 4 and 5, we present alternative score-driven models with score-driven location, scoredriven seasonality, and constant or score-driven scale.We show that each of those models can be written in the form of Equations ( 1) and (2).Hence, the minimum MSE signal smoothing formulas of Equations ( 5)-( 7) can be used, in order to straightforwardly perform signal smoothing for the nonlinear score-driven time series models.

Score-driven location with constant scale
In this section, we present minimum MSE signal extraction for the score-driven location model (Harvey 2013), where the error terms of the signal and noise components are correlated, the signal component s t may be non-stationary, and the noise component n t is independent and identically distributed (i.i.d.).
Model specification-The score-driven location model for fy t g T t¼1 is: where where k is the log-scale parameter and t $ tðÞ is an i.i.d.error term.We assume that the degrees of freedom > 2 (hence, the second moment of t exists).In this model, b m ðLÞ of Equation ( 2) is normalized to one.The log conditional density of y t jF tÀ1 y t jðy 1 , :::, y tÀ1 , s 1 Þ is: where ln ðxÞ is the natural logarithm function, H represents the time-invariant parameters, CðxÞ is the gamma function, and exp ðxÞ is the exponential function.
For the I(0) case, the signal is specified as a QAR(p) model (Harvey 2013, p. 63) as follows: where the roots of ð1 À a 1 z À ::: À a p z p Þ ¼ 0 lie outside the unit circle, and Eðs t Þ ¼ 0: Alternatively, for the I(1) case, the signal is specified as follows (Harvey 2013, p. 76): where Eðs t Þ ¼ 0; a 1 ¼ 1 and a 2 , :::, a p are zeros.For all score-driven models of this article, the signal is equal to the filtered signal, s t ¼ Eðs t jy 1 , :::, y tÀ1 Þ, because the signal is determined by the history of past observations of the dependent variable.Equations ( 10) and ( 11) indicate the elements of matrices D S and D S of Section 3. The score function with respect to s t is: where k is the scaling factor, and l t is the scaled score function (Harvey 2013).Harvey (2013) shows that l t is i.i.d. with Eðl t Þ ¼ 0, and variance The score-driven location models with constant scale are estimated by using the ML method, to which the conditions of Creal, Koopman, and Lucas (2008, 2011, 2013), Harvey (2013) ðy 1 , :::, y p Þ 0 are uncorrelated with u t and v t for t > p, because ðs 1 , :::, s p Þ 0 and ðl 1 , :::, l p Þ 0 are set to zero vectors.(A2) m ¼ 0; therefore, polynomials a p ðÁÞ and b m ðÁÞ share no common zeros.(A3) u t ¼ w 1 l tÀ1 and v t ¼ exp ðkÞ t have zero mean, are covariance stationary, because both l t , and v t are i.i.d. with zero mean; u t and v t are purely nondeterminstic due to the model formulation.(Harvey 2013).Moreover, matrices C W and M of Equations ( 5)-( 7) are invertible.Matrix C U, V with dimensions ðT À pÞ Â T is: where Harvey 2013).The location of C in the first column of C U, V is the second row.Time series fl t g and fv t g are both independent.Hence, all lags and leads of l t and v t are independent, and the remaining elements of C U, V are zero.
4.2.Score-driven location and score-driven scale In this section, the score-driven location model (Harvey 2013) and the score-driven scale model (Harvey 2013) are combined into a score-driven location model with score-driven scale, for which we present the statistical properties of the score functions, and the minimum MSE signal extraction filter.The updating terms of signal and noise are correlated, the signal component may be non-stationary, and the zero mean noise component is conditionally heteroskedastic.
Model specification-The score-driven location model with score-driven scale for fy t g T t¼1 is: where s t and n t are signal and noise, respectively.In this model, b m ðLÞ of Equation ( 2) is normalized to one.We assume that n t jF tÀ1 ¼ v t jF tÀ1 $ t½0, exp ðk t Þ, is heteroskedastic with > 2, where F tÀ1 ¼ ðy 1 , :::, y tÀ1 , s 1 , k 1 Þ: Moreover, t $ tðÞ is i.i.d., and the log conditional density of y t jF tÀ1 is: The signal s t is specified as in Equations ( 10) or ( 11), and the score function is: where k t is the dynamic scaling factor, and l t is the scaled score function.The difference between the score functions of Equations ( 12) and ( 16) is that k t and k t are constant in Equation ( 12), but they are dynamic in Equation ( 16).In Appendix A, we show that, asymptotically at the true values of parameters, l t is white noise with zero mean.The time-varying log-scale k t is specified as follows: where a and b are real-valued parameters, jbj < 1, Eðk t Þ ¼ a=ð1 À bÞ, which is the Beta-t-EGARCH(1,1) model.The updating term z tÀ1 , named score function with respect to log-scale k t is: where the scaling factor is normalized to one, as in the work of Harvey (2013), p. 99).In Appendix A, we show that, asymptotically at the true values of parameters, z t is i.i.d. with zero mean.The score-driven models with score-driven scale are estimated by using the ML method, to which the conditions of Blazsek, Escribano, and Licht (2020) can be applied.
Signal smoothing-In the following we show that, asymptotically at the true values of parameters, the assumptions of McElroy and Maravall (2014, Theorem 1) hold for the score-driven location and score-driven scale model: (A1) d ¼ p þ m ¼ p: Initial values Y Ã ¼ ðy 1 , :::, y p Þ 0 are uncorrelated with u t and v t for t > p, because ðs 1 , :::, s p Þ 0 , ðl 1 , :::, l p Þ 0 , and ðz 1 , :::, z p Þ 0 are set to zero vectors, and all elements of ðk 1 , :::, k p Þ 0 are set to Eðk t Þ ¼ a=ð1 À bÞ: (A2) m ¼ 0; thus, polynomials a p ðÁÞ and b m ðÁÞ share no common zeros.(A3) u t ¼ w 1 l tÀ1 and v t ¼ exp ðk t Þ t have zero mean, are covariance stationary, and are purely nondeterministic.Variables u t and v t have zero mean and are covariance stationary, due to the properties of the score functions; u t and v t are purely nondeterminstic due to the model formulation.
(ii) The conditional variance of l t jF tÀ1 is: By using the law of iterated expectations, where the expectation is given by Equations ( 21) and ( 22).Moreover, C W and M of Equations ( 5)-( 7) are invertible.Variables fu t g and fv t g are correlated.Matrix C U, V with dimensions ðT À pÞ Â T is: where , where Equations ( 21) and ( 22) are used for the computation of E½exp ð2k t Þ: In Appendix B, C 0 is presented, and it is also shown that, asymptotically at the true values of parameters, B, C i for i ¼ 1, :::, J, and D j for j ¼ 2, :::, T, are zeros.

Score-driven trend and seasonality with constant scale
For the score-driven location model, b m ðLÞ is normalized to unity; hence, seasonality is not modeled in the noise component n t .We extend the score-driven location model to the score-driven trend plus score-driven seasonality model, for which we present minimum MSE signal extraction in this article.
Model specification-For score-driven trend plus score-driven seasonality with constant scale, the error terms of signal and noise are correlated, and signal and noise may be non-stationary: where s t is the trend component, q t is the seasonality component, and I t ¼ exp ðkÞ t is the irregular component.We assume that t $ tðÞ is an i.i.d.error term with > 2: Notation s t is used for the trend component, because the objective of this model is to extract the trend from the observed seasonal data series.This implies that noise is the sum of the seasonal and irregular components, as suggested in the work of McElroy (2008).The log conditional density of y t jF tÀ1 ¼ y t jðy 1 , :::, y tÀ1 , s 1 , q 1 Þ is: The signal is specified as in Equations ( 10) or ( 11).In the work of McElroy (2008, p. 990), the methodology is described regarding how the signal plus noise model, y t ¼ s t þ n t , can be applied to models with trend s t plus seasonal q t plus irregular I t components.If the interest is to extract the trend component from the observed data, then n t is defined as n t ¼ q t þ I t : We have studied the possibility of applying a score-driven seasonal component for q t , as defined in the works of Harvey (2013, Section 3.6), and Harvey and Luati (2014), to the minimum MSE signal methods of McElroy (2008), andMcElroy andMaravall (2014).However, the application proves to be not straightforward.Therefore, in this article, we use a seasonality specification, which is simpler than the models of Harvey (2013), and Harvey and Luati (2014), but its application to minimum MSE signals is more straightforward.
Seasonality q t is specified as a QAR(m) model with restricted parameters: where the seasonality period is m > 1, b 1 , :::, b mÀ1 are zeros, jb m j < 1, and Eðq t Þ ¼ 0: Equation ( 28) indicates the elements of D N and D N of Section 3. We show in this section that jb m j < 1 is important for (A2) of McElroy and Maravall (2014, Theorem 1).The score function with respect to ðs t þ q t Þ is: where k is the scaling factor, and l t is the scaled score function.The difference between the score functions of Equations ( 12) and ( 29) is that the derivative of the log-density in Equation ( 12) is with respect to s t and the derivative of the log-density in Equation ( 29) is with respect to The score-driven trend and seasonality with constant scale are estimated by using the ML method, to which the conditions of Creal, Koopman, and Lucas (2008;2011;2013) ðy 1 , :::, y pþm Þ 0 are uncorrelated with u t and v t for t > p þ m, because ðs 1 , :::, s pþm Þ 0 , ðl 1 , :::, l pþm Þ 0 , and ðq 1 , :::, q pþm Þ 0 are set to zero vectors.(A2) Since jb m j < 1, polynomials a p ðÁÞ and b m ðÁÞ share no common zeros.(A3) The noise component is: Variables u t ¼ w 1 l tÀ1 and v t ¼ I t À b m I tÀm þ W m l tÀm have zero mean, are covariance stationary, and are purely nondeterministic.(A4) where we use the following result from Harvey (2013), p. 62 The remaining elements of C V are zeros.Moreover, matrices C W and M of Equations ( 5)-( 7) are invertible.Matrix C U, V with dimensions ðT À pÞ Â ðT À mÞ is: where The location of C 0 in the first column of C U, V is the second row, and the location of D m in the first row of C U, V is the m-th column.For the formulations of C 0 and D m , results from the work of Harvey (2013) are used.The remaining elements of C U, V are zero, because l t and I t , asymptotically at the true values of parameters, are independent time series, and l t is a continuous function of only I t .

Score-driven trend, seasonality, and scale
Model specification-The score-driven trend plus score-driven seasonality with score-driven scale is: where I t ¼ exp ðk t Þ t is the irregular component, and t $ tðÞ is i.i.d. with > 2: Filter s t is Equation ( 10) or ( 11), q t is Equation ( 28), and the score function with respect to ðs t þ q t Þ is: where F tÀ1 ¼ ðy 1 , :::, y tÀ1 , s 1 , q 1 , k 1 Þ, k t is the scaling factor, and l t is the scaled score function l t .The difference between Equations ( 16) and ( 36) is that the derivative of the log-density in Equation ( 16) is with respect to s t and the derivative of the log-density in Equation ( 36) is with respect to ðs t þ q t Þ: Scaled score function l t , asymptotically at the true values of parameters, is white noise with zero mean.Variables k t and z t are defined as in Equations ( 17) and ( 18), respectively.Hence, z t is i.i.d. with zero mean.The score-driven trend plus score-driven seasonality models with score-driven scale are estimated by using the ML method, to which the conditions of Blazsek, Escribano, and Licht (2020) can be applied.
Signal smoothing-In the following we show that the assumptions of McElroy and Maravall (2014, Theorem 1) hold for the score-driven trend, seasonality, and scale model: (A1) d ¼ p þ m: Initial values Y Ã ¼ ðy 1 , :::, y pþm Þ 0 are uncorrelated with u t and v t for t > p þ m, because ðs 1 , :::, s pþm Þ 0 , ðl 1 , :::, l pþm Þ 0 , and ðq 1 , :::, q pþm Þ 0 are set to zero vectors, and each element of ðk 1 , :::, k pþm Þ 0 is set to Eðk t Þ ¼ a=ð1 À bÞ: (A2) Since jb m j < 1, polynomials a p ðÁÞ and b m ðÁÞ share no common zeros.(A3) u t ¼ w 1 l tÀ1 and v t ¼ I t À b m I tÀm þ W m l tÀm have zero mean, are covariance stationary, and are purely nondeterministic.Variables u t and v t have zero mean, and are covariance stationary due to the properties of the score functions.(Harvey 2013).Matrix C V with dimensions ðT À mÞ Â ðT À mÞ is invertible, because it is a full-rank square matrix, with elements (Harvey 2013): where E½exp ð2k t Þ is given by Equations ( 21) and ( 22).Moreover, matrices C W and M are invertible.In Appendix C, we show that C U, V with dimensions ðT À pÞ Â ðT À mÞ is: where C 0 is in the second row and first column of C U, V , and D m is in the first row and m-th column of C U, V : Moreover, 6. Empirical application

Trend extraction from seasonally adjusted US inflation rate
Monthly data from the US inflation rate 100 Â ln ðCPI t =CPI tÀ1 Þ (consumer price index, CPI) are used for the period of January 1948-May 2020, and the pre-sample period is December 1947 that we use for CPI 0 (data source: Federal Research Economic Data, FRED; ticker: CPIAUCSL).Smoothed signal estimation for the US inflation rate is relevant for the US private sector as well as for policymakers at the Federal Reserve.The smoothed US inflation time series may allow the private sector to infer the present and future stance of monetary policy in a better way than the officially reported inflation series (e.g.Erceg and Levin 2003).Moreover, in central banks, it is common for policymakers to monitor one or more measures of core inflation (e.g.Rich and Steindler 2005).The core inflation measure corrects the excessive volatility in the official inflation rate, because the latter is affected by some relative price changes that are not of direct interest to the monetary policy (for alternative measures of core inflation, we refer to: Bryan and Pike (1991); Cristadoro et al. (2005).
The score-driven models are estimated for the seasonally adjusted US inflation rate minus its sample mean, by using the model y t ¼ s t þ n t , which ensures that Eðy t Þ ¼ 0 under the  (Dickey and Fuller 1979) indicates that US inflation rate is I(0), motivating the use of Equation ( 10).The ARCH test (Engle 1982) suggests significant volatility dynamics.For the parameter estimates, robust standard errors are reported in parentheses.The standard errors of ML parameters are estimated by using the Huber Sandwich Estimator (Blasques, van Brummelen, et al. 2021).Ã , ÃÃ , and ÃÃÃ indicate significance at the 10%, 5%, and 1% levels, respectively.
assumption that US inflation rate is I(0).In practice, monthly US inflation rate signals can be obtained after the smoothing procedures, by adding the sample average of monthly US inflation rate to the estimates of s t and ŝt : In Panel (a) of Table 1, the descriptive statistics of US inflation rate are presented.As data are seasonally adjusted, we assume that the information needed for a policy decision is the trend component of the inflation rate, not the seasonal component.In Panel (b) of Table 1, the ML parameter estimates of the QAR(1), QAR(2), and QAR(3) score-driven location specifications with constant and score-driven scales are presented.We use different lag-orders for QAR, to support a correct model specification (Appendix A).In the work of Harvey (2013, p. 75), the use of likelihood-based model selection criteria for score-driven models is suggested.Motivated by that, we compare statistical performances by using the Akaike information criterion (AIC), Bayesian information criterion (BIC), and Hannan-Quinn criterion (HQC).We find the following results: The score-driven location specifications with score-driven scale are superior to the scoredriven location specifications with constant scale, according to AIC, BIC, and HQC (Table 1).For the alternative QAR lag-order specifications for the score-driven location model with constant scale, QAR(2) and QAR(3) are clearly superior to QAR(1), according to AIC, BIC, and HQC.Moreover, the QAR(2) lag-order is supported by BIC and HQC, and the QAR(3) lag-order is supported by AIC.Nevertheless, as the absolute difference between the AICs for QAR(3) and QAR(2) is j À 46:2284 þ 45:2688j ¼ 0:9596 < 2 (Table 1), it could be argued that both models are worthy of consideration (Millar 2011;Burnham and Anderson 2002).In relation to this, we note that the estimates of all time series components for the QAR(2) and QAR(3) location alternatives are very similar.For the alternative QAR lag-order specifications for the score-driven location model with score-driven scale, the QAR(2) lag-order is supported by AIC, BIC, and HQC (Table 1).The use of QAR( 2) is also motivated by the facts that a 2 is not significant for the score-driven QAR(3) location model, and a 2 and a 3 are not significant for the score-driven QAR(3) location plus score-driven scale model (Table 1).In Figure 1, for the QAR(2) location with constant scale, we present the evolution of the seasonally adjusted US inflation rate y t , its filtered signal s t , and its smoothed signal ŝt : In Figure 2, for the QAR(2) location with score-driven scale, we present the evolution of the seasonally adjusted US inflation rate y t , its filtered signal s t , its smoothed signal ŝt , and the time-varying log-scale parameter k t .Figures 1 and 2 indicate that the smoothed signal estimates are similar for the score-driven location models with constant and score-driven scale, which shows the robustness of our signal smoothing procedure.

Trend extraction from not seasonally adjusted (NSA) US inflation rate
It may be the case in practice that trend extraction from a NSA macroeconomic variable is needed for an economic decision.We present an application of the score-driven trend plus scoredriven seasonality model to the NSA US inflation rate for the period of January 1948 to May 2020 (data source: FRED; ticker: CPIAUCNS).Similarly to the application on the seasonally adjusted US inflation rate, signal smoothing is performed for the NSA US inflation rate minus its sample mean, which defines y t in this section.We assume that the period of seasonality is m ¼ 12, which is supported by the local maximum points of the sample periodogram for the NSA US inflation rate (Figure 3).
In Panel (a) of Table 2, the descriptive statistics are presented, from which a relevant result is that NSA US inflation rate is I(0).Motivated by this result, different covariance stationary QAR(p) specifications (Equation ( 10)) are used.In Panel (b) of Table 2, the ML estimates are presented for the score-driven trend plus score-driven seasonality model with constant scale and score-driven scale for the QAR(1) plus QAR(12) and QAR(2) plus QAR( 12) alternatives.
The score-driven trend plus score-driven seasonality specifications with score-driven scale are superior to the score-driven trend plus score-driven seasonality specifications with constant scale, according to AIC, BIC, and HQC (Table 2).For the alternative QAR lag-order specifications for the score-driven trend plus score-driven seasonality model with constant scale, the QAR(1) lagorder is supported by BIC, and the QAR(2) lag-order is supported by AIC and HQC.Hence, the QAR(1) and QAR(2) lag-orders indicate very similar likelihood-based model performances, supported by the fact that the estimates of all time series components for the QAR(1) and QAR(2) alternatives are very similar.For the alternative QAR lag-order specifications for the score-driven trend plus score-driven seasonality model with score-driven scale, the QAR(1) lag-order is supported by AIC, BIC, and HQC (Table 2).
In Figure 4, the evolution of the NSA US inflation rate y t , its filtered signal s t , its smoothed signal ŝt , and the seasonality component q t , for t ¼ 1, :::, T, for the score-driven trend and seasonality model with constant scale, QAR(1) plus QAR(12), is presented.In Figure 5, the evolution of the NSA US inflation rate y t , its filtered signal s t , its smoothed signal ŝt , the seasonality component q t , and the time-varying log-scale parameter k t , for t ¼ 1, :::, T, for the score-driven trend and seasonality model with score-driven scale, QAR(1) plus QAR(12), is presented.We note that the estimates of the time series components for alternative lag-orders of the QAR location model are very similar.The annual seasonality component of the US inflation rate is significant with a time-varying amplitude (Figures 4d and 5d).The smoothed signal for the NSA US inflation rate (Figures 4c and 5c) is similar to the smoothed signal for the seasonally adjusted US inflation rate (see Figures 1c and 2c).This indicates the robustness of the score-driven signal smoothing procedure of the present article.

Conclusions
Signal smoothing is important in statistical applications preceding economic decisions.Motivated by this, in recent decades minimum MSE signal extraction formulas have been developed for a great variety of signal plus noise models in the literature.We have applied results from the literature on signal extraction to score-driven models, to perform signal smoothing for score-driven location, trend, and seasonality models with constant and score-driven scale parameters.We have presented the methodology regarding how minimum MSE signal extraction is applied in a straightforward way to score-driven models with higher than first-order location, trend, and seasonality components with constant and score-driven scale parameters.Real dataset-based results have indicated the robustness of the signal smoothing procedure for seasonally adjusted and NSA US inflation rate time series for the period of January 1948 to May 2020.Our results have suggested the practical use of the following two-step signal smoothing procedure for score-driven models: (i) The score-driven model are estimated by using the ML method.(ii) The ML estimates are substituted into the minimum MSE signal extraction filter.The results have indicated that the novel signal smoothing procedure of this article can be easily applied to complex score-driven signal plus noise models, in order to perform signal smoothing before economic decisions.

¼ E tÀ1
@ ln f ðy t jF tÀ1 , HÞ @s t !Â @s t @H 0 ¼ 0 (A.1)where index t À 1 indicates expectations that are conditional on F tÀ1 : Since @s t =@H 0 6 ¼ 0, E tÀ1 @ ln f ðy t jF tÀ1 , HÞ @s t As a consequence, E tÀ1 ðl t Þ ¼ 0 (i.e.l t is a martingale difference sequence, MDS).Moreover, Thus, z t is a MDS.(ii) Eðl t Þ ¼ 0 and Eðz t Þ ¼ 0, due to the law of iterated expectations.(iii) l t and z t are contemporaneously correlated, as both are functions of v t .(iv) Scaled score function l t is not i.i.d., as it depends on k t .(v) We assume that jk t j < k max < 1 for all t (Blazsek, Escribano, and Licht 2020), which sets an exogenous bound for dynamic scale.The consequence of this assumption is that l t is a bounded function of t (Blazsek, Escribano, and Licht 2020).Therefore, Varðl t Þ < 1, and l t is white noise.If the roots of ð1 À a 1 z À ::: À a p z p Þ ¼ 0 lie outside the unit circle, then s t is covariance stationary for (10).(vi) z t is a bounded function of t (Harvey 2013).Therefore, Varðz t Þ < 1, and z t is white noise.If jbj < 1, then k t is covariance stationary.(vii) Due to jk t j < 1, @l t =@k t and @z t =@s t are bounded functions of t (Blazsek, Escribano, and Licht 2020).(viii) @l t =@s t and @z t =@k t are bounded functions of t (Blazsek, Escribano, and Licht 2020).(ix) Scaled score function l t is an F -measurable function of t (White 2001), because l t is a continuous function of t .Scaled score function l t is strictly stationary and ergodic, because l t is an F -measurable function of ð 1 , :::, t Þ, and because t is strictly stationary and ergodic (White 2001).(x) Score function z t is i.i.d., because z t is a continuous function of t , and because t is i.i.d.(White 2001).(xi) Score function z t is an F -measurable function of t (White 2001), because z t is a continuous function of t (Harvey 2013).Score function z t is strictly stationary and ergodic, because z t is an F -measurable function of t , and because t is strictly stationary and ergodic (White 2001).

Appendix B
The following results are valid asymptotically at the true values of parameters, under the assumption of correct model specification.Matrix C U, V with dimensions ðT À pÞ Â T is represented as: where J ¼ ðT À p À 2Þ: In the following, we formulate the elements B, C 0 , C i for i ¼ 1, :::, J, and D j for j ¼ 2, :::, T: , where Equations ( 21) and ( 22) are used for the computation of E½exp ð2k t Þ: Third, C i ¼ Cov½w 1 l tþi , exp ðk t Þ t ¼ E½w 1 l tþi exp ðk t Þ t for 1 i J, for which we use: which is true because l t is a MDS.Hence, C i ¼ 0 due to the law of iterated expectations.Fourth, D j ¼ Cov½w 1 l tÀj , exp ðk t Þ t ¼ 0 for j ¼ 2, :::, T, because of the following arguments.We use: and we use the law of iterated expectations (White 2001) as follows: which holds because Eðjv t jÞ < 1 (White 2001).We use Equation (B.5), and the law of iterated expectations for Equation (B.4), and we obtain that D j ¼ 0.

Appendix C
The following results are valid asymptotically at the true values of parameters, under the assumption of correct model specification.Matrix C U, V with dimensions ðT À pÞ Â ðT À mÞ is represented as: where J ¼ ðT À p À 2Þ: We formulate the elements B, C 0 , C i for i ¼ 1, :::, J, and D j for j ¼ 2, :: In the following, we present (i), (ii), and (iii): We write the following conditional mean: We use the law of iterated expectations: which holds due to the MDS property of l t , and Eðjl tÀ1 jÞ < 1 (White 2001).By using the law of iterated expectations, Eðl tÀ1 I tÀm Þ ¼ 0: (iii) Eðl tÀ1 l tÀm Þ is zero because l t is a MDS with finite variance. Second, , where E½exp ð2k t Þ is given by Equations ( 21) and ( 22), due to the following arguments.In the following, we present (i), (ii), and (iii): (i) Covðl t , I t Þ ¼ E½exp ð2k t Þ=ð þ 1Þ, where E½exp ð2k t Þ is given by Equations ( 21) and ( 22).(ii) For Covðl t , I tÀm Þ, we use the following conditional expectation: We use the law of iterated expectations (White 2001): which holds because Eðjl t jÞ < 1, and l t is a MDS.By using the law of iterated expectations, Eðl t I tÀm Þ ¼ 0: Hence, Covðl t , I tÀm Þ ¼ 0 for all m.(iii) Covðl t , l tÀm Þ is zero because l t is a MDS with finite variance.

Figure 1 .
Figure 1.Seasonally adjusted US inflation for score-driven location (constant scale), QAR(2).Note: y t is monthly seasonally adjusted US inflation rate minus its sample mean.

Figure 3 .
Figure 3. Periodogram for monthly NSA US inflation rate.Notes: The x-axis corresponds to Fourier frequencies 2pj=T for j ¼ 1, :::, T=2, and the ticks on the x-axis correspond to j.Three local maximum values correspond to j ¼ 73, 145, and 218.These values divided by T ¼ 869 are 0.0840, 0.1669, and 0.2509, respectively.As monthly data are used, we expect that the peaks in the periodogram shall occur close to the frequencies 2pk=12 for k ¼ 1, 2, :::, 6: The values of k=12 for k ¼ 1, 2, and 3 are 0.0833, 0.1667, and 0.2500, respectively, which are in close correspondence to the local maximum-based estimates of j/T, and support the use of annual seasonality (m ¼ 12).

Table 1 .
Descriptive statistics of seasonally adjusted US inflation rate, and ML estimates for the score-driven location models.(a).Descriptive statistics of monthly US inflation rate, 100 Â ln ðCPI t =CPI tÀ1 Þ, seasonally adjusted QAR); exponential generalized ARCH (EGARCH).The null hypothesis of the Shapiro-Wilk test (Shapiro and Wilk 1965) is normal distribution.For the ADF tests with constant and with constant plus linear time trend, BICbased optimal lag selection is used.For the ARCH test, 5 lags are used.The Shapiro-Wilk test rejects normal distribution for US inflation rate.The ADF test

Table 2 .
Descriptive statistics of NSA US inflation rate, and ML estimates for the score-driven trend and seasonality models.(a).Descriptive statistics of monthly US inflation rate, 100 Â ln ðCPI t =CPI tÀ1 Þ, not seasonally adjusted The null hypothesis of the Shapiro-Wilk test is normal distribution.For the ADF tests with constant and with constant plus linear time trend, BIC-based optimal lag selection is used.For the ARCH test, 5 lags are used.For the parameter estimates, robust standard errors are reported in parentheses.The standard errors of ML parameters are estimated by using the Huber Sandwich Estimator.ÃÃÃ indicates significance at the 1% level.