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Fractional trends in unobserved components models
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\paragraph{\bf Abstract.} We develop a generalization of unobserved components models that allows for a wide range of long-run dynamics by modelling the permanent component as a fractionally integrated process. The model allows for cointegration, does not require stationarity, and can be cast in state space form. We derive the Kalman filter estimator for the common fractionally integrated component and establish consistency and asymptotic (mixed) normality of the maximum likelihood estimator. We apply the model to extract a common long-run component of three US inflation measures, where we show that the $I(1)$ assumption is likely to be violated for the common trend.
\paragraph{\bf Keywords.} long memory, unobserved components, fractional cointegration, Kalman filter, state space models
\paragraph{\bf JEL-Classification.}
C32, C51, E31
Unobserved components (UC) models are widely used to decompose time series into latent components of different persistence. Applications in economics include, among others, trend-cycle decompositions, the analysis of long-run equilibrium relations, testing for mean reversion e.g.\ in asset returns, and forecasting KimNel1999, KooShe2015.
Despite their wide spread, current UC models exhibit two major limitations. First, they require a priori assumptions about the integration order of a series and, therefore, an endogenous treatment of the long-run dynamic characteristics is infeasible. And second, they restrict the long-run component to be $I(0)$, $I(1)$, or $I(2)$. Statistical inference about the degree of persistence of a long-run component is then limited to prior unit root testing, ignoring the non-standard behavior of economic series that exhibit long memory and hindering the estimation of the integration order on a continuous support jointly with the other parameters of the model. Furthermore, model selection uncertainty from prior unit root testing is not taken into account. Finally, misspecification of the integration order may pollute the estimates of permanent and transitory components and bias the variance estimates for the permanent and transitory shocks.
While for the Beveridge-Nelson decomposition a generalization to ARFIMA processes was derived by AriMar2004 and Pro2016, and low-frequency transformations that allow for fractional integration have been proposed by MueWat2018, UC models lack a generalization to fractionally integrated processes. Deriving such a generalization is particularly challenging: It requires to study the convergence properties of the Kalman filter through which the unobserved components are estimated when fractional integration is allowed. In addition, to enable feasible estimation for time series of length $n$ with $n$ large, a modification of the Kalman filter is necessary, as the state vector of fractionally integrated processes is of dimension $n+1$, thus making the standard Kalman filter inapplicable from a computational perspective. Moreover, the asymptotic theory of the maximum likelihood estimator, that is utilized to estimate the model parameters, has to be derived. So far, asymptotic results are only available for the $I(1)$ case considered in ChaMilPa2009, where in contrast to our model the integration order is assumed to be known. Providing the theoretical analysis required for fractionally integrated UC models together with a computationally feasible estimator for the latent components is the core of this paper.
We contribute to the literature by deriving a fractionally integrated unobserved components model that allows for a flexible treatment of the long-run dynamic characteristics of multivariate stochastic processes by letting the common integration order to take values on a set of positive real numbers including zero. Since we model a $p$-dimensional vector of observable random variables $\{y_t\}_{t=1}^n$ as a linear function of a scalar latent variable $x_t$ that is fractionally integrated of order $b$, our model exhibits $p-1$ fractional cointegration relations. Furthermore, our model can be used to decompose a set of variables into long- and short-run components, where the latter components are $I(0)$.
The model is cast in state space form and allows for asymptotically stationary and nonstationary data. Although an exact state space representation of our model exists, estimating a latent fractionally integrated component via the Kalman filter is computationally infeasible for time series with sample size $n$ large. Therefore, we derive a modified version of the Kalman filter that is based on a truncated state space representation of our fractionally integrated unobserved components model while correcting the observable variables for the approximation error that results from the truncation. Our modified Kalman filter yields the same prediction error and likelihood function as the standard Kalman filter that is based on the full state space representation of a fractionally integrated process but greatly reduces the computing time by keeping the state dimension manageable. E.g.\ for our application in section (ref), the modified Kalman filter is found to be about $150$ times faster than the standard Kalman filter.
The second main technical contribution of our paper is to establish the asymptotic theory for the maximum likelihood estimator of our fractionally integrated unobserved components model. Since the asymptotic properties of the objective function depend on the fractional integration order $b_0$ of the data-generating process and differ for $b_0 < 1/2$ and $b_0 > 1/2$, we consider the asymptotically stationary case and the nonstationary case separately, where in each case the objective function of the maximum likelihood estimator uniformly converges. While a central limit theorem for martingale difference sequences holds for $b_0 < 1/2$ and yields asymptotic normality of the maximum likelihood estimator, the nonstationary case is more involved. Here, we first show that the prediction error variance of the Kalman filter converges. Next, we derive a functional central limit theorem for the relevant partial sums of the score function that include fractional processes. From the functional central limit theorem the convergence rates of the estimates follow directly. Finally, we prove that the maximum likelihood estimator is asymptotically normally distributed, while a rotation of the parameter estimators that corresponds to the cointegrating matrix converges at rate $n^{b_0}$ to a mixed normal distribution, thus reflecting the behavior of cointegration models. From these results, it follows for the model parameters that standard inference results remain valid when a fractionally integrated component is introduced.
As an empirical application, we consider the estimation of unobserved long-run inflation by extracting a common fractional component from a set of price measures for the US. For inflation, there exists substantial evidence suggesting that the series are fractionally integrated HasWol1995, TscWebWe2013. We confirm such findings and estimate the integration order of unobserved long-run inflation to be $0.476$. We also show that misspecifying the integration order to be one yields estimated fundamental shocks that are antipersistent, which violates one important assumption of unobserved components models.
The structure of the paper is as follows. Section (ref) details the fractionally integrated unobserved components model and discusses the estimation of the conditional expected value of the scalar latent variable that is allowed to be fractionally integrated. Section (ref) considers the maximum likelihood estimator for our model. By generalizing the proofs of ChaMilPa2009 for a common $I(1)$ component to the fractional case, we are able to show consistency, to derive the convergence rates for different parameters and to establish a central limit theorem for the maximum likelihood estimator. In section (ref) the model is applied to extract a common long-run component from different US inflation measures. Section (ref) concludes. All proofs are collected in the appendix.
In this section we first derive the fractionally integrated unobserved components model and state the necessary assumptions for identification. Next, we cast the model in state space form, from which we derive the Kalman filter estimator for the latent common long-run component, thereby generalizing the permanent-transitory decomposition of ChaMilPa2009. Furthermore, since the Kalman filter estimator based on the exact state space representation is computationally infeasible for long time series, we propose a modified Kalman filter estimator that is based on a finite ARMA approximation of the fractionally integrated process but directly corrects for the resulting approximation error. In corollary (ref) we show that the modified estimator yields the same prediction error as the estimator that is based on the exact state space representation and, therefore, has the same likelihood but keeps the state dimension manageable.
To begin with, consider the unobserved components model
where $y_t$ is a $p$-dimensional observable time series, $x_t$ is a scalar latent variable that is fractionally integrated of order $b$, $x_t \sim I(b)$, $b \in D$, $D = \{d \in \mathbb{R}\ | \ 0 \leq d < 3/2,\ d \neq 1/2\}$, $\beta$ is a $p\times1$ vector of factor loadings that are unobserved, $u_t \sim \mathrm{NID}(0, \Sigma)$ and $\eta_t\sim \mathrm{NID}(0, 1)$ are iid errors of dimension $p$ and $1$ that are independent and $\varSigma$ is diagonal and has full rank. We collect the parameters in $\theta = (\beta', (\operatorname{vech} \Sigma)', b)' \in \varTheta$. The model may be interpreted as a system where $p$ observable variables $y_t$ are driven by one common, fractionally integrated stochastic trend $x_t$, such that the whole system is $I(b)$ and $p-1$ cointegration relations exist. The true parameters of the data-generating process are denoted as $\beta_0$, $\Sigma_0$, and $b_0$. They are collected in $\theta_0=(\beta_0', (\operatorname{vech} \varSigma_0)', b_0)' \in \varTheta$. We exclude the singular point $b_0=1/2$ since inference is different for $b_0 < 1/2$, where the maximum likelihood estimator is asymptotically Gaussian, and $b_0>1/2$, where a rotation of the parameter estimator for $\beta$ is asymptotically mixed normal, as will be shown in section (ref). The same restriction applies to other cointegrated models JohNie2012. Since we impose $\operatorname{Var}(\eta_t)=1$, $\varSigma$ diagonal and of full rank, the model is identified up to a sign for $\beta$. Therefore we restrict the first entry to be positive for unique identification.
The fractional difference operator $\Delta^b$ is defined as
and a $+\,$--subscript amounts to a truncation of an operator at $t \leq 0$, i.e.\ for an arbitrary process $z_t$, $\Delta^b_+ z_t = \sum_{j=0}^{t-1}\pi_j(b) L^j z_t$ Joh2008. For $b \in \mathbb{N}_0$ the fractional long-run component nests the standard integer integrated specifications, whereas $b \in D$ adds flexibility to the weighting of past shocks. Throughout the paper, we adopt the type II definition of fractional integration MarRob1999 that assumes zero starting values for all fractional processes, and, as a consequence, allows for a smooth treatment of the asymptotically stationary ($b < 1/2$) and the nonstationary ($b > 1/2$) case. Due to the type II definition the inverse fractional difference $\Delta_+^{-b}$ exists and is given by $\Delta^{-b}_+z_t = (1-L)^{-b}_+z_t = \sum_{j = 0}^{t-1}\varphi_{j}(b)z_{t-j}$, where $\varphi_j(b)=\pi_j(-b)$ for all $j$. Finally, we make use of the fractional lag operator introduced in Joh2008 that is defined as $L_b = 1-\Delta_+^b$ and nests the standard lag operator $L_1 = L$ for $b = 1$. {Note that $L_bz_t$ preserves the integration order of a random variable $z_t$ since $b \in D$ is restricted to be non-negative.
Let $\mathbbm{1}(b \geq 1)$ be an indicator function that becomes one if $b \geq 1$ and zero otherwise and let $d = b - \mathbbm{1}(b \geq 1)$ denote the mean-reverting fraction of a long memory process. Define $\Delta^{-d}_+ =\sum_{j=0}^{t-1} \varphi_j(d) L^j$ and $\Delta^{d}_+ = \sum_{j=0}^{t-1} \pi_j(d) L^j$ as a function of $d$, such that $\Delta^{-b}_+=(1-L)_+^{-\mathbbm{1}(b \geq 1)}\sum_{j=0}^{t-1} \varphi_j(d)L^j$ distinguishes between an integer integration order and the fractionally integrated polynomial with $d \in [0, 1)$. For notational convenience we omit $d$ in the binomial expansion of the fractional difference operators $\Delta_+^d$, $\Delta_+^{-d}$ and denote $\pi_j$, $\varphi_j$ as the $j$-th coefficient of $\Delta_+^d$, $\Delta_+^{-d}$ if not stated different explicitly. Then $x_t$ in (ref) is represented as
Given the parameters $b$, $\beta$, and $\varSigma$, the exact state space representation of our model (ref) is given by
where
${Z}=
$ and where $\eta_t=0$ for all $t\leq 0$ due to (ref).
Let $\mathcal{F}_t$ be the $\sigma$-field generated by the observable variables $y_1$, ..., $y_t$. Furthermore, let $z_{t|s}=\mathrm{E}_\theta(z_t |\mathcal{F}_s)$ for $z=x, \alpha$, and ${P}_{t|s} = \mathrm{Var}_\theta({\alpha}_t|\mathcal{F}_s)$ with ${\omega}^{(i,j)}_{t}$ as its $(i, j)$-th entry for $s=t-1$. The $\theta$-subscript denotes that expectations are taken given a parameter vector $\theta$, and $\operatorname{E}_{\theta_0}(y_t | \mathcal{F}_{t-1}) = \operatorname{E}(y_t | \mathcal{F}_{t-1})$. Additionally, let ${\alpha}_{t|t-1}^{(j)}$ denote the $j$-th entry of ${\alpha}_{t|t-1}$. The prediction and updating steps of the Kalman filter for model (ref) given the observable data and the parameter vector $\theta$ are
The following theorem states the conditional expectation of the latent variable $x_t$ given $\mathcal{F}_{t-1}$ and generalizes the results of ChaMilPa2009 for $I(1)$ stochastic trends to the fractional domain.
The proof of theorem (ref) is contained in appendix (ref). There, and in the proofs that follow, we denote $w_t$ as any $I(0)$ process that is a function of the underlying NID distributed shocks $u_{1},...,u_t$, and $\eta_1,...,\eta_t$. Since $\operatorname{E}_{\theta_0}(y_t |\mathcal{F}_{t-1}) = \operatorname{E}(y_t |\mathcal{F}_{t-1})$, and thus $v_t(\theta_0) = y_t - \operatorname{E}(y_t | \mathcal{F}_{t-1})$, it follows that $(v_t(\theta_0), \mathcal{F}_t)$ is a martingale difference sequence (MDS).
Theorem (ref) illustrates that the Kalman filter estimator $x_{t+1|t}$ can be decomposed into a linear combination of $y_{t+1}$ that is $I(b_0)$ and an additive component $z_{t+1}(\theta)$ where the latter is the prediction error for the fractionally differenced univariate process $\Delta_+^b \frac{\beta' \varSigma^{-1} }{\beta' \varSigma^{-1}\beta } y_{t+1} $ given the filtration $\mathcal{F}_t$. The integration order of this prediction error is given by the following lemma.
The proof is included in appendix (ref). Thus, the Kalman filter estimator $x_{t+1|t}$ is always $I(b_0)$. The prediction error $v_{t+1}(\theta)$ combines errors from $\beta \neq \beta_0$ and errors from $b \neq b_0$. It is $I(b_0)$ for $\beta \neq \beta_0$, since $\left(I - \frac{\beta \beta' \varSigma^{-1}}{\beta' \varSigma^{-1}\beta} \right)\beta_0 x_{t+1} \neq 0$, whereas $\beta = \beta_0$ yields $v_{t+1}(\theta)=\left( I - \frac{\beta_0 \beta_0' \varSigma^{-1}}{\beta_0'\varSigma^{-1}\beta_0} \right)u_{t+1} + \beta_0 z_{t+1}(\theta)\sim I(b_0 - b)$ by lemma (ref). Finally, $v_{t+1}(\theta_0) \sim I(0)$.
Although a finite-order state space representation of the system in (ref) exists since a fractionally integrated process of type II exhibits a finite-order autoregressive representation of length $n-1$, estimating such a system is only computationally feasible when $n$ is small. To estimate $\alpha_t$ the Kalman filter computes the inverse of the $(n+1)\times(n+1)$ covariance matrix $P_{t|t-1}$ for $t=1,...,n$ sequentially, which makes the filter inapplicable for large $n$. As a solution, ChaPal1998 suggest to truncate the Wold representation of a fractionally integrated process after $m$ lags before the model is cast in state space form, and provide consistency results for $b_0 < 1/2$. HarWei2018 find that a purely fractionally integrated trend is well approximated by finite ARMA processes in several simulation studies. For optimization purposes their approach is particularly convenient since it maps from the fractional integration order $b$ to its related ARMA coefficients and, therefore, optimization is conducted over $b$.
Nonetheless, the literature lacks consistency results for finite approximations of fractionally integrated processes in state space form when $b_0 > 1/2$, and we expect any estimator that truncates the fractionally integrated process at lag $m$, $m<n$, to become inconsistent as soon as $b_0 > 1/2$, $b_0 \neq 1$, since the variance of the truncated sum $(1-L)^{-\mathbbm{1}(b_0 \geq 1)} \sum_{j=m+1}^{n-1}\varphi_j(d_0) \eta_{n-j}$ diverges as $n \to \infty$.
As a solution, we include a correction for the resulting approximation error that allows us to contribute to the literature on fractionally integrated processes in state space form by deriving consistency results for the maximum likelihood estimator when $b_0 \in D$. To obtain a computationally feasible representation, we approximate the fractionally integrated process by a finite-order ARMA process, but directly correct for the resulting approximation error. We base our theoretical analysis on ARMA($1, m$) approximations of $x_t$, where the moving average polynomial truncates the stable part of the Wold representation of a fractionally integrated process, whereas the AR polynomial controls for integration orders greater or equal to one. As will be shown in this section, the modified Kalman filter yields the same likelihood function as the one that is based on the exact state space representation of a fractionally integrated process.
Let $\tilde{y}_t$ denote an approximate version of (ref) and (ref) that is obtained by truncating the fractional polynomial $ \sum_{i=0}^{t-1}\varphi_i \eta_{t-i}$ after lag $m$,
such that $(1-L)^{\mathbbm{1}(b \geq 1)}(\tilde{x}_t - x_t) = -\sum_{i=m+1}^{t-1}\varphi_i \eta_{t-i}$. \\ The system matrices and variables of the approximate state space representation are denoted with tilde, i.e.\ $\tilde{T}$, $\tilde{Z}$, $\tilde{R}$, $\tilde{\alpha}_{t}$, $\tilde{v}_t(\theta)$, $\tilde{P}_{t|s}$, and $\tilde{\omega}^{(i,j)}_{t}$. Hence, $\tilde{T} = T^{({1:(m+1), 1:(m+1)})}$ consists of the upper $m+1$ columns and rows of $T$, $\tilde{Z}= Z^{(\cdot, 1:(m+1))}$ holds the first $m+1$ columns of $Z$, $\tilde{R} = R^{(1:(m+1), \cdot)}$ consists of the first $m+1$ rows of $R$ and the $(m+1)$ vector $\tilde{\alpha}_t$ is given by $\tilde{\alpha}_t =
'$. $\tilde{P}_{t|s}$, $\tilde{v}_t(\theta)$ are defined accordingly. The Kalman filter equations (ref) to (ref) hold equivalently if denoted with tilde.
In the following theorem we state the conditional expectation $\tilde{x}_{t+1|t}$ of the truncated model as a function of $x_{t+1|t}$ and an approximation error.
The prediction error $v_{t+1}(\theta)$ can be decomposed into the prediction error of the truncated model plus the approximation error
Note that the approximation error $\epsilon_t(\theta)$ is the Kalman filter estimate for $x_{t} - \tilde{x}_{t} = (1-L)^{-\mathbbm{1}(b \geq 1)} \sum_{i=m+1}^{t-1}\varphi_i \eta_{t-i}$ given $\mathcal{F}_{t-1}$ and, therefore, it is $\mathcal{F}_{t-1}$-measurable and can be calculated given the formula in theorem (ref). Consequently, the results from theorem (ref) for the exact representation carry over if $y_{t}$ is corrected for the approximation error, as the following corollary states.
From corollary (ref) it follows that the prediction errors of the exact representation (ref) using $\{y_t\}_{t=1}^n$ and the truncated model (ref) together with the approximation-corrected $\{\ddot{y}_t\}_{t=1}^n$ are identical and have the same conditional likelihood given $\theta$. Hence, maximizing the likelihood of the approximation-corrected truncated model solves the same optimization problem as for the exact state space representation but requires a smaller number of state estimates from the Kalman filter if $m < n$. The modified Kalman filter outperforms the standard Kalman filter from a computational perspective whenever $p \ll n$, as it requires to invert the $np\times np$ matrix $\varSigma_{Y_n}$ once, whereas the Kalman filter based on the full representation of (ref) sequentially inverts the $(n+1) \times (n+1)$ matrix $P_{t|t-1}$ for each $t=1,...,n$. E.g.\ for our application in section (ref), the modified Kalman filter is about $150$ times faster than the standard Kalman filter.
Although we base our theoretical analysis on ARMA($1, m$) approximations of $x_t$, including further lags of the autoregressive polynomial may improve the approximation quality in finite samples, as HarWei2018 show, and, therefore, speed up the parameter optimization. Nonetheless, the asymptotic results remain unaffected by an extended AR polynomial since correcting for the approximation error yields an exact representation of a fractionally integrated process anyway. For notational convenience we therefore stick to the simplest ARMA({$1, m$}) approximation in section (ref), whereas in our empirical application in section (ref) we use ARMA($4$, $4$) approximations for a faster convergence of the estimator.
Having shown that an exact representation (ref) together with $\{y_t\}_{t=1}^n$ yields the same conditional likelihood of the prediction error as a truncated, approximation-corrected model $\eqref{tru:1}$ together with $\{\ddot{y}_t\}_{t=1}^n$ for a given $\theta$, we turn to the estimation of the unknown parameters $\theta$ in the subsequent section, where we focus on the exact state space representation of (ref). For the asymptotic results to carry over to the truncated, approximation-corrected model it is required that $\epsilon_t(\theta) < \infty$, and therefore the truncation parameter is required to depend on the sample size $n$, $m = m(n)$, whenever $b > 1/2$.
In this section we derive the maximum likelihood (ML) estimator for the unknown parameters $\theta$ in the unobserved components model (ref) with a common fractional trend and determine the asymptotic properties of the ML estimator. With respect to the latter, two major difficulties have to be tackled. First, as it already becomes clear from theorem (ref) and lemma (ref), $z_t(\theta)$ depends on $b_0 - b$ and is nonstationary for $b_0 - b \geq 1/2$. We tackle this issue by first establishing consistency of the ML estimator for $b$, where we show that the estimator is nested in the ARFIMA optimization problem considered in Nie2015. There, consistency of the estimator for $b$ is shown by splitting $D$ into different intervals and showing that the relevant parameter space reduces to $D_3(\kappa_3)=D \cap \{b: b-b_0 \geq -1/2 + \kappa_3\}$, $0<\kappa_3<1/2$, where the objective function of the estimator converges uniformly. Consequently, $z_t(\theta)$ and the partial derivative of $v_t(\theta)$ w.r.t.\ $b$ converge to stationary processes.
The second difficulty arises from the partial derivative of $v_t(\theta)$ w.r.t.\ $\beta$ that is $I(b_0)$, which implies that the convergence rate of the ML estimator for $\beta$ depends on $b_0$ for $b_0 \in (1/2, 3/2)$. Consequently, we consider the asymptotically stationary case $b_0 \in [0, 1/2)$ and the nonstationary case $b_0 \in (1/2, 3/2)$ separately. For both cases we show that the ML estimator of $\theta$ converges to a normal distribution, whereas in the latter case a certain rotation of the parameters is asymptotically mixed normally distributed.
The section is organized as follows. We first state the log likelihood of the state space model (ref) together with its first and second derivative and comment on the convergence of the prediction error variance $F_t$ in (ref). Next, we show consistency of the ML estimator for $b$. Finally, we derive the asymptotic distribution for the ML estimator of $\theta$ for the asymptotically stationary case $b_0 \in [0, 1/2)$ and the nonstationary case $b_0 \in (1/2, 3/2)$ separately, including a discussion on the cointegration properties implied by the model.
The log likelihood of our state space system is given by
where $F^{[n]}=\lim_{t \to \infty} \operatorname{Var}_\theta(v_t(\theta) | \mathcal{F}_{t-1})$ is the steady state variance of the prediction error that depends on the fixed system dimension $n$ due to the type II definition of long memory. The existence of a steady state $F^{[n]}$ is shown in lemma (ref) in appendix (ref). The derivation of the asymptotic properties of the ML estimator requires convergence of the steady state variance $F^{[n]}$ as $n \to \infty$. This is shown in lemma (ref) in appendix (ref), where special care is taken w.r.t.\ the state dimension increasing with $n$.
An analytical solution for the score and Hessian matrix was derived in ChaMilPa2009 and is given by
\begingroup \allowdisplaybreaks and
Having stated the log likelihood together with its derivatives, we turn to the estimation of $b$. By theorem (ref) the prediction error has the decomposition $v_{t+1}(\theta)= (I - \frac{\beta \beta' \varSigma^{-1}}{\beta'\varSigma^{-1}\beta}) y_{t+1} + \beta z_{t+1}(\theta)$. Since the second term of $z_{t+1}(\theta)$ is $I(b_0-b)$ by lemma (ref), the prediction error is $I(b_0)$ whenever $\beta\neq\beta_0$ and $I(b_0-b)$ in case of $\beta=\beta_0$. However, since the first term in $v_{t+1}(\theta)$ is invariant with respect to $b$, only the second term $\beta z_{t+1}(\theta)$ matters w.r.t.\ estimating $b$. The latter term is asymptotically stationary if $b_0 - b < 1/2$, such that a law of large numbers can be applied to obtain uniform convergence of the objective function for $b$. For $b_0 - b \geq 1/2$, $z_{t+1}(\theta)$ is nonstationary, and the rate of convergence of the objective function (ref) depends on $b_0-b$. Thus, the objective function of the ML estimator for $b$ does not converge uniformly on $D$. For ARFIMA models Nie2015 shows consistency of the conditional sum-of-squares (CSS) estimator for $b$, and the CSS estimator has the same limit distribution as the maximum likelihood estimator under Gaussianity HuaRob2011. Thus, by showing that our objective function of the ML estimator for $b$ is asymptotically nested in the ARFIMA objective function considered in Nie2015 and that our setup satisfies assumptions A to D in Nie2015, we prove that consistency for the ML estimator of $b$ carries over from the CSS estimator. The following theorem summarizes the results.
The proof is contained in appendix (ref).
Theorem (ref) implies that the relevant parameter space for $b$ asymptotically reduces to the neighborhood of $b_0$, implying that $z_{t+1}(\hat\theta)$ is asymptotically stationary and the objective function for the ML estimator of $b$ converges uniformly.
Next we turn to the asymptotic analysis of the maximum likelihood estimator for $\theta$. To derive its asymptotic properties, we follow the well-established approach used for stationary models and apply a first order Taylor expansion to the score vector, which yields
where $\hat{\theta}_n$ is the maximum likelihood estimator for $\theta_0$, and $H_n(\theta_n)$ denotes the Hessian with rows evaluated at mean values between $\hat{\theta}_n$ and $\theta_0$. Given that $s_n(\hat{\theta}_n)=0$ if $\hat{\theta}_n$ is an interior solution, we write
where $\nu_n$ is a scaling matrix and $A$ is a rotation matrix that will be defined in (ref) below. Again following ChaMilPa2009, the score vector (ref) evaluated at the true parameter value $\theta_0$ is given by
where $F^{{[n]}}_0$ is $F^{{[n]}}$ evaluated at $\theta = \theta_0$.
It is easy to see that the only stochastic component in $s_n(\theta_0)$ is $v_t(\theta_0)$ and its derivative evaluated at $\theta_0$. From the decomposition of $v_t(\theta_0)$ derived in theorem (ref), one can obtain its derivatives stated in the following lemma.
The proof of lemma (ref) is contained in appendix (ref). As the lemma shows, $\partial v_t(\theta)' / \partial \beta$ at $\theta_0$ is the only source of fractional integration in the gradient $s_n(\theta_0)$, whereas $\partial v_t(\theta)' / \partial \operatorname{vec} \varSigma$, $\partial v_t(\theta)' / \partial b$ at $\theta_0$ are $I(0)$. Similar to the $I(1)$ case studied in ChaMilPa2009 the partial derivative $\partial v_t(\theta)' / \partial \beta$ at $\theta_0$ is a process of dimension $(p\times p)$ that is driven by one common fractionally integrated trend $x_t$, such that $\partial v_t(\theta)' / \partial \beta$ at $\theta_0$ is cointegrated. Defining the $(p \times p)$-dimensional projection matrix
as ChaPal1998 do for the $I(1)$-case, allows to write $\partial v_t(\theta)' / \partial \beta\rvert_{\theta=\theta_0} = - P_x x_t + a_\beta^0(u_t,\eta_t)$. While for each column in $\partial v_t(\theta)' / \partial \beta\big\rvert_{\theta=\theta_0}$ the dimension of the cointegration space is $p-1$, $c\beta_0$ is the only common cointegrating vector for all $p$ columns, where $c$ is any nonzero constant, eliminating the single common trend from all $p^2$ derivatives, $\beta_0'\partial v_t(\theta)' / \partial \beta\big\rvert_{\theta=\theta_0} \sim I(0)$. Thus, the projection matrix satisfies $\beta_0' P_x = 0$. Furthermore, $ P_x \varSigma^{-1}_0 \beta_0 = 0$ holds. From the latter equation it follows that $P_x \varSigma^{-1}_0$ is relevant for determining the cointegration space for $y_t=\beta_0x_t+u_t$. To deal with the singularity in $P_x$, we follow the approach of ChaMilPa2009 and define $\varGamma_0$ as a $p \times (p-1)$ matrix for which
Note that $P_x = \varSigma_0^{-1} \varGamma_0 \varGamma_0'$. Thus, $\varGamma_0' \varSigma_0^{-1}$ determines the $p-1$-dimensional cointegration space for $y_t$. From the left equation in (ref) it follows that the cointegration vectors for $y_t$ and for the partial derivatives $\partial v_t(\theta)' / \partial \beta\rvert_{\theta=\theta_0}$ are orthogonal. For a broad discussion of the cointegrating properties we refer to ChaMilPa2009. In addition, note that the derivatives $\frac{\partial v_t(\theta)'}{\partial \theta} = -\frac{\partial x_{t|t-1}\beta'}{\partial \theta}$ are $\mathcal{F}_{t-1}$-measurable since $x_{t|t-1}$ is $\mathcal{F}_{t-1}$-measurable.
Next, we study the asymptotic properties of $v_t(\theta)$ and $\frac{\partial v_t(\theta)'}{\partial \theta}$ at $\theta = \theta_0$. From the Kalman recursions, in particular (ref) and (ref) which contain random components, it follows that $v_t(\theta)$ is normally distributed since the recursions are linear and the errors $\eta_t$ and $u_t$ are assumed to be NID. Furthermore, $(v_t(\theta_0), \mathcal{F}_t)$ is a martingale difference sequence (MDS) by construction. Moreover, the MDS is asymptotically stationary since its conditional variance $\operatorname{Var}( v_t(\theta_0) | \mathcal{F}_{t-1})$ converges asymptotically, $\lim\limits_{n \to \infty}\lim\limits_{t \to \infty} \operatorname{Var}( v_t(\theta_0) | \mathcal{F}_{t-1})= F_0$, as shown in lemma (ref) in the appendix, so that $F_0$ is the asymptotic variance for the MDS $v_t(\theta_0)$. Since $v_t(\theta_0)$ adapted to $\mathcal{F}_t$ is uncorrelated, normally distributed due to the NID errors as argued above, and has a finite asymptotic variance, we have $v_t(\theta_0) \stackrel{d}{\longrightarrow} \mathrm{NID}(0, F_0^{[n]})$ as $t \to \infty$ for given $n$ and given the adaption to $\mathcal{F}_t$. It follows from the results of Mui1982 on the asymptotic properties of the Wishart distribution that
as $n \to \infty$ where $K$ is the commutation matrix.
As shown in lemma (ref) in appendix (ref), $\left(\frac{\partial v_t(\theta)'}{\partial \theta}\Big\rvert_{\theta = \theta_0} F_0^{{[n]}^{-1}} v_t(\theta_0), \mathcal{F}_t \right)$ is a MDS since the partial derivative is $\mathcal{F}_{t-1}$-measurable. Moreover, both terms in the score vector (ref), $\sum_{t=1}^n \left(v_t(\theta_0) v_t(\theta_0)' - F_0^{[n]}\right)$ and $\sum_{t=1}^n \frac{\partial v_t(\theta)'}{\partial \theta}\Big\rvert_{\theta = \theta_0} F_0^{{[n]}^{-1}} v_t(\theta_0)$, become independent asymptotically. Thus, we obtain comparable results as ChaMilPa2009.
For $b_0 < 1/2$ the asymptotic properties of the ML estimator for ARFIMA processes in the time domain have already been established Ber1995, Rob2006. In the asymptotically stationary case, we can show that their results carry over to unobserved components models.
To derive the asymptotic distribution of the ML estimator for $b_0 < 1/2$, we use a central limit theorem (CLT) for MDS that applies to $\partial v_t(\theta)'/\partial \theta \big\rvert_{\theta=\theta_0}F_0^{[n]^{-1}}v_t(\theta_0)$ since the partial derivatives at $\theta_0$ are asymptotically stationary. Furthermore we show convergence in distribution for the first term in (ref). Lemma (ref) in appendix (ref) summarizes the results for both terms. A martingale CLT for the gradient (ref) then yields $\frac{1}{\sqrt{n}} s_n(\theta_0) \stackrel{d}{\longrightarrow} N(0, \mathcal{J}_0)$, where $\mathcal{J}_0$ is the limiting information matrix Dav2000. Asymptotic independence of both stochastic terms in (ref) facilitates the computation of $\mathcal{J}_0$. Finally, from Dav2000 a CLT for the ML estimator $\hat{\theta}_n$ follows as shown in theorem (ref).
The proof is contained in appendix (ref).
Having shown that the ML estimator is asymptotically normal for $b_0 < 1/2$, we turn to the nonstationary case $b_0 \in (1/2, 3/2)$. Then the usual MDS CLT does not apply since by lemma (ref) the derivative of $v_t(\theta)$ at $\theta_0$ is a nonstationary process. Inference for a broad class of (potentially) nonstationary models is considered in Woo1994, where sufficient conditions for consistency and asymptotic (mixed) normality of the ML estimator are derived. ChaMilPa2009 extend this setup by including a rotation matrix $A$. Their setup also nests our fractional trend model and allowed ParPhi2001 to study the asymptotic behavior of the NLS estimator for nonlinear cointegration models. It requires to consider the following three sufficient conditions:
The random matrices $M$ and $N$ and the nonstochastic matrices $A$ and $\nu_n$ will be defined below in (ref), and $\mu_n$ in the proof of lemma (ref). As in the $I(1)$ case considered in ChaMilPa2009, under conditions ML1 to ML3, equation (ref) converges as $n \to \infty$
Showing that ML1 to ML3 hold, such that (ref) follows, is the subject of the remaining section, where we proceed as follows. To distinguish between $I(b_0)$ and $I(0)$ processes we first derive an expression for the rotation matrix $A$. Lemma (ref) contains a functional central limit theorem (FCLT) for the different components in $A' s_n(\theta_0)$, which directly yields the entries of the scaling matrix $\nu_n$. Finally, in lemmas (ref) to (ref) we prove that ML1 to ML3 hold and thus (ref). Theorem (ref) summarizes the results and defines $M$, $N$.
As lemma (ref) shows, the partial derivative w.r.t.\ $\beta$ at $\theta_0$ is the only source of fractional integration in the partial derivatives of $v_t(\theta)$ at $\theta_0$, whereas the partial derivatives w.r.t.\ $\operatorname{vec} \varSigma$ and $b$ are $I(0)$. Again following ChaMilPa2009, to distinguish between $I(0)$ and $I(b_0)$ components, let the rotation matrix be defined as $A =
$, where $A_N$ is $k \times (p-1)$, $A_S$ is $k \times (1+p(p+1)/2)$ and $A_D$ is $k \times 1$, where $k=p + p(p+1)/2 +1$ is the dimension of $\theta$. The scaling matrix $\nu_n$ adjusts for different convergence rates
From lemma (ref) and the properties of $\varGamma_0$ in (ref) is easy to see that
whereas $A_S'\frac{\partial v_t(\theta)'}{\partial\theta}\big\rvert_{\theta = \theta_0}$, $A_D'\frac{\partial v_t(\theta)'}{\partial\theta}\big\rvert_{\theta = \theta_0}$ are $I(0)$.
To derive the distribution properties of $M$, $N$ in (ref) we define the partial sums
and
Since multiplication with $A_S'$ and $A_D'$ eliminates the nonstationary part of $\frac{\partial v_t(\theta)'}{\partial \theta}\Big\rvert_{\partial \theta = \theta_0}$ and $\frac{\partial v_t(\theta)'}{\partial \theta}\Big\rvert_{\theta = \theta_0} F_0^{{[n]}^{-1}} v_t(\theta_0)$ is a MDS, the FCLT of ChaMilPa2009 carries over directly for $U_n(r)$, $W_n(r)$ and $Y_n(r)$. For $X_n(r)$ that contains nonstationary fractionally integrated common components we extend their FCLT in the following lemma where $\Rightarrow$ denotes weak convergence.
The proof is contained in appendix (ref). Denoting in the sequel $W(1)$ by $W$ and $Y(1)$ by $Y$, one has
With the FCLT of lemma (ref) at hand, lemmas (ref) to (ref) prove that the conditions ML1 to ML3 hold. They are contained in appendix (ref). The following theorem summarizes the results by stating the asymptotic properties of the maximum likelihood estimator.
Define $
' = \left[\operatorname{Var} (Z) + \operatorname{Var} (W) \right]^{-1}(Z - W). $ Then it follows from theorem (ref) that
ChaMilPa2009 conclude from their counterpart of theorem (ref) that
To show this, multiply (ref) by $\beta_0/(\beta_0' \varSigma_0^{-1} \beta_0)^{1/2}$ and then insert (ref) to obtain
Using $P_x' = \varGamma_0 \varGamma_0' \varSigma_0^{-1}$, the second term converges to zero in probability for $n\to\infty$ and $b_0>1/2$ since from (ref) one has $ \varGamma_0' \varSigma_0^{-1} \left(n^{b_0}(\hat{\beta} - \beta_0)\right) = O_p(1)$.
From theorem (ref) it follows directly that the maximum likelihood estimator for $\theta$ is consistent and asymptotically normal. As in the $I(1)$ model of ChaMilPa2009, the estimator for $\theta$ converges at rate $\sqrt{n}$ with one particular exception. $\varGamma_0'\varSigma_0^{-1}\hat{\beta}$ converges at rate $n^{b_0}$ and is mixed normally distributed. Recall that the rotation $\varGamma_0'\varSigma_0^{-1}$ is the cointegrating matrix as it projects out the common fractional trend $\varGamma_0'\varSigma_0^{-1}\beta_0x_t = 0$. Therefore, the faster convergence rate for the cointegrating matrix in error-correction models carries over to the fractionally integrated unobserved components model. Additionally, theorem (ref) shows that the standard inference results, which were shown to be valid for nonstationary $I(1)$ trends in state space models by ChaMilPa2009, remain valid when the persistence of the common component is generalized to the nonstationary fractional domain. Due to (ref), (ref), and (ref) the information matrix equality holds asymptotically. Thus, an estimate for the parameter covariance matrix can be obtained from the negative inverse of the Hessian matrix computed in the numerical optimization.
In a nutshell, the ML estimator is consistent for $b \in D$. It converges to the normal distribution as $n \to \infty$ whenever $b_0 < 1/2$, as shown in theorem (ref). For $b_0 \in (1/2, 3/2)$ theorem (ref) states that the ML estimator is asymptotically normally distributed where a particular rotation of the parameter vector exhibits an asymptotically mixed normal distribution. Thus, $t$-ratios for parameter significance and asymptotic tests such as the likelihood ratio test, the Wald test, and the LM test, remain valid in the fractionally integrated UC model within the two distinct intervals in $D$. Therefore, our results for the nonstationary region generalize the statement of ChaMilPa2009 for the $I(1)$ case. Based on simulation results, HarWei2018 report good finite sample performance of the ML estimator for fractionally integrated UC models.
We apply our fractional UC model to extract a common long-run component from three inflation measures for the US, the consumer price index (CPI), the personal consumption expenditures index (PCI), and the producer price index (PPI). The literature has so far only considered an $I(1)$ common component in US inflation DomGom2006, StoWat2016 that was interpreted as long-run or core inflation. We contribute to the literature by investigating whether the $I(1)$ assumption for the long-run component holds. Furthermore, we show how estimates for the long-run component $x_t$ together with its fundamental shocks $\eta_t$ are affected if fractional integration is allowed for. If the $I(1)$ assumption for the long-run component is violated in the $I(1)$ UC model, then the asymptotic results of ChaMilPa2009 are not applicable. In that case the fractional UC model provides valid inferential results, as it covers integration orders $b \in D$.
The data was downloaded from the Federal Reserve Bank of St.\ Louis (mnemonics: CPIAUCSL, PCEPI, WPSFD49207), is in monthly frequency and spans from 1961:1 to 2018:12. The three series were generated via log differences
where $i \in \{CPI, PCI, PPI\}$ indexes the inflation measures. Since all three series intend to measure price growth for the US, we model them as a function of one common scalar long-run component $x_t$, which in our case is a fractionally integrated trend, and three uncorrelated idiosyncratic components $u_t$
This implies a cointegration rank $r = p - 1 = 2$ among the inflation measures, which is confirmed by the sequential likelihood ratio test for fractional time series of JohNie2012 that clearly rejects the null hypothesis for $r=1$ (p-value $0.001$) but fails to reject for $r=2$ (p-value $0.102$). Furthermore, we allow for $\operatorname{Var}(\eta_t) = \sigma_\eta^2 \neq 1$ and restrict $\beta^{CPI}$ to one for unique identification of $x_t$. Since the standard errors of the three inflation measures differ considerably, we allow for $\beta_{PCI} \neq 1$ and $\beta_{PPI} \neq 1$.
We enrich our ARMA approximation of the fractionally integrated process $x_t$ by additional AR coefficients, which does not affect the asymptotic properties of the ML estimator but reduces the approximation error. Since choosing the same lag order for the AR and the MA polynomial is computationally efficient, as any AR polynomial of length less or equal to $m$ does not affect the dimension of the state vector, we use ARMA($m$, $m$) approximations in the following. As HarWei2018 demonstrate in a simulation study, setting $m \geq 3$ yields an approximation error that is hardly visible. Therefore, we consider ARMA($4, 4$) approximations in the following. Since the Wold representation of an ARMA process $a(L)\tilde{x}_t = b(L)\eta_t$ is given by $\tilde{x}_t = a(L)^{-1}b(L) \eta_t = \psi(L)\eta_t$ the approximation error becomes
and is again $\mathcal{F}_{t}$-measurable.
Technically, for a fixed $b$, the ARMA coefficients in $a(L)$, $b(L)$, and thus $\psi(L)$, are obtained beforehand by minimizing the mean squared error between the Wold representations of $\tilde{x}_t$ and $x_t$. A continuous function that maps from the integration order $b$ to the ARMA coefficients is then constructed by first optimizing over a grid of $b$ and second smoothing the ARMA coefficients over $b$ using splines. Hence, optimization of the likelihood for the fractionally integrated UC model is conducted over the scalar fractional integration order $b$ and does not involve the estimation of any parameters in $a(L)$, $b(L)$. This procedure keeps the dimension of the parameter vector $\theta$ small during the optimization. Further details together with simulation results are contained in HarWei2018.
Starting values for the ML estimator of $\theta$ are obtained by drawing $1000$ combinations of initial values for $b$, $\beta$, and $\varSigma$ from uniform distributions with appropriate support and maximizing the likelihood while ignoring the approximation error. As HarWei2018 show, this procedure already yields quite precise estimates for the unknown parameters and is computationally fast. The optimized parameters corresponding to the largest likelihood are then taken as starting values for the approximation-corrected ML estimator. For an unconstrained optimization, we use a matrix logarithm parametrization for the covariance matrices. Standard errors are denoted in parentheses.
For the loadings we estimate $\hat{\beta} =
'$, which reflects the heterogeneous volatility of the three inflation measures. The integration order estimate $\hat{b}=0.476$ $(0.030)$ is in line with the literature, where e.g.\ \cite{HasWol1995} estimate an integration order of $0.41$ for US CPI inflation from 1969:1 to 1992:12, while \cite{Bai1996} estimates $\hat{b}=0.47$ for US CPI inflation from 1948:1 to 1990:7. Hence, there is substantial evidence for long-run inflation being mean-reverting and integrated of order around $1/2$. Our estimated integration order of $0.476$ implies that a unit shock still has more than $14%$ of its initial impact on inflation after one year, and more than $4%$ of its initial impact after ten years. The variance estimates for the fundamental shocks $\eta_t$, $u_t$ are $\log \hat{\sigma}_{\eta}^2 = -3.275$ $(0.065)$, $\log \hat{\sigma}_{u_{CPI}}^2 = -4.374$ $(0.098)$, $\log \hat{\sigma}_{u_{PCI}}^2 = -5.839$ $(0.232)$, and $\log \hat{\sigma}_{u_{PPI}}^2 = -1.782$ $(0.058)$, implying $\hat{\sigma}_{\eta}^2=0.0378$, $ \hat{\sigma}_{u_{CPI}}^2 = 0.013$, $\hat{\sigma}_{u_{PCI}}^2=0.003$, and $\hat{\sigma}_{u_{PPI}}^2 =0.168$. These estimates reflect the relatively high idiosyncratic volatility of the producer price index series, compared to CPI and PCI. The log likelihood is $311.278$. Our results furthermore indicate that the $I(1)$ assumption for the long-run component is likely to be violated.
As a benchmark we also report results based on the fractionally cointegrated VAR (FCVAR) model of JohNie2012. Note that the two models are not nested, since they specify the fundamental shocks differently. For the FCVAR model, we estimate an integration order $\hat{b}_{FCVAR}=0.394$ $(0.025)$ that is somewhat smaller than the one obtained from our fractionally integrated unobserved components model but provides additional evidence against the $I(1)$ assumption for inflation. The smaller estimated integration order for the FCVAR model may be explained by the findings of SunPhi2004 who show that an additive $I(0)$ term can downward-bias the estimated integration order when the $I(0)$ term is not properly included in the model. Furthermore, we can calculate an estimate for $\beta$ from the orthogonal complement of the cointegrating vector of the FCVAR model and obtain $\hat{\beta}_{FCVAR} =
' $. Again, the results obtained from the FCVAR model slightly differ from the fractionally integrated unobserved components model but point to a similar direction.
Figure (ref) sketches the dynamics of the estimated common fractionally integrated component $\hat{x}_t$ and the idiosyncratic disturbances $\hat{u}_t$ together with two standard deviations (dashed). As one can see, the common component captures the dynamics of the three inflation measures well. Due to the long memory property, mean-reversion can take quite a long time, as the 1970s and the second half of the 1980s show. The disturbance terms seem to be $I(0)$, such that the long-run dynamics of the three inflation measures are well described by one common fractionally integrated trend component and, therefore, two fractional cointegration relations exist. As the figure shows, $u_t$ may be heteroskedastic and even autocorrelated. These features could be included into the model and we leave this challenge open for future research.
We compare our results with the $I(1)$ UC model that was studied in ChaMilPa2009 by estimating the latter as a benchmark. While we obtain similar estimates for the loadings in $\beta$, the log likelihood of the $I(1)$ UC model is $234.322$ and hence clearly smaller than in the fractionally integrated setup. Figure (ref) plots the long-run component estimate from the $I(1)$ UC model for US inflation together with the fractional trend estimate on the left-hand side. The other graph shows the periodogram for the two fundamental shock series that drive the long-run components and are assumed to follow Gaussian white noise processes in both models.
As the graphs show, the two trend estimates are very similar, although the solid line was generated by an $I(1)$ filter, that is an unweighted sum of past shocks, whereas the dashed line was generated by a fractional filter with $b=0.476$ that assigns decreasing weights to $\hat{\eta}_{t-h}$ as $h$ increases. The similarity of the two processes results from a violation of the white noise assumption for the fundamental shocks of the $I(1)$ UC model: As the periodogram shows, these shocks exhibit a zero at the origin, which indicates anti-persistence, whereas the periodogram of the fundamental shocks for the fractional unobserved components model does not show such violations of the white noise assumption. In addition, the exact local Whittle estimator (with $m=n^{0.65}$ as in ShiPhi2005) suggests an integration order of $-0.486$ for the fundamental shocks of the $I(1)$ trend (and $0.00$ for those of the $I(d)$ trend). Applying an $I(1)$ filter to an anti-persistent shock series with integration order $-0.486$ produces a series that is integrated of order $0.514$, instead of an $I(1)$ trend.
Estimating a misspecified $I(1)$ common trend model for US inflation therefore pollutes the fundamental shock estimates and leads to wrong conclusions about their persistence. Since inflation shocks are misleadingly assumed to exhibit a permanent impact, the $I(1)$ model produces incorrect impulse responses, whereas the $I(d)$ model captures the mean-reverting nature of inflation via the impulse response function correctly.
Since the Gaussian white noise assumption for the fundamental shocks is crucial for consistency and asymptotic normality of the ML estimator of ChaMilPa2009, a violation may yield inconsistent parameter estimates and incorrect inference. Thus, for US inflation we find that a fractional common component should be considered instead of an $I(1)$ trend component. In general, the fundamental shocks of the permanent component should be checked for (anti-)persistence.
We expect further consequences in the general multivariate $I(d)$ case that carry over from $I(1)$ UC models: If additional unobserved components are added to the model that correlate with the fundamental shocks, as e.g.\ in the correlated $I(1)$ UC model of MorNelZi2003 or the simultaneous UC model of Web2011, a violation of the $I(1)$ assumption may produce spurious cycles and bias the estimates for the latent components.
We propose a multivariate fractionally integrated unobserved components model and derive a computationally efficient modification of the Kalman filter to estimate a single, fractionally integrated common component. Furthermore, we show consistency and assess the asymptotic distribution of the maximum likelihood estimator for integration orders $b \in D = \{d \in \mathbb{R}\ | \ 0 \leq d < 3/2,\ d \neq 1/2\}$, thereby generalizing the asymptotic results of ChaMilPa2009 for a common $I(1)$ component. As we show, the maximum likelihood estimator is asymptotically normally distributed whenever $b_0 < 1/2$. For $b_0 \in (1/2, 3/2)$ the maximum likelihood estimator is also asymptotically normal, however a particular rotation of the parameter vector, corresponding to the cointegrating matrix, exhibits an asymptotically mixed normal distribution with rate $n^{b_0}$. We apply our fractionally integrated unobserved components model to extract a long-run component from three US inflation series and obtain an estimated integration order of $0.476$ for the long-run component. Due to a violation of the $I(1)$ assumption the widely applied $I(1)$ unobserved components model yields anti-persistent long-run shocks, while those from our fractionally integrated model appear to be in line with the model assumptions.
Future research could generalize our results to multiple common long-run components, potentially exhibiting different integration orders. Furthermore, a trend-cycle decomposition that allows for autocorrelated idiosyncratic shocks may yield new insights with regard to common trends and cycles for macroeconomic time series. Finally, settings with dependent shocks, such as the correlated unobserved components model of MorNelZi2003 and the simultaneous unobserved components model of Web2011, could be considered.
The authors thank Uwe Hassler, Ulrich M\"uller, Morten {\O}.\ Nielsen, Christoph Rust, the participants of the econometric seminar in Nuremberg, the department seminar at the Christian Albrechts University Kiel, the DAGStat conference 2019 in Munich, the workshop on high-dimensional time series in economics and finance 2019 in Vienna, the Annual Meeting of the German Statistical Society 2019 in Trier, the Annual Meeting of the German Economic Society 2019 in Leipzig, the Seminar on International Economic Policy at the University of Zurich, the International Conference on Computational and Financial Econometrics 2019 in London, the Symposium in Honor of Michael Hauser at WU Vienna, and the Standing Field Committee in Econometrics of the German Economic Society for many valuable comments. This work was supported by the German Research Foundation (DFG) via the projects TS283/1-1 and WE4847/4-1.