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Band-Pass Filtering with High-Dimensional Time Series
\affil[1]{ University of L'Aquila, Italy} \affil[2]{ Einaudi Institute for Economics and Finance, Rome, Italy} \affil[3]{ University of Rome “Tor Vergata", Italy}
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Nowcasting deals with the real time assessment of the underlying growth rate of the economy, with the aid of the available indirect information provided by a large number of more timely macroeconomic monthly indicators. A broad and representative measure of aggregate economic activity is offered by gross domestic product (GDP), which is however available only quarterly and with a publication delay. On the contrary, it is desirable to carry out the above assessment at least with monthly frequency.
There is a large literature on the GDP nowcasting problem. Bridge models, see \shortciteA{baffigi2004bridge} and MIxed frequency DAta Sampling models ghysels2007midas,kuzin2013pooling,foroni2014comparison, aim at nowcasting quarterly GDP using monthly indicators. Rather than targeting quarterly GDP, it can be deemed more relevant, for the purpose of monitoring the current state of the economy, to nowcast GDP at the monthly frequency. Low-dimensional exact dynamic factor models have been proposed by \shortciteA{mariano2003new}, \shortciteA{camacho2010introducing}, \shortciteA{frale2011euromind}, and \shortciteA{aruoba2016improving} for this task. More recent contributions have looked at new High-dimensional approximate factor models forni2000generalized,stock2002forecasting,stock2002macroeconomic,bai2002determining,hallin2020time play a pivotal role for nowcasting. They are based on a solid representation theory and provide the way of distilling the comovements in a large set of macroeconomic time series, without incurring in the curse of dimensionality. For applications to nowcasting see \shortciteA{giannone2008nowcasting}, \shortciteA{Altissimo2010new} and \shortciteA{banbura2013now}, among others.
This paper considers the problem of constructing monthly indicators of the medium-to-long-run (M2LR) component of GDP growth at different horizons, namely quarter-on-quarter and year-on-year growth. The reference to the M2LR component of economic growth is meant to exclude high-frequency uninteresting variation, consisting of fluctuations with periodicity less than or equal to one year.
In the unrealistic setting in which a doubly infinite sample of GDP growth measure is available at the monthly frequency, the assessment of the M2LR component of economic growth can be made by the ideal band-pass filter. The difficulty lies not only in the fact that monthly GDP is not observed; an obvious limitation of the ideal band-pass filter is its two-sided nature; more precisely, it is an infinite, two-sided, symmetric moving average of the time series. \shortciteA{baxter1999measuring}, who popularized band-pass filtering in macroeconomics, derive an approximation to the ideal band-pass filter with finite impulse response, which however leaves unaddressed estimation in real time. \shortciteA{christiano2003band} have addressed this problem by proposing forecast extensions, i.e., by replacing the missing future observations with conditional forecasts. This paper argues that forecasts extensions are not needed, and that the relevant information concerning the M2LR component of economic growth can be distilled in real time from a large cross-section of macroeconomic time series, by extracting smooth common factors and projecting GDP growth on the space spanned by them.
We build our approach on \shortciteA{Altissimo2010new}, who made an essential contribution to the above research question through the development of the New EuroCoin (NE) method. NE is a real-time estimate of the euro area GDP growth, cleaned of short-run oscillations. More precisely the ideal target for NE is the component of the GDP growth rate obtained by removing the fluctuations of a period shorter than or equal to 1 year. NE avoids the end-of-sample bias typical of two-sided filters by projecting the target onto suitable linear combinations of a large set of monthly variables. Such linear combinations are designed to discard irrelevant information (idiosyncratic and high-frequency noise) and retain relevant information (common, cyclical and long-run waves).
Against this background, this paper makes several contributions to the above literature, which can be regarded as a development and a refinement of \shortciteA{Altissimo2010new}. From the methodological standpoint, we derive the estimators of the common component and the common factors from best linear prediction principles. In particular, we show that in a finite cross-section the minimum mean square linear estimators of the components depend on data via a finite number of cross-sectional averages of the original series, represented by the generalized principal components of the respective covariance matrices. As the dimension of the cross-section diverges, the estimation error vanishes. The new results enable to clarify some aspects of the theory, among which the relation between the standard stock2002forecasting and generalized principal components forni2005generalized.
Secondly, while NE focuses only on quarterly growth horizon, the methodology is extended so as to consider all possible horizons for growth; in particular, we think that making available a measure of underlying growth at an annual rate is relevant for practitioners.
From the empirical standpoint, we provide a novel application dealing with nowcasting the M2LR component of U.S. GDP, using the popular FRED-MD dataset mccracken2016fred, consisting of 122 monthly macroeconomic time series, for estimating the smooth common factors. The outcome is a monthly indicator of underlying growth that we label US COIN (U.S. COincident INdicator). We then address the question of evaluating our indicator, by assessing the accuracy of the nowcasts of the M2LR component of GDP growth. This is not straightforward, as the target measure is not observable. Thus, we provide a discussion concerning the construction of an oracle measure that we elect as the nowcast target. A comparative validation of US COIN vis-a-vis alternative estimation strategies is then performed, on the basis of a pseudo-real time nowcasting exercise. Finally, we propose a bootstrap procedure for evaluating the finite sample estimation uncertainty of the indicator.
The plan of the paper is the following. The next section reviews the specification of the dynamic factor model and outlines the nowcasting problem. Section (ref) deals with optimal linear estimation of the common component and its M2LR component, and establishes the consistency of the estimator. In section (ref) the smooth factors are used to construct the indicator of M2LR growth at the monthly frequency. The estimation of the spectral density of the series and of the components is considered in section (ref), where we propose a parametric bootstrap method for assessing the finite-sample estimation error uncertainty. Our US COIN indicator is presented in section (ref), which provides details on the dataset and the construction of the benchmark measure of M2LR growth used to assess the methodology. Finally, in section (ref) we draw our conclusions.
Let $\{y_t, t\in \mathbb{Z}\}$ denote the logarithm of GDP in month $t$ and let $\Delta y_t = y_t-y_{t-1}$ denote its monthly growth rate. We do observe neither $y_t$ nor $\Delta y_t$, but quarterly GDP, $\{Y_\tau, \tau = 1, 2, \ldots\}$, is available every third month, at the end of times $3\tau$, $\tau = 1, 2, \ldots$. Its logarithm, $y_\tau = \ln Y_\tau$, is an approximation to the sum of the logarithms of three consecutive months, $y_\tau \simeq y_{3\tau}+y_{3\tau-1}+y_{3\tau-2}$, where the left hand side is a systematic sample of the process $(1+L+L^2)y_t$, so that every third value is observed. Here $L$ denotes the lag operator, $L^ky_t=y_{t-k}$, and $\Delta = 1-L$.
The rationale is that the M2LR component of $\Delta y_t$ can be retrieved from its projection on a fixed, but unknown, number of smooth common factors, extracted from a high-dimensional cross-section of monthly time series. The factors are the drivers of the M2LR component of an $n$-dimensional time series $\{\mathbf{x}_{t}, t\in \mathbb{Z}\}$, which is assumed to be covariance stationary for any $n$. The projection raises a problem of temporal disaggregation, as we observe the logarithmic change $$
$$ where the left hand side is a systematic sample of $(1+L+L^2)^2 \Delta y_t$.
The purpose of this section is to expose the methodology superintending the construction of the indicator of the M2LR component of GDP growth. The next subsection deals with the dynamic factor model formulated at the monthly frequency for the vector $\mathbf{x}_t$, whose elements, without loss of generality are standardized so that $\mbox{E}(x_{it})=0$ and $\mbox{Var}(x_{it})=1,$ where $x_{it}$ is the $i$-th element of the vector $\mathbf{x}_t$, $i= 1, \ldots, n$.
Further, we denote the cross-covariance function of $\mathbf{x}_t$ by $\mbox{\boldmath $\Gamma$}_k^x = \mbox{E}(\mathbf{x}_{t}\mathbf{x}_{t-k}')$, and the spectral density matrix by $\mathbf{\Sigma}^x(\theta)=\frac{1}{2\pi}\sum_{k=-\infty}^\infty \mbox{\boldmath $\Gamma$}_k^x e^{-\imath \theta k}$, where $\theta \in [-\pi, \pi]$ is the frequency in radians.
The approximate dynamic factor model represents $\mathbf{x}_t$ as the sum of two orthogonal components,
where $\mbox{\boldmath $\chi$}_t$ and $\mbox{\boldmath $\xi$}_t$ denote respectively the $n$ dimensional common and idiosyncratic components, with $\mbox{E}(\mbox{\boldmath $\chi$}_t)=\0$, $\mbox{E}(\mbox{\boldmath $\xi$}_t)=\0$, $\mbox{E}(\mbox{\boldmath $\chi$}_{t}\mbox{\boldmath $\chi$}_{t-k}') = \mbox{\boldmath $\Gamma$}_k^\chi$, $\mbox{E}(\mbox{\boldmath $\xi$}_{t}\mbox{\boldmath $\xi$}_{t-k}') = \mbox{\boldmath $\Gamma$}_k^\xi$ and $ \mbox{E}(\mbox{\boldmath $\chi$}_{t}\mbox{\boldmath $\xi$}_s') =\0, \forall t,s \in \mathbb{Z}$. The idiosyncratic processes $\{\xi_{it}\}$ are weakly dependent across the cross-sectional dimension. The above representation implies the following additive decomposition of the cross-covariance function, $\mbox{\boldmath $\Gamma$}_k^x = \mbox{\boldmath $\Gamma$}_k^\chi+\mbox{\boldmath $\Gamma$}_k^\xi$. As for the spectral density matrix: $\mathbf{\Sigma}^x(\theta)=\mathbf{\Sigma}^\chi(\theta)+\mathbf{\Sigma}^\xi(\theta),$ where we have defined $\mathbf{\Sigma}^x(\theta)=\frac{1}{2\pi}\sum_{k=-\infty}^\infty \mbox{\boldmath $\Gamma$}_k^x e^{-\imath \theta k}$, $\mathbf{\Sigma}^\chi(\theta)=\frac{1}{2\pi}\sum_{k=-\infty}^\infty \mbox{\boldmath $\Gamma$}_k^\chi e^{-\imath \theta k}$ and $\mathbf{\Sigma}^\xi(\theta)=\frac{1}{2\pi}\sum_{k=-\infty}^\infty \mbox{\boldmath $\Gamma$}_k^\xi e^{-\imath \theta k}$.
Let $\lambda_j^x(\theta), z=x,\chi, \xi,$ denote the $j$-th largest dynamic eigenvalue of $\mathbf{\Sigma}^x(\theta)$ and let $\mathbf{p}_j^x(\theta)$ be the corrisponding eigenvectors, such that $\mathbf{\Sigma}^x(\theta)\mathbf{p}_j^x(\theta)=\lambda_j^x(\theta)\mathbf{p}_j^x(\theta)$, $j=1,\ldots, n$, and the eigendecomposition $\mathbf{\Sigma}^x(\theta) = \sum_{j=1}^n \lambda_j^x(\theta)\mathbf{p}_j^x(\theta)\bar{\mathbf{p}}_j^x(\theta)$ holds, where $\bar{\mathbf{p}}_j^{x}$ is the complex conjugate transpose of $\mathbf{p}_j^x$. Similarly, consider the eigen-decompositions $\mathbf{\Sigma}^\chi(\theta) = \sum_{j=1}^n \lambda_j^\chi(\theta)\mathbf{p}_j^\chi(\theta)\bar{\mathbf{p}}_j^\chi(\theta)$, $\mathbf{\Sigma}^\xi(\theta) = \sum_{j=1}^n \lambda_j^\xi(\theta)\mathbf{p}_j^\xi(\theta)\bar{\mathbf{p}}_j^\xi(\theta)$.
We also denote with $\mu_j^x$, $\mu_j^\chi$ and $\mu_j^\xi$ the $j$-th largest eigenvalue of $\mbox{\boldmath $\Gamma$}_0^{x}$ $\mbox{\boldmath $\Gamma$}_0^{\chi}$ and $\mbox{\boldmath $\Gamma$}_0^{\xi}$, respectively, and define $\mathbf{S} = \left[ \mathbf{s}_1, \ldots, \mathbf{s}_j, \ldots, \mathbf{s}_n\right]$ the $n\times n$ orthogonal matrix whose columns are the eigenvectors of $\mbox{\boldmath $\Gamma$}_0^x$, $\mbox{\boldmath $\Gamma$}_0^x\mathbf{s}_j = \mu_j^x \mathbf{s}_j$. With obvious notation the eigendecomposition of the covariance matrix of $\mathbf{x}_t$ is $\mbox{\boldmath $\Gamma$}_0^x = \mathbf{S} \mathbf{M}_x \mathbf{S}'$, $\mathbf{M}_x = \mbox{diag}\{\mu_{1}^x, \ldots, \mu_{n}^x\}$. Finally, denote $\mathbf{M}_\chi = \mbox{diag}\{\mu_{1}^\chi, \ldots, \mu_{n}^\chi\}$.
As in \shortciteA{forni2005generalized} we make the following assumption.
Assumptions a) and c) identify the common component as a dynamic linear combination of $q$ common factors, where $q$ is the number of diverging eigenvalues, and the idiosyncratic component, which is both serially and contemporaneously weakly dependent. Assumption b) requires that the first $q+1$ eigenvalues are distinct. The space spanned by the common component has finite dimension $r$.
Under Assumption (ref) it can be shown forni2000generalized,forni2005generalized,forni2009opening that $\mathbf{x}_t$ has a static factor model representation, driven by $r$ common factors. This follows from the representation of the common component, $\mbox{\boldmath $\chi$}_t = \mathbf{B}(L)\mathbf{u}_t$, where $\mathbf{B}(L) = \mathbf{L} [\mathbf{D}(L)]^{-1}\mathbf{K}$, $\mathbf{L}$ is an $n\times r$ matrix of factor loadings, $\mathbf{D}(L) = \mathbf{I} - \sum_{j=1}^p \mathbf{D}_j L^j$ is an $r \times r$ matrix lag polinomial with roots outside the unit circle, $\mathbf{K}$ is $r\times q$, and $\mathbf{u}_t\sim \mbox{WN}(\0, \mathbf{I}_q)$ are the common shocks, also referred to as the dynamic factors.
The following proposition derives the minimum mean square linear estimator of the common component based on a finite cross-section.
Note that minimizing the mean square error matrix means minimizing its operator norm: if $\mathbf{A}$ is an $n\times m$ matrix, $\|\mathbf{A}\|_\Gamma$ is the largest singular value of $(\mbox{\boldmath $\Gamma$}_0^{x})^{-1/2'}\mathbf{A}$. The columns of the matrix $\mathbf{Z}_\chi$ are formed by the first $r$ generalized eigenvectors of the two matrices $\mbox{\boldmath $\Gamma$}_0^\chi$ and $\mbox{\boldmath $\Gamma$}_0^x$, satisfying $\mathbf{Z}_{\chi}'\mbox{\boldmath $\Gamma$}_0^\chi\mathbf{Z}_\chi = \mathbf{M}_\chi^{*}$ and $\mathbf{Z}_{\chi}'\mbox{\boldmath $\Gamma$}_0^x\mathbf{Z}_\chi = \mathbf{I}_r$. Equation ((ref)) is equivalently written $\tilde{\mbox{\boldmath $\chi$}}_t = \mbox{\boldmath $\Gamma$}_0^\chi\mathbf{Z}_\chi \mathbf{Z}_\chi'\mathbf{x}_t$.
The next proposition shows the consistency, as $n\rightarrow\infty$, of the estimator of the common component.
The asymptotic approximation of the estimator of the common component can be written
where $\mathbf{f}_t=\mathbf{Z}_\chi' \mathbf{x}_t$ are the generalized principal components, and we denoted the loadings matrix $\mathbf{A}_{\chi} = \mbox{\boldmath $\Gamma$}_0^x \mathbf{Z}_\chi$, while $\mathbf{P}_{\chi} = \mbox{\boldmath $\Gamma$}_0^x \mathbf{Z}_\chi \mathbf{Z}_{\chi}'$. The space of the common components is spanned by the $r$ columns of $\mathbf{A}_\chi=\mbox{\boldmath $\Gamma$}_0^x \mathbf{Z}_\chi.$ The vectors are orthonormal in $\mathcal{M}_x^n$, the $n$-dimensional Hilbert space endowed with the inner product $ \|\mathbf{x}\|_\Gamma = (\mathbf{x}'[\mbox{\boldmath $\Gamma$}_0^x]^{-1}\mathbf{x})^{1/2}$, as $\mathbf{A}_\chi'[\mbox{\boldmath $\Gamma$}_0^x]^{-1}\mathbf{A}_\chi = \mathbf{I}_r$. The null space is spanned by $\mbox{\boldmath $\Gamma$}_0^x \mathbf{Z}_\xi$, where $\mathbf{Z}_{\xi}$ are the $n-r$ generalized eigenvectors satisfying $\mathbf{Z}_{\xi}'\mbox{\boldmath $\Gamma$}_0^x\mathbf{Z}_\chi = \0$.
The matrix $\mathbf{P}_\chi$ is the $n \times n$ projection matrix with rank $r$, projecting $\mathbf{x}_t$ into the space of the common component. The matrix is idempotent, $\mathbf{P}_\chi^2 = \mathbf{P}_{\chi}$, and it is not symmetric. The $r\times n$ matrix $\mathbf{Z}_\chi'$ is the weak, or generalized, inverse of $\mathbf{A}_\chi$, satisfying the two properties: (i) $\mathbf{A}_\chi \mathbf{Z}_\chi'\mathbf{A}_\chi = \mathbf{A}_{\chi}$ and (ii) $\mathbf{Z}_\chi'\mathbf{A}_\chi \mathbf{Z}_\chi'= \mathbf{Z}_\chi'$. Hence, we can write, $\mathbf{A}_{\chi}^- = \mathbf{Z}_{\chi}'$, $\mathbf{P}_{\chi} = \mathbf{A}_\chi\mathbf{A}_\chi^-$. See \shortciteA{rao1974projectors} and \shortciteA[ch. 7]{seber2008matrix}. According to the terminology in \shortciteA{seber2008matrix}, $\mathbf{Z}_\chi'$ is a minimum norm reflexive $g$-inverse. It is also a least square generalized inverse, see \shortciteA{rao1971generalized}.
It is useful to compare our estimator with the standard principal components analysis (PCA) estimator of $\mbox{\boldmath $\chi$}_t$ stock2002forecasting, stock2002macroeconomic. The latter arises from the projecting of $\mathbf{x}_t$ on the space of the common component in a metric space endowed with the inner product $ \|\mathbf{x}\| = (\mathbf{x}'\mathbf{x})^{1/2}$. Let $\mathbf{S}_r$ denote the $n\times r$ matrix whose columns are the $r$ eigenvectors of $\mbox{\boldmath $\Gamma$}_0^x$ corresponding to the largest eigenvalues. The PCA estimator is $\hat{\mbox{\boldmath $\chi$}_t}^{_{PCA}} = \mathbf{S}_r\mathbf{S}_r'\mathbf{x}_t$. Denoting $\mathbf{M}_r = \mbox{diag}(\mu_1, \ldots, \mu_r)$, $$
$$ where the second expression is for comparison with $$Var(\tilde{\boldmath $\chi$_t}) = \boldmath $\Gamma$_0^x \mathbf{Z}_\chi \mathbf{M}_\chi^{*2} \mathbf{Z}_\chi'\boldmath $\Gamma$_0^x.$$
The estimation mean square error matrix is
which shows that $\mathbf{S}_r \mathbf{M}_r^{-1}\mathbf{S}_r'$ is an alternative Moore-Penrose inverse of $\mbox{\boldmath $\Gamma$}_0^\chi$ and that the two solutions are asymptotically equivalent.
The $r$ generalized principal components distill the co-movements of the $n$ macroeconomic time series that are pervasive. However, their variability contains high frequency trigonometric components that do not contribute to the M2LR growth component of GDP. \shortciteA{Altissimo2010new} consider the additive decomposition of the common component into a low-pass component, $\mbox{\boldmath $\phi$}_t$, a high-pass component, $\mbox{\boldmath $\psi$}_t$,
where $\mbox{E}(\mbox{\boldmath $\phi$}_t) = \0$, and $\mbox{E}(\mbox{\boldmath $\phi$}_t\mbox{\boldmath $\phi$}_{t-k}')=\mbox{\boldmath $\Gamma$}_k^\phi$. The cross-covariance function of $\mbox{\boldmath $\phi$}_t$ is defined as $$ \mbox{\boldmath $\Gamma$}_k^\phi = \sum_{k=1}^q \int_{-\theta_c}^{\theta_c} \lambda_k^x(\theta) \mathbf{p}_k^x(\theta)\bar{\mathbf{p}}_k^x(\theta) e^{\imath \theta k} d\theta, $$ and $0<\theta_c<\pi$ is the cut-off frequency, e.g., $\theta_c = \pi/6$. The high-pass component is assumed orthogonal to $\mbox{\boldmath $\phi$}_t$ and its cross-covariance function is $\mbox{\boldmath $\Gamma$}_k^\psi = \mbox{\boldmath $\Gamma$}_k^\chi - \mbox{\boldmath $\Gamma$}_k^\phi$.
The orthogonal decomposition in ((ref)) is achieved via the following orthogonal decomposition of the common shocks of the representation $\mbox{\boldmath $\chi$}_t = \mathbf{B}(L)\mathbf{u}_t$: $$\mathbf{u}_t = \frac{(1+L)^s}{\varphi(L)}\mbox{\boldmath $\eta$}_t + \sqrt{\varsigma}\frac{(1-L)^s}{\varphi(L)}\mbox{\boldmath $\zeta$}_t, $$ where $\mbox{\boldmath $\eta$}_t\sim\mbox{WN}(\0,\mathbf{I}_q), \mbox{\boldmath $\zeta$}_t\sim\mbox{WN}(\0, \mathbf{I}_q), \mbox{E}(\mbox{\boldmath $\eta$}_t\mbox{\boldmath $\zeta$}_t')=\0,$ $\varphi(L)$ is a scalar polynomial satisfying $$|\varphi(z)|^2 = |1+z|^{2s}+\varsigma |1-z|^{2s}, |z|\geq 1,$$ and $\varsigma$ is related to the cutoff frequency $\theta_c$ by $\varsigma = \left(\frac{1+\cos\theta_c}{1-\cos\theta_c}\right)^s.$ See \shortciteA{proietti2008model} for details. The common shocks are therefore decomposed into a low-pass component and a high-pass one. The former has spectral density $w_\phi(\theta)\mathbf{I}_q$, where $$w_\phi(\theta) =\frac{|1+e^{-\imath \theta}|^{2s}}{|\varphi(e^{-\imath \theta})|^2},$$ This decreases monotonically from 1 to 0 as $\theta$ goes from 0 to $\pi$, and taking the value $1/2$ at the frequency $\theta_c$. As $s\rightarrow \infty$, it tends to the ideal low-pass box-car spectrum $I(|\theta|<\theta_c)$, where $I(\cdot)$ is the indicator function.
Then, the low-pass and high-pass components of $\mbox{\boldmath $\chi$}_t$ are respectively defined as $$\mbox{\boldmath $\phi$}_t = \mathbf{B}(L) \frac{(1+L)^s}{\varphi(L)}\mbox{\boldmath $\eta$}_t, \;\;\;\;\mbox{\boldmath $\psi$}_t = \sqrt{\varsigma}\mathbf{B}(L) \frac{(1-L)^s}{\varphi(L)}\mbox{\boldmath $\zeta$}_t.$$ The former would be estimated on the basis of $\mbox{\boldmath $\chi$}_t$ as $w_\phi(L) \mbox{\boldmath $\chi$}_t$ whittle1963prediction; here $w_\phi(z) = \frac{|1+z|^{2s}}{|\varphi(z)|^2}$ represents the optimal Wiener-Kolmogorov filter whittle1963prediction for estimating $\mbox{\boldmath $\phi$}_t$. An approximation to the target low-pass component could be obtained by applying the filter for $s=2$ to ((ref)). This can be done by applying the state space methodology, as in \shortciteA{proietti2008model}, see also Appendix (ref). However, for $s$ large (e.g., larger than 6) the estimates become computationally unstable.
We hereby show that rather than applying a two sided filter to the estimated common component, we can estimate $\mbox{\boldmath $\psi$}_t$ by cross-sectional averaging. In the sequel we denote the $j$-th largest eigenvalue of $\mbox{\boldmath $\Gamma$}_0^\phi$ and $\mbox{\boldmath $\Gamma$}_0^\psi$ respectively by $ \mu_{j}^\phi$ and $\mu_j^{\psi}$. The following assumption enables the identification of $\mbox{\boldmath $\phi$}_t$ as $n\rightarrow \infty$.
The space of the low-pass component is a subspace of that of the common component. The following proposition characterizes the estimator of the low-pass component.
The linear combinations $\mathbf{f}_{\phi t} = \mathbf{Z}_\phi'\mathbf{x}_t$ are the $r_\phi$ smooth generalized principal components Altissimo2010new. Notice that the solution can be equivalently written $\tilde{\mbox{\boldmath $\phi$}}_t = \mbox{\boldmath $\Gamma$}_0^\phi\mathbf{Z}_\phi \mathbf{Z}_\phi'\mathbf{x}_t$. The columns of the matrix $\mathbf{Z}_\phi$ are formed from the $r_\phi$ generalized eigenvectors of $(\mbox{\boldmath $\Gamma$}_0^\phi, \mbox{\boldmath $\Gamma$}_0^x)$, satisfying $\mbox{\boldmath $\Gamma$}_0^\phi \mathbf{Z}_\phi = \mbox{\boldmath $\Gamma$}_0^x \mathbf{Z}_\phi\mathbf{M}_\phi^*,$ or equivalently $\mathbf{Z}_\phi' \mbox{\boldmath $\Gamma$}_0^\phi \mathbf{Z}_\phi = \mathbf{M}_\phi^*$ and $\mathbf{Z}_\phi'\mbox{\boldmath $\Gamma$}_0^x \mathbf{Z}_\phi = \mathbf{I}_r$. For $n\rightarrow\infty$, the sum space of $\mbox{\boldmath $\phi$}_t$ and $\mbox{\boldmath $\psi$}_t$ is the space of the common component. The consistency of $\tilde{\mbox{\boldmath $\phi$}}_t$ is considered in the next proposition:
The limiting solution can be written $\tilde{\mbox{\boldmath $\phi$}}_t = \mathbf{A}_\phi \mathbf{f}_{\phi t}$, where $\mathbf{f}_{\phi t} = \mathbf{Z}_\phi'\mathbf{x}_t$ are the smooth generalized principal components, or $\tilde{\mbox{\boldmath $\phi$}}_t= \mathbf{P}_\phi\mathbf{x}_t$, where $\mathbf{P}_\phi = \mbox{\boldmath $\Gamma$}_0^x \mathbf{Z}_\phi\mathbf{Z}_\phi'$ is idempotent projection matrix of rank $r_\phi$.
Notice that $\mathbf{I}_n = \mathbf{Q}_\phi+\mathbf{Q}_\psi+\mathbf{Q}_\xi$, where $\mathbf{Q}_\phi$ was defined by Prop. (ref), $\mathbf{Q}_\psi = \mathbf{M}_x^{-\frac{1}{2}}\mathbf{S}'\mbox{\boldmath $\Gamma$}_0^\psi\mathbf{S}\mathbf{M}_x^{-\frac{1}{2}}$, and $\mathbf{Q}_\xi = \mathbf{M}_x^{-\frac{1}{2}}\mathbf{S}'\mbox{\boldmath $\Gamma$}_0^\xi\mathbf{S}\mathbf{M}_x^{-\frac{1}{2}}.$
As $n\rightarrow \infty$, $\mathbf{Q}_\phi \rightarrow\mathbf{P}^*_\phi = \mathbf{Z}_\phi^*\mathbf{Z}_\phi^{*\prime}$, the orthogonal projection matrix into the subspace generated by $\mbox{\boldmath $\phi$}_t^*$. The subspace spanned by $\mbox{\boldmath $\phi$}_t$ is a linear transformation and the projection of $\mathbf{x}_t^*$ onto this subspace is $\mathbf{S}\mathbf{M}_x^{\frac{1}{2}}\mathbf{Z}_\phi^*\mathbf{Z}_\phi^{*\prime}\mathbf{x}_t^*$. We have $\mathbf{Z}_\phi^{*\prime}\mathbf{x}_t^* = \mathbf{Z}_\phi'\mathbf{x}_t$ and $\mathbf{S}\mathbf{M}_x^{\frac{1}{2}}\mathbf{Z}_\phi^*=\mbox{\boldmath $\Gamma$}_0^x\mathbf{Z}_\phi$. Hence $\mbox{\boldmath $\Gamma$}_0^x \mathbf{Z}_\phi \mathbf{Z}_\phi'\mathbf{x}_t$ is the projection of $\mathbf{x}_t$ onto the subspace spanned by $\mbox{\boldmath $\phi$}_t$ and $\mathbf{P}_\phi = \mbox{\boldmath $\Gamma$}_0^x\mathbf{Z}_\phi \mathbf{Z}_\phi'$ is the corresponding projection matrix.
The smooth generalized principal components are now used to construct the indicator of M2LR GDP growth at the monthly frequency. The model for unobserved monthly GDP growth assumes that $\Delta y_t$ depends on the information available in real time only via the smooth generalized principal components, that is,
Given the estimate of $\mathbf{f}_{\phi t}$ obtained in section ((ref)), the vector of loadings $\mbox{\boldmath $\vartheta$}$ can be estimated from the quarterly GDP growth rates, $g_t\approxeq (1+L+L^2)^2 \Delta y_t$, which are observed every third month. The approximation is due to the fact that the sum of the logarithms of monthly GDP is only a first order Taylor approximation of the logarithm of the sum of GDP across the three months representing the quarter.
Filtering both sides of ((ref)) by $(1+L+L^2)^2$ and taking a systematic sample with step 3, yields the approximate quarterly model
where $\mathbf{F}_\tau$ is a systematic sample of $\mathbf{F}_t = (1+L+L^2)^2 \mathbf{f}_{\phi t}$ and $\epsilon_{\tau}^*$ is the MA(1) process $\epsilon_{\tau}^* = \eta^*_\tau +b \eta^*_{\tau-1}$, $b = .221$, $\eta^*_\tau \sim \mbox{WN}(0, 19\sigma^2)$.
As for the M2LR monthly indicator of growth at the annual horizon, letting $a_t = (1+L+L^2)(1+L+\cdots+L^{11})$, and denoting by $a_\tau$ its systematic sample, we have
where $\mathbf{F}_\tau^*$ is a systematic sample of $\mathbf{F}_t^* = (1+L+L^2)(1+L+\cdots+L^{11})\mathbf{f}_{\phi t}$ and $\epsilon_\tau^\dag$ is an MA(4) process with spectral density $$\sigma_{\epsilon^\dag}(\theta) = 12\sum_{j=0}^2 \frac{\sin^2(3\theta_j/2)\sin^2(12\theta/2)}{\sin^4(\theta_j/2)}\sigma^2,\;\;\; \theta_j = \frac{\theta + 2\pi j}{3}, j=0,1,2.$$.
Our methodology aims at producing estimates of the M2LR component of GDP growth at three horizons: the monthly horizon, via $\tilde{\Delta y}_t = \tilde{\mu}+\tilde{\mbox{\boldmath $\vartheta$}}' \tilde{\mathbf{f}}_{\phi t}$, the quarterly horizon, via $\tilde{g}_t = 9\tilde{\mu}+\tilde{\mbox{\boldmath $\vartheta$}}' \tilde{F}_{t}$, and the annual horizon, via $\tilde{a}_t = 36\tilde{\mu}+\tilde{\mbox{\boldmath $\vartheta$}}' \tilde{\mathbf{F}}_t^*$.
The estimation of $\mbox{\boldmath $\phi$}_t$ requires selecting $\mbox{$r$}_{\phi}$ and computing the generalized eigenvectors of the matrix $\tilde{\mbox{\boldmath $\Gamma$}}_0^{\phi}$ in the metric of $\tilde{\mbox{\boldmath $\Gamma$}}_0^x$. While the latter is estimated from the observed time series $\{\mathbf{x}_t, t=1, \ldots, T\}$ by sample variance matrix $\tilde{\mbox{\boldmath $\Gamma$}}_0^x = \frac{1}{T} \sum_t \mathbf{x}_t\mathbf{x}_t'$, the former requires inverting the spectral density estimate $\tilde{\mathbf{\Sigma}}^\phi(\theta)$.
The spectral density of $\mathbf{x}_t$ is estimated by the Bartlett estimator $$\tilde{\mathbf{\Sigma}}^x(\theta) = \frac{1}{2\pi} \sum_{k=-M_T}^{M_T}\left(1-\frac{|k|}{M_T+1}\right)\tilde{\mbox{\boldmath $\Gamma$}}_{k}^xe^{-\imath \theta k},$$ where $\tilde{\mbox{\boldmath $\Gamma$}}_k = \frac{1}{T}\sum_{t=k+1}^T \mathbf{x}_t\mathbf{x}_{t-k}$ is the sample crosscovariance matrix at lag $k$. The window parameter is selected in the range $c T^{1/3}<M_T < cT^{1/2}$, given a positive constant $c$. The asymptotic theory for $\tilde{\mathbf{\Sigma}}^x(\theta)$ is derived in \shortciteA{forni2017dynamic}, which shows its uniform consistency with respect to $\theta$ as $T\rightarrow \infty$, under regularity conditions on the process $\mathbf{x}_t$. In our illustrations we used $M_T = 20$, which corresponds to $0.75T^{1/2}$. The spectrum is estimated at $2m+1=151$ equally spaced frequencies in the range $[-\pi, \pi]$, $\theta_h=2\pi h/(2m+1), h=-m, \ldots, m$.
The spectral density of the common component at $\theta_h$ is estimated by performing the eigendecomposition of $\tilde{\mathbf{\Sigma}}^x(\theta_h) = \sum_{k=1}^n \tilde{\lambda}_k(\theta_h) \tilde{\mathbf{p}}_k(\theta_h)\tilde{\mathbf{p}}_k^H(\theta_h)$ and setting $$\tilde{\mathbf{\Sigma}}^\chi(\theta_h) = \sum_{k=1}^q \tilde{\lambda}_k(\theta_h)\tilde{\mathbf{p}}_k(\theta_h)\tilde{\bar{\mathbf{p}}}_k (\theta_h). $$ The variance-covariance matrix of the common component is then estimated by the Riemann sum $$\tilde{\mbox{\boldmath $\Gamma$}}_0^\chi = \frac{1}{2m+1}\sum_{h=-m}^{m} \tilde{\mathbf{\Sigma}}^\chi(\theta_h),$$ whereas that of $\mbox{\boldmath $\phi$}_t$ is estimated by summing across all $2m_c+1$ frequencies in the range $-\theta_c\leq\theta_h\leq \theta_c$, where $\theta_c=\pi/6$, i.e., $$\tilde{\mbox{\boldmath $\Gamma$}}_0^\phi = \frac{1}{2m_c+1}\sum_{h=-m_c}^{m_c} \tilde{\mathbf{\Sigma}}^\chi(\theta_h).$$ The generalized principal components are then estimated as $\tilde{\phi}_t = \tilde{\mathbf{Z}}_\phi'\mathbf{x}_t$, where $\tilde{\mathbf{Z}}_\phi$ are the generalized eigenvectors of $\tilde{\mbox{\boldmath $\Gamma$}}^\phi_0$.
Finally, The loadings $\mbox{\boldmath $\vartheta$}$ are estimated by performing a band-spectrum regression engle1974band of the observed quarterly growth rates $g_\tau$ on $(1+L+L^2)^2 \tilde{\phi}_t$, sampled at quarterly intervals: denoting the Fourier transforms of $g_{\tau} $ and $\tilde{\mathbf{F}}_{\tau}$, respectively by $$J_g(\omega_j) = \frac{1}{\sqrt{2\pi \mathcal{T}}}\sum_{\tau=1}^\mathcal{T} g_{\tau} e^{-\imath \omega_j \tau}, \mathbf{J}_F(\omega_j) = \frac{1}{\sqrt{2\pi \mathcal{T}}}\sum_{\tau=1}^\mathcal{T} \tilde{\mathbf{F}}_\tau e^{-\imath \omega_j \tau},$$ where $\omega_j=2\pi j/\mathcal{T}$ are the Fourier frequencies in the range $\Omega_c = (0,\pi/6)\cup (2\pi-\pi/6, 2\pi)$ and letting $S(\omega_j) = (1+b^2+2 b \cos \omega_j)$ $$\hat{\mbox{\boldmath $\vartheta$}} = \left(\sum_{j\in \Omega_c} \mathbf{J}_F(\omega_j) \mathbf{J}_F(\omega_j)^H/S(\omega_j)\right)^{-1}\sum_{j\in \Omega_c} \mathbf{J}_F(\omega_j) J_g(\omega)/S(\omega_j).$$
The estimation error uncertainty for finite $n$ and $T$ can be evaluated by a parametric bootstrap method, proceeding along the following steps. By Theorem 9.4.4 in \shortciteA{brillinger1981time}, the sampling distribution of the spectral density estimator $\hat{\mathbf{\Sigma}}^x(\theta)$ can be approximated by a complex Wishart distribution with $\nu = T/M_T$ degrees of freedom and scale matrix $\mathbf{\Sigma}^x(\theta)/\nu$, written $\hat{\mathbf{\Sigma}}^x(\theta)\sim W_{C}\left(\nu,\mathbf{\Sigma}^x(\theta) \right)$. In view of the consistency rates in \shortciteA{forni2017dynamic}, we also considered setting the degrees of freedom $T^*= T/[M_T\log(M_T)]$.
Multiple independent draws are taken from $\hat{\mathbf{\Sigma}}^x(\theta_h)$. For each sample the dynamic eigenvalues and eigenvectors are computed, and conditional on $q$ and $r$ the smooth common component are estimated. Let $\hat{\mathbf{f}}_{\phi t}^+ =\mathbf{Z}_\phi^{+'}\mathbf{x}_t$ denote a draw of smooth generalized principal components based on the generalized eigenvectors of the matrix constructed from the simulated spectral density.
Letting $\Re\left(\cdot\right)$ and $\Im\left(\cdot\right)$ denote respectively the real and complex part of the argument, and let $ \hat{\mathbf{\Sigma}}^{x}_{\Re}(\theta_h) = \Re\left(\hat{\mathbf{\Sigma}}^{x}(\theta_h)\right)$ and $ \hat{\mathbf{\Sigma}}^{x}_{\Im}(j) = \Im \left(\hat{\mathbf{\Sigma}}^{x}(\theta_h)\right)$, we form the matrix $\hat{\mathbf{S}}(\theta_h) = \left(
\right)$ where, for $h =-m, -m+1, \ldots, m-1, m,$ $\theta_h = 2\pi h/(2m+1) \in \left[-\pi;\pi\right]$; the resampling algorithm operates as follows.
For $b = 1,2,\dots,B$,
The draws $\tilde{g}_{t}^{(b)}$ and $\tilde{a}_{t}^{(b)}$, $b = 1, \ldots, B$, can be used to estimate the density nowcasts of GDP underlying growth, that we denote by $\tilde{f}_t^{g}(g_t|\mathcal{I}_t)$ and $\tilde{f}_t^{a}(a_t|\mathcal{I}_t)$, respectively, where $\mathcal{I}_t$ is the information set available at time $t$.
The natural testbed for illustrating and assessing our methodology is the FRED-MD monthly database compiled by \shortciteA{mccracken2016fred}. The dataset consists of monthly observations on 122 U.S. macroeconomic time series observed from January 1959 to December 2019, grouped into 8 categories, see Appendix (ref). The original time series are subjected to a stationarity inducing transformation, according to \shortciteA{mccracken2016fred}. The target series, quarterly GDP (Billions of Chained 2012 Dollars, seasonally adjusted) is also made available at the St. Luis FED Economic Data (https://fred.stlouisfed.org/series/GDPC1).
Our methodology produces monthly nowcasts of the M2LR component of GDP growth $g_t$ (quarterly horizon) and $a_t$ (annual horizon), that will be referred to as US COIN (U.S. COincident INdicators) in the sequel. For measuring their accuracy, as already discussed in \shortciteA{Altissimo2010new}, we need to target a measure of M2LR. The latter cannot be observed directly, but it can be estimated by an approximation to the ideal low-pass filter, the \shortciteA[BK]{baxter1999measuring} filter, to the GDP growth rates interpolated at the monthly frequency. The BK filter is a two-sided symmetric filter that results from truncating the ideal low-pass filter by considering 36 (3 years of monthly data) past and future observations, along with the reference time, and rescaling the impulse response weights so that they sum up to 1. The filter is applied to the interpolated monthly proxies of $g_t$ and $a_t$, obtained by applying the Whittaker-Kotel'nikov-Shannon sampling theorem, see \shortciteA[Theorem 7.2.2]{partington1997interpolation}, to the extended series, as detailed in Appendix (ref). Appendix (ref) also discusses the robustness of this measure, vis-a-vis an alternative estimate based on a model-based band-pass filter.
Figure (ref) displays the monthly target measure of M2LR (red line), which provides the basis for the evaluation of our method, both for quarter-on-quarter (q-o-q) and the year-on-year (y-o-y) growth horizons. Notice that the target measure is an oracle, as it uses also future observations. It is the value the M2LR component would take in month $t$, were GDP growth available for the future 3 years.
For evaluating the performance we perform a pseudo-real time rolling nowcasting experiment using, as a test sample the period January 1980-December 2018. We use as initial training sample the first 241 observations (1960:3 - 1980:1) to estimate the M2LR at the end of the sample (January 1980), that we then compare to the benchmark estimate of the M2LR component of GDP growth. The training sample is then moved forward, by adding one observation at the end and removing the first observation, so as to cover the period 1960:4 - 1980:2; US COIN is re-estimated and a new nowcast for February 1980 is made available. Furthermore, the rank $r$ of the variance covariance matrix $\hat{\mbox{\boldmath $\Gamma$}}_{0}$ and of the spectral density $\mathbf{\Sigma}(\theta)$ is determined by means of the BN criterion proposed by \shortciteA{bai2002determining} and the HL criterion proposed by \shortciteA{hallin2007determining} respectively. This is iterated until we reach the end of the test sample (December 2018).
For a comparative assessment, we could think of alternative ways to nowcasts and forecasts the M2LR component of GDP growth, based on extending the quarterly GDP series by out-of-sample predictions. We envisaged three competitors.
To obtain the monthly nowcast for the methods BP and CF, the interpolated monthly GDP growth proxies of $g_t$ and $a_t$ are obtained by applying the Whittaker-Kotel'nikov-Shannon formula, see Appendix (ref); hence, the BK filter with cut-off frequency $\pi/6$ is applied to the demeaned series, and finally the mean is added back to the filtered series to obtain the three nowcasts of the M2LR of GDP, which are compared to the USCOIN nowcast.
For the sake of notation simplicity, we denote the q-on-q and y-on-y M2LRG targets at time $t$ by $\mathcal{C}_t$, and the estimate based on the rolling sample ending at time $s$ by $\hat{\mathcal{C}}_t(s)$.
The ability of the nowcast $\hat{\mathcal{C}}_t(t)$, to approximate the target $\mathcal{C}_t$ is measured by the mean-square nowcast error (MSNE), $\sum_{t=1}^{T} [\hat{\mathcal{C}}_t(t) - \mathcal{C}_t]^2/T$, where $T$ is the length of the test sample.
We also consider the size of the revision after one month, as we move from one rolling sample to the next. This is measured by the mean-square revision error (MSRE), $\sum_{t=1}^{T-1} [\hat{\mathcal{C}}_{t-1}(t) - \hat{\mathcal{C}}_{t-1}(t-1)]^2/T$.
Table (ref) reports the values of the two statistics, as a fraction of the target variance, $\sum_{t=1}^{T} (\mathcal{C}_{t} - \bar{\mathcal{C}})^2/T, \bar{\mathcal{C}} = \sum_t C_t/T$. The US COIN indicator outperforms its competitors at both horizons in terms of closeness to the target measure. Its MSNE for the q-o-q horizon is 12% smaller than SW, the best performing competitor. The Diebold-Mariano diebold2002comparing test of equal predictive accuracy under square loss leads to a rejection of the null at the 10% significance level For the y-o-y horizon it improves slightly over CF. The size of the revision errors is relatively small for all indicators, except for SW, which displays a larger MSRE.
The bootstrap methodology described in section (ref) produces draws $\tilde{g}_{t}^{(b)}$ and $\tilde{a}_{t}^{(b)}$, $b = 1, \ldots, B$, from the distribution of the M2LR indicators conditional on a finite $n$ and finite $T$ panel of macroeconomic indicators and past quarterly GDP data. Let $\tilde{f}_t^{g}(g_t|\mathcal{I}_t)$ and $\tilde{f}_t^{a}(a_t|\mathcal{I}_t)$ denote respectively the density nowcasts of the q-o-q and y-o-y M2LR estimates, where $\mathcal{I}_t$ is the information set available at time $t$. The variability of the nowcasts is essentially due to the finite sample estimation error uncertainty concerning the smooth principal components, which is in turn ascribed to that concerning the spectral density matrix.
In other words, if we had an infinite cross-section observed for $T$ large, then $\mathbf{f}_{\phi t}$ is asymptotically observed and the only parameter uncertainty would arise for the estimation of the loadings of GDP on the generalized principal components, denoted $\mbox{\boldmath $\vartheta$}$ in ((ref)) and ((ref)). However, for finite $n$ and $T$, $f_{\phi t}$ is affected by measurement error, so that $\tilde{\mathbf{f}}_t^{g}(g_t|\mathcal{I}_t)$ and $\tilde{f}_t^{a}(a_t|\mathcal{I}_t)$ enable to quantify the nowcast uncertainty, by considering the dispersion of the projection of GDP growth on the estimated smooth generalized principal components.
Figure (ref) displays the evolution over time of the nowcasting densities of the q-o-q indicator (top figure) and y-o-y indicator (bottom figure). It is a fan chart with the different shades of red corresponding to the deciles of the distribution. It should be remarked that the densities are estimated in real time using rolling samples of 20 years of data. If a recursively updated training set were used instead, the dispersion would be lower at the cost of larger biases, as the spectral density estimates would be less localized.
To examine whether the nowcast densities are properly calibrated, we evaluate the cumulative distribution function of the probability integral transform of $\mathcal{C}_t$, $PIT_{t} = \int_{-\infty}^{\mathcal{C}_t} \tilde{f}_t^{i}(c|\mathcal{I}_t) d c, i=g,a$, and assess the departure from a standard uniform distribution according to \shortciteA{diebold1998evaluating}, using the critical values obtained in \shortciteA{rossi2019alternative}. Figure (ref) shows that the q-o-q nowcasts are well calibrated. The y-o-y nowcasts are marginally violating the 95% bounds. This predictive failure is due to overprediction of GDP growth occurring around 2009 during the initial phase of the great recession; it can be considered a common feature of real time macroeconomic forecasting.
The paper has considered the objective of estimating a monthly indicator of the M2LR component of GDP growth at the quarter-on-quarter and year-on-year horizons. The reference framework was the dynamic factor model, which was used to distill the low frequency variation of a high-dimensional set of macroeconomic variables. After establishing the finite sample properties of the estimator of the smooth factors, the empirical application to nowcasting the underlying US GDP growth has illustrated that the smooth factors offer a sizable advantage in terms of nowcasting accuracy towards a well defined target measure.
In our empirical exercise we have purposively ignored the time series observations after March 2020, following a major structural break, related to the economic effects of the Covid-19 pandemic crisis. See \shortciteA{ng2021modeling} for a discussion of those effects and a potential solution to the issue of estimating the smooth factors in the extended sample, covering the pandemic shock and the subsequent recovery. We leave this important topic to future research.