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Monthly US business cycle indicators: a new multivariate approach based on a band-pass filter

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Abstract

This article proposes a new multivariate method to construct business cycle indicators. The method is based on a decomposition into trend-cycle and irregular. To derive the cycle, a band-pass filter is applied to the estimated trend-cycle. The whole procedure is fully model based. Its performance is evaluated in relation to the approach by Creal et al. (J Appl Econom 25:695–719, 2010). Using the same set of monthly and quarterly US time series as in Creal et al. two monthly business cycle indicators are obtained for the US. They are represented by the cycles of real GDP and the industrial production index. Both indicators can reproduce previous recessions very well. Series contributing to the construction of both indicators are allowed to be leading, lagging, or coincident relative to the business cycle. Their behavior is assessed by means of spectral concepts after cycle estimation. The proposed method can serve as an attractive tool for policy making, in particular due to its good forecasting performance and quite simple setting without elaborate mechanisms that account for, e.g., volatility changes.

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Notes

  1. For more details on the choice of parameters, see the discussion paper version of this article.

  2. Details on the derivation of \(\mathbf {c}_t\) and \(\mathbf {p}_t\) are provided in Appendix 1.2.

  3. The reader interested in the details about the covariance square root Kalman smoother is referred to the discussion paper version of this article.

  4. An alternative treatment of aggregation patterns relies on the so-called cumulator variables (see Harvey 1989, pp. 306–239). Proietti and Moauro (2006) offer an estimation and signal extraction approach for nonlinear models resulting from the integration of cumulator variables if the models are defined for series in logs.

  5. All computations have been performed with MATLAB R2012b (64-bit) using the SSM Matlab toolbox by Gómez (2012) and procedures written by Víctor Gómez.

  6. For the sake of clarity, the time span in Fig. 2 is restricted to 1972.Q1–2007.Q3.

  7. Instead of the BK filter we could have used an asymmetric band-pass filter, like the one proposed by Christiano and Fitzgerald (2003), to avoid the computation of forecasts. However, it is well known that asymmetric filters induce a phase shift in the estimated cycles relative to the observed series which is an undesirable property. The choice of a symmetric band-pass filter for the comparison can be additionally motivated by the fact that the remaining detrending methods applied in this article are also symmetric filters.

  8. Note that the range of the phase angle is constrained to the interval \([-\pi , -\pi ]\).

  9. The confidence bounds for the estimates of the phase angle and the mean phase angles have been constructed as described in Koopmans (1974, pp. 285–287), and Fisher and Lewis (1983), respectively. All computations for the lead–lag analysis have been performed with MATLAB R2012b (64-bit) using the Spectran toolbox by Marczak and Gómez (2012).

  10. In the forecasting exercise of this section, we do not use real-time data. As argued earlier, the article tries to highlight appealing properties of the presented methodology by relating it to the CKZ study. To facilitate the comparison with the CKZ model and its results, we use exactly the same dataset as CKZ.

  11. Most of the data have been collected from the FRED database of the Federal Reserve Bank of St. Louis. Consumption (personal consumption expenditure) of nondurable and durable goods, services, and investment (fixed private investment), all given in chained dollars, have been downloaded at the Web site of the US Bureau of Economic Analysis.

  12. It is reminded that in the remaining empirical applications of this article we did not account for aggregation constraints in the case of quarterly time series as our primary aim was to identify the timing of recessions (indicated by peaks and the subsequent troughs of the IPI or real GDP cycle) and not the magnitude of the fluctuations. In this forecasting experiment, on the other hand, aggregation constraints matter more. Since predictions of the year-over-year GDP growth involve published quarterly GDP of the preceding year, for consistency reasons we impose that both the predictions and the observed values follow the same aggregation pattern. It is, however, to be noted that for series in logs aggregation constraints hold approximately.

  13. The US Bureau of Economic Analysis compiles three GDP estimates—1, 2, and 3 months after the previous quarter. We do not consider 1 month of delay since the first estimate is preliminary and may deviate from the revised figure too strongly.

  14. The derivation of the polynomial \(\theta _z(B)\) and the variance \(\sigma ^2_a\) is provided by Gómez (2001, p. 371). Without loss of generality, we set \(\sigma ^2_{b}=1\) for the filter used in this article. Then, for this filter \(\sigma _{n}\) = 437.19 and \(\sigma _{a}\) = 568.58.

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Correspondence to Martyna Marczak.

Appendices

Appendix 1: Models for the time series components

1.1 Appendix 1.1: Reduced-form model for the trend-cycle

The reduced-form model of the trend-cycle is the following equation derived from model (2):

$$\begin{aligned} \Delta ^2 {\varvec{\mu }}_{t+1} = \mathbf {K}\eta _{t-1} + \Delta {\varvec{\zeta }}_{t} \end{aligned}$$
(8)

Taking into account that for any square matrix \(\mathbf {M}\), its square root is defined as matrix \(\mathbf {M}^{1/2}\) satisfying \(\mathbf {M}^{1/2}\mathbf {M}^{1/2'}\) = \(\mathbf {M}\), we let \({\varvec{\zeta }}_{t-1}\) = \(\mathbf {D}^{1/2}_{\zeta }\mathbf {u}_{\zeta ,t}\) and \(\eta _{t-1}\) = \(\sigma _{\eta }u_{\eta ,t}\). It is thereby assumed that \(\mathbf {u}_{\zeta ,t}\) and \(u_{\eta ,t}\) are white noise processes with Var\((\mathbf {u}_{\zeta ,t})\) = \(\mathbf {I}\) and Var\((u_{\eta ,t})\) = 1, respectively. Then, Eq. (8) can be rewritten as

$$\begin{aligned} \Delta ^2 {\varvec{\mu }}_t= & {} \mathbf {K}\eta _{t-2} + ({\varvec{\zeta }}_{t-1} - {\varvec{\zeta }}_{t-2}) \\= & {} \mathbf {K}\sigma _{\eta } u_{\eta ,t-1} + \mathbf {D}_{\zeta }^{1/2} \mathbf {u}_{\zeta ,t} - \mathbf {D}_{\zeta }^{1/2} \mathbf {u}_{\zeta ,t-1}, \end{aligned}$$

By defining \(\mathbf {v}_{t}\) = \([\mathbf {u}'_{\zeta ,t}, u_{\eta ,t}]'\) with Var\((\mathbf {v}_t)\) = \(\mathbf {I}\), the following reduced-form model for \({\varvec{\mu }}_t\) can be obtained:

$$\begin{aligned} \Delta ^2 {\varvec{\mu }}_t= & {} \mathbf {C}_0 \mathbf {v}_t + \mathbf {C}_1 \mathbf {v}_{t-1} \nonumber \\= & {} \mathbf {C}(B) \mathbf {v}_t, \end{aligned}$$
(9)

where B is the backshift operator such that \(B\mathbf {v}_{t}\) = \(\mathbf {v}_{t-1}\), and \(\mathbf {C}(B)\) = \(\mathbf {C}_0 + \mathbf {C}_1B\) is a matrix polynomial in B with

$$\begin{aligned} \mathbf {C}_0 = \begin{bmatrix} \mathbf {D}_{\zeta }^{1/2}&0 \end{bmatrix}, \qquad \mathbf {C}_1 = \begin{bmatrix} - \mathbf {D}_{\zeta }^{1/2}&\mathbf {K}\sigma _{\eta } \end{bmatrix} \end{aligned}$$
(10)

1.2 Appendix 1.2: Models for the cycle and trend

From Eqs. (3) and (4), the reduced-form model for \(\mu _{i,t}\) is

$$\begin{aligned} \delta _{z}(B) \mu _{i,t} = \theta _{z}(B)a_{t}, \end{aligned}$$

where \(\theta _{z}(B)\) is of degree 2d. The Wiener–Kolmogorov filters to estimate \(c_{i,t}\) and \(p_{i,t}\) in Eq. (3) are

$$\begin{aligned} h_{c} = \frac{\sigma ^2_b}{\sigma ^2_a}\frac{(1- B^2)^d(1- F^2)^d}{\theta _{z}(B)\theta _{z}(F)},\qquad h_{p} = \frac{\sigma ^2_p}{\sigma ^2_a} \frac{\delta _{z}(B)\delta _{z}(F)}{\theta _{z}(B)\theta _{z}(F)}, \end{aligned}$$

respectively, where F is the forward operator, \(\sigma ^2_b\) = Var\((b_{t})\), \(\sigma ^2_p\) = Var\((p_{t})\), and \(\sigma ^2_a\) = Var\((a_{t})\).Footnote 14

The fixed band-pass filter can be integrated into a model-based approach. To show this, we first consider the pseudo-covariance generating function (CGF) of \(\mu _{i,t}\). Denoted by \(f_{\mu _i}\), the CGF of \(\mu _{i,t}\) can be decomposed as follows:

$$\begin{aligned} f_{\mu _i}&= h_{c}f_{\mu _i}+(1-h_{c})f_{\mu _i} \\&= f_{c_i}+f_{p_i}, \end{aligned}$$

where \(f_{c_i} = h_{c}f_{\mu _i}\) and \(f_{p_i} = (1-h_{c})f_{\mu _i}\). This decomposition defines the decomposition of \(\mu _{i,t}\) into two orthogonal unobserved components, the cycle, \(c_{i,t}\) and the trend, \(p_{i,t}\), with CGFs \(f_{c_i}\) and \(f_{p_i}\), respectively. The models for \(c_{i,t}\) and \(p_{i,t}\) are obtained from their CGFs.

These considerations can be generalized to the case of multivariate cycle, \(\mathbf {c}_t\), and multivariate trend, \(\mathbf {p}_t\). To this end, we use the CGF of the reduced-form model for \({\varvec{\mu }}_t\) in Eq. (9). We also use multivariate counterparts of the filters \(f_{c}\) and \(f_{p}\). Then, the CGF of \(\mathbf {c}_{t}\), \(\mathbf {f}_c\) can be written as

$$\begin{aligned} \mathbf {f}_{c}= & {} {\varvec{\theta }}_{z}^{-1}(B)(\mathbf {I}- B^2\mathbf {I})^{d}(\mathbf {I}-B\mathbf {I})^{-2}(\mathbf {C}_0 + \mathbf {C}_1 B)\frac{\sigma ^2_b}{\sigma ^2_a}\\&(\mathbf {C}'_0 + \mathbf {C}'_1 F)(\mathbf {I}-F\mathbf {I})^{-2}(\mathbf {I}- F^2\mathbf {I})^{d}{\varvec{\theta }}_{z}^{-1}(F) \\= & {} {\varvec{\theta }}_{z}^{-1}(B)(\mathbf {I}- B\mathbf {I})^{d-2}(\mathbf {I}+B\mathbf {I})^{d}(\mathbf {C}_0 + \mathbf {C}_1 B)\frac{\sigma ^2_b}{\sigma ^2_a}\\&(\mathbf {C}'_0+ \mathbf {C}'_1 F)(\mathbf {I}+F\mathbf {I})^{d}(\mathbf {I}- F\mathbf {I})^{d-2}{\varvec{\theta }}_{z}^{-1}(F), \end{aligned}$$

where \({\varvec{\theta }}_{z}(B) = \theta _{z}(B)\mathbf {I}\), and \(\mathbf {C}_0\) and \(\mathbf {C}_1\) are as in Eq. (10). The model for \(\mathbf {c}_t\) is thus given by Eq. (6). In a similar way, it can be shown that the model for \(\mathbf {p}_t\) resulting from \(\mathbf {f}_p\) is given by Eq. (7).

Appendix 2: State space representations

1.1 Appendix 2.1: Monthly model with the trend-cycle

A state space form for the trend-cycle in Eq. (2) is

$$\begin{aligned} {\varvec{\alpha }}_{t+1}&= \mathbf {T}_{\mu }{\varvec{\alpha }}_{t} + \mathbf {H}_{\mu } \mathbf {v}_{t} \nonumber \\ {\varvec{\mu }}_{t}&= \mathbf {Z}_{\mu }{\varvec{\alpha }}_{t}, \end{aligned}$$
(11)

where \({\varvec{\alpha }}_{t}\) = \([{\varvec{\mu }}^{\prime }_{t}, \beta _{t}]^{\prime }\), \(\mathbf {v}_t\) is defined in Appendix 1.1, and

$$\begin{aligned}&\mathbf {T}_{\mu } = \begin{bmatrix} \mathbf {I}_{k}&\quad \mathbf {K}\\ \mathbf{0}&\quad \mathbf {I}_{r} \\ \end{bmatrix}, \quad \mathbf {H}_{\mu } = \begin{bmatrix} \mathbf {D}^{1/2}_{\zeta }&\quad \mathbf{0} \\ \mathbf{0}&\quad \sigma _{\eta } \\ \end{bmatrix}, \nonumber \\&\mathbf {Z}_{\mu } = \begin{bmatrix} \mathbf {I}_{k}&\mathbf{0} \\ \end{bmatrix}, \quad r=1 \end{aligned}$$
(12)

Then, the state space form for the monthly model is

$$\begin{aligned} {\varvec{\alpha }}_{t+1}&= \mathbf {T}{\varvec{\alpha }}_t + \mathbf {H}\mathbf {u}_t \nonumber \\ \mathbf {y}_t&= \mathbf {Z}{\varvec{\alpha }}_t + \mathbf {G}\mathbf {u}_t, \quad t=1,\ldots ,n, \end{aligned}$$
(13)

where \(\mathbf {u}_t = [\mathbf {v}^{\prime }_t, \mathbf {u}^{\prime }_{\epsilon ,t}]^{\prime }\) and \(\mathbf {u}^{\prime }_{\epsilon ,t}\) is such that \({\varvec{\epsilon }}_t = \mathbf {D}_{\epsilon }^{1/2} \mathbf {u}_{\epsilon ,t}\) with Var\((\mathbf {u}_{\epsilon ,t})\) = \(\mathbf {I}\). Hence, it holds that Var\((\mathbf {u}_t)\) = \(\mathbf {I}\). The matrices in model (13) are given as: \(\mathbf {T}=\mathbf {T}_{\mu }\), \(\mathbf {Z}= \mathbf {Z}_{\mu }\) and

$$\begin{aligned} \mathbf {H}= \begin{bmatrix} \mathbf {D}^{1/2}_{\zeta }&\quad \mathbf{0}&\quad \mathbf{0} \\ \mathbf{0}&\quad \sigma _{\eta }&\quad \mathbf{0} \\ \end{bmatrix}, \quad \mathbf {G}= \begin{bmatrix} \mathbf{0}&\quad \mathbf{0}&\quad \mathbf {D}^{1/2}_{\epsilon } \\ \end{bmatrix} \end{aligned}$$

The initial state vector \({\varvec{\alpha }}_{1}\) = \([{\varvec{\mu }}^{\prime }_{1},\beta _{1}]^{\prime }\) is

$$\begin{aligned} {\varvec{\alpha }}_{1} = \mathbf {A}{\varvec{\delta }}+ \overline{\mathbf {p}}, \end{aligned}$$

where \({\varvec{\delta }}\) has dimension \(k+r\) and is diffuse, \(\mathbf {A}\) is a suitable nonstochastic matrix, and \(\overline{\mathbf {p}}\) has zero mean and a well defined covariance matrix.

1.2 Appendix 2.2: Monthly model including the cycle

For numerical reasons, the model for \(\mathbf {p}_{t}\) in Eq. (7) is implemented in cascade form as

$$\begin{aligned} \mathbf {p}_{t} = \left[ {\varvec{\theta }}_{z}^{-1}(B){\varvec{\delta }}_{z}(B)\right] \mathbf {w}_{t}, \end{aligned}$$
(14)

where \(\mathbf {w}_{t}\) follows the model

$$\begin{aligned} \mathbf {w}_{t} = \left[ (\mathbf {I}-B\mathbf {I})^{-2} \widetilde{\mathbf {C}}(B)\right] \widetilde{\mathbf {v}}_{t} \end{aligned}$$

A state space model for \(\mathbf {w}_{t}\) can be easily derived from (11), namely

$$\begin{aligned} {\varvec{\gamma }}_{t+1}&= \mathbf {T}_{w}{\varvec{\gamma }}_t + \mathbf {H}_{w}\widetilde{\mathbf {v}}_t,\\ \mathbf {w}_t&= \mathbf {Z}_{w}{\varvec{\gamma }}_t, \end{aligned}$$

where \(\mathbf {T}_w = \mathbf {T}_{\mu }\), \(\mathbf {Z}_w = \mathbf {Z}_{\mu }\), and \(\mathbf {H}_w = (\sigma _n/\sigma _a)\mathbf {H}_{\mu }\), and the matrices \(\mathbf {T}_{\mu }\), \(\mathbf {Z}_{\mu }\) and \(\mathbf {H}_{\mu }\) are given in (12). As for \(\mathbf {p}_t\) in Eq. (14), we select the multivariate version of the state space representation used by Gómez and Maravall (1994), which is an extension to the nonstationary case of the approach proposed by Akaike (1974). Thus, the state space representation of (14) is

$$\begin{aligned} {\varvec{\xi }}_t&= \mathbf {T}_v {\varvec{\xi }}_{t-1} + \mathbf {H}_v \mathbf {w}_t \nonumber \\ \mathbf {p}_t&= \mathbf {Z}_v {\varvec{\xi }}_t, \end{aligned}$$
(15)

where \({\varvec{\xi }}_t = [\mathbf {p}^{\prime }_{t}, \mathbf {p}^{\prime }_{t+1|t},\ldots ,\mathbf {p}^{\prime }_{t+q|t})^{\prime }\), \(\theta _z(B)\) = \(1+\sum _{i=1}^{q} \theta _{z,i}B^{i}\), \(q = 2d\) is the degree of both polynomials, \(\theta _z(B)\) and \(\delta _z(B)\),

$$\begin{aligned}&\mathbf {T}_v = \begin{bmatrix} \mathbf{0}&\quad \mathbf {I}&\quad \mathbf{0}&\quad \cdots&\quad \mathbf{0} \\ \mathbf{0}&\quad \mathbf{0}&\quad \mathbf {I}&\quad \cdots&\quad \mathbf{0} \\ \vdots&\quad \vdots&\quad \vdots&\quad \ddots&\quad \vdots \\ \mathbf{0}&\quad - \theta _{z,q}\mathbf {I}&\quad - \theta _{z,q-1}\mathbf {I}&\quad \cdots&\quad - \theta _{z,1}\mathbf {I}\end{bmatrix}, \quad \mathbf {H}_v = \begin{bmatrix} \mathbf {I}\\ V_1 \mathbf {I}\\ \vdots \\ V_q \mathbf {I}\end{bmatrix}, \nonumber \\&\mathbf {Z}_v = \begin{bmatrix} \mathbf {I}&\mathbf{0}&\cdots&\mathbf{0} \end{bmatrix}, \end{aligned}$$
(16)

and \(V_i\), \(i=0,\ldots ,q\), are the coefficients obtained from \(V(B) = \delta _z(B)/ \theta _z(B)\). Thus, the state space model for the cascade form of the model for \(\mathbf {p}_t\) described earlier is

$$\begin{aligned} {\varvec{\varphi }}_{t+1}&= \mathbf {T}_p {\varvec{\varphi }}_{t} + \mathbf {H}_p \widetilde{\mathbf {v}}_{t+1} \nonumber \\ \mathbf {p}_t&= \mathbf {Z}_p {\varvec{\varphi }}_t, \end{aligned}$$
(17)

where \({\varvec{\varphi }}_t = \left[ {\varvec{\xi }}^{\prime }_t,{\varvec{\gamma }}^{\prime }_{t+1}\right] ^{\prime }\) and

$$\begin{aligned} \mathbf {T}_p = \begin{bmatrix} \mathbf {T}_v&\quad \mathbf {H}_v \mathbf {Z}_w \\ \mathbf{0}&\quad \mathbf {T}_w \end{bmatrix}, \quad \mathbf {H}_p = \begin{bmatrix} \mathbf{0} \\ \mathbf {H}_w \end{bmatrix}, \quad \mathbf {Z}_p = \begin{bmatrix} \mathbf {Z}_v&\mathbf{0} \end{bmatrix} \end{aligned}$$

Similarly to (15), the state space form considered for \(\mathbf {c}_t\) in Eq. (6) is

$$\begin{aligned} {\varvec{\chi }}_{t+1}&= \mathbf {T}_c {\varvec{\chi }}_{t} + \mathbf {H}_c \overline{\mathbf {v}}_{t+1} \nonumber \\ \mathbf {c}_t&= \mathbf {Z}_c {\varvec{\chi }}_t, \end{aligned}$$
(18)

where \({\varvec{\chi }}_t = \left[ \mathbf {c}^{\prime }_{t},\mathbf {c}^{\prime }_{t+1|t},\ldots ,\mathbf {c}^{\prime }_{t+q-1|t}\right] '\),

$$\begin{aligned}&\mathbf {T}_c = \begin{bmatrix} \mathbf{0}&\quad \mathbf {I}&\quad \mathbf{0}&\quad \cdots&\quad \mathbf{0} \\ \mathbf{0}&\quad \mathbf{0}&\quad \mathbf {I}&\quad \cdots&\quad \mathbf{0} \\ \vdots&\quad \vdots&\quad \vdots&\quad \ddots&\quad \vdots \\ - \theta _{z,q}\mathbf {I}&\quad - \theta _{z,q-1}\mathbf {I}&\quad - \theta _{z,q-2\mathbf {I}}&\quad \cdots&\quad - \theta _{z,1}\mathbf {I}\end{bmatrix}, \quad \mathbf {H}_c = \begin{bmatrix} \mathbf {I}\\ \mathbf {Q}_1 \\ \vdots \\ \mathbf {Q}_{q-1} \end{bmatrix}, \\&\mathbf {Z}_c = \begin{bmatrix} \mathbf {I}&\mathbf{0}&\cdots&\mathbf{0} \end{bmatrix}, \end{aligned}$$

and \(\mathbf {Q}_i\), \(i=0,\ldots ,q-1\), are the coefficient matrices of the following polynomial

$$\begin{aligned} \mathbf {Q}(B) = {\varvec{\theta }}_z^{-1}(B)(\mathbf {I}-B\mathbf {I})^{d-2}(\mathbf {I}+B\mathbf {I})^d \overline{\mathbf {C}}(B) \end{aligned}$$

Taking models (17) and (18) into account, the state space form for \({\varvec{\mu }}_{t}\) = \(\mathbf {p}_{t} + \mathbf {c}_{t}\) is

$$\begin{aligned} {\varvec{\alpha }}_{t+1}&= \begin{bmatrix} \mathbf {T}_p&\quad \mathbf{0} \\ \mathbf{0}&\quad \mathbf {T}_c \end{bmatrix} {\varvec{\alpha }}_t + \begin{bmatrix} \mathbf {H}_p&\quad \mathbf{0} \\ \mathbf{0}&\quad \mathbf {H}_c \end{bmatrix} \begin{bmatrix} \widetilde{\mathbf {v}}_{t+1} \\ \overline{\mathbf {v}}_{t+1} \end{bmatrix} \\ {\varvec{\mu }}_t&= \begin{bmatrix} \mathbf {Z}_p&\quad \mathbf {Z}_c \end{bmatrix} {\varvec{\alpha }}_t, \end{aligned}$$

where \({\varvec{\alpha }}_t = [{\varvec{\varphi }}'_t,{\varvec{\chi }}'_t]'\). Thus, the state space form for \(\mathbf {y}_{t}\) is

$$\begin{aligned} {\varvec{\alpha }}_{t+1}&= \mathbf {T}{\varvec{\alpha }}_t + \mathbf {H}\mathbf {u}_t \\ \mathbf {y}_t&= \mathbf {Z}{\varvec{\alpha }}_t + \mathbf {G}\mathbf {u}_t, \quad t=1,\ldots ,n, \end{aligned}$$

where \(\mathbf {u}_{t}\) = \([\widetilde{\mathbf {v}}'_{t+1},\overline{\mathbf {v}}'_{t+1},\mathbf {u}_{\epsilon ,t}']'\), Var\((\mathbf {u}_{t})\) = \(\mathbf {I}\), and

$$\begin{aligned}&\mathbf {T}= \begin{bmatrix} \mathbf {T}_{p}&\quad \mathbf{0}\\ \mathbf{0}&\quad \mathbf {T}_{c} \\ \end{bmatrix}, \quad \mathbf {H}= \begin{bmatrix} \mathbf {H}_{p}&\quad \mathbf{0}&\quad \mathbf{0} \\ \mathbf{0}&\quad \mathbf {H}_{c}&\quad \mathbf{0} \\ \end{bmatrix}, \\&\mathbf {Z}= \begin{bmatrix} \mathbf {Z}_{p}&\quad \mathbf {Z}_{c} \end{bmatrix},\quad \mathbf {G}= \begin{bmatrix} \mathbf{0}&\quad \mathbf{0}&\quad \mathbf {D}^{1/2}_{\epsilon } \\ \end{bmatrix} \end{aligned}$$

The initial state vector \({\varvec{\alpha }}_{1}\) = \([{\varvec{\varphi }}'_{1},{\varvec{\chi }}'_{1}]'\), where \({\varvec{\varphi }}_{1}\) and \({\varvec{\chi }}_{1}\) are uncorrelated, is

$$\begin{aligned} {\varvec{\alpha }}_{1} = \begin{bmatrix} \mathbf {A}\\ \mathbf{0} \\ \end{bmatrix}{\varvec{\delta }}+ \begin{bmatrix} \overline{\mathbf {p}} \\ {\varvec{\chi }}_{1} \\ \end{bmatrix} \end{aligned}$$

Appendix 3

See Tables 4 and 5.

Table 4 Release information for the time series used in the pseudo-real-time nowcasting of quarterly GDP growth
Table 5 Information updating scheme in the pseudo-real-time nowcasting of GDP growth for quarter \(\tau \)

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Marczak, M., Gómez, V. Monthly US business cycle indicators: a new multivariate approach based on a band-pass filter. Empir Econ 52, 1379–1408 (2017). https://doi.org/10.1007/s00181-016-1108-2

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