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Bayesian estimation of the long-run trend of the US economy

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Abstract

The main purpose of this paper is to scrutinize the long-run trend of the US real GDP during the post-war period. In the empirical analysis, we introduce multivariate unobserved components models that accommodate time-varying volatility bounded by an economically reasonable range. After accounting for the cointegration relationship among the real GDP, consumption, and investment, we find the following. (i) There was an abrupt and significant downturn in the long-run growth rate in 2007. (ii) The annualized long-run growth rate of the real GDP declined to approximately 1.69%, decreasing from a peak of nearly 3.21% during the 2000s. (iii) The bounded volatility assumption enables a trend-cycle decomposition of the US real GDP that matches the NBER’s recession dates.

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Notes

  1. Gerlach et al. (2000) and Huber et al. (2019) design time-varying parameters by using mixture innovations. The merit of their approaches is that both gradual and abrupt changes in the model parameters can be estimated within a unified model framework. For the estimation, Gerlach et al. (2000) propose to integrate all latent state variables, except for those associated with the mixture innovations, by employing a Kalman filter-based algorithm. The estimation algorithm described by Gerlach et al. (2000) is computationally demanding due to the integration step. Huber et al. (2019) developed a new model and a corresponding estimation algorithm to address the computational bottleneck. In future research, it could be worthwhile to model \(\mu _t\) with the mixture innovation models reported by Gerlach et al. (2000) and Huber et al. (2019).

  2. In addition to the aforementioned studies, Sinclair (2009), who studied how US output and the unemployment rate are related in the short run and long run, assume that individual cycle components follow AR(2) processes.

  3. The trend-cycle decomposition of the US real GDP is highly sensitive to the zero correlation assumption. See Morley et al. (2003) for empirical evidence.

  4. Notably, conventional Bayesian estimation methods applied to stochastic volatility models, such as the model developed by Kim et al. (1998), are not applicable to our nonlinear state-space model due to the inequality restriction on the volatility process in Eq. (9).

  5. This study uses quarterly data because the U.S. Bureau of Economic Analysis (https://www.bea.gov/), which announces the US national accounting data, does not provide monthly observations of the real GDP and investment.

  6. We verified that priors more diffuse than those used in this study do not change the main empirical results.

  7. The mean of the inverse Wishart distribution is \( \mathrm{diag}(0.7^2,0.7^2,0.7^2,3^2 ) \over 6 - 4 - 1 \).

  8. The expected duration of each state is computed by \(1 \over {1 - \pi }\), where \(\pi \) represents the corresponding transition probability. The standard deviation of the prior beta distribution is \(0.0139 = \sqrt{ (98 + 2) \over (98 + 2)^2 (98 + 2 +1) }\). Thus, at approximately 68% probability, the prior covers a transition probability of [0.966, 0.994] and an expected state duration of \([30 \ \text {quaters},166 \ \text {quaters}]\).

  9. The annualized standard deviation of the real consumption growth during the 1970s was 1.87%.

  10. The empirical result shown in Table 1 is robust to using different initial values.

  11. We evaluate \(g(Y_{\tau +h} \vert \theta _M^{(r)}, Y_{1:\tau },M)\) using the Kalman filter algorithm. In the computation, \(\Sigma _\tau ^{(r)}\) was also included in the conditional information set. For notation simplicity, we suppress the time-varying covariance matrix in the likelihood function.

  12. In conventional structural break models, one should decide in advance how many structural breaks exist before and after the prediction period. To overcome this drawback, Koop and Potter (2007) developed an advanced model that does not need to know the number of breaks prior to the estimation and prediction. Interested readers should refer to Koop and Potter (2007) and the references therein.

  13. The posterior means of the MUC-SB1 and MUC-SB2 models are similar to the estimates reported in other related studies, such as Gordon (2014) and Fernald et al. (2016), who predict long-run growth of approximately 1.5%. Antolin-Diaz et al. (2017) estimate the time-varying growth rate of the real GDP using a dynamic factor model with a large data set. These authors claim that the long-run growth rate is near 2%, which is similar to the result of the MUC-RW model.

  14. One main obstacle to obtaining reliable estimates of the structural breaks lies in the lack of information regarding the low-frequency trend component. Bai et al. (1998) theoretically demonstrate that the confidence interval of a structural break date does not decrease even asymptotically as the sample size increases; however, the confidence interval is inversely related to the dimension of time-series data that share a common break point.

  15. Smoothed estimates at time t can be significantly affected by events in subsequent time periods.

  16. Fernald et al. (2016) argues that low productivity growth is the main reason for the recent slow growth of the US economy and is expected to last over the long run because the high growth in education attainment that underlies high productivity is no longer achievable.

  17. Notably, \(m_t = \mu _t\) in MUC-RW and \(m_t = s_t\) in MUC-SB1 and MUC-SB2.

  18. \(g_{\theta }(Y_t \vert \beta _{t}^{(n)}, m_{t}^{(n)}, h_{t}^{(n)}) = 1\) under the proposed sampling scheme. No observations are available at time \(t=0\). Therefore, a user-specific importance distribution or an unconditional distribution can be used to generate all elements of \(\beta _0\) at \(t=0\).

  19. We confirm via simulations that the Bayesian method developed here performs well as a practical matter even under the degenerate transition and measurement equations of our UC models and provides reliable estimates of the break date and other latent variables.

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Correspondence to Jaeho Kim.

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Appendix A: Bayesian estimation

Appendix A: Bayesian estimation

The proposed UC model is cast into the following state space representation to gain the joint smoothing distribution of the latent state variables:

$$\begin{aligned} Y_t= & {} D + H\beta _{t} \end{aligned}$$
(A.1)
$$\begin{aligned} \beta _t= & {} {\tilde{\mu }}_{t} + F\beta _{t-1} + G\epsilon _t, \end{aligned}$$
(A.2)

where \(Y_t = [ y_t \; c_t \; i_t]'\), \(D = [0 \; {\bar{c}} \; {\bar{i}}]'\), \(\beta _t = [x_t \; z_t \; z_{t-1} \; g_{c,t} \; g_{c,t-1} \; g_{i,t} \; g_{i,t-1}]'\), \({\tilde{\mu }}_{t} = [\mu _{t} \; 0 \; 0 \; 0 \; 0 \; 0 \; 0]'\), \(\epsilon _t = [u_{t} \; e_t \; \eta _{c,t}\; \eta _{i,t}]'\),

$$\begin{aligned} H = \begin{bmatrix} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ \gamma _c &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{}0 \\ \gamma _i &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{}0 \end{bmatrix}, \ F = \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} \phi _1 &{} \phi _2 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \gamma _{c,1} &{} \gamma _{c,2} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \gamma _{i,1} &{} \gamma _{i,2} \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \end{bmatrix}, G = \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 &{} 0\end{bmatrix}. \end{aligned}$$

Throughout this appendix, we adhere to the following notations of the latent variables: \(\beta _{\kappa :\tau } = \{\beta _\kappa ,\beta _{\kappa +1},\ldots ,\beta _\tau \}\), and \(m_{\kappa :\tau } = \{m_\kappa ,m_{\kappa +1},\ldots ,m_\tau \}\) for \(\tau \ge \kappa \). Regarding the stochastic volatility processes of the shocks in \(\epsilon _t\), we use \(h_{\kappa :\tau } = \{h_\kappa ,h_{\kappa +1},\ldots ,h_\tau \}\), where \(h_t = \{h_{u,t},h_{e,t},h_{\eta _c,t},h_{\eta _i,t}\}\). For MUC-RW, \(m_t = \mu _t\) is assumed to follow the random walk process in Eq. (5). In MUC-SB1 and MUC-SB2, \(m_t = s_t\) is assumed to follow the structural break process in Eq. (6).

1.1 Appendix A.1: posterior simulation of the model parameters

Kim and Chon (2018) show that in univariate UC models, the variance in the permanent shock could be severely biased if the autoregressive and long-run mean growth rate parameters are sampled separately without considering the correlation between the permanent and transitory shocks. Therefore, we sample the model parameters of our multivariate UC models simultaneously using their joint posterior distribution as described in Kim and Chon (2018). In this appendix, we focus on the estimation of MUC-SB1 and MUC-SB2. We can easily estimate the model parameters of MUC-RW by slightly modifying the algorithm explained in this appendix.

Consider the following seemingly unrelated regression (SUR) model derived from the transition equations of the proposed UC model conditional on \(\{\beta _{0:T},s_{0:T},h_{0:T} \}\):

$$\begin{aligned} \begin{bmatrix} \Delta x_t^* \\ z_t^* \\ c_t^* \\ i_t^*\end{bmatrix}&= \begin{bmatrix} I_{s_t =1}^* &{}I_{s_t =2}^* &{} \cdots &{} I_{s_t=K}^*&{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \cdots &{} 0 &{} z_{t-1}^* &{} z_{t-2}^* &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0 &{} g_{c,t-1}^* &{} g_{c,t-2}^* &{} 0 &{} 0 \\ 0 &{} 0 &{} \cdots &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} g_{i,t-1}^* &{} g_{i,t-2}^* \end{bmatrix} \begin{bmatrix} \mu _1 \\ \mu _2 \\ \vdots \\ \mu _K \\ \phi _1 \\ \phi _2 \\ \gamma _{c,1} \\ \gamma _{c,2} \\ \gamma _{i,1} \\ \gamma _{i,2} \end{bmatrix} \nonumber \\&\qquad + \begin{bmatrix} u_{t}^* \\ e_{t}^* \\ \eta _{c,t}^* \\ \eta _{i,t}^* \end{bmatrix}, \end{aligned}$$
(A.3)
$$\begin{aligned}&({\tilde{Y}}_t = {\tilde{X}}_t \Gamma + {\tilde{\epsilon }}_t), \nonumber \\&\quad \begin{bmatrix} u_{t}^* \\ e_{t}^* \\ \eta _{c,t}^* \\ \eta _{i,t}^* \end{bmatrix} \sim N\left( \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \begin{bmatrix} \sigma ^2_{u,1} &{} \rho _{u,e} \sigma _{u,1}\sigma _{e,1} &{} \rho _{u,\eta _c} \sigma _{u,1}\sigma _{\eta _{c},1} &{} \rho _{u,\eta _i} \sigma _{u,1}\sigma _{\eta _i,1} \\ \rho _{u,e} \sigma _{e,1}\sigma _{u,1} &{} \sigma ^2_{e,1} &{} \rho _{e,\eta _c} \sigma _{e,1}\sigma _{\eta _c,1} &{} \rho _{e,\eta _i} \sigma _{e,1}\sigma _{\eta _i,1} \\ \rho _{u,\eta _c} \sigma _{\eta _c,1}\sigma _{u,1} &{} \rho _{e,\eta _c} \sigma _{\eta _c,1}\sigma _{e,1} &{} \sigma ^2_{\eta _c,1} &{} \rho _{\eta _c,\eta _i} \sigma _{\eta _c,1}\sigma _{\eta _i,1} \\ \rho _{u,\eta _i} \sigma _{\eta _i,1}\sigma _{u,1} &{} \rho _{e,\eta _i} \sigma _{\eta _i,1}\sigma _{e,1} &{} \rho _{\eta _c,\eta _i} \sigma _{\eta _c,1}\sigma _{\eta _i,1} &{} \sigma ^2_{\eta _i,1} \end{bmatrix}\right) , \nonumber \\&\quad ({\tilde{\epsilon }}_t \sim N(0, \Sigma _1)), \end{aligned}$$
(A.4)

where \(\Gamma = [\mu _1 \; \mu _2 \; \ldots \; \mu _K \; \phi _1 \; \phi _2 \; \gamma _{c,1} \; \gamma _{c,2} \; \gamma _{i,1} \; \gamma _{i,2}]'\); \(\Delta x_t^* = {(x_t - x_{t-1})\over \exp (h_{u,t}/2)}\); \(I_{s_t=k}^* = {I(s_t = k)\over \exp (h_{u,t}/2)}\) for \(k=1,2,\ldots ,K\); \(z_{t-j}^* = {z_{t-j}\over \exp (h_{e,t}/2)}, \ g_{c,t-j}^* = {g_{c,t-j}\over \exp (h_{\eta _c,t}/2)}, \ g_{i,t-j}^* = {g_{i,t-j}\over \exp (h_{\eta _i,t}/2)} \) for \(j=0,1,2\). With a normal prior distribution of \(\Gamma \) and an inverse Wishart prior of \(\Sigma _1\), we can obtain the conditional posterior distribution of \(\Gamma \) as follows:

$$\begin{aligned} \Gamma \vert Y_{1:T},\beta _{1:T},s_{1:T}, \theta _{-\Gamma } \sim N({{\bar{\Gamma }}}, {{\bar{\Omega }}_{\Gamma }})I_{\Gamma }, \end{aligned}$$
(A.5)

where \({\bar{\Gamma }} = {\bar{\Omega }}_\Gamma ({\underline{\Omega }}_\Gamma ^{-1}{\underline{\Gamma }} + {\tilde{X}}'(\Sigma _0^{-1} \otimes I){\tilde{X}} {\hat{\Gamma }})\), \({\bar{\Omega }}_\Gamma = ({\underline{\Omega }}_\Gamma ^{-1} + {\tilde{X}}'(\Sigma _0^{-1} \otimes I){\tilde{X}})^{-1}\), and \({\hat{\Gamma }} = ({\tilde{X}}'(\Sigma _0^{-1} \otimes I){\tilde{X}})^{-1} {\tilde{X}}'(\Sigma _0^{-1} \otimes I){\tilde{Y}}\); \(\theta _{-\Gamma }\) is the set of all model parameters, except for \(\Gamma \). \(\{ {\underline{\Gamma }}, {\underline{\Omega }}_{\Gamma }\}\) are the prior mean and covariance matrices of \(\Gamma \). The data matrices \(\{{\tilde{X}},{\tilde{Y}}\}\) are obtained by vertically stacking \(\{{\tilde{X}}_t,{\tilde{Y}}_t\}\). The indication function \(I_{\Gamma }\) assumes a value of 1 if both the regime identification and stationarity conditions of \(\{\mu ,\phi ,\gamma _c,\gamma _i\}\) hold and 0 otherwise. We carry out the posterior simulation od \(\Gamma \) using the multivariate normal distribution given in Eq. (A.5) with rejection sampling.

Conditional on \(\{\beta _{0:T},s_{0:T},h_{1:T} \}\) and \(\Gamma \), it is relatively simple to draw other model parameters. Conditional on \(\Gamma \), the covariance matrix \(\Sigma _1\) at \(t=1\) is simulated from a posterior inverse Wishart distribution with a conjugate prior \(IW(\Lambda _0,\nu _0)\), where \(\Lambda _0\) and \(\nu _0\) are the prior scale and degree of freedom parameters. We simulate the posterior samples of the transition probabilities of \(s_t\) from posterior beta distributions. The samples of \(\{\sigma _{h,u}^2\), \(\sigma _{h,e}^2\), and \(\sigma _{h,\eta _c}^2, \sigma _{h,\eta _i}^2\}\) are drawn from posterior inverse gamma distributions using conventionally used conjugate priors. All posterior sampling steps are standard. Therefore, we do not describe the derivation of the posterior distributions here.

The sampling scheme of the cointegration parameters \(\{ {\bar{c}}, \lambda _c,{\bar{i}}, \lambda _i\}\) is based on the following Cholesky decomposition:

$$\begin{aligned} \begin{aligned} \begin{bmatrix} \eta _{c,t} \\ \eta _{i,t} \end{bmatrix}&= P A_t A_t^{-1} \begin{bmatrix} u_t \\ e_t\\ \eta _{c,t} \\ \eta _{i,t} \end{bmatrix} = P A_t \begin{bmatrix} \nu _{1,t} \\ \nu _{2,t} \\ \nu _{3,t} \\ \nu _{4,t} \end{bmatrix} \\&= \begin{bmatrix} A_{t,3,1} \ \nu _{1,t} + A_{t,3,2} \ \nu _{2,t} \\ A_{t,4,1} \ \nu _{1,t} + A_{t,4,2} \ \nu _{2,t} \end{bmatrix} + \begin{bmatrix} A_{t,3,3} \ \nu _{3,t} \\ A_{t,4,3} \ \nu _{3,t} + A_{t,4,4} \ \nu _{4,t} \end{bmatrix}, \ \ \begin{bmatrix} \nu _{3,t} \\ \nu _{4,t} \end{bmatrix} \ \sim N(0,I_2) \end{aligned} \end{aligned}$$
(A.6)

where \(P = \begin{bmatrix} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1\end{bmatrix}\) is a selection matrix, \(A_t\) is a (\(4 \times 4\)) lower triangular matrix such that \(A_tA_t' = \Sigma _t\), \(A_{t,i,j}\) is the (ij) element of \(A_{t}\), and \(\Sigma _t\) is the time-varying variance–covariance matrix of \([u_t \; e_t \; \eta _{c,t} \; \eta _{i,t}]'\). Given all other parameters, \(u_t\) (\(=y_t - x_t\)), and \(e_t\) (\(=z_t - \phi _1z_{t-1} - \phi _2z_{t-2})\), we can obtain \(A_t\), and the normalized errors \(\{ \nu _{1,t},\nu _{2,t}\}\). We can derive the posterior mean and covariance matrix of \(\{ {\bar{c}}, \lambda _c,{\bar{i}}, \lambda _i\}\) with a conjugate prior based on the following multiple regression representation:

$$\begin{aligned} \begin{aligned} \begin{bmatrix} c_t^* \\ i_t^* \end{bmatrix} - \begin{bmatrix} A_{t,3,1} \ \nu _{1,t} + A_{t,3,2} \ \nu _{2,t} \\ A_{t,4,1} \ \nu _{1,t} + A_{t,4,2} \ \nu _{2,t} \end{bmatrix}&= \begin{bmatrix} x_{c,0}^* &{} x_{c,1}^* &{} 0 &{} 0 \\ 0 &{} 0 &{} x_{i,0}^* &{} x_{i,1}^* \end{bmatrix} \ \begin{bmatrix} {\bar{c}} \\ \lambda _c \\ {\bar{i}} \\ \lambda _i \end{bmatrix} \\&\quad + \begin{bmatrix} A_{t,3,3} \ \nu _{3,t} \\ A_{t,4,3} \ \nu _{3,t} + A_{t,4,4} \ \nu _{4,t} \end{bmatrix}, \end{aligned} \end{aligned}$$
(A.7)

where \(c_t^* = (c_t -\gamma _{c,1} c_{t-1}-\gamma _{c,2} c_{t-2})\), \(i_t^* = (i_t -\gamma _{i,1} i_{t-1}-\gamma _{i,2} i_{t-2})\), \(x_{c,0}^* = (1-\gamma _{c,1} - \gamma _{c,2})\), \(x_{i,0}^* = (1-\gamma _{i,1} - \gamma _{i,2})\), \(x_{c,1}^* = (x_t -\gamma _{c,1} x_{t-1}-\gamma _{c,2} x_{t-2})\), \(x_{i,1}^* = (x_t -\gamma _{i,1} x_{t-1}-\gamma _{i,2} x_{t-2})\).

1.2 Appendix A.2: posterior simulation of the latent state variables

To draw the trend and cycle components of the UC model, we apply the Kalman smoothing algorithm described by Carter and Kohn (1994) conditional on \(m_{0:T}\)Footnote 17 and \(h_{1:T}\). Notably, the smoothing algorithm developed by Frühwirth-Schnatter (1994) can also be applied in the posterior sampling step. Because of the degenerate measurement and transition equations, we draw only the trend component \(x_t\) from the smoothing algorithm and recover the cycle components using Eqs. (1), (2), and (3). We do not explain the Gibbs sampling step in detail because it is a standard method.

This appendix derives a particle Gibbs (PG) sampling algorithm to draw \(m_{0:T}\) and \(h_{1:T}\) from their posterior joint distribution conditional on \(\beta _{0:T}\). Our estimation method is based on a study by Lindsten et al. (2015).

1.2.1 A.2.1. Sequential Monte Carlo method

Suppose that we have the following approximate filtering density for the latent variables:

$$\begin{aligned} p_{\theta }(m_{t-1},h_{t-1} \vert Y_{1:t-1},\beta _{1:t-1}) \approx \sum _{n = 1}^{N} {\hat{\omega }}_{t-1}^{(n)} \delta _{\{m_{t-1}^{(n)},h_{t-1}^{(n)}\}}(m_{t-1},h_{t-1}) \end{aligned}$$
(A.8)

where \(\{ m_{t-1}^{(n)},h_{t-1}^{(n)} \}_{n =1}^{N}\) and \(\{{\hat{\omega }}_{t-1}^{(n)} \}_{n=1}^N\) denote N particles of the latent variables and their corresponding normalized importance weights at time \(t-1\), and \(\delta _{\{m_{t-1}^{(n)},h_{t-1}^{(n)}\}}(m_{t-1},h_{t-1} )\) is a Dirac measure of each particle in \(\{ m_{t-1}^{(n)},h_{t-1}^{(n)}\}_{n=1}^N\). For forward filtering, an ancestor index denoted by \(a_t^{(n)}\) is drawn using the following importance weights prior to sampling a new particle \(\{m_t^{(n)},h_{t}^{(n)}\}\):

$$\begin{aligned} Pr(a_t^{(n)} = r) = {\hat{\omega }}_{t-1}^{(r)}, \end{aligned}$$

where \(r = 1,2,\ldots ,N\). Then, an importance distribution is used to generate the new particle \(\{ m_t^{(n)},h_t^{(n)}\}\) conditional on \(\{m_{t-1}^{(a_t^{(n)})},h_{t-1}^{(a_t^{(n)})}\}\), which is the \(a_t^{(n)}\)-th particle in the particle swarm at time \(t-1\).

To construct an importance distribution, we employ the transition densities of \(h_t\) and \(m_t\) as follows:

$$\begin{aligned} q(m_t ,h_t\vert m_{t-1},h_{t-1}) = f_{\theta }(m_t \vert m_{t-1})f_{\theta }(h_t \vert h_{t-1}). \end{aligned}$$
(A.9)

The corresponding unnormalized importance weight is given byFootnote 18 as follows:

$$\begin{aligned} {\bar{\omega }}_t^{(n)} = f_{\theta }(\beta _t \vert \beta _{t-1}, m_{t}^{(n)}, h_{t}^{(n)})I(h_{u,t}), \end{aligned}$$

where \(I(h_{u,t})=1\) if the stochastic volatility of the permanent shock is within the prespecified bounds and \(I(h_{u,t})=0\) otherwise. After sampling the ancestor index and the new particle and evaluating the unnormalized importance weight of all N particles, we obtain \(\{a_{t}^{(n)}\}_{n=1}^{N}\), \(\{s_t^{(n)},h_t^{(n)} \}_{n=1}^N\), and \(\{{\bar{\omega }}_t^{(n)} \}_{n=1}^N\). The importance weight of \(\{m_t^{(n)},h_t^{(n)}\}\) is computed through self-normalization as follows:

$$\begin{aligned} {\hat{\omega }}_t^{(n)} = {{\bar{\omega }}_t^{(n)} \over \sum _{i=1}^N {\bar{\omega }}_t^{(i)}} \end{aligned}$$
(A.10)

for \(n=1,2,\ldots ,N\). Notably, \(Y_t\) is irrelevant for the importance weight given by \(\beta _t\) and \(\beta _{t-1}\) because there is no measurement error in Eq. (A.1). By continuously performing the particle sampling steps up to time T, we can collect \(M_{0:T} = \{m_{0}^{(n)},m_{1}^{(n)},\ldots ,m_{T}^{(n)}\}_{n=1}^N\); \(H_{1:T} = \{h_{1}^{(n)},\ldots ,h_{T}^{(n)}\}_{n=1}^N\); \(A_{1:T} = \{a_{1}^{(n)},a_{2}^{(n)},\ldots ,a_{T}^{(n)}\}_{n=1}^N\). This algorithm is referred to as the sequential Monte Carlo (SMC) method. Recall that \(h_{1}^{(n)}=1\).

1.2.2 A.2.2. Particle Gibbs sampler with ancestor sampling

Among \(\{M_{0:T},H_{1:T},A_{1:T}\}\) generated by the SMC method, we sample one particular particle path known as a reference particle trajectory. The reference particle trajectory is sampled by first drawing the index \(b_T \in \{1,2,\ldots ,N \}\) and then sequentially recovering \(b_{t-1} = a_t^{(b_t)}\) for \(t=T,\ldots ,1\). Here, the index \(b_t\) indicates the locations of the individual particles in the chosen reference particle trajectory as follows:

$$\begin{aligned} \{m_0^{(b_0)},m_1^{(b_1)},\ldots ,m_T^{(b_T)},h_1^{(b_1)},\ldots ,h_T^{(b_T)} \}. \end{aligned}$$

For notational simplicity, we label the reference particle trajectory \(\{m_{0:T}^{(b_{0:T})},h_{1:T}^{(b_{1:T})}\}\). The probability of \(b_T = r\) for \(r = 1,2,\ldots ,N\) is determined by the importance weight at time T as follows:

$$\begin{aligned} Pr(b_T = r \vert \theta , \beta _{0:T}, M_{0:T},A_{1:T}) = {\hat{w}}_T^{(r)}. \end{aligned}$$
(A.11)

Then, we update the remaining particles and the ancestor indices denoted by \(\{ {M_{0:T}^{(-b_{0:T})}},\) \({H_{1:T}^{(-b_{1:T})}}, A_{1:T}^{(-b_{0:T})}\}\) conditional on \(\{m_{0:T}^{(b_{0:T})},h_{1:T}^{(b_{1:T})}, b_{0:T}\}\) that we obtain in the previous step. The update is simple. \(N-1\) particles and the corresponding ancestor indices are sampled by treating the accepted \(\{m_{t}^{(b_{t})},h_{t}^{(b_{t})}, b_{t-1}\}\) as the N-th particle and the N-th ancestor index for \(t=0,1,\ldots ,T\). This step is called the conditional sequential Monte Carlo (CSMC) method. A detailed exposition of its implementation is provided in the outline of the proposed PG algorithm.

The ancestor sampling method updates the ancestor index \(b_{t-1} \in \{1,2,\ldots ,N \}\) conditional on \(\{ m_{t}^{(b_{t})},h_{t}^{(b_t)}, b_t \}\). The sampling procedure is carried out for the entire period. To construct a valid MCMC step, consider the following target density for the ancestor sampling, which is the backward kernel at time t:

$$\begin{aligned} \Psi (b_{t-1}) \propto f_{\theta }(m_{t}^{(b_{t})} \vert m_{t-1}^{(b_{t-1})}) f_{\theta }(h_{t}^{(b_{t})} \vert h_{t-1}^{(b_{t-1})}) {\hat{\omega }}_{t-1}^{(b_{t-1})}. \end{aligned}$$
(A.12)

Because the normalization constant of the backward kernel is unknown, we employ the importance sampling method again to draw \(b_{t-1}\). The normalized importance weights of \(l = 1,2,\ldots ,N\) are given by

$$\begin{aligned} {\hat{\tau }}_t^{(l)} = {{\bar{\tau }}^{(l)}_{t} \over \sum _{j=1}^{N} {\bar{\tau }}^{(j)}_{t} }, \end{aligned}$$

where

$$\begin{aligned} {\bar{\tau }}^{(l)}_{t} = f_{\theta }(m_{t}^{(b_{t})} \vert m_{t-1}^{(l)}) f_{\theta }(h_{t}^{(b_{t})} \vert h_{t-1}^{(l)}) {\hat{\omega }}_{t-1}^{(l)}. \end{aligned}$$
(A.13)

Importance sampling is performed conditional on the previously accepted reference particle trajectory. This step refers to the conditional importance sampling (CIS) algorithm.

The PG algorithm derived in this appendix is an extension of the PG algorithm developed by Lindsten et al. (2014) to estimate a nonlinear switching state space model. The uniform ergodicity of the suggested algorithm is demonstrated in Lindsten et al. (2014). The following PG algorithm provides new posterior samples of \(m_{0:T}\) and \(h_{1:T}\) conditional on previously accepted \(\{m_{0:T}^{(b_{0:T})},h_{1:T}^{(b_{1:T})},b_{0:T}\}\) and \(\beta _{0:T}\):Footnote 19

\({\underline{\mathbf{Conditional ~SMC ~with ~Ancestor ~Sampling}}}\)

  • For \(t=0\),

    1. (i-1)

      Draw \(\{m_0^{(n)}\}_{n=1}^{N-1}\) from the \(f_\theta (m_0)\) and set \(h_0 = 1\).

    2. (i-2)

      Update \(\{m_0^{(b_0)},h_0^{(b_0)}\}\) using CIS algorithm.

      Set \(\{m_0^{(N)},h_0^{(N)}\} = \{m_0^{(b_0)},h_0^{(b_0)}\}\).

    3. (i-3)

      Set \(\{{\hat{\omega }}_0^{(n)} = {1\over N}\}_{n=1}^{N}\).

  • Iterate steps (ii-1), (ii-2), (ii-3), (ii-4), and (ii-5) for \(t=1,2,\ldots ,T\).

    1. (ii-1)

      Draw ancestor indices, \(\{ a_t^{(n)}\}_{n=1}^{N_p-1}\) with the normalized probabilities, \(\{ {\hat{\omega }}_{t-1}^{(n)} \}_{n=1}^{N_p}\).

    2. (ii-2)

      Draw \(\{m_t^{(n)},h_t^{(n)}\}_{n=1}^{N-1}, \) from \(f_{\theta }(m_t \vert m_{t-1}^{(a_t^{(n)})})\) and \(f_{\theta }(h_t \vert h_{t-1}^{(a_t^{(n)})})\).

    3. (ii-3)

      Run CIS algorithm to update \(b_{t-1}\).

      Set \(a_t^{(N)} = b_{t-1}\) and \(\{m_t^{(N)},h_t^{(N)}\} = \{m_t^{(b_t)},h_t^{(b_t)}\}\).

    4. (ii-5)

      Calculate the unnormalized weights \({\bar{\omega }}^{(n)}_{t}\) and obtain the normalized weights \({\hat{\omega }}_t^{(n)} = {{\bar{\omega }}^{(n)}_{t} \over \sum _{j=1}^N {\bar{\omega }}^{(j)}_{t} }\) of \(n = 1,2,\ldots ,N\).

  • Perform steps (iii-1), and (iii-2) for \(t=T\).

    1. (iii-1)

      Draw \(b_T^*\) with \(\{ {\hat{\omega }}_{T}^{(n)} \}_{n=1}^N\) and construct \(b_{0:T-1}^*\) using \(b_{t-1}^* = a_t^{(b_t^*)}\) for \(t = T,T-1,\ldots ,1\).

    2. (iii-2)

      Construct a new reference particle trajectory \(\{h_{1,1:T}^{(b_{1:T}^*)}\}\).

      Set \(\{h_{i,1:T}^{(b_{1:T})},b_{0:T}\} = \{h_{1,1:T}^{(b_{1:T}^*)},b_{0:T}^*\}\).

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Kim, J., Chon, S. Bayesian estimation of the long-run trend of the US economy. Empir Econ 62, 461–485 (2022). https://doi.org/10.1007/s00181-021-02024-4

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