1. Introduction
 i
 introduce and study the SSR model, and
 ii
 propose a method for approximate frequentist estimation and inference.
2. The Model
3. Properties of the Process
3.1. The Unobserved Components
3.2. The Observed Process
4. LikelihoodBased Estimation and Inference
 1.
 $\frac{1}{\sqrt{T}}}{\mathit{S}}_{T}\left({\theta}^{*}\right)\stackrel{D}{\to}N(0,\phantom{\rule{0.166667em}{0ex}}{\mathsf{\Omega}}_{S})$ as $T\to \infty $, with ${\mathsf{\Omega}}_{S}>0$,
 2.
 $\frac{1}{T}}{\mathit{I}}_{T}\left({\theta}^{*}\right)\stackrel{P}{\to}{\mathsf{\Omega}}_{I$ as $T\to \infty $, with ${\mathsf{\Omega}}_{I}>0$, and
 3.
 ${max}_{h,i,j=1,\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}{d}_{\theta}}{sup}_{\theta \in \mathcal{N}\left({\theta}^{*}\right)}{\partial}^{3}{\ell}_{T}\left(\theta \right)/\partial {\theta}_{h}\partial {\theta}_{i}\partial {\theta}_{j}\le {c}_{T}$,
5. The Incomplete Data Framework
5.1. The State Space Form and the Optimal Filtering Problem
5.2. The Sample Score and Observed Information as Smoothing Problems
6. Particle FilterBased Approximations
6.1. Particle Filtering
Algorithm 1: Locally Optimal Particle Filter. 
Given a parameter $\theta \in \mathsf{\Theta}$, initialize by setting ${x}_{0}^{\left(i\right)}:={x}_{0}$ and ${\overline{w}}_{0}^{\left(i\right)}:=1/N$ for $i=1,\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}N$. For $t=0,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}T$:

6.2. The Approximate Sample Score and Observed Information Matrix
7. Particle FilterBased Estimation and Inference
Algorithm 2: Stochastic Approximation. 
Choose the initial parameter ${\theta}_{0}\in \mathsf{\Theta}$, the particle counts ${\left\{{N}_{j}\right\}}_{j=1}^{\infty}$, the step sizes ${\left\{{\gamma}_{j}\right\}}_{j=1}^{\infty}$ and weighting matrices ${\left\{{B}_{j}\right\}}_{j=1}^{\infty}$. For $j=0,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}K$:

8. Model Diagnostics
9. Simulation Study
10. An Illustration
11. Conclusions
Acknowledgments
Conflicts of Interest
Appendix A. Auxiliary Results
 i
 $\int {p}_{\omega}({y}_{t}\mid {x}_{t}){p}_{\lambda}({x}_{t}\mid {x}_{t1})\phantom{\rule{0.166667em}{0ex}}\mathrm{d}{x}_{t}>0$ for all ${x}_{t1}\in {\mathcal{R}}^{p}$, and
 ii
 $\underset{{x}_{t}\in {\mathcal{R}}^{p}}{sup}}{p}_{\omega}({y}_{t}\mid {x}_{t})<\infty $,
 i
 ${p}_{\omega}({y}_{t}\mid {x}_{t}){p}_{\lambda}({x}_{t}\mid {x}_{t1})\ll {p}_{\theta}({x}_{t}\mid {x}_{t1},\phantom{\rule{0.166667em}{0ex}}{y}_{t})$ for all ${x}_{t1}\in {\mathcal{R}}^{p}$,
 ii
 $\underset{{x}_{t1},\phantom{\rule{0.166667em}{0ex}}{x}_{t}\in {\mathcal{R}}^{p}\times {\mathcal{R}}^{p}}{sup}}\frac{{p}_{\omega}({y}_{t}\mid {x}_{t}){p}_{\lambda}({x}_{t}\mid {x}_{t1})}{{p}_{\theta}({x}_{t}\mid {x}_{t1},\phantom{\rule{0.166667em}{0ex}}{y}_{t})}>0$, and
 iii
 ${p}_{\theta}({x}_{t}\mid {x}_{t1},\phantom{\rule{0.166667em}{0ex}}{y}_{t})>0$ for all ${x}_{t1}\in {\mathcal{R}}^{p}$,
Appendix B. Main Results
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1.  We use the Choleski factorization to ensure positive definiteness of the covariance matrices ${\mathsf{\Omega}}_{u}$, $\mathsf{\Lambda}$ and ${\mathsf{\Omega}}_{\mathsf{\Phi}}$. Thus, we estimate the parameters B, A, ${\mathsf{\Omega}}_{u}={C}_{u}{C}_{u}^{\prime}$, $\mu $, $\mathsf{\Phi}$, ${\mathsf{\Omega}}_{\mathsf{\Phi}}={C}_{\mathsf{\Phi}}{C}_{\mathsf{\Phi}}^{\prime}$ and $\mathsf{\Lambda}={C}_{\mathsf{\Lambda}}{C}_{\mathsf{\Lambda}}^{\prime}$ using Algorithm 2 and transform the covariances to the original parametrization. We obtain standard errors via the $\delta $method. 
2.  Because we initialize at the true parameter value, the parameter sequences stabilize within the first 100 realizations. Using $K=600$ iterations is sufficient to reduce the impact of the approximation error. 
3.  The choice of weight matrix is based on handtuning the convergence speed of Algorithm 2 by running a small number of trialanderror runs with $N=50$ particles and constant step size $\gamma =1$. 
4.  Obtained via a Bloomberg LP Terminal using the ticker codes ‘GDBR10 Index’ and ‘GGGB10YR Index’. 
5.  The difference between computing the classic standard errors with $N=1000$ and $N=\mathrm{10,000}$ particles is negligible. 
6.  Particle filterbased approximate robust standard errors have been suggested in Doucet and Shephard (2012), but we do not pursue this idea further in the present context. 
Parameter  Estimate  Std.err.  Parameter  Estimate  Std.err. 

${B}_{1}$  $1.0000$    $\mu $  $0.3449$  $0.5526$ 
${B}_{2}$  $1.0000$    $\varphi $  $1.0085$  $0.0152$ 
${A}_{1}$  $0.0154$  $3.4\times {10}^{5}$  ${\omega}_{\varphi}^{2}$  $0.0306$  $0.0031$ 
${A}_{2}$  $1.0000$    ${\omega}_{\eta}^{2}$  $360.1600$  $33.6250$ 
${\omega}_{u\phantom{\rule{0.166667em}{0ex}}11}^{2}$  $2.1063$  $0.1974$  ${\omega}_{\eta ,\phantom{\rule{0.166667em}{0ex}}\nu}$  $22.2400$  $0.8119$ 
${\omega}_{u\phantom{\rule{0.166667em}{0ex}}12}$  $3.5924$  $0.3435$  ${\omega}_{\nu}^{2}$  $1.7880$  $2.0728$ 
${\omega}_{u\phantom{\rule{0.166667em}{0ex}}22}^{2}$  $6.6327$  $0.6214$ 
Univariate tests for AR 12:  ${\tilde{\mathit{e}}}_{\mathit{t},\phantom{\rule{0.166667em}{0ex}}1}$  $\mathit{F}\left(2227\right)=1.4523$  $\mathit{p}=0.2362$ 
${\tilde{\mathit{e}}}_{\mathit{t},\phantom{\rule{0.166667em}{0ex}}2}$  $\mathit{F}\left(2227\right)=1.2086$  $\mathit{p}=0.3005$  
Multivariate test for AR 12:  $F\left(8448\right)=1.6084$  $p=0.1200$  
Univariate tests for ARCH:  ${\tilde{e}}_{t,\phantom{\rule{0.166667em}{0ex}}1}$  $F\left(1227\right)=4.7008$  $p=0.0312$ 
${\tilde{e}}_{t,\phantom{\rule{0.166667em}{0ex}}2}$  $F\left(1227\right)=0.58861$  $p=0.4438$ 
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