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Supplementary Appendix for
'Selective linear segmentation for detecting relevant parameter changes'

Email addresses:
(Arnaud Dufays),  (Aristide Houndetoungan),  (Alain Coën).

Section A is devoted to the proof of the consistency of our estimator (see Proposition 1 in the paper). We also discuss the theoretical consequences of imposing bounded eigenvalues and how to calibrate the spike and slab prior to mimic the SELO penalty function. Section B shows that the criterion used to select the best model stands as the marginal likelihood of a linear model with specific prior distributions. It also develops the posterior and the predictive distributions of the Bayesian model. Section C contains the proof of the consistency of the marginal likelihood criterion as well as its link with the Bayesian information criterion. The time-varying parameter (TVP) model used as an alternative process is specified in Section D. The section also provides how the TVP model is estimated and additional graphics of the parameter dynamics. Finally, Section E briefly discusses the Bayesian alternatives to the selective segmentation approach.

Keywords: change-point, structural change, time-varying parameter, model selection, Hedge funds.

JEL Classification: C11, C12, C22, C32, C52, C53.

§ PROOFS OF THE CONSISTENCY OF THE PENALTY FUNCTION

In this appendix, we proof Proposition <ref>. To do so, we first state and prove two Lemmas.

Under the conditions <ref>-<ref> and let,

\begin{align}
f_{T}(\bbeta) = \dfrac{1}{T}||\by- \bX_{\btau} \bbeta||_2^2 + \sum_{j=2}^m\sum_{k=1}^{K}\pen(\Delta\beta_{jk}|a_k,\lambda)
\end{align}

Then of every $\nu \in (0,1)$, there exists a constant $C_0 > 0$ such that

$$ \liminf_{T \rightarrow \infty} ~ \mathbf{P} \left[ \argmin_{||\bbeta- \bbeta^*||_2 \leq C\sqrt{\frac{Km\sigma^2}{T}}} f_{T}(\bbeta) \subseteq \left\{\bbeta  \in \Re^{Km\times 1}; ||\bbeta- \bbeta^*||_2 < C\sqrt{\frac{Km\sigma^2}{T}} \right\} \right] > 1-\nu$$

for all $C \geq C_0$.

The proof is given in Appendix <ref>.

Let $C>0$ and $f_{T}$ as defined by Equation (<ref>). Under the conditions <ref>-<ref>,

$$\liminf_{T \rightarrow \infty} ~ \mathbf{P}  \left[ \argmin_{||\bbeta- \bbeta^*||_2 \leq C\sqrt{\frac{Km\sigma^2}{T}}} f_{T}(\bbeta) \subseteq \left\{\bbeta  \in \Re^{Km\times 1}; \bbeta_{A^c} = 0\right\} \right] = 1$$

where $A^c = \{(j,k), j=1,\dots,m \text{ and } k=0,\dots,K-1\} \backslash A$ is the complement of $A$ in $\{(j,k), j=1,\dots,m \text{ and } k=0,\dots,K-1\}$, $\bbeta_{A^c} \in \Re^{|A^c|\times 1}$ is the $|A^c|$-dimensional sub-vector of $\bbeta$ containing components subscripted by $A^c$.

See Appendix <ref> for the proof.

§.§ Proof of Lemma 1

We consider the objective function

f_{T}(\bbeta) = \dfrac{1}{T}||\by- \bX_{\btau} \bbeta||_2^2 + \sum_{j=2}^m\sum_{k=1}^{K}\pen(\Delta\beta_{jk}|a_k,\lambda)

Let $\alpha_T = \sqrt{\frac{Km\sigma^2}{T}}$ and $\nu \in (0,1)$. To prove the lemma <ref>, It suffices to show that

$$ \mathbf{P} \left( f_{T}(\bbeta^*) < \inf_{||\bu||_2 = 1} f_{T}(\bbeta^* + C\alpha_T\bu) \right) = 1 -\nu $$

for C>0 sufficiently large and for any $T$ sufficiently large. In other words, we shall show that  $H_{T}(\bu) = f_{T}(\bbeta^* + C\alpha_T \bu)  - f_{T}(\bbeta^*)$ is positive for any $T$ when $C$ is large enough and for all $||\bu||_2 = 1$, where $\bu = (u_{11}, u_{12} ~\dots, u_{1K}, ~\dots, ~ u_{mK}) \in \REAL^{mK\times 1}$.

We can easily show that

\begin{align*}
H_{T}(\bu) &= \dfrac{1}{T}\left(C^2\alpha_T^2||\bX_{\btau}\bu||_2^2 - 2C\alpha_T\bepsilon'\bX_{\btau}\bu  \right) + \\
& \sum_{j=2}^m\sum_{k=1}^{K}\left(\pen(\Delta\beta_{jk}^* + C\alpha_T u_{jk}|a_k,\lambda) - \pen(\Delta\beta_{jk}^*|a_k,\lambda)\right)\\
H_{T}(\bu) &\geq \dfrac{1}{T}\left(C^2\alpha_T^2||\bX_{\btau}\bu||_2^2 - 2C\alpha_T\bepsilon'\bX_{\btau}\bu\right) +\\
& \sum_{(j,k) \in \mathbf{D}(\bu)}\left(\pen(\Delta\beta_{jk}^{*+}|a_k,\lambda) - \pen(\Delta\beta_{jk}^*|a_k,\lambda)\right)
\end{align*}

where $\Delta\beta_{jk}^{*+}= \Delta\beta_{jk}^* + C\alpha_T u_{jk}$ and

$ \mathbf{D}(\bu) = \left\{(j,k); j\geq 2 ~ \text{and} ~ \pen(\Delta\beta_{jk}^{*+}|a_k,\lambda) - \pen(\Delta\beta_{jk}^*|a_k,\lambda) <0 \right\}$.

For any $(j,k) \in \mathbf{D}(\bu)$, clearly $\Delta\beta_{jk}^* \ne 0$, otherwise $\pen(\Delta\beta_{jk}^{*+}|a_k,\lambda) - \pen(\Delta\beta_{jk}^*|a_k,\lambda) \geq 0$. Thus, if $C>0$ is sufficiently large and fixed, as $\lim_{T \to \infty} C\alpha_T = 0$, we can consider that $\Delta\beta_{jk}^{*+}$ and $\Delta\beta_{jk}^{*}$  have the same sign for $T$ sufficiently large; that is $0 \not\in (c_T^- , c_T^+)$, where
$c_T^- = \min(\Delta\beta_{jk}^{*+},\Delta\beta_{jk}^*)$ and $c_T^+ = \max(\Delta\beta_{jk}^{*+},\Delta\beta_{jk}^*)$. By the fact that $\pen(x|a_k,\lambda)$ is a concave function on $x\in (-\infty, 0]$ and on $x \in [0, +\infty)$, thus also on $(c_T^- , c_T^+)$, we can establish the following conditions using the mean value theorem.

$$ \dfrac{\pen(\Delta\beta_{jk}^{*+}|a_k,\lambda) - \pen(\Delta\beta_{jk}^*|a_k,\lambda)}{C\alpha_T u_{jk}} \leq \max\left(\pen'(\Delta\beta_{jk}^{*+}|a_k,\lambda),\pen'(\Delta\beta_{jk}^*|a_k,\lambda)\right) $$

$$ \dfrac{\pen(\Delta\beta_{jk}^{*+}|a_k,\lambda) - \pen(\Delta\beta_{jk}^*|a_k,\lambda)}{C\alpha_T u_{jk}} \geq \min\left(\pen'(\Delta\beta_{jk}^{*+}|a_k,\lambda),\pen'(\Delta\beta_{jk}^*|a_k,\lambda)\right) $$

where $\pen'$ stands for the $\pen$ first derivative.

Let us note that $\forall ~(j,k) \in \mathbf{D}(\bu)$, $\Delta\beta_{jk}^* > 0 \implies u_{jk}<0$ and $\Delta\beta_{jk}^* < 0 \implies u_{jk}>0$ so that $\pen(\Delta\beta_{jk}^{*+}|a_k,\lambda) - \pen(\Delta\beta_{jk}^*|a_k,\lambda) <0$ holds. Applying the mean value theorem in both cases, we end up with a common condition given by

\begin{align*}
\pen(\Delta\beta_{jk}^{*+}|a_k,\lambda) - \pen(\Delta\beta_{jk}^*|a_k,\lambda)&\geq -C\alpha_T|u_{jk}||\pen'(\Delta\beta_{jk}^* + C\alpha_T u_{jk}|a_k,\lambda)|\\
&\geq -\frac{C\lambda \alpha_T a_k\zeta}{\ln(2)(\rho^2 - 2\rho C \alpha_T)}\\
&\geq -\frac{C\lambda \alpha_T a_{max}\zeta}{\ln(2)(\rho^2 - 2\rho C \alpha_T)}
\end{align*}

where the last two inequalities come from the $|\pen'(\Delta\beta_{jk}^* + C\alpha_T u_{jk}|a_k,\lambda)|$ minimization with respect to $\Delta\beta_{jk}^*$. Then

\begin{align*}
H_{T}(\bu) &\geq \underbrace{\dfrac{C^2\alpha_T^2||\bX_{\btau}\bu||_2^2}{T}}_{Q_1} - \underbrace{\dfrac{2C\alpha_T\bepsilon'\bX_{\btau}\bu }{T}}_{Q_2}  -\underbrace{\frac{CKm\lambda\alpha_T a_{max}\zeta}{\ln(2)(\rho^2 - 2\rho C \alpha_T)}}_{Q_3}
\end{align*}

Focusing on each term, we can show that

Q_1 \equiv  \dfrac{C^2\alpha_T^2||\bX_{\btau}\bu||_2^2}{T} = C^2\alpha_T^2\bu'\dfrac{\bX_{\btau}'\bX_{\btau}}{T}\bu \geq C^2\alpha_T^2\lambda_{T,\min}$$

where $\lambda_{T,\min}$ is the smallest eigenvalue of $\dfrac{\bX_{\btau}'\bX_{\btau}}{T}$.

To show this condition, we can decompose  $\dfrac{\bX_{\btau}'\bX_{\btau}}{T}$ into $\boldsymbol{U}\boldsymbol{\Lambda}\boldsymbol{U}'$ (by <ref>). Moreover, any vector of $Km$ dimension can be decomposed into a linear combination of the eigenvectors (i.e., $\displaystyle \boldsymbol{u} = \boldsymbol{U}\boldsymbol{\omega}$). Note that $\displaystyle \boldsymbol{u}'\boldsymbol{u} = \boldsymbol{\omega}'\boldsymbol{U}'\boldsymbol{U}\boldsymbol{\omega} = \boldsymbol{\omega}'\boldsymbol{\omega} = \sum_{i=1}^{Km}\omega_i^2 = 1$.

Thus $\displaystyle \bu'\dfrac{\bX_{\btau}'\bX_{\btau}}{T}\bu = \boldsymbol{\omega}'\boldsymbol{U}'\boldsymbol{U}\boldsymbol{\Lambda}\boldsymbol{U}'\boldsymbol{U}\boldsymbol{\omega} = \boldsymbol{\omega}'\boldsymbol{\Lambda}\boldsymbol{\omega} = \sum_{i=1}^{Km}\omega_i^2\lambda_i$ $\geq \lambda_{T,\min}$.

The second term is given by

\begin{align*}
Q_2 \equiv  \dfrac{2C\alpha_T\bepsilon'\bX_{\btau}\bu }{T} &\leq \dfrac{2C\alpha_T|\bepsilon'\bX_{\btau}\bu |}{T}\\
&\leq \dfrac{2C\alpha_T||\bepsilon'\bX_{\btau}||_2||\bu||_2}{T} ~ \text{by Cauchy-Schwartz}\\
&\leq \frac{2C\alpha_T^2}{\sqrt{Km\sigma^2}}\sqrt{\frac{(\bepsilon' (\bX_{\btau}\bX_{\btau}') \bepsilon)}{T}} \\
&\leq \mathcal{O}_p(C\alpha_T^2) \quad (By~ \ref{Xfullrank} ~ and ~ \ref{exogeneity}).
\end{align*}

To show that $\sqrt{\frac{(\bepsilon' (\bX_{\btau}\bX_{\btau}') \bepsilon)}{T}}=\mathcal{O}_p(1)$, we rely on the spectral theorem to decompose $\bX_{\btau}\bX_{\btau}'$ into two orthogonal matrices and a diagonal matrix of eigenvalues. With this decomposition, we can show that   $\epsilon' \frac{\bX_{\btau}\bX_{\btau}'}{T} \epsilon \leq \text{max}_i \lambda_i \frac{\epsilon'\epsilon}{T}$ which is $\mathcal{O}_p(1)$ under Assumption <ref> and the fact that the variance is bounded.

The last term $Q_3$ is defined by

\begin{align*}
Q_3 &\equiv \frac{CKm\lambda\alpha_T a_{max}\zeta}{\ln(2)(\rho^2 - 2\rho C \alpha_T)} \\
&= C\alpha_T^2 \dfrac{\lambda \zeta\dfrac{a_{\max}}{(Km)^{-1}\alpha_T^3}}{\ln(2)\left(\left(\dfrac{\rho}{\alpha_T}\right)^2 - 2C \left(\dfrac{\rho}{\alpha_T}\right)\right)}
\end{align*}

By <ref>, $\displaystyle \left(\frac{\rho}{\alpha_T}\right)^2 - 2C \left(\frac{\rho}{\alpha_T}\right) \to \infty$ and by  <ref>, $\displaystyle \lim_{T \to \infty} \lambda \zeta\frac{a_{\max}}{(Km)^{-1}\alpha_T^3} < \infty$.
$Q_3 = o(C\alpha_T^2)$.

Combining the conditions on $Q_1$, $Q_2$ and $Q_3$ we establish that
$$H_{T}(\bu) \geq C^2\alpha_T^2\lambda_{T,\min} + \mathcal{O}_p(C\alpha_T^2) + o(C\alpha_T^2)$$
It follows that there exists $C_0>0$ is large such that for all $C > C_0$, $\displaystyle \mathbf{P}\left(\inf_{||\bu||_2 =1}  H_{T}(\bu) > 0\right) = 1-\nu$, for $T$ sufficiently large.

§.§ Lemma 2

Let $\bbeta \in \Re^{Km\times 1}$ such that $||\bbeta - \bbeta^*||_2 < C\sqrt{\frac{Km\sigma^2}{T}}$. We consider $\tilde{\bbeta} \in \Re^{Km\times 1}$, where $\tilde{\bbeta}_{A^c} = 0$ and $\tilde{\bbeta}_{A} = \bbeta_{A}$.
We can notice that

\begin{align*}
||\bbeta - \tilde{\bbeta}||_2 &=  ||\bbeta_{A^c} - \tilde{\bbeta}_{A^c}||_2 = ||\bbeta_{A^c} -  \bbeta^*_{A^c}||_2\\
||\bbeta - \tilde{\bbeta}||_2 &< C\alpha_T
\end{align*}

On the other hand

\begin{align*}
||\bbeta^* - \tilde{\bbeta}||_2 &=  ||\bbeta^*_{A} - \tilde{\bbeta}_{A}||_2 =  ||\bbeta^*_{A} -  \bbeta_{A}||_2\\
||\bbeta^* - \tilde{\bbeta}||_2 &< C\alpha_T
\end{align*}

Let us define $\displaystyle G_{T}(\bbeta) = f_{T}(\bbeta) - f_{T}(\tilde{\bbeta})$. Similarly to the proof of the lemma <ref>, it suffices to show that $G_{T}(\bbeta,\tilde{\bbeta}) > 0$.

\begin{align*}
G_{T}(\bbeta) &=  \dfrac{1}{T}\left(||\by - \bX_{\btau}\bbeta||_2^2 - ||\by - \bX_{\btau}\tilde{\bbeta}||_2^2\right) + \sum_{\substack{
(j,k) \in A^c \\
j \geq  2}}\pen(\Delta\beta_{jk}|a_k,\lambda) \\
&= \dfrac{1}{T}\left(||\by - \bX_{\btau}\tilde{\bbeta} - \bX_{\btau}(\bbeta-\tilde{\bbeta})||_2^2 -||\by - \bX_{\btau}\tilde{\beta}||_2^2\right) + \sum_{\substack{
(j,k) \in A^c \\
j \geq  2}}\pen(\Delta\beta_{jk}|a_k,\lambda)  \\
&= (\bbeta - \tilde{\bbeta})'\dfrac{\bX_{\btau}'\bX_{\btau}}{T}(\bbeta - \tilde{\bbeta}) - 2(\bbeta - \tilde{\bbeta})'\dfrac{\bX_{\btau}'(\by - \bX_{\btau}\tilde{\bbeta})}{T}  + \sum_{\substack{
(j,k) \in A^c \\
j \geq  2}}\pen(\Delta\beta_{jk}|a_k,\lambda)  \\
&= (\bbeta - \tilde{\bbeta})'\dfrac{\bX_{\btau}'\bX_{\btau}}{T}(\bbeta - \tilde{\bbeta}) - 2(\bbeta - \tilde{\bbeta})'\dfrac{\bX_{\btau}'\boldsymbol{\epsilon}}{T} - 2(\bbeta - \tilde{\bbeta})'\dfrac{\bX_{\btau}'\bX_{\btau}}{T}  (\bbeta^* - \tilde{\bbeta} ) \\
& \quad + \sum_{\substack{
(j,k) \in A^c \\
j \geq  2}}\pen(\Delta\beta_{jk}|a_k,\lambda)  \\
&= (\bbeta - \tilde{\bbeta})'\dfrac{\bX_{\btau}'\bX_{\btau}}{T}(\bbeta - \tilde{\bbeta}) - 2\alpha_T\dfrac{(\bbeta - \tilde{\bbeta})'}{\sqrt{Km\sigma^2}}\dfrac{\bX_{\btau}'\boldsymbol{\epsilon}}{\sqrt{T}} - 2(\bbeta - \tilde{\bbeta})'\dfrac{\bX_{\btau}'\bX_{\btau}}{T}  (\bbeta^* - \tilde{\bbeta} ) \\
& \quad + \sum_{\substack{
(j,k) \in A^c \\
j \geq  2}}\pen(\Delta\beta_{jk}|a_k,\lambda)
\end{align*}

By <ref>, <ref> and <ref>, $\dfrac{\bX_{\btau}'\bX_{\btau}}{T} = \mathcal{O}_p(1)$ and $\dfrac{\bX_{\btau}'\boldsymbol{\epsilon}}{\sqrt{T}} = \mathcal{O}_p(1)$ (as it is a martingale difference sequence following assumption <ref>). Moreover $||\bbeta - \tilde{\bbeta}||_2 < C\alpha_T$ and
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$$ G_{T}(\bbeta) =\mathcal{O}_p\left(||\bbeta - \tilde{\bbeta}||_2\alpha_T\right) + \sum_{\substack{
(j,k) \in A^c \\
j \geq  2}}\pen(\Delta\beta_{jk}|a_k,\lambda) $$

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$\forall~ (j,k) \in A^c$, $\Delta\beta^*_{jk} = 0$ and $\Delta\beta_{jk}$ is strictly  positive or negative.
Thus, $-C\alpha_T \leq \beta_{jk} < 0$ or $0 < \Delta\beta_{jk} < C\alpha_T$, since $|\Delta\beta_{jk}| \leq ||\bbeta - \bbeta^*||_2 < C\alpha_T$.
In both cases, we end up with

\begin{align*}
\dfrac{\pen\left(C\alpha_T|a_k,\lambda \right)}{C\alpha_T} &\leq \dfrac{\pen\left(\Delta\beta_{jk}|a_k,\lambda\right)}{|\Delta\beta_{jk}|}\\
\displaystyle \pen(\Delta\beta_{jk}|a_k,\lambda) &\geq \frac{\lambda}{\ln(2)C\alpha_T}\ln\left(\frac{C\alpha_T}{C\alpha_T + a_k\zeta } + 1\right)|\Delta\beta_{jk}|\\
\displaystyle \pen(\Delta\beta_{jk}|a_k,\lambda) &\geq \frac{\lambda}{\ln(2)C}\ln\left(\frac{C\alpha_T}{C\alpha_T + a_{\max}\zeta } + 1\right)|\Delta\beta_{jk}|
\end{align*}

for any $T$ sufficiently large. Thus

\begin{align*}
\sum_{\substack{
(j,k) \in A^c \\
j \geq  2}}\pen(\Delta\beta_{jk}|a_k,\lambda) \geq \frac{\lambda}{\ln(2)C}\ln\left(\frac{C}{C + a_{\max}\zeta \sqrt{\frac{T}{K m \sigma^2}}} + 1\right)||\bbeta - \tilde{\bbeta}||_2.
\end{align*}

Furthermore, by <ref> $\displaystyle a_{\max} = \mathcal{O}_p\left(\sqrt{\frac{mK\sigma^2}{T}} \frac{\sigma_2}{T}\right)$. Thereby

$$a_{\max}\zeta \sqrt{\frac{T}{K m \sigma^2}} \overset{p}{\to} 0 \text{ and }\displaystyle \liminf_{T \to \infty}  \left(\frac{C}{C + a_{\max}\zeta \sqrt{\frac{T}{K m \sigma^2}}} + 1\right) > 0$$

It follows  that, there exists $\tilde{C} > 0$ such that

$$ \dfrac{G_{T}(\bbeta,\tilde{\bbeta})}{||\bbeta - \tilde{\bbeta}||_2} \geq \tilde{C}\lambda + \mathcal{O}_p\left(\sqrt{\dfrac{Km\sigma^2}{T}}\right) $$

Thereby the result follows.

§.§ Proof of the Proposition <ref>

The theorem is immediately given by the lemmas (<ref>) and (<ref>), in the sense that there exists a sequence of local minima $\hat{\bbeta}$ of $f_{T}(\bbeta)$ such that $||\hat{\bbeta} - \bbeta^*|| = \mathcal{O}_p\left(T^{-\frac{1}{2}}\right)$ (since $m$, $K$ and $\sigma^2$ are constants) and $\hat{\bbeta}_{A^c}  = \boldsymbol{0} \in \REAL^{|A^c|\times 1}$. Thus, as $T^{-\frac{1}{2}} \to 0$, it follows that $||\hat{\bbeta}_{A} - \bbeta_{A}^*||_2 =  o_P(1)$.

§.§ Consequence of bounded eigenvalues

We show that bounded eigenvalues of the matrix $\frac{\bX_{\btau}'\bX_{\btau}}{T}$ implies a fixed number of regimes. Note first that

\begin{eqnarray}
\bX_{\btau}'\bX_{\btau} & = & \sum_{t=1}^T (\1{t} \otimes \bx_t)(\1{t} \otimes \bx_t)', \\
& = & \sum_{t=1}^T (\bx_t  \bx_t') \otimes (\1{t} \1{t}'),
\end{eqnarray}

where we define $\1{t} = (\1{t > \tau_0},\1{t > \tau_1},\ldots,\1{t > \tau_{m-1}})'$. Let us define $n_i = \sum_{t=1}^T \1{t > \tau_{i-1}}$, i.e., the number of observations from the beginning of regime $i$ to the end of the sample. Working with $\bx_t \equiv 1$, we have that

\begin{eqnarray}
\frac{\bX_{\btau}'\bX_{\btau}}{T} & = & \frac{1}{T} \sum_{t=1}^T (\1{t} \1{t}'),\\
& = & \frac{1}{T}  \begin{pmatrix} n_1 & n_2 & n_3 & \ldots & n_m \\ n_2 & n_2 & n_3 & \ldots & n_m \\ n_3 & n_3 & n_3 & \ldots & n_m \\ & & &\ldots & \\n_m & n_m & n_m & \ldots & n_m \end{pmatrix}
\end{eqnarray}

in which $n_1 = T$. It leads to the following determinant, when $m>1$,

\begin{eqnarray}
|\frac{\bX_{\btau}'\bX_{\btau}}{T}| & = &T^{-m} n_m  \prod_{i=1}^{m-1}(n_{i}-n_{i+1}), \\
& = & \frac{n_m}{T}  \prod_{i=1}^{m-1} \frac{(n_{i}-n_{i+1})}{T}.
\end{eqnarray}

Note that $n_i = T - \tau_{i - 1} = \tau_{m} - \tau_{i - 1}$, for $i = 1, \dots, m$. Thus,

\begin{eqnarray}
|\frac{\bX_{\btau}'\bX_{\btau}}{T}| &= & \prod_{i=1}^{m} \frac{\tau_{i} - \tau_{i - 1}}{T}\\
&= & \prod_{i=1}^{m} \delta_{\tau_i} > 0
\end{eqnarray}

where $\delta_{\tau_i} = \dfrac{\tau_i - \tau_{i - 1}}{T}$.

It shows that the number of segments cannot increase with $T$ otherwise the determinant tends to zero. Let us assume that $m = \mathcal{O}(T^q)$ with $q>0$. It is clear that when T tends to $\infty$, there exists $r \in \mathbb{N}$ such that $\delta_{\tau_r}= \mathcal{O}(T^x)$ where $x < 0$. If such a $r$ does not exist, this would imply that $\forall i, ~ \delta_{\tau_i}$ does not drift to 0 as $T \rightarrow \infty$ and then $m = \mathcal{O}(1)$ because $\displaystyle\inf_{i}\{\delta_{\tau_i}\}m \leq 1$. But, because $m = \mathcal{O}(T^q)$ with $q>0$, there exists $r \in \mathbb{N}$ such that $\delta_{\tau_r} = \mathcal{O}(T^x)$ where $x < 0$. Therefore, we have that

\begin{eqnarray}
|\frac{\bX_{\btau}'\bX_{\btau}}{T}| &= & \delta_{\tau_r} \prod_{\substack{i = 1 \\ i \ne r}}^{m} \delta_{\tau_i} \to 0,
\end{eqnarray}

which contradicts Assumption <ref>. Hence $m = \mathcal{O}(1)$. As a result  <ref> implies that $m < \infty$. In addition, <ref> also implies that $\displaystyle\min_{i}\{\delta_{\tau_i}\} > 0$; that is $\delta_{\tau_i}$ does not drift to 0 as $T \rightarrow \infty$.

§.§ Approximation of the penalty function with mixture of normal densities

To derive the DAEM algorithm, a mixture of two normal densities has been assumed for the mean parameter. We now provide a simple mixture  approximation of the SELO penalty. Note that in practice, one can use the output of the DAEM algorithm as a starting point to optimize the function of Equation (<ref>). Due to the mixture approximation and the continuity of the SELO penalty function, the starting point would be in general very close to the value that globally minimizes the function (<ref>). We now use a mixture of two normal densities that can be understood as a spike and slab prior in the Bayesian paradigm. In particular, we calibrate a spike and slab prior <cit.> to the SELO penalty function. Given a mean parameter $\beta$, the spike and slab prior is specified as,

\begin{align}
\begin{split}\label{eq:standardSS}
\beta & \sim  \mathcal{N}(0,r_{z}), \\
z & \sim  \text{Bernoulli}(1-\omega),
\end{split}
\end{align}

where $r_0<r_1$ such that the spike distribution arises with probability $P[z=0|\omega]=\omega$. By marginalizing out $z$, we get a mixture of two normal densities given by

\begin{align}
\begin{split}
f(\beta|\omega,r_{0},r_{1}) & = \omega  (\frac{1}{ r_0})^{\frac{1}{2}}f_Z(\frac{\beta }{\sqrt{ r_0}}) + (1-\omega)(\frac{1}{ r_1})^{\frac{1}{2}}f_Z(\frac{\beta}{\sqrt{ r_1}}).
\end{split}
\end{align}

in which $f_Z(x)$ denotes the Normal density function evaluated at $x$ and with expectation and variance equal to 0 and 1, respectively. The calibration is done as follows:

\begin{align}
\begin{split}
c \equiv \frac{r_1}{r_0}  & = 10000, \\
\omega& = \frac{(\exp(\lambda) - 1)}{(\sqrt{c}+(\exp(\lambda) - 1))}, \\
r_0 & = \frac{a^2}{8}\frac{1-c^{-1}}{|\ln(\exp(\lambda) -1)|}
\end{split}
\end{align}

We now detail how we come up with this simple calibration. Given $\beta$, note that the probability of being in the slab component is equal to

\begin{align}
\begin{split}
Pr(z = 1|\beta,\omega,r) & = \frac{1}{\frac{\omega}{(1-\omega)} \frac{(\frac{1}{r_0})^{\frac{1}{2}}f_Z(\frac{\beta}{\sqrt{ r_0}})}{(\frac{1}{ r_1})^{\frac{1}{2}}f_Z(\frac{\beta }{\sqrt{ r_1}})} + 1}. \\
\end{split}
\end{align}

To mimic the SELO penalty function, we impose the following constraints on the spike and slab hyper-parameters.

* As standards in the spike and slab literature, we fix $c \equiv \frac{r_1}{r_0} = 10000$ <cit.>.

* The SELO function imposes a penalty equal to $\pen(a)-\pen(0) = \lambda y$ with $y=0.99$. To fix the same penalty value (neglecting $y$ because $y\approx 1$), we set

\begin{align}
\begin{split}
(-\lambda) & = \ln f(\beta=a|\omega,r_{z}) - \ln f(\beta=0|\omega,r_{z}) , \\
& = \ln \frac{f(\beta=a|\omega,r_{1},z=1)}{f(\beta=0|\omega,r_{1},z=1)} + \ln \frac{Pr(z = 1|\beta=0,\omega,r)}{Pr(z = 1|\beta=a,\omega,r)}, \\
& = -\frac{a^2}{2r_1}  + \ln \frac{Pr(z = 1|\beta=0,\omega,r)}{Pr(z = 1|\beta=a,\omega,r)}, \\
& \approx \ln \frac{Pr(z = 1|\beta=0,\omega,r)}{Pr(z = 1|\beta=a,\omega,r)},  \text{ because } r_1>>a^2, \\
& \approx \ln Pr(z = 1|\beta=0,\omega,r), \text{ because } Pr(z = 1|\beta=a,\omega,r)\approx 1,\\
& = -\ln (\frac{\omega}{(1-\omega)} \sqrt{c}+ 1).
\end{split}
\end{align}

* Finally, we impose that $Pr(z = 1|\beta,\omega,r)=Pr(z = 0|\beta,\omega,r)$ when $\beta = \frac{a}{2}$ (this is called the intersection point in [70]). This means that the slab component starts to dominate when $|\beta|>\frac{a}{2}$. It leads to the constraints:

\begin{align}
\begin{split}
0 & = (2\pi)^{-\frac{1}{2}} [\frac{\omega}{\sqrt{r_0}} \exp(-\frac{a^2}{8r_0}) - \frac{1-\omega}{\sqrt{r_1}} \exp(-\frac{a^2}{8r_1})], \\
r_0 & = \frac{a^2}{8}\frac{1-c^{-1}}{|\ln(\exp(\lambda) -1)|}.
\end{split}
\end{align}

Figure <ref> shows the penalty imposed by the SELO function and by the calibrated spike and slab prior for several values of $\lambda$ and $a$. We observe that the spike and slab prior provides a good approximation of the penalty function.

[$\lambda=1$, $a=0.01$]
[$\lambda=1$, $a=0.1$]

[$\lambda=5$, $a=0.01$]
[$\lambda=5$, $a=0.1$]

[$\lambda=10$, $a=0.01$]
[$\lambda=10$, $a=0.1$]

Penalty imposed by the SELO function in dotted black line and the spike and slab prior in blue for several values of $\lambda$ and $a$. Vertical lines are the intersection points of the spike and slab densities (at $|x|=\frac{a}{2}$).

§ MARGINAL LIKELIHOOD (<REF>) FOR THE LINEAR MODEL

Let us derive the criterion (<ref>). We first define $X_1 = \tilde{\bX}_{\tau_0}$, $X_2=X_{\btau}^{\hat{A}}$ and $M_{X_1} = \bM_{\tilde{\bX}_{\tau_0}}$. Given the prior distributions in Equation (<ref>), the marginal likelihood is given by,

\begin{eqnarray*}
f(\by|\ba,\lambda,\btau) &= & \int \int (2\pi)^{\frac{-(T+k_{\hat{A}})}{2}}(\sigma^{2})^{-\frac{(T+2+k_{\hat{A}})}{2}}|g_{\hat{A}} (X_2)' M_{X_1} X_2)|^{1/2}\\
& & \hspace{-2cm} \exp \frac{-1}{2\sigma^2}\{\underbrace{(y-X_1 \bbeta_1 - X_2\Delta \bbeta)'(y-X_1 \bbeta_1 - X_2\Delta \bbeta) + \Delta \bbeta'g_{\hat{A}} (X_2)' M_{X_1} X_2) \Delta \bbeta}_{B}\}d(\bbeta_1,\Delta \bbeta) d\sigma^2.
\end{eqnarray*}

Focusing on the expression in the exponential, we have

\begin{align*}
B & =   (y-X_1 \bbeta_1 - X_2\Delta \bbeta)'(y-X_1 \bbeta_1 - X_2\Delta \bbeta) + \Delta \bbeta'g_{\hat{A}} \underbrace{ (X_2' M_{X_1} X_2)}_{\Sigma_X} \Delta \bbeta,\\
B& =  (y- X_2\Delta \bbeta)'(y- X_2\Delta \bbeta) + \Delta \bbeta'g_{\hat{A}} \underbrace{ (X_2' M_{X_1} X_2)}_{\Sigma_X} \Delta \bbeta + \bbeta_1'X_1'X_1\bbeta_1 -2\bbeta_1'X_1'(y- X_2\Delta \bbeta),\\
B& =  (y- X_2\Delta \bbeta)'(y- X_2\Delta \bbeta) + \Delta \bbeta'g_{\hat{A}}\Sigma_X\Delta \bbeta + (\bbeta_1-\bar{\bbeta_1})'\Omega^{-1}(\bbeta_1-\bar{\bbeta_1}) - \bar{\bbeta_1}'\Omega^{-1}\bar{\bbeta_1},
\end{align*}

where $\Omega^{-1}=X_1'X_1$, $\bar{\bbeta_1}=(X_1'X_1)^{-1}X_1' (y- X_2\Delta \bbeta)$ and $\bar{\bbeta_1}'\Omega^{-1}\bar{\bbeta_1} = (y- X_2\Delta \bbeta)'X_1(X_1'X_1)^{-1}X_1' (y- X_2\Delta \bbeta) = (y- X_2\Delta \bbeta)'P_{X_1}(y- X_2\Delta \bbeta)$. The marginal likelihood can be simplifies as

\begin{align*}
&f(\by|\ba,\lambda,\btau) =  |X_1'X_1|^{-\frac{1}{2}} \int \int (2\pi)^{\frac{-(T+k_{\hat{A}}-K)}{2}}(\sigma^{2})^{-\frac{(T+2+k_{\hat{A}}-K)}{2}}|g_{\hat{A}}\Sigma_X|^{1/2}\\
& \quad \exp \frac{-1}{2\sigma^2}\{\underbrace{(y- X_2\Delta \bbeta)'(y- X_2\Delta \bbeta) + \Delta \bbeta'g_{\hat{A}}\Sigma_X\Delta \bbeta - (y- X_2\Delta \bbeta)'P_{X_1}(y- X_2\Delta \bbeta)}_{C}\}d(\Delta \bbeta) d\sigma^2.
\end{align*}

Again, focusing on the expression of the exponential, we obtain

\begin{eqnarray*}
C & = & (y- X_2\Delta \bbeta)'(y- X_2\Delta \bbeta) + \Delta \bbeta'g_{\hat{A}}\Sigma_X\Delta \bbeta - (y- X_2\Delta \bbeta)'P_{X_1}(y- X_2\Delta \bbeta),\\
& = & y'[I_T-P_{X_1}]y + \Delta \bbeta'[g_{\hat{A}} X_2'M_{X_1}X_2 + X_2'X_2 - X_2'P_{X_1}X_2]\Delta \bbeta -2\Delta \bbeta'X_2'[I_T - P_{X_1}]y,\\
& = & y'M_{X_1}y + \Delta \bbeta'[(1+g_{\hat{A}})X_2'M_{X_1}X_2]\Delta \bbeta -2\Delta \bbeta'X_2'M_{X_1}y,\\
& = & y'M_{X_1}y + (\Delta \bbeta-\bar{\bmu})'\bar{\Sigma}^{-1}(\Delta \bbeta-\bar{\bmu}) - \bar{\bmu}'\bar{\Sigma}^{-1}\bar{\bmu},
\end{eqnarray*}

where $\bar{\Sigma}^{-1} = (1+g_{\hat{A}})X_2'M_{X_1}X_2 = (1+g_{\hat{A}})\Sigma_X$ and $\bar{\bmu} = \bar{\Sigma} X_2'M_{X_1}y$, $\bar{\bmu}'\bar{\Sigma}^{-1}\bar{\bmu} = (1+g_{\hat{A}})^{-1}y'M_{X_1}X_2 [X_2'M_{X_1}X_2]^{-1}X_2'M_{X_1}y$.

Eventually, we find the following marginal likelihood

\begin{eqnarray*}
f(\by|\ba,\lambda,\btau) &= & (2\pi)^{\frac{-(T-K)}{2}} |X_1'X_1|^{-\frac{1}{2}} |g_{\hat{A}} \Sigma_X|^{1/2} |(1+g_{\hat{A}})\Sigma_X|^{\frac{-1}{2}}  \int (\sigma^{2})^{-\frac{(T+2-K)}{2}}\\
& & \exp \frac{-1}{2\sigma^2}\{y'M_{X_1}y-(1+g_{\hat{A}})^{-1}y'M_{X_1}X_2 [X_2'M_{X_1}X_2]^{-1}X_2'M_{X_1}y\} d\sigma^2,\\
& = & (\pi)^{\frac{-(T-K)}{2}} \Gamma(\frac{T-K}{2})|X_1'X_1|^{-\frac{1}{2}} \\
& & (\frac{g_{\hat{A}}}{1+g_{\hat{A}}})^{k_{\hat{A}}/2} [y'M_{X_1}y-(1+g_{\hat{A}})^{-1}y'M_{X_1}X_2 [X_2'M_{X_1}X_2]^{-1}X_2'M_{X_1}y]^{-\frac{T-K}{2}}, \\
& = & (\pi)^{\frac{-(T-K)}{2}} \Gamma(\frac{T-K}{2})|X_1'X_1|^{-\frac{1}{2}} \\
& & (\frac{g_{\hat{A}}}{1+g_{\hat{A}}})^{k_{\hat{A}}/2} [\frac{g_{\hat{A}}}{1+g_{\hat{A}}}y'M_{X_1}y+\frac{1}{(1+g_{\hat{A}})}[\tilde{y}'\tilde{y}-\tilde{y}'X_2[X_2'M_{X_1}X_2]^{-1}X_2\tilde{y}]]^{-\frac{T-K}{2}}, \\
& = & (\pi)^{\frac{-(T-K)}{2}} \Gamma(\frac{T-K}{2})|X_1'X_1|^{-\frac{1}{2}} (\frac{g_{\hat{A}}}{1+g_{\hat{A}}})^{k_{\hat{A}}/2} [\frac{g_{\hat{A}}}{1+g_{\hat{A}}}s_{X_1}+\frac{1}{(1+g_{\hat{A}})}s_{X_1,X_2}]^{-\frac{T-K}{2}} ,
%& = & (\pi)^{\frac{-(T-K)}{2}} |X_1'X_1|^{\frac{1}{2}} (\frac{g_{\hat{A}}}{1+g_{\hat{A}}})^{\frac{k_{\hat{A}}}{2}} \Gamma(\frac{T-K}{2}) (y'M_{X_1}y-(1+g_{\hat{A}})^{-1} y'X_2 \Sigma_X^{-1} X_2'y)^{-\frac{T-K}{2}}\\
%& \propto & (\frac{g_{\hat{A}}}{1+g_{\hat{A}}})^{\frac{k_{\hat{A}}}{2}} (y'M_{X_1}y-(1+g_{\hat{A}})^{-1} y'X_2 (X_2'X_2)^{-1} X_2'y)^{-\frac{T-K}{2}}
\end{eqnarray*}

where the penultimate equality comes from the Frisch-Waugh theorem.

§.§ Posterior distribution

\begin{eqnarray*}
f(\bbeta_1,\Delta \bbeta, \sigma^2|\by,\btau) & \propto & (2\pi)^{\frac{-(T+k_{\hat{A}})}{2}}(\sigma^{2})^{-\frac{(T+2+k_{\hat{A}})}{2}}|g_{\hat{A}} (X_2)' M_{X_1} X_2)|^{1/2}\\
& & \exp \frac{-1}{2\sigma^2} \left(\underbrace{(y-X_1 \bbeta_1 - X_2\Delta \bbeta)'(y-X_1 \bbeta_1 - X_2\Delta \bbeta) + \Delta \bbeta'g_{\hat{A}} (X_2)' M_{X_1} X_2 \Delta \bbeta}_{\text{Exp}}\right)
\end{eqnarray*}

Focusing on the expression of the exponential, we have

\begin{eqnarray*}
\text{Exp} & = & (y- X_2\Delta \bbeta)'(y- X_2\Delta \bbeta) + \Delta \bbeta'g_{\hat{A}}\Sigma_X\Delta \bbeta + (\bbeta_1-\bar{\bbeta_1})'\Omega^{-1}(\bbeta_1-\bar{\bbeta_1}) - \bar{\bbeta_1}'\Omega^{-1}\bar{\bbeta_1},\\
& = & y'M_{X_1}y - \bar{\bmu}'\bar{\Sigma}^{-1}\bar{\bmu} + (\Delta \bbeta-\bar{\bmu})'\bar{\Sigma}^{-1}(\Delta \bbeta-\bar{\bmu}) + (\bbeta_1-\bar{\bbeta_1})'\Omega^{-1}(\bbeta_1-\bar{\bbeta_1}),\\
& = & \frac{g_{\hat{A}}}{1+g_{\hat{A}}}s_{X_1}+\frac{1}{(1+g_{\hat{A}})}s_{X_1,X_2} + (\Delta \bbeta-\bar{\bmu})'\bar{\Sigma}^{-1}(\Delta \bbeta-\bar{\bmu}) + (\bbeta_1-\bar{\bbeta_1})'\Omega^{-1}(\bbeta_1-\bar{\bbeta_1}),\\
\end{eqnarray*}

where $\bar{\Sigma}^{-1} = (1+g_{\hat{A}})X_2'M_{X_1}X_2 = (1+g_{\hat{A}})\Sigma_X$ and $\bar{\bmu} = \bar{\Sigma} X_2'M_{X_1}y$, $\bar{\bmu}'\bar{\Sigma}^{-1}\bar{\bmu} = (1+g_{\hat{A}})^{-1}y'M_{X_1}X_2 [X_2'M_{X_1}X_2]^{-1}X_2'M_{X_1}y$ and $\Omega^{-1}=X_1'X_1$, $\bar{\bbeta_1}=(X_1'X_1)^{-1}X_1' (y- X_2\Delta \bbeta)$. The posterior distribution can be decomposed as

\begin{eqnarray*}
f(\bbeta_1,\Delta \bbeta, \sigma^2|\by,\btau) & = & f(\sigma^2|\by,\btau) f(\Delta \bbeta|\by,\btau,\sigma^2) f(\bbeta_1|\by,\btau,\sigma^2,\Delta \bbeta)\\
& \propto & (\sigma^{2})^{-\frac{(T+2-K)}{2}} \exp \frac{-1}{\sigma^2}\{\frac{\frac{g_{\hat{A}}}{1+g_{\hat{A}}}s_{X_1}+\frac{1}{(1+g_{\hat{A}})}s_{X_1,X_2}}{2}\}\\
& & (\sigma^{2})^{-\frac{(k_{\hat{A}})}{2}}|g_{\hat{A}} (X_2)' M_{X_1} X_2)|^{1/2} \exp \frac{-1}{2\sigma^2}\{(\Delta \bbeta-\bar{\bmu})'\bar{\Sigma}^{-1}(\Delta \bbeta-\bar{\bmu})\}\\
& & (\sigma^{2})^{-\frac{(K)}{2}}\exp \frac{-1}{2\sigma^2}\{(\bbeta_1-\bar{\bbeta_1})'\Omega^{-1}(\bbeta_1-\bar{\bbeta_1})\}.\\
\end{eqnarray*}

It gives the following posterior distribution

\begin{eqnarray*}
\sigma^2|\by,\btau & \sim & \mathcal{IG}(\frac{T-K}{2},\frac{\frac{g_{\hat{A}}}{1+g_{\hat{A}}}s_{X_1}+\frac{1}{(1+g_{\hat{A}})}s_{X_1,X_2}}{2}),\\
\Delta \bbeta|\by,\btau,\sigma^2 & \sim & \NORM((1+g_{\hat{A}})^{-1}[X_2'M_{X_1}X_2]^{-1}X_2'M_{X_1}y, \sigma^2 (1+g_{\hat{A}})^{-1}[X_2'M_{X_1}X_2]^{-1}),  \\
\bbeta_1|\by,\btau,\sigma^2,\Delta \bbeta & \sim & \mathcal N((X_1'X_1)^{-1}X_1' (y- X_2\Delta \bbeta), \sigma^2(X_1'X_1)^{-1}).
\end{eqnarray*}

§.§ Predictive density

In Appendix <ref>, we derive the following posterior distributions:

\begin{eqnarray*}
\sigma^2|\by,\btau & \sim & \mathcal{IG}(\underbrace{\frac{T-K}{2}}_{a_{\sigma^2}},\underbrace{\frac{\frac{g_{\hat{A}}}{1+g_{\hat{A}}}s_{X_1}+\frac{1}{(1+g_{\hat{A}})}s_{X_1,X_2}}{2}}_{b_{\sigma^2}}),\\
\Delta \bbeta|\by,\btau,\sigma^2 & \sim & \NORM(\underbrace{(1+g_{\hat{A}})^{-1}[X_2'M_{X_1}X_2]^{-1}X_2'M_{X_1}y}_{\bmu_{\Delta \bbeta }}, \underbrace{\sigma^2 (1+g_{\hat{A}})^{-1}[X_2'M_{X_1}X_2]^{-1}}_{\bSigma_{\Delta \bbeta}}),  \\
\bbeta_1|\by,\btau,\sigma^2,\Delta \bbeta & \sim & \mathcal N(\underbrace{(X_1'X_1)^{-1}X_1' (y- X_2\Delta \bbeta)}_{\bmu_{\bbeta}}, \underbrace{\sigma^2(X_1'X_1)^{-1}}_{\bSigma_{\bbeta}}).
\end{eqnarray*}

Given these results, we can derive the joint posterior distribution of the variable $\bpsi  = \begin{pmatrix} \bbeta_1 \\ \Delta \bbeta \end{pmatrix}$. In particular, a standard algebraic calculus leads to

\begin{align}
\begin{split}
\bpsi|\by,\btau,\sigma^2 & \sim \NORM\left(\begin{pmatrix} \hat{\bbeta}_1 - \bB \mu_{\Delta \bbeta } \\ \bmu_{\Delta \bbeta }\end{pmatrix}, \begin{pmatrix}\bSigma_{\bbeta}^{-1} & \bSigma^{-1}_{\beta}\bB \\ \bB'\bSigma^{-1}_{\beta}& [\bSigma_{\Delta \bbeta }^{-1}+\bB'\bSigma_{\bbeta}^{-1}\bB]\end{pmatrix}^{-1}\right), \\
& \sim \NORM\left( \bmu_{\bpsi},\bSigma_{\bpsi} \right)
\end{split}
\end{align}

with $\hat{\bbeta}_1 = (X_1'X_1)^{-1}X_1'y$ and $\bB= (X_1'X_1)^{-1}X_2$. Consequently, the predictive density is given by

\begin{align}
\begin{split}
y_{T+1}|\by,\btau,\sigma^2 & \sim \NORM\left(\bx_{T+1}'\bmu_{\bpsi},\sigma^2(\bx_{T+1}'\bSigma_{\psi}\bx_{T+1} + 1)\right).
\end{split}
\end{align}

Since $\sigma^2|\by,\btau$ follows an inverse gamma distribution, the predictive distribution of $y_{T+1}|\by$ is a student distribution. Its density is given by

\begin{align}
\begin{split} \label{eq:student}
f(y_{T+1}|\by,\btau) & = \frac{b_{\sigma^2}^{a_{\sigma^2}}}{\Gamma(a_{\sigma^2})}(2\pi (\bx_{T+1}'\bSigma_{\bpsi}\bx_{T+1} + 1))^{-\frac{1}{2}} \\
& \quad \int (\sigma^2)^{-(a_{\sigma^2}+1+0.5)} \exp \left(-\frac{1}{\sigma^2}[\frac{(y_{T+1}-\bx_{T+1}'\bmu_{\bpsi})^2 (\bx_{T+1}'\bSigma_{\bpsi}\bx_{T+1} + 1)^{-1} + 2b_{\sigma^2}}{2}] \right)d\sigma^2, \\
& = \frac{b_{\sigma^2}^{a_{\sigma^2}}}{\Gamma(a_{\sigma^2})}(2\pi (\bx_{T+1}'\bSigma_{\bpsi}\bx_{T+1} + 1))^{-\frac{1}{2}} \Gamma(a_{\sigma^2}+0.5) \\
& \quad \left(\frac{(y_{T+1}-\bx_{T+1}'\bmu_{\bpsi})^2(\bx_{T+1}'\bSigma_{\bpsi}\bx_{T+1} + 1)^{-1} + 2b_{\sigma^2}}{2}\right)^{-(a_{\sigma^2}+0.5)},\\
\end{split}
\end{align}

The final expression in Equation (<ref>) is equivalent to a student density with expectation $\bx_{T+1}'\bmu_{\bpsi}$, scale parameter $\frac{b_{\sigma^2}}{a_{\sigma^2}}(\bx_{T+1}'\bSigma_{\psi}\bx_{T+1} + 1)$ and degree of freedom equal to $2a_{\sigma^2}$.

§ CONSISTENCY OF THE CRITERION

To prove the theorem, we focus on the ratio of the criterion for two different models $s=(a_s,\lambda_s)$ and $j=(a_j,\lambda_j)$ where $s$ is considered as the true model. To simplify the notation, we denote by $\bX_z$ the explanatory variable included by model z (i.e., $\bX_z = \bX_{\btau}^{\hat{A}_z}$) for $z=s,j$ and $g_{\hat{A}} = g = \frac{1}{w(T)}$ and write the marginal likelihood as $f(\by|a_z,\lambda_z)$ instead of $f(\by|a_z,\lambda_z,\btau)$. We need to show that $\frac{f(\by|a_j,\lambda_j)}{f(\by|a_s,\lambda_s)} \rightarrow_p 0 $ for any $j\neq s$. In particular, we have

\begin{equation} \label{eq:BF}
\begin{split}
\frac{f(\by|a_j,\lambda_j)}{f(\by|a_s,\lambda_s)} & = \underbrace{\frac{(\frac{g}{1+g})^{k_{\hat{A}_j}/2}}{(\frac{g}{1+g})^{k_{\hat{A}_s}/2}}}_{C_{js}} \underbrace{[\frac{\frac{g}{1+g}s_{\tilde{\bX}_{\tau_0}}+\frac{1}{(1+g)}s_{\tilde{\bX}_{\tau_0},\bX_j}}{\frac{g}{1+g}s_{\tilde{\bX}_{\tau_0}}+\frac{1}{(1+g)}s_{\tilde{\bX}_{\tau_0},\bX_s}}]^{-\frac{T-K}{2}}}_{D_{js}}.
\end{split}
\end{equation}

Focusing on the first term, it is easy to show that

\begin{equation*}
\begin{split}
C_{js} & = \frac{(1+g)^{k_{\hat{A}_s}/2}}{(1+g)^{k_{\hat{A}_j}/2}} g^{\frac{k_{\hat{A}_j} - k_{\hat{A}_s}}{2}} \\
& = \frac{(1+w(T)^{-1})^{k_{\hat{A}_s}/2}}{(1+w(T)^{-1})^{k_{\hat{A}_j}/2}} w(T)^{\frac{k_{\hat{A}_s}- k_{\hat{A}_j}}{2}} \\
& = \mathcal{O}(w(T)^{\frac{k_{\hat{A}_s}- k_{\hat{A}_j}}{2}}).
\end{split}
\end{equation*}

When $T \ra \infty$, we have

\begin{equation*}
\begin{split}
C_{js} & = 0 \text{ when } k_{\hat{A}_s} < k_{\hat{A}_j}, \\
& = 1 \text{ if } k_{\hat{A}_s} = k_{\hat{A}_j}, \\
&  \ra + \infty \text{ when } k_{\hat{A}_s} > k_{\hat{A}_j}.
\end{split}
\end{equation*}

We now discuss three possible cases.

* $k_{\hat{A}_s} < k_{\hat{A}_j}$ and the model $j$ does not nest the model $s$. In such case, the term $C_{js} \ra 0$. The second term also tends to zero since we have

\begin{equation*}
\begin{split}
D_{js} & = [\frac{\frac{g}{1+g}s_{\tilde{\bX}_{\tau_0}}+\frac{1}{(1+g)}s_{\tilde{\bX}_{\tau_0},\bX_s}}{\frac{g}{1+g}s_{\tilde{\bX}_{\tau_0}}+\frac{1}{(1+g)}s_{\tilde{\bX}_{\tau_0},\bX_j}}]^{\frac{T-K}{2}}\\
& = [\frac{g s_{\tilde{\bX}_{\tau_0}}+ s_{\tilde{\bX}_{\tau_0},\bX_s}}{g s_{\tilde{\bX}_{\tau_0}}+ s_{\tilde{\bX}_{\tau_0},\bX_j}}]^{\frac{T-K}{2}}
\end{split}
\end{equation*}

Using he fact that $M_j$ does not nest $M_s$ and the Frisch-Waugh theorem (see also Lemma A.1 in [31]), we have that $\lim_{T \ra \infty} \frac{s_{\tilde{\bX}_{\tau_0},\bX_j}}{T} = \sigma^2 + b_j$ with $b_j>0$. Combining with the fact that $g\ra 0$, we end up with a limit of $D_{js}$ given by

\begin{equation*}
\begin{split}
\lim_{T \ra \infty} D_{js} & = [\frac{\sigma^2}{\sigma^2  + b_j}]^{\frac{T-K}{2}} \ra 0.
\end{split}
\end{equation*}

* The model $j$ does not nest the true model but $K_j < K_s$. In such case, the term $C_{js} \ra + \infty$. However, we can show that $\lim_{T \ra \infty} C_{js} w(T)^{-\frac{(K_s-K_j)}{2} + \frac{K_s-K_j}{T-K}} \ra 1$. Indeed, we have that

\begin{equation*}
\begin{split}
\lim_{T \ra \infty} C_{js} w(T)^{-\frac{(K_s-K_j)}{2} + \frac{K_s-K_j}{T-K}} & = \frac{(1+w(T)^{-1})^{k_{\hat{A}_s}/2}}{(1+w(T)^{-1})^{k_{\hat{A}_j}/2}} w(T)^{\frac{K_s-K_j}{T-K}}.
\end{split}
\end{equation*}

Let us define $q_T = w(T)^{\frac{K_s-K_j}{T-K}}$. We can compute the limit as follows $\lim_{T \ra \infty} w(T)^{\frac{K_s-K_j}{T-K}} = \lim_{T \ra \infty} \exp \ln q_T$. The limit of $\ln q_T$ is given by

\begin{equation*}
\begin{split}
\lim_{T \ra \infty} q_T & =\lim_{T \ra \infty} \frac{K_s-K_j}{T-K} \ln w(T), \\
& = \lim_{T \ra \infty} \frac{w'(T)}{w(T)} ~~( = 0~~\text{ by assumption}).
\end{split}
\end{equation*}

We conclude that $\lim_{T \ra \infty} w(T)^{\frac{K_s-K_j}{T-K}} = 1$. Now, we need to show that $D_{js}w(T)^{\frac{(K_s-K_j)}{2} - \frac{K_s-K_j}{T-K}} \ra 0$. In fact, we have

\begin{eqnarray*}
D_{js}w(T)^{\frac{\frac{(K_s-K_j)}{2}(T-K-2)}{T-K}}  & = & \lim_{T \ra \infty} (\underbrace{\frac{\sigma^2}{\sigma^2 + b_j}}_{a<1})^{\frac{T-K}{2}}  w(T)^{\frac{(K_s-K_j)}{2}}\\
&  =& \lim_{T \ra \infty}  \frac{w(T)^{\frac{(K_s-K_j)}{2}}}{a^{-\frac{T-K}{2}}},
\end{eqnarray*}

By applying $\left\lceil \frac{(K_s-K_j)}{2}\right\rceil$ times the Hospital's rule, we find that $a^{\frac{T-K}{2}}$ dominates and so $D_{js}w(T)^{\frac{\frac{(K_s-K_j)}{2}(T-K-2)}{T-K}} \ra 0$.

* We now consider the last case in which the model $j$ nests the true model $s$. Consequently, we have $K_s < K_j$ and the term $C_{js} \ra 0$. Regarding the other term, we can express it as

\begin{equation*}
\begin{split}
D_{js} & = [\frac{g s_{\tilde{\bX}_{\tau_0}}+ s_{\tilde{\bX}_{\tau_0},\bX_s}}{g s_{\tilde{\bX}_{\tau_0}}+ s_{\tilde{\bX}_{\tau_0},\bX_j}}]^{\frac{T-K}{2}},\\
& = \underbrace{[\frac{s_{\tilde{\bX}_{\tau_0},\bX_s}}{s_{\tilde{\bX}_{\tau_0},\bX_j}}]^{\frac{T-K}{2}}}_{Q_1} \underbrace{[\frac{A_s+ w(T)}{A_j+ w(T)}]^{\frac{T-K}{2}}}_{Q2},
\end{split}
\end{equation*}

where $A_i = \frac{s_{\tilde{\bX}_{\tau_0}}}{s_{\tilde{\bX}_{\tau_0},\bX_i}}$ for $i=j,s$. It is clear that the first term $Q_1$ has a limiting distribution related to the likelihood ratio test. In fact, we have that

\begin{eqnarray*}
\frac{T-K}{2} \ln \frac{s_{\tilde{\bX}_{\tau_0},\bX_s}}{s_{\tilde{\bX}_{\tau_0},\bX_j}} & = & \underbrace{\frac{T-K}{2T}}_{\ra \frac{1}{2}} \underbrace{T\ln \frac{s_{\tilde{\bX}_{\tau_0},\bX_s}}{s_{\tilde{\bX}_{\tau_0},\bX_j}}}_{\ra_d \chi^2(\Delta_{js})},\\
& \ra_d & \text{Gamma}(\frac{\Delta_{js}}{2},1),
\end{eqnarray*}

in which $\Delta_{js} = |K_s-K_j|$. Since $Y \sim \text{Gamma}(\frac{\Delta_{js}}{2},1)$ is $\mathcal{O}_p(1)$, we have that $C_{js} \exp Y \ra_p 0$.

Focusing on the second term $Q_2$, using assumption (iii), we have that

\begin{equation*}
\begin{split}
\frac{T-K}{2} \ln [\frac{A_s+ w(T)}{A_j+ w(T)}] & = \frac{T-K}{2}\ln [1 + \frac{A_s - A_j}{A_j+ w(T)}], \\
& = \mathcal{O}_p(\frac{T}{w(T)}).\\
& \ra_p  [0,\infty).
\end{split}
\end{equation*}

Since $C_{js} \ra 0$, we conclude that $C_{js} Q_1 Q_2 \ra_p 0$.

§.§ Convergence to the BIC

The BIC of a linear regression model with $K$ parameters is given by

\begin{align}
\begin{split}
BIC(K) &  = -\frac{T}{2} \ln(\frac{s_{\bX}}{T})	-\frac{K}{2} \ln T, \\
& = \underbrace{-\frac{T}{2} \ln(s_{\bX})	-\frac{K}{2} \ln T}_{BIC^*(K)} + \frac{T}{2}\ln T,
\end{split}
\end{align}

in which $s_{\bX}$ denotes the sum of squared residuals given the $T \times K$ dimensional exogenous variables $\bX$ evaluated at the OLS estimates. In this appendix, we show that the logarithm of the marginal likelihood and the BIC$^*(\alpha k_{\hat{A}})$ converges in probability to 0 when $g_{\hat{A}} = \frac{1}{T^{\alpha}} $ with $\alpha>1$. In particular, the marginal likelihood is given by

\begin{equation}
\begin{split}
f(\by|\ba,\lambda,\btau) & = (\frac{g_{\hat{A}}}{1+g_{\hat{A}}})^{k_{\hat{A}}/2} [\frac{g_{\hat{A}}}{1+g_{\hat{A}}}s_{\tilde{\bX}_{\tau_0}}+\frac{1}{(1+g_{\hat{A}})}s_{\tilde{\bX}_{\tau_0},\tilde{\bX}_{\btau}^{\hat{A}}}]^{-\frac{T-K}{2}}.
\end{split}
\end{equation}

We have the following results:

\begin{align}
\begin{split}
f(\by|\ba,\lambda,\btau) & = T^{-\frac{\alpha k_{\hat{A}}}{2}} [\frac{1}{T^{\alpha}}s_{\tilde{\bX}_{\tau_0}}+\frac{T^{\alpha}-1}{T^{\alpha}}s_{\tilde{\bX}_{\tau_0},\tilde{\bX}_{\btau}^{\hat{A}}}]^{-\frac{T-K}{2}}, \\
& = T^{-\frac{\alpha k_{\hat{A}}}{2}} [s_{\tilde{\bX}_{\tau_0},\tilde{\bX}_{\btau}^{\hat{A}}}]^{-\frac{T-K}{2}} [\frac{T^{\alpha}-1}{T^{\alpha}}]^{-\frac{T-K}{2}} [\frac{1}{T^{\alpha}-1} \frac{s_{\tilde{\bX}_{\tau_0}}}{s_{\tilde{\bX}_{\tau_0},\tilde{\bX}_{\btau}^{\hat{A}}}}+ 1]^{-\frac{T-K}{2}}, \\
& = T^{-\frac{\alpha k_{\hat{A}}}{2}} [s_{\tilde{\bX}_{\tau_0},\tilde{\bX}_{\btau}^{\hat{A}}}]^{-\frac{T-K}{2}}  \underbrace{[1-\frac{1}{T^{\alpha}}]^{-\frac{T-K}{2}}}_{C_1} \underbrace{[\frac{1}{T^{\alpha}-1} \frac{s_{\tilde{\bX}_{\tau_0}}}{s_{\tilde{\bX}_{\tau_0},\tilde{\bX}_{\btau}^{\hat{A}}}}+ 1]^{-\frac{T-K}{2}}}_{C_2}, \\
\ln f(\by|\ba,\lambda,\btau) & = -\frac{T - K}{T}\frac{T}{2}\ln s_{\tilde{\bX}_{\tau_0},\tilde{\bX}_{\btau}^{\hat{A}}} - \frac{\alpha k_{\hat{A}}}{2}\ln T + \ln C_1 + \ln C_2
\end{split}
\end{align}

We now show that the two quantities, i.e. $C_1$ and $C_2$, tends to 1 when $T\ra +\infty$:

\begin{align}
\begin{split}
C_1 & = \exp \left(-\frac{T-K}{2} \ln [1-\frac{1}{T^{\alpha}}] \right), \\
& \ra_p 1 \text{ since } \alpha>1, \\
C_2 & = \exp \left(-\frac{T-K}{2} \ln	[\frac{1}{T^{\alpha}-1} \frac{s_{\tilde{\bX}_{\tau_0}}}{s_{\tilde{\bX}_{\tau_0},\tilde{\bX}_{\btau}^{\hat{A}}}}+ 1]	\right), \\
& \ra_p 1 \text{ since } \alpha>1 \text{ and } \frac{s_{\tilde{\bX}_{\tau_0}}}{s_{\tilde{\bX}_{\tau_0},\tilde{\bX}_{\btau}^{\hat{A}}}}=\mathcal{O}_p(1),
\end{split}
\end{align}

It follows that

\begin{align}
\begin{split} \ln f(\by|\ba,\lambda,\btau) -  \left(-\frac{T - K}{T}\frac{T}{2} \ln s_{\tilde{\bX}_{\tau_0},\tilde{\bX}_{\btau}^{\hat{A}}}	-\frac{\alpha k_{\hat{A}}}{2} \ln T\right) &\overset{p}{\rightarrow} 0\\
\ln f(\by|\ba,\lambda,\btau) -  \left(-\frac{T}{2} \ln s_{\tilde{\bX}_{\tau_0},\tilde{\bX}_{\btau}^{\hat{A}}}	-\frac{\alpha k_{\hat{A}}}{2} \ln T\right) &\overset{p}{\rightarrow} 0 \\
\dfrac{f(\by|\ba,\lambda,\btau)}{\exp\left(-\frac{T}{2} \ln s_{\tilde{\bX}_{\tau_0},\tilde{\bX}_{\btau}^{\hat{A}}}	-\frac{\alpha k_{\hat{A}}}{2} \ln T\right)} &\overset{p}{\rightarrow} 1
\end{split}
\end{align}

As a result, the models selected using the marginal likelihood are equivalent (asymptotically) to those of the BIC with $\alpha k_{\hat{A}}$ parameters. In addition, the posterior probabilities computed using the marginal likelihood converge to the posterior probabilities that would have been computed using the BIC with $\alpha k_{\hat{A}}$ parameters (since the term $\frac{T}{2}\ln T$ cancels out).

§ TIME-VARYING PARAMETER MODEL

We also consider a standard time-varying parameter process (see [64]). The model specification is given by

\begin{eqnarray}
y_t & = & \bx_t' \bbeta_0 + \bx_t' \text{diag}(\bomega)\bbeta_t + \sigma_t \epsilon_t,\label{eq:Bitto1}\\
\bbeta_t|\bbeta_{t-1} & \sim & \NORM(\bbeta_{t-1},I_K), \\
\ln \sigma_t^2 & = &  \ln \sigma_{t-1}^2 + \eta_t, \text{for } t>0, \\
\ln \sigma_0^2 & \sim & N(0,1), \label{eq:Bitto2}
\end{eqnarray}

where $\bomega=(\omega_1,...,\omega_K)'$, $\eta_t \sim \NORM(0,q)$ with $q\sim \mathcal{IG}(\underline{q}_1=3,\underline{q}_2=20)$, $(\bbeta_0',\bomega')' \sim \NORM(0,I_{2K})$ and $K$ stands for the number of explanatory variables. We also define $\ln \sigma_{1:T} = (\ln \sigma_{1}^2, \ldots, \ln \sigma_{T}^2)'$ and $y_{1:T}$, $\bbeta_{0:T}$ analogously. In order to take into account the autocorrelation structure, we use the same lag orders as exposed in Table <ref>. The other explanatory variables consist in an intercept and the seven factors.

The model parameters can be estimated with a standard Markov-chain Monte Carlo <cit.>. In particular,  the model parameters are estimated using an MCMC algorithm which consists of the following steps:

* Sampling $\ln \sigma_{1:T}$ using the offset mixture approach of [48]. In particular, given $\bbeta_{0:T}$ and $\bomega$, we compute $y_t^* = \ln(\nu_t^2 + c)$, for all $t=1,\ldots,T$ in which $\nu_t = y_t- \bx_t (\bbeta_0 + \text{diag}(\omega_1,...,\omega_K)\bbeta_t)$ and $c=0.0001$. The model boils down to a standard TVP model with time-varying intercept since we have

\begin{eqnarray}
y_t^* & = & \ln \sigma_t^2  + \ln \epsilon_t^2,\\
\ln \sigma_t^2 & = &  \ln \sigma_{t-1}^2 + \eta_t.
\end{eqnarray}

Approximating the distribution $\ln \epsilon_t^2$ with a 8-component mixture of normal distributions, we sample the time-varying variance from a high-dimensional multivariate normal distribution using the sampler 'all without a loop' (AWOL) as suggested in [56] <cit.>.

* Sampling $\ln \sigma_0^2|\ln \sigma_1^2,q \sim N((q^{-1}+1)^{-1}\frac{\ln \sigma_1^2}{q},(q^{-1}+1)^{-1})$.

* Sampling $q|\ln \sigma_{0:T} \sim IG(\underline{q}_1+\frac{T}{2},\underline{q}_2 + \sum_{t=1}^{T}\frac{(\ln \sigma_t^2-\ln \sigma_{t-1}^2)^2}{2}$).

* Sampling $\bbeta_{1:T}|y_{1:T},\bbeta_0,\bomega,\ln \sigma_{1:T}$ using a Kalman filter. Note that this step could also be carried out with the AWOL approach.

* Sampling $(\bbeta_0',\bomega')'|y_{1:T},\bbeta_{1:T},\ln \sigma_{1:T}$ from a multivariate normal distribution. In fact, conditioning to $\bbeta_{1:T}$ and $\ln \sigma_{1:T}$, the model boils down to a standard regression model since we have

\begin{eqnarray}
y_t & = & \bx_t' \bbeta_0 + (\bx_t'\text{diag}(\bbeta_t)) \bomega + \sigma_t \epsilon_t,\\
& = & \bx_t' \bbeta_0 + \mathbf{z}_t'\bomega + \sigma_t \epsilon_t,
\end{eqnarray}

in which $\mathbf{z}_t=(\bx_t'\text{diag}(\bbeta_t))'$.

§.§ Time-varying parameters computed with the FIA returns

[SELO - PMKT]
[TVP - PMKT]

[SELO -SMB]
[TVP - SMB]

[SELO - TERM]
[TVP - TERM]

[SELO - DEF]
[TVP - DEF]

FIA returns - Selective segmentation (SELO) model and Time-varying parameter (TVP) model. Posterior medians (black) and the 90% credible intervals (dotted black lines) of the model parameters over time. For the SELO method, we take the break uncertainty into account using the MCMC algorithm presented in Section <ref>.

[SELO - PTFSBD]
[TVP - PTFSBD]

[SELO - PTFSFX]
[TVP - PTFSFX]

[SELO - PTFSCOM]
[TVP - PTFSCOM]

[SELO - UMD]
[TVP - UMD]

FIA returns - Selective segmentation (SELO) model and Time-varying parameter (TVP) model. Posterior medians (black) and the 90% credible intervals (dotted black lines) of the model parameters over time. For the SELO method, we take the break uncertainty into account using the MCMC algorithm presented in Section <ref>.

[SELO - PTFSIR]
[TVP - PTFSIR]

[SELO - PTFSSTK]
[TVP - PTFSSTK]

[SELO - CPI]
[TVP - CPI]

[SELO - NAREIT]
[TVP - NAREIT]

FIA returns - Selective segmentation (SELO) model and Time-varying parameter (TVP) model. Posterior medians (black) and the 90% credible intervals (dotted black lines) of the model parameters over time. For the SELO method, we take the break uncertainty into account using the MCMC algorithm presented in Section <ref>.

§ BAYESIAN ALTERNATIVES TO THE SELECTIVE SEGMENTATION METHOD

The DGP J exhibits 100 explanatory variables and one CP in ten parameters. To capture which model parameters are experiencing a break, we are aware of four Bayesian alternatives that are [40], [27], [43] and [24]. We now discuss why these alternatives do not work when 100 explanatory variables are involved.

§.§ [40]

The model of [40] stands for a particular case of the general mixture state space model developed in [37]. They explain how to do  inference for a Gaussian state space model with a latent variable $K_t$ that determines the state of the model parameters.

Modelling breaks in the mean parameters, their approach allows  estimating the following state space model:

\begin{eqnarray}
y_t & = & \beta_{t,1} + \beta_{t,2} x_{t,2} + \ldots + \beta_{t,N} x_{t,N} + \sigma \eta_t,  \label{eq:obs}\\
\beta_{t,i} & = & \beta_{t-1,i} + \gamma_{i,K_{t,i}} \nu_{t,i}, ~\text{for } i=1,\ldots,N, \text{ and } t>1, \label{eq:TVP}
\end{eqnarray}

in which $N$ is the number of explanatory variables, $\eta_t  \sim N(0,1)$, $K_t=\{K_{t,1},\ldots,K_{t,N}\}$ and $\nu_t = (\nu_{t,1},\ldots,\nu_{t,N})' \sim N(0,I_{N})$ (with $I_N$, the identity matrix of dimension $N$). To capture breakpoints, [40] suggest to set the states of the latent variable $K_{t,i}$ to $\{0,1\}$ such that we have $\gamma_{i,0}=0$ (i.e., no break when $K_{t,i}=0$) and $\gamma_{i,1}\in \Re^{+}$ (i.e., break in the $i$th parameter when $K_{t,i}=1$). The model parameters are given by $\btheta=\{\gamma_{1,1},\ldots,\gamma_{N,1},\beta_{1,1},\ldots,\beta_{1,N},\sigma\}$.

To efficiently estimate the model, [40] relies on the algorithm of [37]. The main contribution of [37] is to marginalize out the mean parameters $\beta_{1:T,1:N}$ and to provide an analytical formula for the latent variable distribution $f(K_t|y_{1:T},K_{\neq t},\btheta)$ from which $K_t$ is sampled in the MCMC algorithm. To normalize the posterior distribution $f(K_t|y_{1:T},K_{\neq t},\btheta)$, it requires to sum over all the possible values of $K_t$. Because $K_t=\{K_{t,1},\ldots,K_{t,N}\}$ and $K_{t,i}=\{0,1\}~\forall i\in[1,N]$, the number of possible values for $K_t$ amounts to $2^{N}$. Consequently, it increases geometrically with the number of explanatory variables. This is why [16] on page 9 argue that the number of explanatory variables should be small (i.e., at least below fourteen) otherwise some structure on the break dynamic should be accounted for. With 100 regressors involved in DGP J, it is infeasible to compute the latent variable distribution $f(K_t|y_{1:T},K_{\neq t},\btheta)$ because its normalization requires to sum over $2^{100}$ values.

§.§ [27]

[27] relies on the approach of [21] for finding which parameters are experiencing a break. The method consists of estimating all the possible models given several number of breakpoints. Then, the best specification is selected by maximizing the marginal likelihood that is computed, for each model, using the method of [20]. Considering DGP J and its 100 exogenous variables, the number of models to estimate reaches $\sum_{i=0}^{\bar{m}} 2^{100 i}$ in which $\bar{m}$ is the maximum number of breaks that can experience a parameter. In our context, the approach is computationally infeasible even when the upper bound of the number of break is equal to 1.

§.§ [43]

[43] propose a threshold approach to approximate the MCMC inference of mixture state space models. It generalizes the method of [40] because it is not limited by the number of explanatory variables. As illustrated in Appendix D of [25], the approximation makes the MCMC inference depending on the starting value and the estimated breakpoints are unstable from one estimation to another. Consequently, in-sample results and forecasting exercises will also depend on starting values. Because the question on how to choose the starting values is not addressed in the paper,  the method does not provide reproducible results. However, it could be useful for exploring the space in order to find a promising starting value to be used in the MCMC algorithm of [40].

§.§ [24]

[24] rely on the standard CP model (see, e.g., [21], [63] or [54]) to capture which parameters are time-varying when a break is detected. They specify the model parameters in first-difference with respect to the previous regime. By doing so, shrinkage priors can be used to infer which parameters are time-varying. The two main contributions of the paper are  i) the introduction of a shrinkage prior that is a 2-component mixture of Uniform distributions (hereafter, 2MU) and ii) a method that operates for models exhibiting the path dependence issue such as ARMA and GARCH processes.

The new shrinkage prior mimics the standard information criteria such as the AIC and the BIC because one hyper-parameter of the 2MU distribution acts like a penalty on the log-likelihood. Consequently, the 2MU prior can be seen as a Bayesian alternative to the popular $L_0$ penalty functions used in classical statistics. However, the 2MU prior is not suited for high-dimensional regressions because it is not continuous. To mitigate this problem, [24] propose a sequential Monte Carlo algorithm, which is known to explore multi-modal distributions more efficiently than MCMC algorithms based on a single chain <cit.>. Unfortunately, this algorithm is computationally intensive. While it takes around 10 minutes on a 6-CORE i5-8400 (2.8 Ghz) for estimating a series from DGPs A to F of our paper, it runs for 2.5 hours on the same computer for estimating one series similar to DGP J but with only 20 explanatory variables. Consequently, it is computationally infeasible to estimate the model of [24] on 100 series from DGP J as we do with the selective segmentation approach. As a comparison, the selective segmentation method requires ??? minutes on the same computer for detecting which parameters are time-varying in one simulated series from DGP J.


##  Tuple-Circle Loss

Contrastive learning, an approach in self-supervised learning, allows the
model to learn rich representative features. Contrastive representation
learning aims to learn such a feature space where similar sample pairs stay
close while dissimilar ones (called negative samples $x^{-}$) are far apart
see Fig.LABEL:fig:crossmodal_learning. In a single modality setting, each
sample in a dataset is treated as a different instant (called anchor sample
$x$). Similar counterparts (called positive samples $x^{+}$) for training are
created by random augmentations, e.g., rotating an image, gaussian blur. We
consider an image-lidar pair acquired from the same scene and their augmented
counterparts as a dataset in our setting and the pixel and point corresponding
to the same physical locations as positive pair. It is ideal to have a loss
function to deal with multiple positive and negative samples in a cross-modal
learning setting because samples from different modalities can construct
different pairs. Many loss functions have been proposed for contrastive
learning. Sun et al. [Sun2020_Circleloss] proposed a unified loss function
from a pair similarity optimization viewpoint on deep feature learning. Based
on the unified loss function, he came up with Circle Loss (see
Eq.eq:orig_circle_loss) by adding self-paced weights and margins. where
$\alpha_{j}^{-}=\gamma\max(s_{j}^{-}+m,0)$ and
$\alpha_{i}^{+}=\gamma\max(1+m-s_{i}^{+},0)$ are self-paced weights. $L$ and
$K$ are the total numbers of the neigative samples and positive samples.
$\bigtriangleup^{-}=m$ and $\bigtriangleup^{+}=1-m$ are the negative pair and
positive margin. There are only three hyper-parameters of circle loss in which
$\gamma$ is a scale factor, $m$ controls the radius of the decision boundary.
$s^{+}=f\left(x,x^{+}\right)$ represents similarity between features of
positive samples, and $s^{-}=f\left(x,x^{-}\right)$ denotes similarity between
features of negative samples. Our preliminary test shows that self-pace weight
plays a vital role in the convergence of cross-modality learning. The weights
of negative pairs $\alpha_{j}^{-}$ decrease while the distance between the
negative pairs increases. This behavior guarantees that the features from the
two different modalities will not be pushed too far between each other before
the two models converge to the same features space. The self-paced weights of
positive samples $\alpha_{i}^{+}$ also have similar effects. As the two
positive samples become close to each in the features, the weights decrease to
allow optimization to take care of other pairs. However, similar to other
approaches[Chopra2005_CVPR, Sohn2016_NIPS, Van2018_arXiv], circle loss also
does not consider the properties of cross-modality learning and uses a unified
representation to represent the features (same-sized vectors). We propose
Tuple-Circle loss to generalize the circle loss to cross-modality feature
learning. In a cross-modality setting, samples from different modalities can
share some common information but may also contain modality-specific
information [Liu2021_ICCV, Jiang2021_LiDARNet, Bousmalis2016_NIPS]. Separate
treatment of the common and modality-specific information can make learning
easier. To learn features across K modalities, we represent the features as a
k-tuple to separate the information $x_{f}=\left[x_{t1},x_{t2},\dots
x_{tk}\right]$ in which $k-1$ features vectors are shared with other modality
and $1$ vector is for private feature learning. For inter similarity
measurement $s^{+}_{inter}=f\left(x,x^{+}\right)$, for cross-modality
similarity, we have $s^{+}_{crossk}=f\left(x_{tk},x_{tk}^{+}\right)$.
Therefore, our final loss function can be represented as: where
$S_{inter}^{+}andS_{inter}^{-}$ are sum of similarity measurement within
modality as described in Eq. eq:inter_measure and
$S_{cross}^{+}andS_{cross}^{-}$ sum of similarity measurement for cross
modality as shown in Eq. eq:cross_measure. In an Image-Point Cloud cross
modality setting, we represent the features as
$x_{f}=\left[x_{sh},x_{pr}\right]$ where $x_{sh}$ denotes shared features
between image and point cloud and $x_{pr}$ represents modality-specific
private features in point cloud or image. During training, we use one set of a
positive pair$(x,x^{+})$ and $N-1$ negative pairs
$\left[(x,x^{-}_{1}),\dots,(x,x^{-}_{N-1})\right]$ from one modality. For
inter-modal learning, we have: For cross modality, we can have $4$ cross-
modality positive pairs and $2N-2$ negative pairs by combination: In this
paper, we use cosine similarity to measure the similarity between two
features. For inter modality, we use the whole vector to compute cosine
similarity $s^{+}=\frac{x_{f}x_{f}^{+}}{|x_{f}||x_{f}^{+}|}$ and for cross
modalities, we only use the shared part of the feature vector to compute the
similarity $s^{+}_{sh}=\frac{x_{sh}x^{+}_{sh}}{|x_{sh}||x^{+}_{sh}|}$.

##  Models

We develop a dual U-Net structure framework(see Fig.LABEL:fig:framework) for
cross-modality dense feature learning and describe the network structure next.
There are several approaches to learning from LiDAR data. One way is to
project LiDAR data onto a spherical plane and process further using the 2D CNN
network. By using this point cloud representation, we are able to unify both
models under 2D CNN architecture[Goodfellow2016_DeepLearning]. However, the
projection process leads to the loss of 3D information of the point cloud. For
generality of our model and allow our model to be applied to point clouds from
other 3D sensors, we chose PointNet/PointNet++ as the model for point cloud
feature learning. PointNet/PointNet++ was designed to process raw point clouds
directly, and several architectures for 3D data[Zhang2019_Access] are designed
on a similar structure and applied to our method. We first explore the vanilla
multi-scale group (MSG) version of PointNet++, which was designed for the
Point Cloud semantic segmentation task [Qi2017_NIPS]. The model has an encoder
consisting of a farest-sampling layer and sets abstraction layers, and the
decoder is comprised of feature propagation layers. However, the model has
difficulty producing good point-level features. A potential reason for this is
that the feature propagation layers in the decoder create point features by
interpolating the neighborhood features, and there is no learnable weight for
the interpolation. This limits the description ability of the feature. We make
a simple modification by adding a set-abstraction layer before each feature
propagation layer to overcome this limitation. As shown in
Fig.LABEL:fig:framework(b), we create new features for up-sampled points by
grouping their neighborhood in the old point clouds and passing it to a
multilayer perceptron (MLP). After that, we concatenate the new features with
the interpolated features and feed them to the MLP like standard feature
propagation layers described in [Qi2017_NIPS]. This modification allows the
set-abstraction layers to provide more neighbor information and learn-able
weights for interpolation. We propose a 2D CNN network with a U-Net structure
for image feature learning. The encoder is a ResNet having four ResNet
layers[He2016_CVPR], while the decoder is an inverse of the encoder that
replaces the first convolution layer with the deconvolution layer to up-sample
the feature. Convolutional neural networks (CNNs) are inherently unable to
handle non-trivial geometric transformations due to the fixed geometric
structures in their building modules[Bronstein2017_ispm]. The PointNet++, on
the other hand, is more flexible for modeling geometric transformations and
has a broader receptive field view. To make the 2D model more flexible, we
replace the second convolutional layers of the ResNet block in the encoder
with deformable convolutional layers[Dai2017_ICCV]. The same modification is
made in the decoder model.


# High-resolution spectroscopy of the $\nu_{3}$ antisymmetric C-H stretch of
C2H${}_{2}^{+}$ using leak-out action spectroscopy

Stephan Schlemmera and Eline Plaara and Divita Guptaa and Weslley Guilherme
Dias de Paiva Silvaa and Thomas Salomon a and Oskar Asvanya CONTACT O. Asvany.
Email<EMAIL_ADDRESS>a I. Physikalisches Institut, Universität zu
Köln, Zülpicher Str. 77, 50937 Köln, Germany

###### Abstract

The antisymmetric C-H stretching vibration $\nu_{3}$ (${}^{2}\Pi$ $\leftarrow$
${}^{2}\Pi$) of ionized acetylene, C2H${}_{2}^{+}$, has been revisited using a
cryogenic 22-pole ion trap machine. Two action spectroscopic techniques, the
novel leak-out spectroscopy (LOS) method and the more established laser-
induced reactions (LIR) method, are applied and compared. Mass selectivity and
cryogenic temperatures down to 4 K enabled the observation of uncontaminated
spectra in which the $\Lambda$-doubling components of this open-shell molecule
are mostly well resolved, leading to a slight refinement of the spectroscopic
parameters.

###### keywords:

ion traps; rovibrational spectroscopy; action spectroscopy; carbocations;
molecular cations; Lambda doubling

††articletype: Dieter Gerlich Special Issue

##  1 Introduction

Given the importance of molecular cations to astrochemistry, these ions have
been investigated early on with high-resolution rovibrational spectroscopy.
Using a cooled discharge tube in conjunction with a tunable difference-
frequency mixing laser source, many of these important species, in particular
carbocations, were probed for the first time in the group of Takeshi Oka in
Chicago [1, 2, 3, 4]. One of these fundamental ions is cationic acetylene,
C2H${}_{2}^{+}$ ($\tilde{X}^{2}\Pi_{u}$), for which the antisymmetric C-H
stretch $\nu_{3}$ (${}^{2}\Pi_{u}$ $\leftarrow$ ${}^{2}\Pi_{u}$) has been
recorded in rotational resolution [5, 6]. A strong perturbation for higher
rotational levels of the excited state was observed [6], requiring the
application of ground state combination differences as an analytical tool.
Thus the spectroscopic parameters of C2H${}_{2}^{+}$, as well as those of its
deuterated siblings HCCD+ and C2D${}_{2}^{+}$ [7, 6] were obtained for the
first time.

Due to the possibility of cryogenic cooling and mass selectivity, it became
apparent in the 1990s that action spectroscopy performed in multipole ion
traps has several advantages over conventional spectroscopy. It was in
particular Dieter Gerlich, who pushed forward with the development of his
22-pole ion trap [8]. This enabled cooling of trapped molecular ions for ion
chemistry [9] and spectroscopic applications, first in Freiburg, and later in
Chemnitz. Together with Stephan Schlemmer, they developed the action
spectroscopy method laser induced reactions (LIR), and first applied it to the
electronic spectroscopy of N${}_{2}^{+}$ [10]. A few years later, exploiting
the endothermic reaction C2H${}_{2}^{+}$ \+ H2 \+ h$\nu$ $\rightarrow$
C2H${}_{3}^{+}$ \+ H, LIR was also proven to work well for vibrational
spectroscopy, as demonstrated for the $\nu_{3}$ band of C2H${}_{2}^{+}$ [11].
Because of the large bandwidth of the applied tunable IR laser
($\Delta\nu=0.24$ cm-1), and also due to the relatively high trap temperature
of $T=90$ K, the $\Lambda$-doubling and the rotational details of the Q-branch
were unfortunately not resolved (see Fig. 1 in [11]). Nevertheless, LIR laid
the ground for the first direct (but low-resolution) investigation of the cis-
bending vibration $\nu_{5}$ of this molecule [12, 13], which is now known in
high-resolution by ZEKE-spectroscopy [14, 15].

In this paper, we re-investigate the $\nu_{3}$ band of C2H${}_{2}^{+}$ using
action spectroscopic methods, but this time applying cryogenic buffer gas
cooling to the lowest possible temperatures and using high-resolution light
sources. Apart from exploiting LIR, we apply an action spectroscopy method
called leak-out-spectroscopy (LOS), recently developed in the Cologne
laboratories [16]. As LOS does not require a reaction partner, but only an
inert collision partner (typically a noble gas atom), it is more generally
applicable than LIR. In addition, LOS is very effective and can be applied
down to a trap temperature of 4 K. We compare the spectra obtained by both
action spectroscopic methods, and refine the spectroscopic parameters of the
$\nu_{3}$ band first reported 40 years ago.

##  2 Experimental

The rovibrational transitions of C2H${}_{2}^{+}$ were measured in COLTRAP, a 4
K 22-pole ion trap instrument which has been described in detail previously
[17, 18]. Ions were generated in a storage ion source by electron impact
ionization ($E_{e}=32.5$ eV) of a 50:50 acetylene (C2H2; Messer $\geq$ 99.6%
purity) - helium (Linde $\geq$99.999% purity) gas mixture. The initially
produced ions are pulsed into a quadrupole mass analyzer and selected for mass
$m=26$ u. The remaining ions (on the order of several ten thousands) are
guided into a 22-pole ion trap mounted on a 4 K cold head. Efficient trapping
and thermalization of the ions is achieved through collisions with Helium
buffer gas which is continuously introduced at low densities. Additionally, in
order to provide a proper collisional partner for the novel LOS approach and
to avoid extensive freeze-out at the trap surfaces, a 1:3 mixture of Neon
diluted in Helium is pulsed into the trap through a piezoelectrically actuated
valve at the beginning of each trapping cycle.

To detect the rovibrational transitions of C2H${}_{2}^{+}$, we used two
different action spectroscopy methods, one of them being the recently
developed and very sensitive LOS technique. The LOS method is described in
detail in a recent publication [16] and illustrated here in Fig. 1. LOS is
based on the escape of a trapped ion after collision-induced transfer of its
internal energy to kinetic energy. During the trapping time, the ion ensemble
is irradiated for 100 ms by an IR beam traversing the trap. If the radiation
frequency is resonant with a rovibrational transition of C2H${}_{2}^{+}$, the
ion is excited into the $\nu_{3}$ state. A vibrationally inelastic collision
between the excited C2H${}_{2}^{+}$ ion and a Neon atom then leads to a
redistribution of the internal excitation and partly to kinetic energy of both
collision partners. Then, the accelerated C2H${}_{2}^{+}$ ion has enough
energy to pass the exit potential barrier (of about 100 mV) which keeps the
other, thermalized ions trapped, and to fly out towards the ion detector. The
trapping cycle is repeated at a frequency of 1 Hz and the escaping
C2H${}_{2}^{+}$ ions are counted as a function of the laser wavenumber to
record the rovibrational spectrum.

Figure 1:  Principle of leak-out-spectroscopy (LOS), shown here for
C2H${}_{2}^{+}$ in collision with Ne atoms: (1) Several ten thousand
C2H${}_{2}^{+}$ ions are held in a 4 K ion trap, and resonantly excited into a
vibrationally excited state by laser photons. (2) Neon atoms are co-added to
the trap volume and collide with the vibrationally excited ions. (3) After the
collision, the vibration-to-translation energy transfer (V-T-process)
accelerates both collision partners. (4) The accelerated ions that overcome
the trap barrier are guided into the detector and are counted as a function of
the laser wavenumber to produce the rovibrational spectrum.

In former measurement campaigns, we used LIR as the action spectroscopic
method, described in detail in references [11, 12, 13] for the case of
C2H${}_{2}^{+}$. In brief, the slightly endothermic reaction given in the
introduction is exploited, and the enhancement of the reaction product
C2H${}_{3}^{+}$ signals the absorption of a photon by C2H${}_{2}^{+}$. The
number of product ions C2H${}_{3}^{+}$ is thus detected as a function of the
laser wavenumber to record the rovibrational spectrum. Typically, the neutral
reaction partner H2 is continuously added to the trap leading to a constant
number density. In order to avoid freeze-out of H2, the trap is operated at
slightly elevated temperatures above 10 K.

In both action spectroscopy campaigns, we used continuous-wave (cw) high-
resolution optical parametric oscillators (OPOs), operating in the 3 $\mu$m
spectral region, as IR light sources. For the LOS campaign, we used a
commercial Aculight Argos Model 2400 C. The IR beam entered the vacuum
environment via a 0.6 mm thick diamond window (Diamond Materials GmbH),
traversed the 22-pole trap, and exited the vacuum system via a CaF window
after which it was absorbed by a power meter. The power was on the order of
200 mW. The irradiation time was controlled by a laser shutter (Thorlabs model
SH05). The frequency of the IR radiation has been measured by a wavemeter
(Bristol model 621A) with an accuracy of 0.001 cm-1 (in case it is well
adjusted). We did additional calibration measurements with neutral C2H2
contained in an absorption cell, whose exact frequencies are given in the
HITRAN database [19]. After the calibration, we shifted our data up by 0.007
cm-1. With this procedure, the accuracy of the data is expected to be on the
order of 0.001 cm-1.

##  3 Rovibrational spectrum obtained with LOS

C2H${}_{2}^{+}$ is a linear open-shell molecule with ground state
$\tilde{X}^{2}\Pi_{u}$. The coupling of the electron spin ($S=1/2$) and
orbital ($\Lambda=1$) angular momenta leads to two spin-orbit fine structure
components, the lower-lying ${}^{2}\Pi_{3/2}$ (F1) and the ${}^{2}\Pi_{1/2}$
(F2) manifold, which is about 30 cm-1 higher in energy. In addition, the
physical rotation of the molecule lifts the degeneracy of the orbital angular
momentum, leading to $\Lambda$-doubling, which is resolvable in our
measurements for higher rotational quantum numbers.

The overview LOS-spectrum of C2H${}_{2}^{+}$ is shown in Fig. 2, together with
a spectroscopic simulation performed with PGOPHER [20]. The measured Q-branch
is shown in more detail in Fig. 3. Unfortunately, because of the limited
tuning range of the OPO module, only part of the P-branch could be recorded.
As expected, the lower-lying F1 component is much stronger than F2 at our
cryogenic conditions, as seen by the color-coded simulation in Fig. 2. The
inset in Fig. 2 shows the F1 line $J^{\prime}\leftarrow J"=6.5\leftarrow 5.5$,
with resolved $\Lambda$-doubling (separated by $\approx$0.0025 cm-1),
exhibiting the expected 3:1 ratio between the two ortho-para components. By
analyzing such resolved lines, in particular those in the Q-branch, a kinetic
ion temperature of about $T_{kin}\approx 20$ K is obtained, while the
simulation in Fig. 2 gives an estimation of the rotational temperature of
about $T_{rot}\approx 22-26$ K.

The list of $\nu_{3}$-lines measured with LOS is given in the supplementary
material. There, the lines measured with LOS are compared to those measured by
the Oka group [6], showing that our cryogenic cooling permits to resolve many
$\Lambda$-doublets for the first time (see also Fig. 3). On average, the lines
from the Oka measurements are only about 0.0007 cm-1 lower than our LOS data,
indicating a good calibration of both data sets.

Figure 2:  Rovibrational spectrum of the $\nu_{3}$ antisymmetric C-H stretch
band of C2H${}_{2}^{+}$, recorded using the novel leak-out spectroscopy (LOS)
method. The upper part is the measured high-resolution spectrum (in green),
while the bottom part is a simulation based on the fitted parameters given in
Table 1 at 26 K. In the simulation, the lower-energy F1 and the higher-energy
F2 components are color-coded in black and red, respectively. The inset shows
the F1 component of the J’ $\leftarrow$ J” = 6.5 $\leftarrow$ 5.5 transition
in which the $\Lambda$-type doubling (e and f) is resolved. The grey trace in
the upper part shows an overview spectrum also obtained by LOS with a low-
resolution pulsed OPO (Laser Vision). The green trace is provided as a
separate data file in the supplementary material.

##  4 LOS versus LIR

Fig. 3 shows a zoom of the Q-branch region comparing the measurements done
with LOS and LIR. Please observe that in the LIR measurements the action
signal is given by the ion counts of the accumulated reaction products
C2H${}_{3}^{+}$, measured in a small time window of a few ms after the
irradiation/trapping cycle, while for LOS the targeted ions C2H${}_{2}^{+}$
are directly detected during the irradiation/trapping cycle, lasting about 100
ms. In the Cologne group, two measurement campaigns have been performed with
LIR to investigate C2H${}_{2}^{+}$, one in 2008 and another in 2014 (both
unpublished). The LIR-measurement presented in Fig. 3 is the one done in 2008,
performed in a different ion trap machine (a 10 K trap machine called LIRTRAP,
see e.g. [12]), and using a home-built OPO. The nominal ion trap temperature
was about 13.5 K. It should be noted here that LIR can also operate at
temperatures below the condensation temperature of the reaction gas (H2 in
this case), by simply pulsing the reaction gas into the trap as done in later
works [21, 22]. However, this was not implemented in the LIR measurement shown
in Fig. 3.

Figure 3:  Comparison of measurements of the $\nu_{3}$ Q-branch of
C2H${}_{2}^{+}$ using the methods LOS and LIR. With LIR, the laser traversing
the trap is promoting the reaction C2H${}_{2}^{+}$ \+ H2 $\rightarrow$
C2H${}_{3}^{+}$ \+ H at about $T$ = 13.5 K, and the C2H${}_{3}^{+}$ products
created during the trapping time are counted as a function of the laser
wavenumber. In this Figure, the product counts have been multiplied by a
factor of 10 for better comparison. For LOS, the trap is held at a nominal
temperature of 4 K, and one simply counts the C2H${}_{2}^{+}$ ions which are
kicked out of the ion trap after the resonantly excited ions collide with Ne
atoms. Higher ion counts have been reached for LOS and therefore signal-to-
noise ratio achieved in the LOS measurements is better than for LIR.

Both action methods detect the two fine-structure components F1 and F2, and
nicely resolve the $\Lambda$-doubling for lines with J’ $\leftarrow$ J” = 4.5
$\leftarrow$ 4.5 and higher in F1. The LOS measurement exhibits a somewhat
lower kinetic temperature ($T_{kin}\approx 20$ K or even lower) compared to
the LIR measurement ($T_{kin}\approx 27$ K), as determined from the Doppler
widths of well-separated lines. The discrepancy to the nominal temperatures
($T=4$ K and $T=13.5$ K, respectively) is assumed to be caused by heating
effects when the ions undergo collisions in the RF field of the ion trap [23].
This can be particularly severe by the use of the heavy collision partner Neon
in the case of LOS, while LIR uses only the light collision partners He and
H2. Also, the rotational temperature obtained with LOS is somewhat lower, on
the order of $T_{rot}\approx 22$ K, as determined from the fading of the
R-branch transitions in Fig 2. In summary, the very good signal-to-noise ratio
renders LOS the preferred action spectroscopic method in the case of
C2H${}_{2}^{+}$. The overall advantage of LOS, however, is that it can be
generally applied to any cation or anion, without the need of a suitable low-
temperature reaction.

##  5 Spectroscopic constants

The assigned rovibrational transitions observed with our high-resolution LOS
method were fitted using a ${}^{2}\Pi$ $\leftarrow$ ${}^{2}\Pi$ linear top
Hamiltonian as implemented in Western’s PGOPHER program [20] to determine the
spectroscopic parameters of C2H${}_{2}^{+}$ for the ground and the $\nu_{3}$ =
1 vibrational states. The determined parameters comprise the rotational
constants ($B$), the spin-orbit coupling constants ($A$), $\Lambda$-doubling
constants ($q,p$), the quartic centrifugal distortion constants ($D$) of the
ground and vibrationally excited states, as well as the $\nu_{3}$ band origin.

As already pointed out by Jagod et al [6], the $\nu_{3}$ vibrationally excited
state is heavily perturbed, most likely by Renner-Teller states involving the
$\nu_{4}$ and $\nu_{5}$ bending vibrations, thus, making a fit of the
unperturbed ground state via combination differences mandatory. We followed
this recipe. To better determine the distortion constant $D_{0}$ and the
$\Lambda$-doubling constants, we included 82 lines from Oka with higher
rotational quantum numbers where the $\Lambda$-doubling has been resolved.
These were the lines with frequencies below 3100 cm-1 (P-branch) and above
3158 cm-1 (R-branch) [6]. The fit, with 68 combination differences, has a
reasonably good quality with an rms of obs-calc values of 0.0022 cm-1 (more
than half of the used lines are from Jagod et al [6] with an uncertainty of
0.003 cm-1) and a corresponding weighted rms of 0.71. The derived
spectroscopic parameters from this ground state combination difference fit are
given in Table 1. All used transition frequencies and fit residuals are listed
in the supporting information material. The fit results from Jagod et al [6]
are very close to ours and are only marginally improved by our data; they are
repeated in Table 1.

Table 1:  Spectroscopic parameters of C2H${}_{2}^{+}$. Those of the ground
state have been obtained by using ground state combination differences of our
data and those of Jagod et al [6]. All listed values are given in cm-1 with
their respective uncertainties provided in parenthesis. The results previously
reported by Jagod et al [6] are also included in the Table.

| this work | Jagod et al [6]
---|---|---
Parameter | ground | $\nu_{3}$ | ground | $\nu_{3}$
$\nu$ |  |  | 3135. | 97376(48) |  |  | 3135. | 9813(10)
$B$ | 1. | 104636(16) | 1. | 099521(23) | 1. | 10463(2) | 1. | 09843(7)
$A$ | -30. | 871(16) | -30. | 42744(79) | -30. | 91(2) | -30. | 466(2)
$q$ | -0. | 332(13)$\times 10^{-3}$ | -0. | 352(65)$\times 10^{-3}$ | -0. | 359(16)$\times 10^{-3}$ | -0. | 359$\times 10^{-3}$ _a_
$p$ | -0. | 63(13)$\times 10^{-3}$ | -0. | 63$\times 10^{-3}$ _a_ |  |  |  |
$D$ | 1. | 600(19)$\times 10^{-6}$ | 1. | 600$\times 10^{-6}$ _a_ | 1. | 57(2)$\times 10^{-6}$ | 1. | 57$\times 10^{-6}$ _a_

a Fixed to the ground state value

In the second step, the ground state spectroscopic parameters were fixed, and
the parameters of the $\nu_{3}$ band were fitted. Due to the mentioned heavy
perturbation, occurring at higher rotational quantum numbers in both F1 and F2
series, only our data have been taken into account. Also, some of our lines
with high obs-calc values were excluded from the fit, which are marked in the
list given in the supplementary information. Our spectroscopic parameters of
the $\nu_{3}$ state are also given in Table 1. Due to the improved measurement
precision of our novel approach, these are refined compared to the results of
Jagod et al [6]. There are also some obvious shifts in the values (e.g. our
band origin is about 0.008 cm-1 lower), which we attribute to the ambiguity
when fitting a semi-rigid molecular model to a perturbed system.

##  6 Summary and Conclusion

LOS is a novel action spectroscopic method [16, 24, 25] which needs further
validation. In this work on C2H${}_{2}^{+}$, it is compared with a more
established action spectroscopic method, LIR [11], showing very consistent
spectral features and a slightly better signal-to-noise ratio. This is an
encouraging result, as spectra of hitherto unexplored molecules, obtained with
LOS, sometimes exhibited unidentified lines (see the example of HCCCH+ below),
as well as hot bands in case the trap is operated at temperatures beyond 4 K.
This work thus demonstrates the reliability of LOS, and shows that former
observations of unidentified lines are most probably intrinsic to the molecule
under investigation and not caused by our novel method.

The overall advantage of LOS over other action methods is its independence
from e.g. chemical reactions or temperature ranges, so that it can be
generally applied to any cation or anion. Also, ions with higher masses can be
probed by LOS, simply by applying heavier non-reactive collision partners such
as N2, Ar, Kr or Xe. The low-temperature operation can be preserved by pulsing
in He-mixtures of these gases into the trap. Investigations along these
directions are currently being conducted in the Cologne laboratories. Only
recently, LOS opened the way to investigate the larger sibling of
C2H${}_{2}^{+}$, HCCCH+ (also $\tilde{X}^{2}\Pi_{u}$). Similar to this work,
we probed the antisymmetric C-H stretching vibration $\nu_{3}$, predicted to
be in the 3200 cm-1 region [26]. Although we detected strong and somewhat
regular P- and R-branches with a suitable rotational constant, as well as some
$\Lambda$-doubled lines with a typical 3:1 ratio, unfortunately, this mode
seems even more heavily perturbed than that of C2H${}_{2}^{+}$. Again, the
reason for these perturbations is most probably a variety of Renner-Teller
states involving bending vibrations, of which there are three for HCCCH+. The
perturbations caused by these, as well as the plethora of unidentified
interleaved lines, make the analysis of this band very challenging. For
instance, a clear identification of the Q-branch and the two fine-structure
components (F1 and F2) was not possible yet. High-level ab initio calculations
of its ground state, in particular of the spin-orbit splitting constant $A$,
would be of great help.

The rotationally resolved spectrum of the antisymmetric $\nu_{3}$ C-H
stretching vibration of C2H${}_{2}^{+}$ recorded in the laboratory of Dieter
Gerlich was the first action spectrum involving a classical chemical reaction.
This marked the starting point of a rapidly developing field where many
mechanisms have been invented as a possible action to make the spectrum
_visible_. Professor Gerlich opened this avenue and invented a number of those
enabling techniques. LOS is the latest in this row and revisiting the spectrum
of C2H${}_{2}^{+}$ demonstrates the advances of the field w.r.t. sensitivity
and spectral resolution. LOS allows to record background-free spectra of mass-
selected species of seemingly unlimited choice. Kicking out all ions by _one_
particular transition, thus, addressing, e.g., a specific structural isomer,
nuclear spin isomer or just an isobaric species, allows for the analysis and
preparation of selected samples for subsequent experiments. For instance, this
will enable the study of isomer-selected reactions in the future.

## Acknowledgements

The LIR-measurement of the Q-branch depicted in Fig. 3 has been conducted in
the year 2008 in collaboration with Sabrina Gärtner and Jürgen Krieg.
Additional LIR measurements have been performed by Pavol Jusko in 2014 (not
shown in this work). The current work has been supported by an ERC advanced
grant (MissIons: 101020583), the Deutsche Forschungsgemeinschaft (DFG) via SFB
956 (project ID 184018867), sub-project B2, and the Gerätezentrum ”Cologne
Center for Terahertz Spectroscopy” (DFG SCHL 341/15-1). W.G.D.P.S. thanks the
Alexander von Humboldt foundation for funding through a postdoctoral
fellowship. The authors gratefully acknowledge the work done by the electrical
and mechanical workshops of the I. Physikalisches Institut over the last
years.

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  * [3] J. Tang and T. Oka, Journal of Molecular Spectroscopy 196 (1), 120–130 (1999).
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