Silviase/jgov_v1.4_full_ann huggingface dataset:
## 3 研究情報基盤の整備

研究情報基盤は研究活動に不可欠ないわば生命線としての性格を有するとされており、情報通信技術の急速な進展に対応して、研究情報基盤の整備を進めることは、我が国の研究開発の国際競争力を確保する上で重要である。一方で、研究成果としての論文やデータ等の共有・利活用を促進させることにより、より効率的・効果的にイノベーションを創出するオープンサイエンスの動きが世界的に活発になっている。このような状況に対応するため、政府は、研究機関間のネットワークの整備・高度化、データベースの構築・提供等を進めている。

## (i) ネットワークの整備

現代社会の基幹システムを構成しているコンピュータネットワークは研究開発の現場において開発された後、様々な分野に応用されてきたものであり、先端的な研究開発を進めるに当たりネットワークの性能の一層の向上が求められている。

情報通信研究機構は、構築・運営している新世代通信網テストベッド（JGN-X）により、新世代ネットワーク技術などの研究開発・実証実験を推進している(第3章第1節2（2）参照)。

国立情報学研究所は、我が国の大学等の学術研究及び教育活動全般を支える基盤として学術情報ネットワーク（SINET $ ^{1} $ ）を運用している。大量データを利活用する先端研究や機関間の情報共有による連携研究ニーズの高まりに伴い、平成26年末現在で、国内の約800の大学・研究機関が接続するとともに、海外の学術ネットワークとの連携も密に図っている。

農林水産省は、農林水産関連の研究機関を相互に接続する農林水産省研究ネットワーク（MAFFIN $ ^{2} $ ）を構築・運営しており、平成26年度現在で89機関が接続している。MAFFINはフィリピンと接続しており、海外との研究情報流通のバックボーンともなっている。

## (データベースの構築・提供)

閲覧・複写・貸出し等による論文等の原文献提供サービスは、各機関が持つ原文献の書誌情報、所在情報等をデータベース化することにより、コンピュータを利用して、増大する情報を迅速、正確かつ容易な検索を可能とする。また、近年、電子ジャーナルの普及、電子書籍の導入等、コンテンツ自体の電子的利用も進んでおり、データベースの構築・提供の重要性が増している。

我が国で発行される全ての出版物が納本されている国立国会図書館は、収集・保管している資料に関するデータベースを作成し、ウェブサイト $ ^{3} $ で情報を提供している。

国立情報学研究所は、効果的・効率的な研究開発活動の促進に向け、イノベーション創出に必要な学術情報を体系的に収集、使いやすいように整備し、インターネット上で公開している。例えば、全国の大学図書館等が所蔵する学術図書・雑誌の目録所在情報データベースや国内の学術論文データベースを構築・提供しているほか、大学等が教育研究成果を保存・公開する機関リポジトリ $ ^{4} $ の構築において、共用リポジトリのシステムを提供し支援するとともに、機関リポジトリ間のデータベース連携（JAIRO $ ^{5} $ ）を図っている。

科学技術振興機構は、国内外の科学技術に関する文献、特許、研究者等に関する基本情報のデー
タベースを整備し、各情報を関連付けて提供するサービス（Ｊ－ＧＬＯＢＡＬ）を行っており、平成26年度は、登載した論文等が引用した文献情報についても提供できるように機能強化を図った。科学技術に関する文献の日本語抄録等を作成してデータベースを整備し、インターネットにより有料で提供等を行う文献情報検索サービス（ＪＤｒｅａｍⅢ）については、平成26年度から新聞・雑誌記事や国内外のデータベースから関連度の高い情報を横断的に表示できる機能を追加した。また、我が国の研究成果を世界に向けて発信する機能を強化するため、学協会の学会誌・論文誌における論文の投稿から査読・審査、公開までの工程を電子化して行う科学技術情報発信・流通総合システム（Ｊ－ＳＴＡＧＥ²）を整備・提供しており、平成26年度からは、登載論文が引用した文献について、ＤＯＩ³を持つ場合は参照できるよう、当該ＵＲＬを自動で付与する改善を行った。さらに、国内文献にＤＯＩを付与するジャパンリンクセンター（ＪａＬＣ⁴）は、システムの改善により、論文以外のコンテンツ（書籍や研究データ）にもＤＯＩを付与できるように強化した。

農林水産省は、国内で発行されている農林水産関係学術誌の論文等の書誌データベース（JASI）等、農林水産関係の文献情報や図書資料類の所在情報を構築・提供している。また、研究開発型の独立行政法人、国公立試験研究機関や大学の農林水産分野の研究報告等をデジタル化した全文情報データベース、衛星画像データベース、試験研究機関で実施中の研究課題データベース等を構築・提供している。

平成26年度の主な研究情報基盤関連施策の概要は、第2-4-10表のとおりである。

■第2-4-10表／主な研究情報基盤関連施策（平成26年度）

府 省 名	実 施 機 関	施 策 名
国 会	国立国会図書館	国立国会図書館科学技術関係資料収集整備
総 務 省	情報通信研究機構	最先端の研究開発テストベッドネットワーク（JGN-X）の構築
文部科学省	科学技術振興機構	基本的な科学技術情報の整備と活用促進（J-GLOBAL等）
ライフサイエンスデータベース統合推進事業（NBDC）
科学技術論文の電子化・国際化、発信・流通促進（J-STAGE等）
海洋研究開発機構	情報基盤業務費
国立情報学研究所	学術情報ネットワークの整備（SINET4）
厚生労働省	国立感染症研究所	感染症情報センター経費
生物学的製剤の安全性情報収集、解析、評価に係る研究事業費
農林水産省	農林水産技術会議事務局	農林水産研究情報総合センターの運営（JASI、MAFFIN等）
国土交通省	国土地理院	地球地図プロジェクトの推進（地球地図第3版整備のための技術開発）
環境省	自然環境局生物多様性センター	生物多様性情報の収集・管理・提供の推進
文部科学省特許庁	科学技術振興機構工業所有権情報・研修館	特許・文献情報統合検索システムの整備
特許庁関係府省	工業所有権情報・研修館	リサーチツール特許データベース（RTDB）の提供
# Sources of the Great Recession: A Bayesian Approach of a Data-Rich DSGE model with Time-Varying Volatility Shocks

Hirokuni Iiboshi, Tatsuyoshi Matsumae and Shin-Ichi Nishiyama

December 2014

Economic and Social Research Institute

Cabinet Office

Tokyo, Japan

The views expressed in “ESRI Discussion Papers” are those of the authors and not those of the Economic and Social Research Institute, the Cabinet Office, or the Government of Japan. (Contact us: https://form.cao.go.jp/esri/en_opinion-0002.html)
# Sources of the Great Recession: A Bayesian Approach of a Data-Rich DSGE model with Time-Varying Volatility Shocks $ ^{*} $

Hirokuni Iiboshi $ \dagger $

Tatsuyoshi Matsumae $ ^{\dagger} $ and Shin-Ichi Nishiyama $ ^{\S} $

December 2014

## Abstract

In order to investigate sources of the Great Recession (Dec. 2007 to Jun. 2009) of the US economy in the latter portion of the first decade of the 2000s, we modified the standard New Keynesian dynamic stochastic general equilibrium (DSGE) model by embedding financial frictions in both the banking and the corporate sectors. Furthermore, the structural shocks in the model are assumed to possess stochastic volatility (SV) with a leverage effect. Then, we estimated the model using a data-rich estimation method and utilized up to 40 macroeconomic time series in the estimation. In light of a DSGE model, we suggest the following three empirical evidences in the Great Recession: (1) the negative bank net-worth shock gradually spread before the corporate net worth shock burst; (2) the data-rich approach and the structural shocks with SV found the contribution of the corporate net worth shock to a substantial portion of the macroeconomic fluctuations after the Great Recession, which is unlike the standard DSGE model; and (3) the Troubled Asset Relief Program (TARP) would work to bail out financial institutions, whereas balance sheets in the corporate sector would still not have stopped deteriorating. Incorporating time-varying volatilities of shocks into the DSGE model makes their credible bands narrower than half of the constant volatilities, which result implies that it is a realistic assumption based on the dynamics of the structural shocks. It is plausible that tiny volatilities (or uncertainty) in ordinary times change to an extraordinary magnitude at the turning points of business cycles.

Keywords: New Keynesian DSGE model, Data-rich approach, Bayesian estimation, financial friction, stochastic volatility, leverage effect.

JEL Classification: E32, E37, C32, C53.
conventional VAR model in which it is difficult to interpret their coefficients from economic theory. Instead, a DSGE model expresses the co-movement of multi-variables from a causal association and a theoretical coherence, based on a micro-founded dynamic model, following the spirit of new neo-classical synthesis. That is, converting from a conventional VAR model to a DSGE model in a systematic component part of a DFM indicates that

$$ comovement\left(systematic\ variation\right)=genetic\ correlation, $$

$$ \Rightarrow comovement=causal association $$

which induces a resolution of the drawback of a DFM.

## (3) Synergetic effect of combination

According to Stock and Watson (2002a,b), the consistency of the DFM suggests that increasing the number of observations, series $ X_{t} $ is expected to increase the certainty that idiosyncratic components not explained by an economic system are removed from the data using a DFM. Accordingly, increasing the number of observations could improve the accuracy of measuring the co-movement $ S_{t} $ and the exogenous structural shocks $ \varepsilon_{t} $ . If the estimation of the structural shocks $ \varepsilon_{t} $ successfully explains actual business cycles, the indication will be the validity of the structural shocks and, in addition, that of the DSGE model. And in a data-rich framework, the same data set is applicable even for DSGE models with different model variables, so the possibility of model selection among many alternative models emanates. This possibility implies that the data-rich approach is expected to contribute to the evaluation and selection among DSGE models from the point of view of the validity of the structural shocks and the marginal likelihood (or the Bayes factor).

#### 2.2.2 Dynamic Factor Model (DFM)

Recently, the estimation method of DFMs was developed and applied to many fields of macroeconomics and finance. The DFM, which is a statistical model estimating the common factors of business cycles, was proposed by Sargent and Sims (1977) and empirically applied by Stock and Watson (1989), who extracted one unobserved common factor of the business fluctuation from many macroeconomic time series using the Kalman filter. $ ^{2} $

The DFMs are represented by state-space models composed of the following three linear equations. When $ F_{t} $ denotes the $ N \times 1 $ vector of the unobserved common factor, and when $ X_{t} $ denotes the $ J \times 1 $ vector of a massive panel of macroeconomic and financial data, note that $ J \gg N $
macroeconomic indicators $ X_{t} $ and which consists of common factors $ S_{t} $ with economic concepts. $ ^{3} $ A transition equation (2.9) indicates an AR(1) process of common factors (or model concepts) $ S_{t} $ with structural shocks $ \varepsilon_{t} $ and also dynamics converging to a rational expectation equilibrium determined by a macroeconomic model. From (2.8) and (2.9), we can see that the model variable $ S_{t} $ is realized from the inter-correlation of the data indicators $ X_{t} $ whose movement is coherent with each economic concept derived from the economic theory. In contrast, measurement errors $ e_{t} $ fluctuate from only specific factors of each observable variable $ X_{t} $ but do not depend on economic theory and other data indicators. $ ^{4} $

Structural shocks $ \varepsilon_{t} $ and disturbance terms $ \nu_{t} $ of measurement errors $ e_{t} $ follow normal distributions, i.e. $ \varepsilon_{t} \sim \mathrm{i.i.d.}~N(\mathbf{0}, \mathbf{Q}(\boldsymbol{\theta})) $ and $ \nu_{t} \sim \mathrm{i.i.d.}~N(\mathbf{0}, \mathbf{R}) $ , respectively. And their variance covariance matrix $ \mathbf{Q}(\boldsymbol{\theta}) $ and R are positive, definite, and diagonal matrices. Coefficients $ \Psi $ of the AR(1) process (2.10) are also diagonal matrices. These indicate that the measurement errors $ e_{t} $ are independent from each other in terms of cross section but dependent with their lag variables in terms of time series restriction. Matrices $ \mathbf{G}(\boldsymbol{\theta}) $ , $ \mathbf{E}(\boldsymbol{\theta}) $ and $ \mathbf{Q}(\boldsymbol{\theta}) $ are functions of the structural parameters $ \theta $ .

### 2.3 Data Rich DSGE models with Stochastic Volatility

#### 2.3.1 Stochastic Volatility in a Data-Rich DSGE models

In the previous subsection, the structural shocks $ \varepsilon_{t} $ of the DSGE model are assumed to follow i.i.d. normal distribution. Since the adoption of the data-rich approach is expected to improve the accuracy of the structural shocks, the next task of the approach is thought to specify the structural shocks. In this paper, we used a relaxation of the assumption of

where $ \hat{y}_{t} $ is the model variable of output gap, $ \hat{\pi}_{t} $ is the model variable of inflation.
worth back to the household. The total number of transfers from exiting entrepreneurs to the household will be $ (1 - \gamma_{t+1}^{E})n_{t}^{E} $ . Accordingly, the net transfer, $ \Xi_{t+1}^{E} $ that the household will receive from entrepreneurs at period $ t+1 $ is $ (1 - \gamma_{t+1}^{E} - \xi^{E})n_{t}^{E} $ .

#### 3.1.2 Individual Entrepreneur's Problem

Each entrepreneur produces homogenous intermediate goods, $ y_{t}(j) $ , and they are perfectly competitive when selling their products to retailers. The production function for the intermediate goods is given by

$$ y_{t}(j)=\omega_{t}(j)A_{t}k_{t}(j)^{\alpha}l_{t}(j)^{1-\alpha}, $$

where $ k_{t}(j) $ is capital inputs and $ l_{t}(j) $ is labour inputs. The total factor productivity shock (hereafter, TFP shock), $ A_{t} $ , is common across all entrepreneurs. However, following Carlstrom and Fuerst (1997) and BGG (1999), we assume each entrepreneur is subject to an idiosyncratic shock, $ \omega_{t}(j) $ , which is a private information to entrepreneur j and assumed to be an i.i.d. shock with a mean equal to one, i.e., $ E[\omega_{t}(j)] = 1 $ .

The balance sheet statement of each entrepreneur at the end of period t can be expressed as

$$ q_{t}k_{t+1}(j)\quad=\quad b_{t}^{E}(j)+n_{t}^{E}(j), $$

where $ q_{t} $ is the real price of capital, $ k_{t+1}(j) $ is the capital that will be used for production in period $ t+1 $ but purchased at the end of period t, $ b_{t}^{E}(j) $ is the real debt issued at period t, and $ n_{t}^{E}(j) $ is the net worth at period t. With the assumption of risk-neutrality and finite planning horizon, the net worth itself is never enough in financing the cost of the capital purchase, and, therefore, each entrepreneur will rely on external financing in equilibrium.

The income statement for entrepreneur j is specified as follows:

$$ n_{t}^{E}(j)=p_{t}^{m c}(j)y_{t}(j)-w_{t}l_{t}(j)-\frac{R_{t-1}^{E}(j)}{\pi_{t}}b_{t-1}^{E}(j)+q_{t}(1-\delta)k_{t}(j) $$

where $ p_{t}^{mc}(j) $ is the real price of intermediate goods j, $ R_{t-1}^{E}(j)/\pi_{t} $ is the real rate of borrowing cost, $ R_{t-1}^{E}(j) $ is the nominal borrowing rate and $ \pi_{t} $ is the inflation rate, and $ \backslash\delta $ is the capital depreciation rate.

Each entrepreneur entering period t maximizes her discounted cash flow by choosing capital inputs, labour inputs, and debt issuance, subject to (3.1), (3.2), and (3.3). $ ^{7} $ The FOCs for each entrepreneur j are given as follows:

$$ w_{t}\ =\ (1-\alpha)\frac{p_{t}^{mc}(j)y_{t}(j)}{l_{t}(j)} $$

$$ \begin{array}{r c l}{E_{t}\left[\gamma_{t+1}^{E}\frac{R_{t}^{E}(j)}{\pi_{t+1}}\right]}&{=}&{E_{t}\left[\gamma_{t+1}^{E}\frac{\alpha p_{t+1}^{m c}(j)y_{t+1}(j)/k_{t+1}(j)+(1-\delta)q_{t+1}}{q_{t}}\right].}\\ \end{array} $$
$$ V_{t}^{F}(m)\geq\lambda b_{t}^{E}(m). $$

In other words, the net present value of the banking business needs to always exceed the reservation value retained by the banker. $ ^{8} $

Now, assuming the incentive constraint (3.16) to be binding each period and by maximizing the objective function (3.15) subject to the constraint (3.16), Gertler and Kiyotaki (2010) showed that the value function of the banker can be expressed as follows:

$$ V_{t}^{F}(m)=\nu_{t}b_{t}^{E}(m)+\eta_{t}n_{t}^{F}(m) $$

where

$$ \begin{array}{r c l}{\nu_{t}}&{\equiv}&{E_{t}\left[\left(1-\gamma_{t+1}^{F}\right)\beta\left(r_{t+1}^{F}(m)-r_{t+1}\right)+\beta\gamma_{t+1}^{F}\frac{b_{t+1}^{E}(m)}{b_{t}^{E}(m)}\nu_{t+1}\right]}\end{array} $$

$$ \eta_{t}\quad\equiv\quad E_{t}\left[\left(1-\gamma_{t+1}^{F}\right)+\beta\gamma_{t+1}^{F}\frac{n_{t+1}^{F}(m)}{n_{t}^{F}(m)}\eta_{t+1}\right]. $$

Now, from the incentive constraint (3.16) and the value function (3.17), it follows that

$$ \frac{b_{t}^{E}(m)}{n_{t}^{F}(m)}\leq\frac{\eta_{t}}{\lambda-\nu_{t}}\equiv\phi_{t} $$

which states that the leverage ratio of banker m cannot exceed the (time-varying) threshold $ \phi_{t} $ . By the assumption that the incentive constraint is binding for every period in equilibrium, the asset and the net worth by banker m have the following relationship:

$$ b_{t}^{E}(m)=\phi_{t}n_{t}^{F}(m). $$

#### 3.2.3 Aggregation

Gertler and Karadi (2011) and Gertler and Kiyotaki (2010) showed that time-varying threshold $ \phi_{t} $ does not depend on banker-specific factors and is common among all bankers. Consequently, from eq. (3.14), the aggregate asset and net worth in the banking sector can be expressed as follows:

$$ b_{t}^{E}=\phi_{t}n_{t}^{F} $$

where $ b_{t}^{E} \equiv \int_{0}^{1} b_{t}^{E}(m) dm $ and $ n_{t}^{F} = \int_{0}^{1} n_{t}^{F}(m) dm $ . Now, from an individual banker's net worth transition (3.14) and taking into account the entrance and exit of bankers, the aggregate net worth transition equation of banking sector is given by the following:

$$ n_{t+1}^{F}=\gamma_{t+1}^{F}\left[\left(\overline{r}_{t+1}^{F}-r_{t+1}\right)b_{t}^{E}+r_{t+1}n_{t}^{F}\right]+\xi^{F}n_{t}^{F} $$

where $ \overline{r}_{t+1}^{F} $ stands for the average of the realized risk-adjusted returns, $ r_{t+1}^{F}(m) $ , among all bankers. From the optimal debt contract specified in (3.8) and using the aggregate condition in (3.9), $ \overline{r}_{t+1}^{F} $ is related to the borrowing rate, the external finance premium, and the inflation rate, as follows:

$$ \overline{r}_{t+1}^{F}=\frac{R_{t}^{E}}{\pi_{t+1}s_{t}}. $$
ratio, $ b_{t}^{E}/n_{t}^{F} $ , as observable variables in the measurement equation (2.11). The first seven series are generally used in a large amount of literature estimating DSGE models (see, for instance, Smets and Wouters, 2003 and 2007). $ ^{13} $ Using the four remaining financial time-series as observable variables is the featured difference of our DSGE model compared with existing models. These four actual series were selected for matching the model variables corresponding to the two financial frictions. The entrepreneur's nominal borrowing rate (8), $ R_{t}^{E} $ , is the yield on Moody's Baa-rated corporate bonds, which is de-trended via the Hodrick-Prescott filter for the same reason of inflation and the interest rate. To measure the external financial premium (9), $ s_{t} $ , we employed the charge-off rates for all banks' credits and issuer loans, measured as an annualized percentage of uncollectible loans. The charge-off rate was deemed to be consistent with our model variable (10 and 11). The two leverage ratios, $ q_{t}k_{t}/n_{t}^{E} $ and $ b_{t}^{E}/n_{t}^{F} $ , were calculated as their total asset divided by their net worth, respectively. We took the natural logarithm for both leverage ratios, and then either demeaned for the entrepreneur's leverage ratio, or detrended the banking sector leverage ratio with the Hodrick-Prescott filter, based on taking into account the Basel Capital Accord Revision.

In Cases B and D, which adopted the data-rich approach, indicating one model variable corresponding to four actual series, we employed an additional 29 series, which consisted of 18 series of key macroeconomics and 11 series of the banking sector, with the existing 11 series in Cases A and C. We selected 18 data indicators for the six key model variables used in a standard DSGE model such as (1) output, (2) consumption, (3) investment, (4) inflation, (5) real wage, and (6) labour input, but not the nominal interest rate, as mentioned in the Data Appendix, along the same lines of Boivin and Giannoni (2006). On the other hand, 11 series of the banking sector were selected along the lines concentrated on banking, which is a departure from previous work dealing with the data-rich framework. Three additional banking indicators include the following: (i) the core capital leverage ratio, (ii) the domestically chartered commercial banks' leverage ratio, and (iii) the leverage ratio of brokers and dealers, and these were selected as data indicators, corresponding to the model variable, such as the banking sector leverage ratio. Notice that as an observable variable of the leverage ratio, we used the inverse of the commonly-used ratio, i.e., bank asset over bank equity. $ ^{14} $ As data indicators for the external financial premium, we selected three kinds of loan charge-off rates based on different institutions, which were transformed as percentage deviations from trends using the same detrending methods described above.
First we will examine real activities. Figures 4 and 5 show the historical decompositions of the real GDP and the gross private domestic investment, respectively. Because the decompositions of these real activities' variables have similar properties, we discuss them as a whole. Although the signs of the contribution of each shock are the same in every case of the two variables at each period, we can see that the sizes of the contribution of shocks are different depending on the cases. In Case A (standard DSGE model), the TFP shock accounted for a large portion of the sources of the Great Recession (light blue shade), while the decline of the bank net-worth impacted a small part of drops. And the positive corporate net-worth increased and contributed to raising these variables by a significant portion during the recession, in this case. On the other hand, the remaining three cases showed that the positive effect of the corporate net-worth shock were small, and that the bank net-worth shock accounted for a bigger portion of the downturn of these real activities in the period. Even in the period of the US economy's recovery, Cases A and C show a different picture from what Cases B and D show. The main source of blocking the recovery is derived from the negative TFP shocks in Cases A and C, whereas TARP worked and prominently improved the bank's balance sheet, so the positive bank net-worth shock contributed to raising the real activities in the remaining cases. $ ^{17} $ In addition, decompositions of these cases suggest that a deterioration of the corporate sector's balance sheet is likely to be the main source blocking a recovery after the recession. $ ^{18} $

## [Insert Figure 4 and 5 around here]

In Figure 6, Moody’s bond index (corporate Baa) is decomposed as a corporate borrowing rate. According to the figure, a sharp rise of the rate might be derived from the mainly negative bank net-worth shock as well as from a fall of the TFP shock, whereas the positive firm net-worth shock contributed to the downward rate in the recession. And then, the firm net-worth shock turned to be remarkably negative, seriously deteriorating its balance sheet and accounting for a large portion of the rise of the rate after the recession. On the other hand, TARP might work well and make the bank net-worth shock become positive, which would contribute to the downward borrowing rate after 2010Q1. In particular, we can see these findings in Cases B and D.

## [Insert Figure 6 around here]

Figure 7 depicts the decomposition of the commercial banks’ leverage ratio, defined as the inverse of the commonly-used ratio, i.e., the bank’s asset over the bank’s net-worth. As
(or the uncertainty) in ordinary times changed to an extraordinary magnitude at the turning points of the business cycles. We also estimated that the monetary policy shock had an opposite leverage effect of SV, which implies that tightening a policy makes interest rates more volatile.

## A Appendix

### A.1 Sampling Stochastic Volatility with Leverage

Step 4 of MCMC procedure described in Section 4 employs the algorithm of Omori et al. (2007) which is the extension of Kim et al. (1998) toward a SV model with leverage effect. This subsection is based on Justiniano and Primiceri (2008) who employed Kim et al. (1998) for drawing the stochastic volatilities.

Accoding to Omori et al. (2007), the key idea of MCMC algorithm of a SV model with leverage effect is to obtain a draw from an approximate linear and Gaussian state space form such as

$$ \left(\begin{array}{c}{\sigma_{t}^{*}}\\ {h_{t+1}}\\ \end{array}\right)=\left(\begin{array}{c}{h_{t+1}}\\ {\mu+\phi(h_{t}-\mu)}\\ \end{array}\right)+\left(\begin{array}{c}{z_{t}^{*}}\\ {\nu_{t}}\\ \end{array}\right), $$

$$ \left\{\left(\begin{array}{c}{z_{t}^{*}}\\ {\nu_{t}}\\ \end{array}\right)\mid d_{i,t},u_{i t}=k,\rho_{i},\omega_{i}\right\}=\left(\begin{array}{c}{m_{k}+v_{k}\zeta_{t}}\\ {d_{t}\rho\omega\left(a_{k}+b_{k}v_{k}\zeta_{t}\right)\mathbf{e x p}(m_{k}/2)+\omega\sqrt{1-\rho^{2}}\zeta_{t}^{*}}\\ \end{array}\right), $$

$$ \left(\begin{array}{c}{\zeta_{t}}\\ {\zeta_{t}^{*}}\end{array}\right)\sim i.i.d.N(0,I_{2}), $$

where $ \sigma_{i,t}^{*} = \log \sigma_{i,t} = h_{i,t} + z_{i,t}^{*} $ , $ h_{it} = \log \sigma_{i,t} $ , and $ z_{it}^{*} = \log (z_{it}^{2}) $ . And $ d_{it} $ , and $ \eta_{i,t} $ are denoted as

$$ \begin{align*}d_{i,t}&=\mathbf{I}(z_{i,t}\geq0)-\mathbf{I}(z_{i,t}<0),\\\eta_{i,t}&=(h_{i,t}-\mu)-\phi(h_{i,t-1}-\mu),\end{align*} $$

where, $ I(\cdot) $ is an indicator function which indicates $ d_{i,t}=1 $ when $ z_{i,t}>0 $ , or otherwise: $ d_{i,t}=-1 $ .

Suppose that the MCMC algorithm has implemented iteration g, generating samples $ \Phi_{i}^{(g)} = (\phi_{i}, \rho_{i}, \omega_{i}) $ and $ \boldsymbol{H}^{T,(g)} $ . In iteration $ g + 1 $ , the following four steps are used to a set of new draws.

Step 1: Draw the structural shocks $ \varepsilon_{t}^{(g+1)} $

In order to generate a new sample of stochastic volatilities, we need to obtain a new sample of structural shocks. This can be done using simulation smoother debeloped by de Jong and Shephard (1995) whose algorithm is described in Appendix A2. We obtain a new draw of structural shocks from eq.(A.12) of Appendix A2.

Step 2: Draw the stochastic volatilities $ H^{T(g+1)} $ with leverage effect

With a draw of Shocks in hand, nonlinear measurement equations (2.2) in Section 2.1, which is represented as eq.(A.3) for each structural shock, can be easily converted in linear one such as eq.(A.4) by squaring and taking logarithms of every element. This induces the following approximating state space representation (A.4) and (A.5).

$$ \varepsilon_{i,t}=\sigma_{i,t}z_{i,t},\mathbf{i}=1,2,\cdots,\mathbf{M}, $$

$$ \tilde{\varepsilon}_{i,t}=2h_{i,t}+z_{i,t}^{*}, $$