# Building Multivariate Time-Varying Smooth Transition Correlation GARCH Models, with an Application to the Four Largest Australian Banks

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## Abstract

**:**

## 1. Introduction

`mtvgarch`, includes, among other things, the estimation routines as well as the necessary specification and evaluation tests. The code is maintained in a private GitHub repository and can be obtained upon request.

## 2. The MTV Model

## 3. The Three Stages of Model Building

## 4. Specification of the MTV Model

#### 4.1. Specification of the Univariate Variance Equations

#### 4.2. Specification of Time-Varying Correlations

_{0}: ${\gamma}_{1}=0$. The form of the expansion depends on the order of the exponent in (6).

_{0}: ${\mathsf{\rho}}_{A1}={\mathsf{\rho}}_{A2}={\mathbf{0}}_{N(N-1)/2}$.2

_{0}: ${\mathsf{\rho}}_{A1}={\mathbf{0}}_{N(N-1)/2}$. This version of the test is more powerful than the former in the case that time-variation in the correlations is monotonic. However, and especially with longer time horizons, this may not always be the case, and the square term of the expansion is able to capture at least some nonmonotonic changes.

## 5. Estimation of the MTV Model

- AN1.
- In (4), ${\alpha}_{i0}>0$, either ${\alpha}_{i1}>0$ and ${\alpha}_{i1}+{\kappa}_{i1}\ge 0$ or ${\alpha}_{i1}\ge 0$ and ${\alpha}_{i1}+{\kappa}_{i1}>0$, ${\beta}_{i1}\ge 0$, and ${\alpha}_{i1}+{\kappa}_{i1}/2+{\beta}_{i1}<1$ for $i=1,\dots ,N$.
- AN2.
- The parameter subspaces $\{{\alpha}_{i0}\times {\alpha}_{i1}\times {\kappa}_{i1}\times {\beta}_{i1}\}$, $i=1,\dots ,N$, are compact, the whole space ${\mathsf{\Theta}}_{h}$ is compact, and the true parameter value ${\mathsf{\theta}}_{h}^{0}$ is an interior point of ${\mathsf{\Theta}}_{h}$.
- AN3.
- ${\mathsf{\zeta}}_{t}\sim $ iid$N(\mathbf{0},{\mathit{I}}_{N})$.

## 6. Evaluation of the MTV Model

## 7. Big Four Results

#### 7.1. Main Features of the Australian Banking Sector 1990–2020

#### 7.2. Modelling the Error Variances

#### 7.3. Modelling the Error Correlations

## 8. Conclusions

`mtvgarch`, which is maintained in a private GitHub repository and contains all the econometric tools necessary for building MTV-GARCH models.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Test Statistics

#### Appendix A.1. Test Statistic for TVV-Model Specification

**Lemma**

**A1.**

**Proof.**

#### Appendix A.2. Test Statistic for MTV-GARCH Model Evaluation

- Compute the $SS{R}_{0}={\sum}_{t=1}^{T}{({\widehat{\zeta}}_{t}^{2}-1)}^{2}$.
- Regress ${\widehat{\zeta}}_{t}^{2}-1$ on $({\mathit{r}}_{1t},{\mathit{r}}_{2t})$, and form the sum of squared residuals $SS{R}_{1}$.
- Compute the test statistic $LM=T\frac{SS{R}_{0}-SS{R}_{1}}{SS{R}_{0}}$.

- Regress ${\mathit{r}}_{2t}$ on ${\mathit{r}}_{1t}$ and obtain residuals ${w}_{t}$. When ${\mathit{r}}_{2t}$ has more than one variable, run the regression for each of them separately and, thereby, obtain a set of residuals ${\mathit{w}}_{t}$.
- Regress $\mathbf{1}$ on $({\widehat{\zeta}}_{t}^{2}-1){\mathit{w}}_{t}$ and form the sum of squared residuals $SSR$.
- Compute the test statistic $L{M}_{R}=T-SSR$.

#### Appendix A.3. Test of Constant Correlations

**Lemma**

**A2.**

**Proof.**

#### Appendix A.4. Test for an Additional Transition in the Correlations

## Appendix B. Simulations of Test Statistics

#### Appendix B.1. Tests of GARCH Equations

**Table A1.**Specification stage for the deterministic component in volatilities of each of the four banks. $\tilde{\alpha}$ and $\tilde{\beta}$ are the initial estimates used for calibrating the test statistic distribution. The rolling window method allows the GARCH intercept to adjust to target the unconditional variance in a window of size 400. The ‘calm period’ selects the continuous period from November 2003 to October 2007, which has very little visible variation in the baseline volatility. For comparison, the GARCH estimates from the entire sample period are reported along with the final estimates from the TV-GARCH model.

$\tilde{\mathsf{\alpha}}$ | $\tilde{\mathsf{\beta}}$ | Persistence | Kurtosis | ||
---|---|---|---|---|---|

Rolling window 400 | ANZ | 0.090 | 0.836 | 0.926 | 3.38 |

CBA | 0.087 | 0.850 | 0.937 | 3.43 | |

NAB | 0.095 | 0.817 | 0.912 | 3.36 | |

WBC | 0.085 | 0.858 | 0.943 | 3.45 | |

Calm period | ANZ | 0.073 | 0.852 | 0.925 | 3.24 |

CBA | 0.081 | 0.842 | 0.923 | 3.29 | |

NAB | 0.066 | 0.829 | 0.896 | 3.14 | |

WBC | 0.091 | 0.806 | 0.897 | 3.28 | |

Entire period GARCH only | ANZ | 0.065 | 0.927 | 0.992 | 6.40 |

CBA | 0.089 | 0.890 | 0.979 | 4.83 | |

NAB | 0.104 | 0.867 | 0.971 | 4.85 | |

WBC | 0.075 | 0.911 | 0.986 | 5.08 | |

Entire period TV-GARCH | ANZ | 0.078 | 0.880 | 0.957 | 3.50 |

CBA | 0.091 | 0.860 | 0.950 | 3.61 | |

NAB | 0.107 | 0.825 | 0.931 | 3.62 | |

WBC | 0.084 | 0.878 | 0.962 | 3.70 |

**Figure A1.**Simulated distributions of GARCH estimates and implied persistence and kurtosis measures for a selection of window widths. The baseline ${g}_{t}$ has a single transition. The dotted vertical lines indicate the true values of the parameters $\alpha $, $\beta $, persistence, and kurtosis.

**Figure A2.**Simulated distributions of GARCH estimates and implied persistence and kurtosis measures for a selection of window widths. The baseline ${g}_{t}$ has an asymmetric double transition. The dotted vertical lines indicate the true values of the parameters $\alpha $, $\beta $, persistence, and kurtosis.

**Figure A3.**Simulated distributions of GARCH estimates and implied persistence and kurtosis measures for a selection of window widths. The baseline ${g}_{t}$ has two double transitions. The dotted vertical lines indicate the true values of the parameters $\alpha $, $\beta $, persistence, and kurtosis.

#### Appendix B.2. Evaluation Tests of GARCH Equations

- First step
- The individual TVGARCH models are estimated, assuming the series are uncorrelated.
- Second step
- Estimate the correlation model conditional on the volatility model estimates from the previous step. Then, estimate the TVGARCH models conditional on the correlation estimates.

**Table A2.**Size simulation for the three types of misspecification tests in Amado and Teräsvirta (2017). 2000 replications. $T=2000$, $N=2$. MS1: ${g}_{t}$ additively misspecified, alternative linearised with a first-order term only; MS2-a: GARCH(1,1) vs. GARCH(1,2); MS2-b: GARCH(1,1) vs. GARCH(2,1); MS3: test for remaining ARCH, lag 1.

Standard | Robust | ||||||
---|---|---|---|---|---|---|---|

10% | 5% | 1% | 10% | 5% | 1% | ||

CCC two-step | MS1 | 0.146 | 0.085 | 0.020 | 0.132 | 0.074 | 0.016 |

MS2-a | 0.122 | 0.064 | 0.012 | 0.101 | 0.048 | 0.013 | |

MS2-b | 0.143 | 0.080 | 0.017 | 0.108 | 0.051 | 0.008 | |

MS3 | 0.125 | 0.061 | 0.010 | 0.104 | 0.054 | 0.010 | |

STCC two-step | MS1 | 0.134 | 0.074 | 0.023 | 0.121 | 0.055 | 0.015 |

MS2-a | 0.123 | 0.059 | 0.015 | 0.101 | 0.045 | 0.013 | |

MS2-b | 0.122 | 0.062 | 0.019 | 0.087 | 0.044 | 0.010 | |

MS3 | 0.115 | 0.058 | 0.015 | 0.100 | 0.050 | 0.011 | |

CCC multi-step | MS1 | 0.145 | 0.083 | 0.022 | 0.133 | 0.073 | 0.014 |

MS2-a | 0.116 | 0.062 | 0.015 | 0.097 | 0.052 | 0.009 | |

MS2-b | 0.133 | 0.069 | 0.018 | 0.100 | 0.046 | 0.010 | |

MS3 | 0.120 | 0.062 | 0.016 | 0.107 | 0.060 | 0.014 | |

STCC multi-step | MS1 | 0.147 | 0.084 | 0.023 | 0.135 | 0.068 | 0.012 |

MS2-a | 0.130 | 0.059 | 0.011 | 0.103 | 0.046 | 0.006 | |

MS2-b | 0.120 | 0.067 | 0.016 | 0.090 | 0.039 | 0.005 | |

MS3 | 0.112 | 0.055 | 0.012 | 0.104 | 0.047 | 0.009 |

#### Appendix B.3. Tests of Correlations

**Table A3.**Size-study: Test of constant correlations. Data are generated as an MTV-CCC with an equicorrelation coefficient of 0.33 (CEC33) and 0.67 (CEC67) and a Toeplitz structure with a correlation coefficient of 0.5 (CTC50) and 0.9 (CTC90). Tests are based on the first-order polynomial approximation. A total of 5000 replications.

CEC33 | CEC67 | CTC50 | CTC90 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

N | T | 1% | 5% | 10% | 1% | 5% | 10% | 1% | 5% | 10% | 1% | 5% | 10% |

2 | 25 | 0.023 | 0.076 | 0.132 | 0.022 | 0.069 | 0.128 | 0.024 | 0.074 | 0.130 | 0.022 | 0.070 | 0.126 |

50 | 0.015 | 0.063 | 0.116 | 0.016 | 0.064 | 0.115 | 0.016 | 0.064 | 0.115 | 0.015 | 0.062 | 0.109 | |

100 | 0.011 | 0.056 | 0.104 | 0.010 | 0.054 | 0.102 | 0.011 | 0.056 | 0.103 | 0.010 | 0.051 | 0.101 | |

250 | 0.012 | 0.055 | 0.108 | 0.010 | 0.054 | 0.107 | 0.011 | 0.055 | 0.106 | 0.009 | 0.053 | 0.108 | |

500 | 0.010 | 0.051 | 0.097 | 0.009 | 0.049 | 0.097 | 0.010 | 0.050 | 0.096 | 0.009 | 0.050 | 0.094 | |

1000 | 0.010 | 0.048 | 0.099 | 0.010 | 0.048 | 0.095 | 0.010 | 0.046 | 0.097 | 0.010 | 0.049 | 0.092 | |

5 | 100 | 0.011 | 0.054 | 0.112 | 0.011 | 0.053 | 0.110 | 0.011 | 0.056 | 0.112 | 0.011 | 0.053 | 0.111 |

250 | 0.014 | 0.054 | 0.099 | 0.012 | 0.051 | 0.099 | 0.013 | 0.053 | 0.100 | 0.012 | 0.051 | 0.101 | |

500 | 0.010 | 0.050 | 0.104 | 0.010 | 0.053 | 0.106 | 0.009 | 0.052 | 0.101 | 0.010 | 0.054 | 0.105 | |

1000 | 0.010 | 0.056 | 0.102 | 0.010 | 0.052 | 0.103 | 0.009 | 0.053 | 0.100 | 0.008 | 0.053 | 0.103 | |

10 | 250 | 0.013 | 0.055 | 0.112 | 0.013 | 0.057 | 0.112 | 0.013 | 0.057 | 0.110 | 0.012 | 0.054 | 0.115 |

500 | 0.009 | 0.049 | 0.101 | 0.010 | 0.049 | 0.104 | 0.008 | 0.053 | 0.103 | 0.010 | 0.050 | 0.103 | |

1000 | 0.011 | 0.052 | 0.102 | 0.011 | 0.054 | 0.105 | 0.011 | 0.053 | 0.099 | 0.012 | 0.056 | 0.103 | |

20 | 1000 | 0.012 | 0.056 | 0.106 | 0.012 | 0.057 | 0.106 | 0.013 | 0.056 | 0.103 | 0.012 | 0.056 | 0.107 |

**Table A4.**Size-study: Test of constant correlations. Data are generated as an MTV-GARCH-CEC with persistence of 0.95 and 0.97, kurtosis of 4 and 6, and an equicorrelation coefficient of 0.33 and 0.67. Tests are based on the first-order polynomial approximation. A total of 2500 replications.

CEC33 | CEC67 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

kurtosis = 4 | kurtosis = 6 | kurtosis = 4 | kurtosis = 6 | |||||||||||

Persistence | N | T | 1% | 5% | 10% | 1% | 5% | 10% | 1% | 5% | 10% | 1% | 5% | 10% |

0.95 | 2 | 500 | 0.012 | 0.056 | 0.108 | 0.016 | 0.056 | 0.106 | 0.016 | 0.070 | 0.122 | 0.016 | 0.092 | 0.122 |

2 | 1000 | 0.009 | 0.044 | 0.103 | 0.009 | 0.042 | 0.097 | 0.011 | 0.045 | 0.093 | 0.009 | 0.044 | 0.097 | |

2 | 2000 | 0.008 | 0.042 | 0.094 | 0.007 | 0.042 | 0.090 | 0.010 | 0.052 | 0.099 | 0.009 | 0.046 | 0.092 | |

5 | 500 | 0.006 | 0.062 | 0.118 | 0.006 | 0.070 | 0.114 | 0.018 | 0.076 | 0.140 | 0.018 | 0.082 | 0.146 | |

5 | 1000 | 0.016 | 0.060 | 0.119 | 0.016 | 0.061 | 0.112 | 0.016 | 0.059 | 0.115 | 0.018 | 0.060 | 0.112 | |

5 | 2000 | 0.010 | 0.058 | 0.108 | 0.008 | 0.051 | 0.102 | 0.016 | 0.060 | 0.116 | 0.010 | 0.052 | 0.098 | |

10 | 500 | 0.016 | 0.058 | 0.118 | 0.020 | 0.064 | 0.114 | 0.020 | 0.068 | 0.116 | 0.024 | 0.080 | 0.128 | |

10 | 1000 | 0.018 | 0.053 | 0.104 | 0.015 | 0.051 | 0.101 | 0.014 | 0.061 | 0.111 | 0.017 | 0.063 | 0.110 | |

10 | 2000 | 0.014 | 0.072 | 0.126 | 0.012 | 0.060 | 0.112 | 0.018 | 0.082 | 0.142 | 0.013 | 0.062 | 0.118 | |

0.97 | 2 | 500 | 0.010 | 0.056 | 0.114 | 0.012 | 0.054 | 0.118 | 0.020 | 0.072 | 0.114 | 0.014 | 0.068 | 0.120 |

2 | 1000 | 0.011 | 0.043 | 0.102 | 0.011 | 0.044 | 0.103 | 0.012 | 0.047 | 0.107 | 0.013 | 0.048 | 0.103 | |

2 | 2000 | 0.009 | 0.046 | 0.094 | 0.007 | 0.042 | 0.089 | 0.010 | 0.056 | 0.108 | 0.012 | 0.050 | 0.093 | |

5 | 500 | 0.004 | 0.066 | 0.124 | 0.012 | 0.056 | 0.104 | 0.012 | 0.088 | 0.152 | 0.018 | 0.086 | 0.164 | |

5 | 1000 | 0.015 | 0.063 | 0.113 | 0.014 | 0.067 | 0.114 | 0.018 | 0.063 | 0.121 | 0.019 | 0.060 | 0.125 | |

5 | 2000 | 0.010 | 0.060 | 0.110 | 0.008 | 0.050 | 0.100 | 0.015 | 0.060 | 0.118 | 0.012 | 0.050 | 0.101 | |

10 | 500 | 0.012 | 0.062 | 0.108 | 0.016 | 0.070 | 0.112 | 0.016 | 0.072 | 0.112 | 0.022 | 0.086 | 0.148 | |

10 | 1000 | 0.016 | 0.053 | 0.100 | 0.015 | 0.056 | 0.107 | 0.015 | 0.063 | 0.113 | 0.018 | 0.057 | 0.110 | |

10 | 2000 | 0.015 | 0.074 | 0.132 | 0.014 | 0.058 | 0.108 | 0.016 | 0.088 | 0.142 | 0.010 | 0.063 | 0.112 |

**Table A5.**Size-study: Test of constant correlations. Data are generated as an MTV-GARCH-CTC with persistence of 0.95 and 0.97, kurtosis of 4 and 6, and a correlation matrix with a Toeplitz structure with a correlation coefficient of 0.5 and 0.9. Tests are based on the first-order polynomial approximation. A total of 2500 replications.

CTC50 | CTC90 | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

kurtosis = 4 | kurtosis = 6 | kurtosis = 4 | kurtosis = 6 | |||||||||||

Persistence | N | T | 1% | 5% | 10% | 1% | 5% | 10% | 1% | 5% | 10% | 1% | 5% | 10% |

0.95 | 2 | 500 | 0.010 | 0.064 | 0.102 | 0.010 | 0.070 | 0.106 | 0.018 | 0.094 | 0.136 | 0.026 | 0.088 | 0.146 |

2 | 1000 | 0.009 | 0.041 | 0.097 | 0.011 | 0.042 | 0.103 | 0.014 | 0.053 | 0.096 | 0.020 | 0.062 | 0.104 | |

2 | 2000 | 0.008 | 0.044 | 0.096 | 0.009 | 0.044 | 0.090 | 0.017 | 0.066 | 0.120 | 0.014 | 0.048 | 0.098 | |

5 | 500 | 0.006 | 0.062 | 0.118 | 0.010 | 0.058 | 0.114 | 0.020 | 0.120 | 0.212 | 0.050 | 0.134 | 0.210 | |

5 | 1000 | 0.014 | 0.060 | 0.112 | 0.018 | 0.064 | 0.113 | 0.027 | 0.076 | 0.134 | 0.034 | 0.093 | 0.144 | |

5 | 2000 | 0.011 | 0.057 | 0.110 | 0.008 | 0.052 | 0.105 | 0.020 | 0.075 | 0.142 | 0.018 | 0.058 | 0.110 | |

10 | 500 | 0.012 | 0.070 | 0.120 | 0.016 | 0.080 | 0.128 | 0.040 | 0.114 | 0.172 | 0.078 | 0.150 | 0.230 | |

10 | 1000 | 0.012 | 0.049 | 0.100 | 0.013 | 0.051 | 0.102 | 0.019 | 0.078 | 0.127 | 0.032 | 0.089 | 0.147 | |

10 | 2000 | 0.019 | 0.072 | 0.127 | 0.014 | 0.059 | 0.111 | 0.033 | 0.110 | 0.178 | 0.018 | 0.077 | 0.140 | |

0.97 | 2 | 500 | 0.014 | 0.066 | 0.114 | 0.018 | 0.068 | 0.116 | 0.016 | 0.082 | 0.134 | 0.030 | 0.104 | 0.164 |

2 | 1000 | 0.009 | 0.044 | 0.101 | 0.008 | 0.042 | 0.099 | 0.016 | 0.051 | 0.112 | 0.022 | 0.063 | 0.119 | |

2 | 2000 | 0.010 | 0.050 | 0.102 | 0.009 | 0.044 | 0.092 | 0.024 | 0.070 | 0.120 | 0.015 | 0.052 | 0.100 | |

5 | 500 | 0.014 | 0.056 | 0.128 | 0.008 | 0.074 | 0.130 | 0.024 | 0.134 | 0.208 | 0.052 | 0.160 | 0.256 | |

5 | 1000 | 0.013 | 0.059 | 0.112 | 0.016 | 0.066 | 0.123 | 0.022 | 0.082 | 0.157 | 0.037 | 0.102 | 0.164 | |

5 | 2000 | 0.014 | 0.062 | 0.112 | 0.010 | 0.052 | 0.101 | 0.028 | 0.086 | 0.145 | 0.020 | 0.066 | 0.116 | |

10 | 500 | 0.018 | 0.080 | 0.128 | 0.022 | 0.088 | 0.130 | 0.040 | 0.114 | 0.172 | 0.100 | 0.188 | 0.278 | |

10 | 1000 | 0.012 | 0.054 | 0.105 | 0.013 | 0.054 | 0.107 | 0.019 | 0.078 | 0.127 | 0.030 | 0.104 | 0.181 | |

10 | 2000 | 0.016 | 0.072 | 0.132 | 0.016 | 0.062 | 0.110 | 0.033 | 0.110 | 0.178 | 0.026 | 0.089 | 0.150 |

## Appendix C. Details of Maximisation by Parts

- Assume $ln{h}_{it}({\mathsf{\theta}}_{hi},{\mathsf{\theta}}_{gi})=0$, $i=1,\dots ,N$, and estimate parameters ${\mathsf{\theta}}_{g}={({\mathsf{\theta}}_{g1},\dots ,{\mathsf{\theta}}_{gN})}^{\prime}$, $i=1,\dots ,N$, equation by equation, assuming ${\mathit{P}}_{t}({\mathsf{\theta}}_{\rho})={\mathit{I}}_{N}$. Denote the estimate ${\mathit{S}}_{t}({\widehat{\mathsf{\theta}}}_{g}^{(1,1)})$. This means that the deterministic components ${g}_{i}(t/T,{\mathsf{\theta}}_{gi})$ have been estimated once, including the intercept ${\delta}_{i0}$ in (2).
- Estimate ${\mathit{P}}_{t}({\mathsf{\theta}}_{\rho})$ given ${\mathsf{\theta}}_{g}={\widehat{\mathsf{\theta}}}_{g}^{(1,1)}$. This requires a separate iteration because ${\mathit{P}}_{t}({\mathsf{\theta}}_{\rho})$ is nonlinear in parameters; see (5) and (6). Denote the estimate ${\mathit{P}}_{t}({\widehat{\mathsf{\theta}}}_{\rho}^{(1,1)})$.
- Re-estimate ${\mathit{S}}_{t}({\mathsf{\theta}}_{g})$ assuming ${\mathit{P}}_{t}({\mathsf{\theta}}_{\rho})={\mathit{P}}_{t}({\widehat{\mathsf{\theta}}}_{\rho}^{(1,1)})$. This yields ${\mathit{S}}_{t}({\widehat{\mathsf{\theta}}}_{g}^{(1,2)})$. Then, re-estimate ${\mathit{P}}_{t}({\mathsf{\theta}}_{\rho})$ given ${\mathsf{\theta}}_{g}={\widehat{\mathsf{\theta}}}_{g}^{(1,2)}$. Iterate until convergence. Let the result after ${R}_{1}$ iterations be ${\mathit{S}}_{t}({\mathsf{\theta}}_{g})={\mathit{S}}_{t}({\widehat{\mathsf{\theta}}}_{g}^{(1,{R}_{1})})$ and ${\mathit{P}}_{t}({\mathsf{\theta}}_{\rho})={\mathit{P}}_{t}({\widehat{\mathsf{\theta}}}_{\rho}^{(1,{R}_{1})})$. The resulting estimates are maximum likelihood ones under the assumption ${\mathit{D}}_{t}({\mathsf{\theta}}_{h},{\mathsf{\theta}}_{g})={\mathit{I}}_{N}$.
- Estimate ${\mathsf{\theta}}_{h}$ from ${\mathit{D}}_{t}({\mathsf{\theta}}_{h},{\widehat{\mathsf{\theta}}}_{g}^{(1,{R}_{1})})$ using ${\mathit{P}}_{t}({\mathsf{\theta}}_{\rho})={\mathit{P}}_{t}({\widehat{\mathsf{\theta}}}_{\rho}^{(1,{R}_{1})})$. This is a standard multivariate conditional correlation GARCH estimation step as in Bollerslev (1990), because ${\mathit{S}}_{t}({\widehat{\mathsf{\theta}}}_{g}^{(1,{R}_{1})})$ is fixed and does not affect the maximum and ${\mathit{P}}_{t}({\widehat{\mathsf{\theta}}}_{\rho}^{(1,{R}_{1})})$ is known. In total, steps 1–4 form the first iteration of the maximisation algorithm. Denote the estimate ${\widehat{\mathsf{\theta}}}_{h}^{(1)}$.
- Estimate ${\mathsf{\theta}}_{g}$ from ${\mathit{S}}_{t}({\mathsf{\theta}}_{g})$ keeping ${\mathit{D}}_{t}({\widehat{\mathsf{\theta}}}_{h}^{(1)},{\widehat{\mathsf{\theta}}}_{g}^{(1,{R}_{1})})$ and ${\mathit{P}}_{t}({\widehat{\mathsf{\theta}}}_{\rho}^{(1,{R}_{1})})$ fixed. This step is analogous to the first part of Step 3. The difference is that ${\mathit{D}}_{t}({\widehat{\mathsf{\theta}}}_{h}^{(1)},{\widehat{\mathsf{\theta}}}_{g}^{(1,{R}_{1})})\ne {\mathit{I}}_{N}$. Denote the estimator ${\mathit{S}}_{t}({\widehat{\mathsf{\theta}}}_{g}^{(2,1)})$.
- Estimate ${\mathit{P}}_{t}({\mathsf{\theta}}_{\rho})$ given ${\mathsf{\theta}}_{g}={\widehat{\mathsf{\theta}}}_{g}^{(2,1)}$ and ${\mathsf{\theta}}_{h}={\widehat{\mathsf{\theta}}}_{h}^{(1)}$. Denote the estimator ${\mathit{P}}_{t}({\widehat{\mathsf{\theta}}}_{\rho}^{(2,1)})$. Iterate until convergence, ${R}_{2}$ iterations. The result: ${\mathit{S}}_{t}({\mathsf{\theta}}_{g})={\mathit{S}}_{t}({\widehat{\mathsf{\theta}}}_{g}^{(2,{R}_{2})})$ and ${\mathit{P}}_{t}({\mathsf{\theta}}_{\rho})={\mathit{P}}_{t}({\widehat{\mathsf{\theta}}}_{\rho}^{(2,{R}_{2})})$.
- Estimate ${\mathsf{\theta}}_{h}$ from ${\mathit{D}}_{t}({\mathsf{\theta}}_{h},{\widehat{\mathsf{\theta}}}_{g}^{(2,{R}_{2})})$ using ${\mathit{P}}_{t}({\mathsf{\theta}}_{\rho})={\mathit{P}}_{t}({\widehat{\mathsf{\theta}}}_{\rho}^{(2,{R}_{2})})$ (${\mathit{S}}_{t}({\widehat{\mathsf{\theta}}}_{g}^{(2,{R}_{2})})$ is fixed). The result: ${\mathsf{\theta}}_{h}={\widehat{\mathsf{\theta}}}_{h}^{(2)}$. This completes the second full iteration.
- Repeat steps 5–7 and iterate until convergence.

## Appendix D. Estimated Transition Equations

## Notes

1 | Available also in https://econ.au.dk/research/researchcentres/creates/research/creates-research-papers/supplementary-downloads/rp-2012-09, accessed on 26 January 2023. |

2 | The operator vecl$(\xb7)$ stacks the subdiagonal elements of its argument matrix. |

3 | See Explanatory Statement, Banking (prudential standard) Determination 2007, Nos 5, 12 and 15. https://www.legislation.gov.au/Details/F2007L04593/ (accessed on 26 January 2023), https://www.legislation.gov.au/Details/F2007L04600/ (accessed on 26 January 2023) and https://www.legislation.gov.au/Details/F2007L04603/ (accessed on 26 January 2023). |

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**Figure 1.**The market capitalisation of the Big Four as percentage of ASX200 (

**left**) and of ASX200 Financials Index (

**right**).

**Figure 4.**The first 100 autocorrelations of squared standardised returns ${\epsilon}_{it}^{2}/{\widehat{g}}_{it}$.

**Figure 5.**Estimated multiplicative component ${\widehat{g}}_{it}^{1/2}$ (solid curve) and the absolute returns $|{\epsilon}_{it}|$ (grey area).

**Table 1.**Univariate estimation results for the four banks. GJR is the GJR-GARCH(1,1) equation, and TV-GJR is the TV-GJR-GARCH equation; standard errors in parentheses.

${\mathsf{\alpha}}_{\mathit{i}0}$ | ${\mathsf{\alpha}}_{\mathit{i}1}$ | ${\mathsf{\kappa}}_{\mathit{i}1}$ | ${\mathsf{\beta}}_{\mathit{i}1}$ | Persistence | Kurtosis | ||
---|---|---|---|---|---|---|---|

ANZ | GJR | $\underset{(0.005)}{0.020}$ | $\underset{(0.007)}{0.039}$ | $\underset{(0.008)}{0.044}$ | $\underset{(0.008)}{0.929}$ | 0.991 | 3.76 |

TV-GJR | $\underset{(0.016)}{0.111}$ | $\underset{(0.005)}{0.015}$ | $\underset{(0.007)}{0.046}$ | $\underset{(0.027)}{0.792}$ | 0.831 | 3.02 | |

CBA | GJR | $\underset{(0.005)}{0.035}$ | $\underset{(0.008)}{0.060}$ | $\underset{(0.011)}{0.063}$ | $\underset{(0.010)}{0.886}$ | 0.977 | 3.66 |

TV-GJR | $\underset{(0.014)}{0.107}$ | $\underset{(0.006)}{0.021}$ | $\underset{(0.010)}{0.065}$ | $\underset{(0.020)}{0.813}$ | 0.867 | 3.06 | |

NAB | GJR | $\underset{(0.009)}{0.065}$ | $\underset{(0.010)}{0.077}$ | $\underset{(0.014)}{0.075}$ | $\underset{(0.014)}{0.850}$ | 0.964 | 3.68 |

TV-GJR | $\underset{(0.019)}{0.152}$ | $\underset{(0.006)}{0.021}$ | $\underset{(0.009)}{0.058}$ | $\underset{(0.030)}{0.731}$ | 0.780 | 3.03 | |

WBC | GJR | $\underset{(0.006)}{0.031}$ | $\underset{(0.007)}{0.045}$ | $\underset{(0.010)}{0.058}$ | $\underset{(0.009)}{0.910}$ | 0.985 | 3.70 |

TV-GJR | $\underset{(0.011)}{0.079}$ | $\underset{(0.004)}{0.015}$ | $\underset{(0.006)}{0.041}$ | $\underset{(0.020)}{0.829}$ | 0.864 | 3.02 |

**Table 2.**Estimation results for the four banks’ time-varying correlations. A total of 90% of the estimated transition is between the dates 18 October 2006 and 28 February 2008. The centre point of the location corresponds to 28 June 2007 with ± two standard error ranges of 11 May–13 August 2007.

${\mathit{P}}_{(1)}$ | ${\mathit{P}}_{(2)}$ | |||||||

ANZ | CBA | NAB | ANZ | CBA | NAB | |||

CBA | $\underset{(0.011)}{0.485}$ | CBA | $\underset{(0.006)}{0.782}$ | |||||

NAB | $\underset{(0.010)}{0.503}$ | $\underset{(0.010)}{0.525}$ | NAB | $\underset{(0.005)}{0.808}$ | $\underset{(0.005)}{0.787}$ | |||

WBC | $\underset{(0.009)}{0.606}$ | $\underset{(0.011)}{0.500}$ | $\underset{(0.011)}{0.492}$ | WBC | $\underset{(0.004)}{0.830}$ | $\underset{(0.005)}{0.818}$ | $\underset{(0.005)}{0.814}$ | |

Transition parameters: | c | $\eta $ | ||||||

$\underset{(0.002)}{0.552}$ | $\underset{(0.162)}{5.020}$ |

${\mathit{P}}_{(1)}$ | ${\mathit{P}}_{(2)}$ | |||||||

ANZ | CBA | NAB | ANZ | CBA | NAB | |||

CBA | $\underset{(0.011)}{0.484}$ | CBA | $\underset{(0.006)}{0.784}$ | |||||

NAB | $\underset{(0.010)}{0.510}$ | $\underset{(0.011)}{0.518}$ | NAB | $\underset{(0.005)}{0.811}$ | $\underset{(0.005)}{0.785}$ | |||

WBC | $\underset{(0.009)}{0.607}$ | $\underset{(0.011)}{0.504}$ | $\underset{(0.011)}{0.490}$ | WBC | $\underset{(0.004)}{0.831}$ | $\underset{(0.005)}{0.816}$ | $\underset{(0.005)}{0.812}$ | |

Transition parameters: | ||||||||

c | $\eta $ | |||||||

ANZ | CBA | NAB | ANZ | CBA | NAB | |||

CBA | $\underset{(0.005)}{0.550}$ | CBA | $\underset{(0.280)}{4.764}$ | |||||

NAB | $\underset{(0.001)}{0.567}$ | $\underset{(0.008)}{0.532}$ | NAB | $\underset{(-)}{7.000}$ | $\underset{(0.341)}{4.514}$ | |||

WBC | $\underset{(0.005)}{0.555}$ | $\underset{(0.004)}{0.547}$ | $\underset{(0.004)}{0.549}$ | WBC | $\underset{(0.308)}{5.198}$ | $\underset{(0.254)}{4.761}$ | $\underset{(0.294)}{4.873}$ |

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**MDPI and ACS Style**

Hall, A.D.; Silvennoinen, A.; Teräsvirta, T.
Building Multivariate Time-Varying Smooth Transition Correlation GARCH Models, with an Application to the Four Largest Australian Banks. *Econometrics* **2023**, *11*, 5.
https://doi.org/10.3390/econometrics11010005

**AMA Style**

Hall AD, Silvennoinen A, Teräsvirta T.
Building Multivariate Time-Varying Smooth Transition Correlation GARCH Models, with an Application to the Four Largest Australian Banks. *Econometrics*. 2023; 11(1):5.
https://doi.org/10.3390/econometrics11010005

**Chicago/Turabian Style**

Hall, Anthony D., Annastiina Silvennoinen, and Timo Teräsvirta.
2023. "Building Multivariate Time-Varying Smooth Transition Correlation GARCH Models, with an Application to the Four Largest Australian Banks" *Econometrics* 11, no. 1: 5.
https://doi.org/10.3390/econometrics11010005