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CNAME

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+frds.io

_sources/algorithms/garch-ccc.rst.txt

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+================
+GARCH(1,1) - CCC
+================
+
+Introduction
+============
+
+The Multivariate GARCH(1,1) model generalizes the univariate :doc:`/algorithms/garch` 
+framework to multiple time series, capturing not only the conditional variances 
+but also the conditional covariances between the series. One common form is the 
+**Constant Conditional Correlation (CCC) model** proposed by Bollerslev (1990).
+
+.. tip:: Check `Examples`_ section for code guide and comparison to Stata.
+
+Return equation
+---------------
+
+The return equation for a :math:`N`-dimensional time series is:
+
+.. math::
+   :label: mv_return_eq
+
+   \mathbf{r}_t = \boldsymbol{\mu} + \boldsymbol{\epsilon}_t
+
+Here, :math:`\mathbf{r}_t` is a :math:`N \times 1` vector of returns, and :math:`\boldsymbol{\mu}` is a :math:`N \times 1` vector of mean returns. :math:`\boldsymbol{\epsilon}_t` is the :math:`N \times 1` vector of shock terms.
+
+Shock equation
+--------------
+
+The shock term is modelled as:
+
+.. math::
+   :label: mv_shock_eq
+
+   \boldsymbol{\epsilon}_t = \mathbf{H}_t^{1/2} \mathbf{z}_t
+
+Here, :math:`\mathbf{H}_t` is a :math:`N \times N` conditional covariance matrix, 
+:math:`\mathbf{H}_t^{1/2}` is a :math:`N \times N` positive definite matrix,
+and :math:`\mathbf{z}_t` is a :math:`N \times 1` vector of standard normal innovations.
+
+Conditional covariance matrix
+-----------------------------
+
+In the CCC-GARCH(1,1) model, the conditional covariance matrix :math:`\mathbf{H}_t` is constructed as:
+
+.. math::
+   :label: mv_volatility_eq
+
+   \mathbf{H}_t = \mathbf{D}_t\mathbf{R}\mathbf{D}_t
+
+where :math:`\mathbf{D}_t=\text{diag}(\mathbf{h}_t)^{1/2}`, 
+and :math:`\mathbf{h}_t` is a :math:`N \times 1` vector whose elements are univariate GARCH(1,1) variances for each time series.
+:math:`\mathbf{R}` is a positive definite constant conditional correlation matrix.
+
+.. admonition:: A bivarite example
+   :class: note
+
+   In a bivariate GARCH(1,1) setting, we have two univariate GARCH(1,1) processes, one for each return series.
+   Specifically, the GARCH(1,1) equations for the conditional variances :math:`h_{1t}` and :math:`h_{2t}` can be written as:
+
+   .. math::
+      :label: garch1_modified
+
+      h_{1t} = \omega_1 + \alpha_1 \epsilon_{1,t-1}^2 + \beta_1 h_{1,t-1}
+
+   .. math::
+      :label: garch2_modified
+
+      h_{2t} = \omega_2 + \alpha_2 \epsilon_{2,t-1}^2 + \beta_2 h_{2,t-1}
+
+   where,
+
+   - :math:`\epsilon_{1,t-1}` and :math:`\epsilon_{2,t-1}` are past shock terms from their respective time series. 
+   - The parameters :math:`\omega_1, \alpha_1, \beta_1, \omega_2, \alpha_2, \beta_2` are to be estimated.
+
+   With these individual variances, the conditional covariance matrix :math:`\mathbf{H}_t` is:
+
+   .. math::
+      :label: conditional_cov_matrix_modified
+
+      \mathbf{H}_t = \begin{pmatrix}
+      h_{1t} & \rho\sqrt{h_{1t} h_{2t}} \\\\
+      \rho\sqrt{h_{1t} h_{2t}} & h_{2t}
+      \end{pmatrix}
+
+   Here, :math:`\rho` is the correlation between the two time series. 
+   It is assumed to be constant over time in the CCC-GARCH framework.
+
+   The constant correlation matrix :math:`\mathbf{R}` simplifies to a :math:`2 \times 2` matrix:
+
+   .. math::
+      :label: bivariate_constant_corr_matrix
+      
+      \mathbf{R} = 
+      \begin{pmatrix}
+         1      & \rho \\\\
+         \rho & 1
+      \end{pmatrix}
+
+Log-likelihood function
+-----------------------
+
+The log-likelihood function for the :math:`N`-dimensional multivariate GARCH CCC model is:
+
+.. math::
+   :label: mv_log_likelihood
+
+   \begin{align}
+   \ell &= -\frac{1}{2} \sum_{t=1}^T \left[ N\ln(2\pi) + \ln(|\mathbf{H}_t|) + \mathbf{\epsilon}_t' \mathbf{H}_t^{-1} \mathbf{\epsilon}_t \right] \\\\
+        &= -\frac{1}{2} \sum_{t=1}^T \left[ N\ln(2\pi) + \ln(|\mathbf{D}_t\mathbf{R}\mathbf{D}_t|) + \mathbf{\epsilon}_t' \mathbf{D}_t^{-1}\mathbf{R}^{-1}\mathbf{D}_t^{-1} \mathbf{\epsilon}_t \right] \\\\
+        &= -\frac{1}{2} \sum_{t=1}^T \left[ N\ln(2\pi) + 2 \ln(|\mathbf{D}_t|) + \ln(|\mathbf{R}|)+ \mathbf{z}_t' \mathbf{R}^{-1} \mathbf{z}_t \right] 
+   \end{align}
+
+where :math:`\mathbf{z}_t=\mathbf{D}_t^{-1}\mathbf{\epsilon}_t` is the vector of standardized residuals. 
+
+This function is maximized to estimate the model parameters.
+
+.. admonition:: A bivariate example
+   :class: note
+
+   In the bivariate case, the log-likelihood function can be specifically written as a function of all parameters.
+
+   The log-likelihood function :math:`\ell` for the bivariate case with all parameters :math:`\Theta = (\mu_1, \omega_1, \alpha_1, \beta_1, \mu_2, \omega_2, \alpha_2, \beta_2, \rho)` is:
+
+   .. math::
+      :label: log_likelihood_bivariate
+
+      \ell(\Theta) = -\frac{1}{2} \sum_{t=1}^T \left[ 2\ln(2\pi) + 2 \ln(|\mathbf{D}_t|) + \ln(|\mathbf{R}|)+ \mathbf{z}_t' \mathbf{R}^{-1} \mathbf{z}_t \right] 
+
+   Here, :math:`\mathbf{z}_t'` is the transpose of the vector of standardized residuals :math:`\mathbf{z}_t`, 
+
+   .. math::
+
+      \mathbf{z}_t = \begin{pmatrix}
+        z_{1,t} \\
+        z_{2,t}
+      \end{pmatrix}
+
+   Further,
+
+   .. math::
+
+      \mathbf{D}_t = \begin{pmatrix}
+      \sqrt{h_{1t}} & 0 \\\\
+      0 & \sqrt{h_{2t}}
+      \end{pmatrix}
+
+   so the log-determinant of :math:`\mathbf{D}_t` is
+
+   .. math::
+
+      \ln(|\mathbf{D}_t|) = \frac{1}{2} \ln(h_{1t} h_{2t}) 
+
+   The log-determinant of :math:`\mathbf{R}` is 
+
+   .. math::
+
+      \ln(|\mathbf{R}|) = \ln(1 - \rho^2)
+
+   Inverse of :math:`\mathbf{R}` is 
+
+   .. math::
+
+      \mathbf{R}^{-1} = \frac{1}{1 - \rho^2} \begin{pmatrix}
+      1 & -\rho \\
+      -\rho & 1
+      \end{pmatrix}
+
+   Lastly, :math:`\mathbf{z}_t' \mathbf{R}^{-1} \mathbf{z}_t` is
+
+   .. math::
+
+      \mathbf{z}_t' \mathbf{R}^{-1} \mathbf{z}_t = \frac{1}{1 - \rho^2} \left[ z_{1t}^2 - 2\rho z_{1t} z_{2t} + z_{2t}^2 \right]
+
+
+   Inserting all of these into :math:`\ell(\Theta)` in equation :math:numref:`log_likelihood_bivariate`:
+
+   .. math::
+      :label: log_likelihood
+
+      \ell(\Theta) = -\frac{1}{2} \sum_{t=1}^T \left[ 2\ln(2\pi) + \ln(h_{1t} h_{2t} (1 - \rho^2)) + \frac{1}{1 - \rho^2} \left( z_{1t}^2 - 2\rho z_{1t} z_{2t} + z_{2t}^2 \right) \right]
+
+Estimation techniques
+=====================
+
+My implementation of :class:`frds.algorithms.GARCHModel_CCC` fits the GARCH-CCC model 
+by simultaneously estimating all parameters via maxmimizing the log-likelihood :math:numref:`log_likelihood`.
+
+General steps are:
+
+1. Use :class:`frds.algorithms.GARCHModel` to estimate the :doc:`/algorithms/garch` model for each of the returns.
+
+2. Use the standardized residuals from the estimated GARCH models to compute correlation coefficient.
+
+3. Use as starting vaues the estimated parameters from above in optimizing the loglikelihood function.
+
+
+References
+==========
+
+- `Engle, R. F. (1982) <https://doi.org/10.2307/1912773>`_, "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation." *Econometrica*, 50(4), 987-1007.
+
+- `Bollerslev, T. (1990) <https://doi.org/10.2307/2109358>`_, "Modelling the Coherence in Short-Run Nominal Exchange Rates: A Multivariate Generalized ARCH Model." *Review of Economics and Statistics*, 72(3), 498-505.
+
+API
+===
+
+.. autoclass:: frds.algorithms.GARCHModel_CCC
+   :exclude-members: Parameters
+
+.. autoclass:: frds.algorithms.GARCHModel_CCC.Parameters
+   :exclude-members: __init__
+   :no-undoc-members:
+
+Examples
+========
+
+Let's import the dataset.
+
+>>> import pandas as pd
+>>> data_url = "https://www.stata-press.com/data/r18/stocks.dta"
+>>> df = pd.read_stata(data_url, convert_dates=["date"])
+
+Scale returns to percentage returns for better optimization results
+
+>>> returns1 = df["toyota"].to_numpy() * 100
+>>> returns2 = df["nissan"].to_numpy() * 100
+
+Use :class:`frds.algorithms.GARCHModel_CCC` to estimate a GARCH(1,1)-CCC. 
+
+>>> from frds.algorithms import GARCHModel_CCC
+>>> model_ccc = GARCHModel_CCC(returns1, returns2)
+>>> res = model_ccc.fit()
+>>> from pprint import pprint
+>>> pprint(res)
+Parameters(mu1=0.02745814255283541,
+           omega1=0.03401400758840226,
+           alpha1=0.06593379740524756,
+           beta1=0.9219575443861723,
+           mu2=0.009390068254041505,
+           omega2=0.058694325049554734,
+           alpha2=0.0830561828957614,
+           beta2=0.9040961791372522,
+           rho=0.6506770477876749,
+           loglikelihood=-7281.321453218112)
+
+These results are comparable to the ones obtained in Stata, and even marginally 
+better based on log-likelihood. In Stata, we can estimate the same model as below:
+
+.. code-block:: stata
+
+    webuse stocks, clear
+    replace toyota = toyota * 100
+    replace nissan = nissan * 100
+    mgarch ccc (toyota nissan = ), arch(1) garch(1)
+   
+The Stata results are:
+
+.. code-block:: stata
+
+    Constant conditional correlation MGARCH model
+
+     Sample: 1 thru 2015                                      Number of obs = 2,015
+     Distribution: Gaussian                                   Wald chi2(.)  =     .
+     Log likelihood = -7282.961                               Prob > chi2   =     .
+
+     -------------------------------------------------------------------------------------
+                         | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
+     --------------------+----------------------------------------------------------------
+     toyota              |
+                 _cons   |   .0277462   .0302805     0.92   0.360    -.0316024    .0870948
+     --------------------+----------------------------------------------------------------
+     ARCH_toyota         |
+                    arch |
+                    L1.  |   .0666384   .0101597     6.56   0.000     .0467257    .0865511
+                         |
+                  garch  |
+                    L1.  |   .9210688   .0119214    77.26   0.000     .8977032    .9444343
+                         |
+                  _cons  |   .0344153   .0109208     3.15   0.002      .013011    .0558197
+     --------------------+----------------------------------------------------------------
+     nissan              |
+                  _cons  |   .0079682   .0349351     0.23   0.820    -.0605034    .0764398
+     --------------------+----------------------------------------------------------------
+     ARCH_nissan         |
+                    arch |
+                    L1.  |   .0851778   .0132656     6.42   0.000     .0591778    .1111779
+                         |
+                  garch  |
+                    L1.  |   .9016613   .0150494    59.91   0.000     .8721649    .9311577
+                         |
+                  _cons  |   .0603765   .0178318     3.39   0.001     .0254269    .0953262
+     --------------------+----------------------------------------------------------------
+     corr(toyota,nissan) |   .6512249   .0128548    50.66   0.000       .62603    .6764199
+     -------------------------------------------------------------------------------------
+
+See `Stata's reference manual <https://www.stata.com/manuals/ts.pdf>`_ for its
+estimation techniques.