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# GARCH models

## Introduction

Generalized autoregressive conditional heteroskedasticity (GARCH) models aim to model the conditional volatility of a time series. Let r_{t} be the dependent variable, for example the returns of a stock in time t. We can model this series as:

r_{t} = \mu + \sigma_{t}\epsilon_{t}


Here mu is the expected value of r_{t}, \sigma_{t} is the standard deviation of r_{t} in time t, and \epsilon_{t} is an error term for time t.

GARCH models are motivated by the desire to model \sigma_{t} conditional on past information. A primitive model might be a rolling standard deviation - e.g. a 30 day window - or an exponentially weighted standard deviation. A windowed model imposes an arbitrary cutoff which does not seem desirable. An EWMA is slightly more attractive, but how to select the weighting parameter \lambda is not immediate.

ARCH/GARCH models are an alterative model which allow for parameters to be estimated in a likelihood-based model. The basic driver of the model is a weighted average of past squared residuals. These lagged squared residuals are known as ARCH terms. Bollerslev (1986) extended the model by including lagged conditional volatility terms, creating GARCH models. Below is the formulation of a GARCH model:

y_{t} \sim N\left(\mu,\sigma_{t}^{2}\right)

\sigma_{t}^{2} = \omega + \alpha\epsilon_{t}^{2} + \beta{\sigma_{t-1}^{2}}


We need to impose constraints on this model to ensure the volatility is over 1, in particular \omega, \alpha, \beta > 0. If we want to ensure stationarity, we also need to ensure \alpha + \beta < 1.

Once we have estimated parameters for the model, we can perform retrospective analysis on volatility, as well as make forecasts for future conditional volatility.

## Example

First let us load some financial time series data from Yahoo Finance:

import numpy as np
import pyflux as pf
import pandas as pd
from datetime import datetime
import matplotlib.pyplot as plt
%matplotlib inline

jpm = DataReader('JPM',  'yahoo', datetime(2006,1,1), datetime(2016,3,10))
returns.index = jpm.index.values[1:jpm.index.values.shape[0]]
returns.columns = ['JPM Returns']

plt.figure(figsize=(15,5));
plt.plot(returns.index,returns);
plt.ylabel('Returns');
plt.title('JPM Returns');

One way to visualize the underlying volatility of the series is to plot the absolute returns \mid{y}\mid:

plt.figure(figsize=(15,5))
plt.plot(returns.index, np.abs(returns))
plt.ylabel('Absolute Returns')
plt.title('JP Morgan Absolute Returns');

There appears to be some evidence of volatility clustering over this period. Let’s fit a GARCH(1,1) model using a point mass estimate z^{MLE}:

model = pf.GARCH(returns,p=1,q=1)
x = model.fit()
x.summary()

GARCH(1,1)
======================================== =================================================
Dependent Variable: JPM Returns          Method: MLE
Start Date: 2006-01-05 00:00:00          Log Likelihood: 6594.7911
End Date: 2016-03-10 00:00:00            AIC: -13181.5822
Number of observations: 2562             BIC: -13158.188
==========================================================================================
Latent Variable           Estimate   Std Error  z        P>|z|    95% C.I.
========================= ========== ========== ======== ======== ========================
Vol Constant              0.0
q(1)                      0.0933
p(1)                      0.9013
Returns Constant          0.0009     0.0065     0.1359   0.8919   (-0.0119 | 0.0137)
==========================================================================================

The standard errors are not shown for transformed variables. You can pass through a transformed=False argument to summary to obtain this information for untransformed variables.

We can plot the GARCH latent variables with :py:func:plot_z:

model.plot_z(figsize=(15,5))

We can plot the fit with :py:func:plot_fit:

And plot predictions of future conditional volatility with :py:func:plot_predict:

model.plot_predict(h=10)

If we had wanted predictions in DataFrame form, we could have used :py:func:predict:.

We can view how well we predicted using in-sample rolling prediction with :py:func:plot_predict_is:

model.plot_predict_is(h=50,figsize=(15,5))

## Class Description

.. py:class:: GARCH(data, p, q, target)

**Generalized Autoregressive Conditional Heteroskedasticity Models (GARCH)**

==================   ===============================    ======================================
Parameter            Type                                Description
==================   ===============================    ======================================
data                 pd.DataFrame or np.ndarray         Contains the univariate time series
p                    int                                The number of autoregressive lags :math:\sigma^{2}
q                    int                                The number of ARCH terms :math:\epsilon^{2}
target               string or int                      Which column of DataFrame/array to use.
==================   ===============================    ======================================

**Attributes**

.. py:attribute:: latent_variables

A pf.LatentVariables() object containing information on the model latent variables,
prior settings. any fitted values, starting values, and other latent variable
information. When a model is fitted, this is where the latent variables are updated/stored.
Please see the documentation on Latent Variables for information on attributes within this
object, as well as methods for accessing the latent variable information.

**Methods**

Adjusts the priors for the model latent variables. The latent variables and their indices
can be viewed by printing the latent_variables attribute attached to the model instance.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
index                int                         Index of the latent variable to change
prior                pf.Family instance          Prior distribution, e.g. pf.Normal()
==================   ========================    ======================================

**Returns**: void - changes the model latent_variables attribute

.. py:method:: fit(method, **kwargs)

Estimates latent variables for the model. User chooses an inference option and the
method returns a results object, as well as updating the model's latent_variables
attribute.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
method               str                         Inference option: e.g. 'M-H' or 'MLE'
==================   ========================    ======================================

See Bayesian Inference and Classical Inference sections of the documentation for the
full list of inference options. Optional parameters can be entered that are relevant
to the particular mode of inference chosen.

**Returns**: pf.Results instance with information for the estimated latent variables

.. py:method:: plot_fit(**kwargs)

Plots the fit of the model against the data. Optional arguments include *figsize*,
the dimensions of the figure to plot.

**Returns** : void - shows a matplotlib plot

.. py:method:: plot_ppc(T, nsims)

Plots a histogram for a posterior predictive check with a discrepancy measure of the
user's choosing. This method only works if you have fitted using Bayesian inference.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
T                    function                    Discrepancy, e.g. np.mean or np.max
nsims                int                         How many simulations for the PPC
==================   ========================    ======================================

**Returns**: void - shows a matplotlib plot

.. py:method:: plot_predict(h, past_values, intervals, **kwargs)

Plots predictions of the model, along with intervals.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
h                    int                         How many steps to forecast ahead
past_values          int                         How many past datapoints to plot
intervals            boolean                     Whether to plot intervals or not
==================   ========================    ======================================

Optional arguments include *figsize* - the dimensions of the figure to plot. Please note
that if you use Maximum Likelihood or Variational Inference, the intervals shown will not
reflect latent variable uncertainty. Only Metropolis-Hastings will give you fully Bayesian
prediction intervals. Bayesian intervals with variational inference are not shown because
of the limitation of mean-field inference in not accounting for posterior correlations.

**Returns** : void - shows a matplotlib plot

.. py:method:: plot_predict_is(h, fit_once, fit_method, **kwargs)

Plots in-sample rolling predictions for the model. This means that the user pretends a
last subsection of data is out-of-sample, and forecasts after each period and assesses
how well they did. The user can choose whether to fit parameters once at the beginning
or every time step.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
h                    int                         How many previous timesteps to use
fit_once             boolean                     Whether to fit once, or every timestep
fit_method           str                         Which inference option, e.g. 'MLE'
==================   ========================    ======================================

Optional arguments include *figsize* - the dimensions of the figure to plot. **h** is an int of how many previous steps to simulate performance on.

**Returns** : void - shows a matplotlib plot

.. py:method:: plot_sample(nsims, plot_data=True)

Plots samples from the posterior predictive density of the model. This method only works
if you fitted the model using Bayesian inference.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
nsims                int                         How many samples to draw
plot_data            boolean                     Whether to plot the real data as well
==================   ========================    ======================================

**Returns** : void - shows a matplotlib plot

.. py:method:: plot_z(indices, figsize)

Returns a plot of the latent variables and their associated uncertainty.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
indices              int or list                 Which latent variable indices to plot
figsize              tuple                       Size of the matplotlib figure
==================   ========================    ======================================

**Returns** : void - shows a matplotlib plot

.. py:method:: ppc(T, nsims)

Returns a p-value for a posterior predictive check. This method only works if you have
fitted using Bayesian inference.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
T                    function                    Discrepancy, e.g. np.mean or np.max
nsims                int                         How many simulations for the PPC
==================   ========================    ======================================

**Returns**: int - the p-value for the discrepancy test

.. py:method:: predict(h, intervals=False)

Returns a DataFrame of model predictions.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
h                    int                         How many steps to forecast ahead
intervals            boolean                     Whether to return prediction intervals
==================   ========================    ======================================

Please note that if you use Maximum Likelihood or Variational Inference, the intervals shown
will not reflect latent variable uncertainty. Only Metropolis-Hastings will give you fully
Bayesian prediction intervals. Bayesian intervals with variational inference are not shown
because of the limitation of mean-field inference in not accounting for posterior correlations.

**Returns** : pd.DataFrame - the model predictions

.. py:method:: predict_is(h, fit_once, fit_method)

Returns DataFrame of in-sample rolling predictions for the model.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
h                    int                         How many previous timesteps to use
fit_once             boolean                     Whether to fit once, or every timestep
fit_method           str                         Which inference option, e.g. 'MLE'
==================   ========================    ======================================

**Returns** : pd.DataFrame - the model predictions

.. py:method:: sample(nsims)

Returns np.ndarray of draws of the data from the posterior predictive density. This
method only works if you have fitted the model using Bayesian inference.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
nsims                int                         How many posterior draws to take
==================   ========================    ======================================

**Returns** : np.ndarray - samples from the posterior predictive density.



## References

Bollerslev, T. (1986). Generalized Autoregressive Conditional Heteroskedasticity. Journal of Econometrics. April, 31:3, pp. 307–27.

Engle, R.F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica. 50(4), 987-1007.

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