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 pandas_datareader import DataReader
from datetime import datetime
import matplotlib.pyplot as plt
%matplotlib inline

jpm = DataReader('JPM',  'yahoo', datetime(2006,1,1), datetime(2016,3,10))
returns = pd.DataFrame(np.diff(np.log(jpm['Adj Close'].values)))
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 plot_z():

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

We can plot the fit with plot_fit():

And plot predictions of future conditional volatility with plot_predict():

model.plot_predict(h=10)

If we had wanted predictions in DataFrame form, we could have used predict():.

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

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

Class Description

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 \(\sigma^{2}\)
q	int	The number of ARCH terms \(\epsilon^{2}\)
target	string or int	Which column of DataFrame/array to use.

Attributes

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

adjust_prior(index, prior)

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

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

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

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

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

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

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

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

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

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

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

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.