Beta-t-EGARCH in-mean regression models¶

Introduction¶

We can expand the Beta-t-EGARCH in-mean model to include exogenous regressors in both the returns and conditional volatility equation:

$y_{t} = \mu + \sum^{m}_{k=1}\phi_{m}{X_{m,t}} + \exp\left(\lambda_{t\mid{t-1}}/2\right)\epsilon_{t}$
$\lambda_{t\mid{t-1}} = \alpha_{0} + \sum^{p}_{i=1}\alpha_{i}\lambda_{t-i} + \sum^{q}_{j=1}\beta_{j}\left(\frac{\left(\nu+1\right)y_{t-j}^{2}}{\nu\exp\left(\lambda_{t-j\mid{t-j-1}}\right) + y_{t-j}^{2}}-1\right) + \sum^{m}_{k=1}\gamma_{m}{X_{m,t}}$
$\epsilon_{t} \sim t_{\nu}$

Developer Note¶

• This model type has yet to be Cythonized so performance can be slow.

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

a = DataReader('JPM',  'yahoo', datetime(2000,1,1), datetime(2016,3,10))
a_returns.index = a.index.values[1:a.index.values.shape[0]]
a_returns.columns = ["JPM Returns"]

spy = DataReader('SPY',  'yahoo', datetime(2000,1,1), datetime(2016,3,10))
spy_returns.index = spy.index.values[1:spy.index.values.shape[0]]
spy_returns.columns = ['S&P500 Returns']

one_day = np.log(1+one_mon)/365

returns = pd.concat([one_day,a_returns,spy_returns],axis=1).dropna()
excess_m = returns["JPM Returns"].values - returns['DGS1MO'].values
excess_spy = returns["S&P500 Returns"].values - returns['DGS1MO'].values
final_returns = pd.DataFrame(np.transpose([excess_m,excess_spy, returns['DGS1MO'].values]))
final_returns.columns=["JPM","SP500","Rf"]
final_returns.index = returns.index

plt.figure(figsize=(15,5))
plt.title("Excess Returns")
x = plt.plot(final_returns);
plt.legend(iter(x), final_returns.columns);


Let’s fit an EGARCH-M model to JPM’s excess returns series, with a risk-free rate regressor:

modelx = pf.EGARCHMReg(p=1,q=1,data=final_returns,formula='JPM ~ Rf')
results = modelx.fit()
results.summary()

EGARCHMReg(1,1)
======================================== =================================================
Dependent Variable: JPM                  Method: MLE
Start Date: 2001-08-01 00:00:00          Log Likelihood: 9621.2996
End Date: 2016-03-10 00:00:00            AIC: -19226.5993
Number of observations: 3645             BIC: -19176.9904
==========================================================================================
Latent Variable           Estimate   Std Error  z        P>|z|    95% C.I.
========================= ========== ========== ======== ======== ========================
p(1)                      0.9936
q(1)                      0.0935
v                         6.5414
GARCH-M                   -0.0199    0.023      -0.8657  0.3866   (-0.0649 | 0.0251)
Vol Beta 1                -0.0545    0.0007     -77.9261 0.0      (-0.0559 | -0.0531)
Vol Beta Rf               -0.0346    1.1161     -0.031   0.9753   (-2.2222 | 2.153)
Returns Beta 1            0.0011     0.0011     1.0218   0.3069   (-0.001 | 0.0032)
Returns Beta Rf           -1.1324    0.0941     -12.0398 0.0      (-1.3167 | -0.948)
==========================================================================================


Let’s plot the latent variables with plot_z():

modelx.plot_z([5,7],figsize=(15,5))


For this stock, the risk-free rate has a negative effect on excess returns. For the effects of returns on volatility, we are far more uncertain. We can plot the fit with plot_fit():

modelx.plot_fit(figsize=(15,5))


Class Description¶

class EGARCHMReg(data, formula, p, q)

Long Memory Beta-t-EGARCH Models

Parameter Type Description
data pd.DataFrame or np.ndarray Contains the univariate time series
formula string Patsy notation specifying the regression
p int The number of autoregressive lags $$\sigma^{2}$$
q int The number of ARCH terms $$\epsilon^{2}$$

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

add_leverage()

Adds a leverage term to the model, meaning volatility can respond differently to the sign of the news; see Harvey and Succarrat (2013). Conditional volatility will now follow:

$\lambda_{t\mid{t-1}} = \alpha_{0} + \sum^{p}_{i=1}\alpha_{i}\lambda_{t-i} + \sum^{q}_{j=1}\beta_{j}u_{t-j} + \kappa\left(\text{sgn}\left(-\epsilon_{t-1}\right)(u_{t-1}+1)\right)$
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
oos_data pd.DataFrame Exogenous variables in a frame for h steps
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
oos_data pd.DataFrame Exogenous variables in a frame for h steps
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¶

Black, F. (1976) Studies of stock price volatility changes. In: Proceedings of the 1976 Meetings of the American Statistical Association. pp. 171–181.

Fernandez, C., & Steel, M. F. J. (1998a). On Bayesian Modeling of Fat Tails and Skewness. Journal of the American Statistical Association, 93, 359–371.

Harvey, A.C. & Chakravarty, T. (2008) Beta-t-(E)GARCH. Cambridge Working Papers in Economics 0840, Faculty of Economics, University of Cambridge, 2008. [p137]

Harvey, A.C. & Sucarrat, G. (2013) EGARCH models with fat tails, skewness and leverage. Computational Statistics and Data Analysis, Forthcoming, 2013. URL http://dx.doi.org/10.1016/j.csda.2013.09. 022. [p138, 139, 140, 143]

Mandelbrot, B.B. (1963) The variation of certain speculative prices. Journal of Business, XXXVI (1963). pp. 392–417

Nelson, D. B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347—370.