# Beta-t-EGARCH in-mean models¶

## Introduction¶

Beta-t-EGARCH in-mean models extend $$Beta$$-$$t$$-$$EGARCH(p,q)$$ models by allowing returns to depend upon a past conditional volatility component. The $$Beta$$-$$t$$-$$EGARCH(p,q)$$ in-mean model includes this effect $$\phi$$ as follows:

$y_{t} = \mu + \phi\exp\left(\lambda_{t\mid{t-1}}/2\right) + \exp\left(\lambda_{t\mid{t-1}}/2\right)\epsilon_{t}$
$\lambda_{t\mid{t-1}} = \alpha_{0} + \sum^{p}_{i=1}\alpha_{i}\lambda_{t-i} + \sum^{q}_{j=1}\beta_{j}u_{t-j}$
$\epsilon_{t} \sim t_{\nu}$

## 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

jpm = DataReader('JPM',  'yahoo', datetime(2006,1,1), datetime(2016,3,10))
returns.index = jpm.index.values[1:jpm.index.values.shape]
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 $$Beta$$-$$t$$-$$EGARCH-M(1,1)$$ model using a BBVI estimate $$z^{BBVI}$$:

model = pf.EGARCHM(returns,p=1,q=1)
x = model.fit('BBVI', record_elbo=True, iterations=1000, map_start=False)
x.summary()

EGARCHM(1,1)
======================================== ==================================================
Dependent Variable: AAPL Returns         Method: BBVI
Start Date: 2006-01-05 00:00:00          Unnormalized Log Posterior: 7016.148
End Date: 2016-11-11 00:00:00            AIC: -14020.2959822
Number of observations: 2734             BIC: -13984.8148561
===========================================================================================
Latent Variable             Median             Mean               95% Credibility Interval
=========================== ================== ================== =========================
Vol Constant                -0.2842            -0.2841            (-0.3474 | -0.2207)
p(1)                        0.9624             0.9624             (0.9579 | 0.9665)
q(1)                        0.1889             0.1889             (0.1784 | 0.2)
v                           8.689              8.6902             (8.0781 | 9.3552)
Returns Constant            0.0001             0.0001             (-0.0094 | 0.0093)
GARCH-M                     0.1087             0.1085             (0.0226 | 0.1942)
===========================================================================================


We can plot the ELBO through BBVI by calling plot_elbo(): on the results object:

x.plot_elbo(figsize=(15,7)) As we can see, the ELBO converges after around 200 iterations. We can plot the model fit through plot_fit():

model.plot_fit(figsize=(15,7)) And plot predictions of future conditional volatility with plot_predict():

model.predict(h=10) We can plot samples from the posterior predictive density through plot_sample():

model.plot_sample(figsize=(15, 7)) And we can do posterior predictive checks on discrepancies of interest:

from scipy.stats import kurtosis
model.plot_ppc(T=kurtosis,figsize=(15, 7))
model.plot_ppc(T=np.std,figsize=(15, 7))  Here it appears our generated samples generate kurtosis that is slightly lower than the data, and a standard deviation that is slightly higher, but we are not too off in both checks.

## Class Description¶

class EGARCHM(data, p, q, target)

Beta-t-EGARCH in-mean Models

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

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
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.