GAS regression models

Introduction

The principle behind score-driven models is that the linear update \(y_{t} - \theta_{t}\), that the Kalman filter relies upon, can be robustified by replacing it with the conditional score of a non-normal distribution. For this reason, any class of traditional state space model has a score-driven equivalent.

For example, consider a dynamic regression model in this framework:

\[p\left(y_{t}\mid\theta_{t}\right)\]

\[\theta_{t} = \boldsymbol{x}_{t}^{'}\boldsymbol{\beta}_{t}\]

\[\boldsymbol{\beta}_{t} = \boldsymbol{\beta}_{t-1} + \boldsymbol{\eta}H_{t-1}^{-1}S_{t-1}\]

Here \(\eta\) represents the learning rates or scaling terms, and are the latent variables which are estimated in the model.

Example

We will use a dynamic t regression to extract a dynamic \(\beta\) for a stock. Using t-distributed errors is more robust than a normality assumption, that could be obtained with a Kalman filter. The \(\beta\) captures the amount of systematic risk in the stock - i.e. the stock’s relationship with the market.

from pandas_datareader import DataReader
from datetime import datetime

a = DataReader('AMZN',  'yahoo', datetime(2012,1,1), datetime(2016,6,1))
a_returns = pd.DataFrame(np.diff(np.log(a['Adj Close'].values)))
a_returns.index = a.index.values[1:a.index.values.shape[0]]
a_returns.columns = ["Amazon Returns"]

spy = DataReader('SPY',  'yahoo', datetime(2012,1,1), datetime(2016,6,1))
spy_returns = pd.DataFrame(np.diff(np.log(spy['Adj Close'].values)))
spy_returns.index = spy.index.values[1:spy.index.values.shape[0]]
spy_returns.columns = ['S&P500 Returns']

one_mon = DataReader('DGS1MO', 'fred',datetime(2012,1,1), datetime(2016,6,1))
one_day = np.log(1+one_mon)/365

returns = pd.concat([one_day,a_returns,spy_returns],axis=1).dropna()
excess_m = returns["Amazon 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=["Amazon","SP500","Risk-free rate"]
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);

We can fit a GAS Regression model with a t() family:

model = pf.GASReg('Amazon ~ SP500', data=final_returns, family=pf.t())

Next we estimate the latent variables. For this example we will use a Maximum Likelihood estimate \(z^{MLE}\):

x = model3.fit()
x.summary()

t GAS Regression
======================================== =================================================
Dependent Variable: Amazon               Method: MLE
Start Date: 2012-01-04 00:00:00          Log Likelihood: 3158.435
End Date: 2016-06-01 00:00:00            AIC: -6308.87
Number of observations: 1101             BIC: -6288.8541
==========================================================================================
Latent Variable           Estimate   Std Error  z        P>|z|    95% C.I.
========================= ========== ========== ======== ======== ========================
Scale 1                   0.0
Scale SP500               0.0474
t Scale                   0.0095
v                         2.8518
==========================================================================================

We can plot the fit with plot_fit():

model.plot_fit(intervals=False,figsize=(15,15))

One of the advantages of using a GASRegression rather than a Kalman filtered Dynamic Linear Regression is that the GASRegression with t errors is more robust to outliers. We do not produce the whole analysis here, but for the same data, the filtered estimates are compared below:

Class Description

class GASReg(data, formula, target, family)

Generalized Autoregressive Score Regression Models (GASReg).

Parameter	Type	Description
data	pd.DataFrame or np.ndarray	Contains the univariate time series
formula	string	Patsy notation specifying the regression
target	string or int	Which column of DataFrame/array to use.
family	pf.Family instance	The distribution for the time series, e.g pf.Normal()

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, oos_data, 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

To be clear, the oos_data argument should be a DataFrame in the same format as the initial dataframe used to initialize the model instance. The reason is that to predict future values, you need to specify assumptions about exogenous variables for the future. For example, if you predict h steps ahead, the method will take the h first rows from oos_data and take the values for the exogenous variables that you asked for in the patsy formula.

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, oos_data, 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

To be clear, the oos_data argument should be a DataFrame in the same format as the initial dataframe used to initialize the model instance. The reason is that to predict future values, you need to specify assumptions about exogenous variables for the future. For example, if you predict h steps ahead, the method will take the 5 first rows from oos_data and take the values for the exogenous variables that you specified as exogenous variables in the patsy formula.

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

Creal, D; Koopman, S.J.; Lucas, A. (2013). Generalized Autoregressive Score Models with Applications. Journal of Applied Econometrics, 28(5), 777–795. doi:10.1002/jae.1279.

Harvey, A.C. (2013). Dynamic Models for Volatility and Heavy Tails: With Applications to Financial and Economic Time Series. Cambridge University Press.