# GAS local linear trend 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 local linear model in this framework:

$p\left(y_{t}\mid\mu_{t}\right)$
$\mu_{t} = \mu_{t-1} + \beta_{t-1} + \eta_{1}{H_{t-1}^{-1}S_{t-1}}$
$\beta_{t} = \beta_{t-1} + \eta_{2}{H_{t-1}^{-1}S_{t-1}}$

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

## Example¶

We will construct a local linear trend model for US GDP using a skew t-distribution. Here is the data:

growthdata = pd.read_csv('http://www.pyflux.com/notebooks/GDPC1.csv')
USgrowth = pd.DataFrame(np.log(growthdata['VALUE']))
USgrowth.index = pd.to_datetime(growthdata['DATE'])
USgrowth.columns = ['Logged US Real GDP']
plt.figure(figsize=(15,5))
plt.plot(USgrowth.index, USgrowth)
plt.ylabel('Real GDP')
plt.title('US Logged Real GDP');


Here can fit a GAS Local Linear Trend model with a Skewt() family:

model = pf.GASLLT(data=USgrowth-np.mean(USgrowth),family=pf.Skewt())


Next we estimate the latent variables. For this example we will use a BBVI estimate $$z^{BBVI}$$:

x = model.fit('BBVI', iterations=20000, record_elbo=True)
10% done : ELBO is 70.1403732024, p(y,z) is 82.4582067151, q(z) is 12.3178335126
20% done : ELBO is 71.5399641383, p(y,z) is 84.3269580596, q(z) is 12.7869939213
30% done : ELBO is 95.3663747496, p(y,z) is 108.551290696, q(z) is 13.1849159469
40% done : ELBO is 124.357073241, p(y,z) is 138.132000673, q(z) is 13.7749274322
50% done : ELBO is 144.111819073, p(y,z) is 158.386802182, q(z) is 14.274983109
60% done : ELBO is 164.792526642, p(y,z) is 179.422645151, q(z) is 14.6301185085
70% done : ELBO is 178.18148403, p(y,z) is 193.190633108, q(z) is 15.0091490782
80% done : ELBO is 206.095112618, p(y,z) is 221.579871841, q(z) is 15.4847592232
90% done : ELBO is 210.854594358, p(y,z) is 226.705793141, q(z) is 15.8511987823
100% done : ELBO is 226.965067448, p(y,z) is 243.29536546, q(z) is 16.3302980111

Final model ELBO is 224.286972026


We can plot the ELBO with plot_elbo(): on the results object:

x.plot_elbo(figsize=(15,7))


We can plot the latent variables with plot_z():

model.plot_z([0,1,3])

model.plot_z([2,4])


The states are stored as an attribute states in the results object. Let’s plot the trend state:

plt.figure(figsize=(15,5))
plt.title("Local Trend for US GDP")
plt.ylabel("Trend")
plt.plot(USgrowth.index[21:],x.states[1][20:]);


This reflects the underlying growth potential of the US economy.

We can also calculate the average growth rate for a forward forecast:

print("Average growth rate for this period is")
print(str(round(100*np.mean(np.exp(np.diff(model.predict(h=4)['Logged US Real GDP'].values)) - 1),3)) + "%")

Average growth rate for this period is
0.504%


## Class Description¶

class GASLLT(data, integ, target, family)

GAS Local Linear Trend Models.

Parameter Type Description
data pd.DataFrame or np.ndarray Contains the univariate time series
integ int How many times to difference the data (default: 0)
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, 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¶

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.