# Dynamic Linear regression models¶

## Introduction¶

Gaussian state space models - often called structural time series or unobserved component models - provide a way to decompose a time series into several distinct components. These components can be extracted in closed form using the Kalman filter if the errors are jointly Gaussian, and parameters can be estimated via the prediction error decomposition and Maximum Likelihood.

We can support Dynamic Linear Regression in the state space framework:

$y_{t} = \boldsymbol{x}_{t}^{'}\boldsymbol{\beta}_{t} + \epsilon_{t}$
$\boldsymbol{\beta}_{t} = \boldsymbol{\beta}_{t-1} + \boldsymbol{\eta}_{t}$
$\epsilon_{t} \sim N\left(0,\sigma_{\epsilon}^{2}\right)$
$\boldsymbol{\eta}_{t} \sim N\left(\boldsymbol{0},\Sigma_{\eta}\right)$

## Example¶

In constructing portfolios in finance, we are often after the $$\beta$$ of a stock which can be used to construct the systematic component of returns. But this may not be a static quantity. For normally distributed returns (!) we can use a dynamic linear regression model using the Kalman filter and smoothing algorithm to track its evolution. First let’s get some data on excess returns. We’ll look at Amazon stock (AMZN) and use the S&P500 as ‘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.index = a.index.values[1:a.index.values.shape]
a_returns.columns = ["Amazon Returns"]

spy = DataReader('SPY',  'yahoo', datetime(2012,1,1), datetime(2016,6,1))
spy_returns.index = spy.index.values[1:spy.index.values.shape]
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["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); Here we define a Dynamic Linear regression as follows:

model = pf.DynReg('Amazon ~ SP500', data=final_returns)


We can also use the higher-level wrapper which allows us to specify the family, although if we pick a non-Gaussian family then the model will be estimated in a different way (not through the Kalman filter):

model = pf.DynamicGLM('Amazon ~ SP500', data=USgrowth, family=pf.Normal())


Next we estimate the latent variables. For this example we will use a maximum likelihood point mass estimate $$z^{MLE}$$:

x = model.fit()
x.summary()

Dynamic Linear Regression
====================================== =================================================
Dependent Variable: Amazon             Method: MLE
Start Date: 2012-01-04 00:00:00        Log Likelihood: 2871.5419
End Date: 2016-06-01 00:00:00          AIC: -5737.0838
Number of observations: 1101           BIC: -5722.0719
========================================================================================
Latent Variable         Estimate   Std Error  z        P>|z|    95% C.I.
======================= ========== ========== ======== ======== ========================
Sigma^2 irregular       0.0003
Sigma^2 1               0.0
Sigma^2 SP500           0.0024
========================================================================================


We can plot the in-sample fit using plot_fit():

model.plot_fit(figsize=(15,15)) The third plot shows $$\beta_{AMZN}$$. Following the burn-in period, the $$\beta$$ hovered just above 1 in 2013, although it became very correlated with market performance in 2014. More recently it has settled down again to hover just above 1. The fourth plot shows the remaining residual component of return (not including $$\alpha$$).

## Class Description¶

class DynLin(formula, data)

Dynamic Linear Regression models

Parameter Type Description
formula string Patsy notation specifying the regression
data pd.DataFrame Contains the univariate time series

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.

simulation_smoother(beta)

Returns np.ndarray of draws of the data from the Durbin and Koopman (2002) simulation smoother.

Parameter Type Description
beta np.array np.array of latent variables

Recommended just to use model.latent_variables.get_z_values() for the beta input, if you have already fit a model.

Returns : np.ndarray - samples from simulation smoother