# lss¶

class quantecon.lss.LinearStateSpace(A, C, G, H=None, mu_0=None, Sigma_0=None)[source]

Bases: object

A class that describes a Gaussian linear state space model of the form:

\begin{align}\begin{aligned}x_{t+1} = A x_t + C w_{t+1}\\y_t = G x_t + H v_t\end{aligned}\end{align}

where $${w_t}$$ and $${v_t}$$ are independent and standard normal with dimensions k and l respectively. The initial conditions are $$\mu_0$$ and $$\Sigma_0$$ for $$x_0 \sim N(\mu_0, \Sigma_0)$$. When $$\Sigma_0=0$$, the draw of $$x_0$$ is exactly $$\mu_0$$.

Parameters:
Aarray_like or scalar(float)

Part of the state transition equation. It should be n x n

Carray_like or scalar(float)

Part of the state transition equation. It should be n x m

Garray_like or scalar(float)

Part of the observation equation. It should be k x n

Harray_like or scalar(float), optional(default=None)

Part of the observation equation. It should be k x l

mu_0array_like or scalar(float), optional(default=None)

This is the mean of initial draw and is n x 1

Sigma_0array_like or scalar(float), optional(default=None)

This is the variance of the initial draw and is n x n and also should be positive definite and symmetric

Attributes:
A, C, G, H, mu_0, Sigma_0see Parameters
n, k, m, lscalar(int)

The dimensions of x_t, y_t, w_t and v_t respectively

Methods

 Convert array_like objects (lists of lists, floats, etc.) into well formed 2D NumPy arrays geometric_sums(beta, x_t) Forecast the geometric sums Pulls off the imuplse response coefficients to a shock in $$w_{t}$$ for $$x$$ and $$y$$ Create a generator to calculate the population mean and variance-covariance matrix for both $$x_t$$ and $$y_t$$ starting at the initial condition (self.mu_0, self.Sigma_0). replicate([T, num_reps, random_state]) Simulate num_reps observations of $$x_T$$ and $$y_T$$ given $$x_0 \sim N(\mu_0, \Sigma_0)$$. simulate([ts_length, random_state]) Simulate a time series of length ts_length, first drawing Compute the moments of the stationary distributions of $$x_t$$ and $$y_t$$ if possible.
convert(x)[source]

Convert array_like objects (lists of lists, floats, etc.) into well formed 2D NumPy arrays

geometric_sums(beta, x_t)[source]

Forecast the geometric sums

\begin{align}\begin{aligned}S_x := E \Big[ \sum_{j=0}^{\infty} \beta^j x_{t+j} | x_t \Big]\\S_y := E \Big[ \sum_{j=0}^{\infty} \beta^j y_{t+j} | x_t \Big]\end{aligned}\end{align}
Parameters:
betascalar(float)

Discount factor, in [0, 1)

betaarray_like(float)

The term x_t for conditioning

Returns:
S_xarray_like(float)

Geometric sum as defined above

S_yarray_like(float)

Geometric sum as defined above

impulse_response(j=5)[source]

Pulls off the imuplse response coefficients to a shock in $$w_{t}$$ for $$x$$ and $$y$$

Important to note: We are uninterested in the shocks to v for this method

• $$x$$ coefficients are $$C, AC, A^2 C...$$

• $$y$$ coefficients are $$GC, GAC, GA^2C...$$

Parameters:
jScalar(int)

Number of coefficients that we want

Returns:
xcoeflist(array_like(float, 2))

The coefficients for x

ycoeflist(array_like(float, 2))

The coefficients for y

moment_sequence()[source]

Create a generator to calculate the population mean and variance-covariance matrix for both $$x_t$$ and $$y_t$$ starting at the initial condition (self.mu_0, self.Sigma_0). Each iteration produces a 4-tuple of items (mu_x, mu_y, Sigma_x, Sigma_y) for the next period.

Yields:
mu_xarray_like(float)

An n x 1 array representing the population mean of x_t

mu_yarray_like(float)

A k x 1 array representing the population mean of y_t

Sigma_xarray_like(float)

An n x n array representing the variance-covariance matrix of x_t

Sigma_yarray_like(float)

A k x k array representing the variance-covariance matrix of y_t

replicate(T=10, num_reps=100, random_state=None)[source]

Simulate num_reps observations of $$x_T$$ and $$y_T$$ given $$x_0 \sim N(\mu_0, \Sigma_0)$$.

Parameters:
Tscalar(int), optional(default=10)

The period that we want to replicate values for

num_repsscalar(int), optional(default=100)

The number of replications that we want

random_stateint or np.random.RandomState/Generator, optional

Random seed (integer) or np.random.RandomState or Generator instance to set the initial state of the random number generator for reproducibility. If None, a randomly initialized RandomState is used.

Returns:
xarray_like(float)

An n x num_reps array, where the j-th column is the j_th observation of $$x_T$$

yarray_like(float)

A k x num_reps array, where the j-th column is the j_th observation of $$y_T$$

simulate(ts_length=100, random_state=None)[source]

Simulate a time series of length ts_length, first drawing

$x_0 \sim N(\mu_0, \Sigma_0)$
Parameters:
ts_lengthscalar(int), optional(default=100)

The length of the simulation

random_stateint or np.random.RandomState/Generator, optional

Random seed (integer) or np.random.RandomState or Generator instance to set the initial state of the random number generator for reproducibility. If None, a randomly initialized RandomState is used.

Returns:
xarray_like(float)

An n x ts_length array, where the t-th column is $$x_t$$

yarray_like(float)

A k x ts_length array, where the t-th column is $$y_t$$

stationary_distributions()[source]

Compute the moments of the stationary distributions of $$x_t$$ and $$y_t$$ if possible. Computation is by solving the discrete Lyapunov equation.

Returns:
mu_xarray_like(float)

An n x 1 array representing the stationary mean of $$x_t$$

mu_yarray_like(float)

An k x 1 array representing the stationary mean of $$y_t$$

Sigma_xarray_like(float)

An n x n array representing the stationary var-cov matrix of $$x_t$$

Sigma_yarray_like(float)

An k x k array representing the stationary var-cov matrix of $$y_t$$

Sigma_yxarray_like(float)

An k x n array representing the stationary cov matrix between $$y_t$$ and $$x_t$$.

quantecon.lss.simulate_linear_model(A, x0, v, ts_length)[source]

This is a separate function for simulating a vector linear system of the form

$x_{t+1} = A x_t + v_t$

given $$x_0$$ = x0

Here $$x_t$$ and $$v_t$$ are both n x 1 and $$A$$ is n x n.

The purpose of separating this functionality out is to target it for optimization by Numba. For the same reason, matrix multiplication is broken down into for loops.

Parameters:
Aarray_like or scalar(float)

Should be n x n

x0array_like

Should be n x 1. Initial condition

vnp.ndarray

Should be n x ts_length-1. Its t-th column is used as the time t shock $$v_t$$

ts_lengthint

The length of the time series

Returns:
xnp.ndarray

Time series with ts_length columns, the t-th column being $$x_t$$