lss

Computes quantities associated with the Gaussian linear state space model.

References

https://lectures.quantecon.org/py/linear_models.html

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

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:

A : array_like or scalar(float)

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

C : array_like or scalar(float)

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

G : array_like or scalar(float)

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

H : array_like or scalar(float), optional(default=None)

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

mu_0 : array_like or scalar(float), optional(default=None)

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

Sigma_0 : array_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_0 : see Parameters

n, k, m, l : scalar(int)

The dimensions of x_t, y_t, w_t and v_t respectively

Methods

convert(x)	Convert array_like objects (lists of lists, floats, etc.) into well formed 2D NumPy arrays
geometric_sums(beta, x_t)	Forecast the geometric sums
impulse_response([j])	Pulls off the imuplse response coefficients to a shock in \(w_{t}\) for \(x\) and \(y\)
moment_sequence()	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
stationary_distributions()	Compute the moments of the stationary distributions of \(x_t\) and \(y_t\) if possible.

convert(x)

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

geometric_sums(beta, x_t)

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:

beta : scalar(float)

Discount factor, in [0, 1)

beta : array_like(float)

The term x_t for conditioning

Returns:

S_x : array_like(float)

Geometric sum as defined above

S_y : array_like(float)

Geometric sum as defined above

impulse_response(j=5)

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:

j : Scalar(int)

Number of coefficients that we want

Returns:

xcoef : list(array_like(float, 2))

The coefficients for x

ycoef : list(array_like(float, 2))

The coefficients for y

moment_sequence()

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_x : array_like(float)

An n x 1 array representing the population mean of x_t

mu_y : array_like(float)

A k x 1 array representing the population mean of y_t

Sigma_x : array_like(float)

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

Sigma_y : array_like(float)

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

replicate(T=10, num_reps=100, random_state=None)

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

Parameters:

T : scalar(int), optional(default=10)

The period that we want to replicate values for

num_reps : scalar(int), optional(default=100)

The number of replications that we want

random_state : int or np.random.RandomState, optional

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

Returns:

x : array_like(float)

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

y : array_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)

Simulate a time series of length ts_length, first drawing

\[x_0 \sim N(\mu_0, \Sigma_0)\]

Parameters:

ts_length : scalar(int), optional(default=100)

The length of the simulation

random_state : int or np.random.RandomState, optional

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

Returns:

x : array_like(float)

An n x ts_length array, where the t-th column is \(x_t\)

y : array_like(float)

A k x ts_length array, where the t-th column is \(y_t\)

stationary_distributions()

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_x : array_like(float)

An n x 1 array representing the stationary mean of \(x_t\)

mu_y : array_like(float)

An k x 1 array representing the stationary mean of \(y_t\)

Sigma_x : array_like(float)

An n x n array representing the stationary var-cov matrix of \(x_t\)

Sigma_y : array_like(float)

An k x k array representing the stationary var-cov matrix of \(y_t\)

Sigma_yx : array_like(float)

An k x n array representing the stationary cov matrix between \(y_t\) and \(x_t\).

quantecon.lss.simulate_linear_model

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:

A : array_like or scalar(float)

Should be n x n

x0 : array_like

Should be n x 1. Initial condition

v : np.ndarray

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

ts_length : int

The length of the time series

Returns:

x : np.ndarray

Time series with ts_length columns, the t-th column being \(x_t\)