# 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)[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: 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 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-convariance 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([max_iter, tol]) 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: beta : scalar(float) Discount factor, in [0, 1) beta : array_like(float) The term x_t for conditioning S_x : array_like(float) Geometric sum as defined above S_y : array_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: j : Scalar(int) Number of coefficients that we want xcoef : list(array_like(float, 2)) The coefficients for x ycoef : list(array_like(float, 2)) The coefficients for y
moment_sequence()[source]

Create a generator to calculate the population mean and variance-convariance 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)[source]

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. 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)[source]

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. 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(max_iter=200, tol=1e-05)[source]

Compute the moments of the stationary distributions of $$x_t$$ and $$y_t$$ if possible. Computation is by iteration, starting from the initial conditions self.mu_0 and self.Sigma_0

Parameters: max_iter : scalar(int), optional(default=200) The maximum number of iterations allowed tol : scalar(float), optional(default=1e-5) The tolerance level that one wishes to achieve mu_x_star : array_like(float) An n x 1 array representing the stationary mean of $$x_t$$ mu_y_star : array_like(float) An k x 1 array representing the stationary mean of $$y_t$$ Sigma_x_star : array_like(float) An n x n array representing the stationary var-cov matrix of $$x_t$$ Sigma_y_star : array_like(float) An k x k array representing the stationary var-cov matrix of $$y_t$$
quantecon.lss.multivariate_normal(mean, cov[, size, check_valid, tol])

Draw random samples from a multivariate normal distribution.

The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or “center”) and variance (standard deviation, or “width,” squared) of the one-dimensional normal distribution.

Parameters: mean : 1-D array_like, of length N Mean of the N-dimensional distribution. cov : 2-D array_like, of shape (N, N) Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling. size : int or tuple of ints, optional Given a shape of, for example, (m,n,k), m*n*k samples are generated, and packed in an m-by-n-by-k arrangement. Because each sample is N-dimensional, the output shape is (m,n,k,N). If no shape is specified, a single (N-D) sample is returned. check_valid : { ‘warn’, ‘raise’, ‘ignore’ }, optional Behavior when the covariance matrix is not positive semidefinite. tol : float, optional Tolerance when checking the singular values in covariance matrix. out : ndarray The drawn samples, of shape size, if that was provided. If not, the shape is (N,). In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.

Notes

The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution.

Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, $$X = [x_1, x_2, ... x_N]$$. The covariance matrix element $$C_{ij}$$ is the covariance of $$x_i$$ and $$x_j$$. The element $$C_{ii}$$ is the variance of $$x_i$$ (i.e. its “spread”).

Instead of specifying the full covariance matrix, popular approximations include:

• Spherical covariance (cov is a multiple of the identity matrix)
• Diagonal covariance (cov has non-negative elements, and only on the diagonal)

This geometrical property can be seen in two dimensions by plotting generated data-points:

>>> mean = [0, 0]
>>> cov = [[1, 0], [0, 100]]  # diagonal covariance


Diagonal covariance means that points are oriented along x or y-axis:

>>> import matplotlib.pyplot as plt
>>> x, y = np.random.multivariate_normal(mean, cov, 5000).T
>>> plt.plot(x, y, 'x')
>>> plt.axis('equal')
>>> plt.show()


Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed.

References

 [1] Papoulis, A., “Probability, Random Variables, and Stochastic Processes,” 3rd ed., New York: McGraw-Hill, 1991.
 [2] Duda, R. O., Hart, P. E., and Stork, D. G., “Pattern Classification,” 2nd ed., New York: Wiley, 2001.

Examples

>>> mean = (1, 2)
>>> cov = [[1, 0], [0, 1]]
>>> x = np.random.multivariate_normal(mean, cov, (3, 3))
>>> x.shape
(3, 3, 2)


The following is probably true, given that 0.6 is roughly twice the standard deviation:

>>> list((x[0,0,:] - mean) < 0.6)
[True, True]

quantecon.lss.simulate_linear_model[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: 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 x : np.ndarray Time series with ts_length columns, the t-th column being $$x_t$$