Source code for statsmodels.tsa.vector_ar.irf

# -*- coding: utf-8 -*-
"""
Impulse reponse-related code
"""

from __future__ import division

import numpy as np
import numpy.linalg as la
import scipy.linalg as L

from scipy import stats

from statsmodels.tools.decorators import cache_readonly
from statsmodels.tools.tools import chain_dot
#from statsmodels.tsa.api import VAR
from statsmodels.compat.python import range
import statsmodels.tsa.tsatools as tsa
import statsmodels.tsa.vector_ar.plotting as plotting
import statsmodels.tsa.vector_ar.util as util

mat = np.array

class BaseIRAnalysis(object):
    """
    Base class for plotting and computing IRF-related statistics, want to be
    able to handle known and estimated processes
    """

    def __init__(self, model, P=None, periods=10, order=None, svar=False,
                 vecm=False):
        self.model = model
        self.periods = periods
        self.neqs, self.lags, self.T = model.neqs, model.k_ar, model.nobs

        self.order = order

        if P is None:
            sigma = model.sigma_u

            # TODO, may be difficult at the moment
            # if order is not None:
            #     indexer = [model.get_eq_index(name) for name in order]
            #     sigma = sigma[:, indexer][indexer, :]

            #     if sigma.shape != model.sigma_u.shape:
            #         raise ValueError('variable order is wrong length')

            P = la.cholesky(sigma)

        self.P = P

        self.svar = svar

        self.irfs = model.ma_rep(periods)
        if svar:
            self.svar_irfs = model.svar_ma_rep(periods, P=P)
        else:
            self.orth_irfs = model.orth_ma_rep(periods, P=P)

        self.cum_effects = self.irfs.cumsum(axis=0)
        if svar:
            self.svar_cum_effects = self.svar_irfs.cumsum(axis=0)
        else:
            self.orth_cum_effects = self.orth_irfs.cumsum(axis=0)

        # long-run effects may be infinite for VECMs.
        if not vecm:
            self.lr_effects = model.long_run_effects()
            if svar:
                self.svar_lr_effects = np.dot(model.long_run_effects(), P)
            else:
                self.orth_lr_effects = np.dot(model.long_run_effects(), P)


        # auxiliary stuff
        if vecm:
            self._A = util.comp_matrix(model.var_rep)
        else:
            self._A = util.comp_matrix(model.coefs)

    def cov(self, *args, **kwargs):
        raise NotImplementedError

    def cum_effect_cov(self, *args, **kwargs):
        raise NotImplementedError

    def plot(self, orth=False, impulse=None, response=None,
             signif=0.05, plot_params=None, subplot_params=None,
             plot_stderr=True, stderr_type='asym', repl=1000,
             seed=None, component=None):
        """
        Plot impulse responses

        Parameters
        ----------
        orth : bool, default False
            Compute orthogonalized impulse responses
        impulse : string or int
            variable providing the impulse
        response : string or int
            variable affected by the impulse
        signif : float (0 < signif < 1)
            Significance level for error bars, defaults to 95% CI
        subplot_params : dict
            To pass to subplot plotting funcions. Example: if fonts are too big,
            pass {'fontsize' : 8} or some number to your taste.
        plot_params : dict

        plot_stderr: bool, default True
            Plot standard impulse response error bands
        stderr_type: string
            'asym': default, computes asymptotic standard errors
            'mc': monte carlo standard errors (use rpl)
        repl: int, default 1000
            Number of replications for Monte Carlo and Sims-Zha standard errors
        seed: int
            np.random.seed for Monte Carlo replications
        component: array or vector of principal component indices
        """
        periods = self.periods
        model = self.model
        svar = self.svar

        if orth and svar:
            raise ValueError("For SVAR system, set orth=False")

        if orth:
            title = 'Impulse responses (orthogonalized)'
            irfs = self.orth_irfs
        elif svar:
            title = 'Impulse responses (structural)'
            irfs = self.svar_irfs
        else:
            title = 'Impulse responses'
            irfs = self.irfs

        if plot_stderr == False:
            stderr = None

        elif stderr_type not in ['asym', 'mc', 'sz1', 'sz2','sz3']:
            raise ValueError("Error type must be either 'asym', 'mc','sz1','sz2', or 'sz3'")
        else:
            if stderr_type == 'asym':
                stderr = self.cov(orth=orth)
            if stderr_type == 'mc':
                stderr = self.errband_mc(orth=orth, svar=svar,
                                         repl=repl, signif=signif,
                                         seed=seed)
            if stderr_type == 'sz1':
                stderr = self.err_band_sz1(orth=orth, svar=svar,
                                           repl=repl, signif=signif,
                                           seed=seed,
                                           component=component)
            if stderr_type == 'sz2':
                stderr = self.err_band_sz2(orth=orth, svar=svar,
                                           repl=repl, signif=signif,
                                           seed=seed,
                                           component=component)
            if stderr_type == 'sz3':
                stderr = self.err_band_sz3(orth=orth, svar=svar,
                                           repl=repl, signif=signif,
                                           seed=seed,
                                           component=component)

        fig = plotting.irf_grid_plot(irfs, stderr, impulse, response,
                                     self.model.names, title, signif=signif,
                                     subplot_params=subplot_params,
                                     plot_params=plot_params,
                                     stderr_type=stderr_type)
        return fig

    def plot_cum_effects(self, orth=False, impulse=None, response=None,
                         signif=0.05, plot_params=None,
                         subplot_params=None, plot_stderr=True,
                         stderr_type='asym', repl=1000, seed=None):
        """
        Plot cumulative impulse response functions

        Parameters
        ----------
        orth : bool, default False
            Compute orthogonalized impulse responses
        impulse : string or int
            variable providing the impulse
        response : string or int
            variable affected by the impulse
        signif : float (0 < signif < 1)
            Significance level for error bars, defaults to 95% CI
        subplot_params : dict
            To pass to subplot plotting funcions. Example: if fonts are too big,
            pass {'fontsize' : 8} or some number to your taste.
        plot_params : dict

        plot_stderr: bool, default True
            Plot standard impulse response error bands
        stderr_type: string
            'asym': default, computes asymptotic standard errors
            'mc': monte carlo standard errors (use rpl)
        repl: int, default 1000
            Number of replications for monte carlo standard errors
        seed: int
            np.random.seed for Monte Carlo replications

        """

        if orth:
            title = 'Cumulative responses responses (orthogonalized)'
            cum_effects = self.orth_cum_effects
            lr_effects = self.orth_lr_effects
        else:
            title = 'Cumulative responses'
            cum_effects = self.cum_effects
            lr_effects = self.lr_effects

        if stderr_type not in ['asym', 'mc']:
            raise TypeError
        else:
            if stderr_type == 'asym':
                stderr = self.cum_effect_cov(orth=orth)
            if stderr_type == 'mc':
                stderr = self.cum_errband_mc(orth=orth, repl=repl,
                                                signif=signif, seed=seed)
        if not plot_stderr:
            stderr = None

        fig = plotting.irf_grid_plot(cum_effects, stderr, impulse, response,
                                     self.model.names, title, signif=signif,
                                     hlines=lr_effects,
                                     subplot_params=subplot_params,
                                     plot_params=plot_params,
                                     stderr_type=stderr_type)
        return fig


[docs]class IRAnalysis(BaseIRAnalysis): """ Impulse response analysis class. Computes impulse responses, asymptotic standard errors, and produces relevant plots Parameters ---------- model : VAR instance Notes ----- Using L├╝tkepohl (2005) notation """ def __init__(self, model, P=None, periods=10, order=None, svar=False, vecm=False): BaseIRAnalysis.__init__(self, model, P=P, periods=periods, order=order, svar=svar, vecm=vecm) if vecm: self.cov_a = model.cov_var_repr else: self.cov_a = model._cov_alpha self.cov_sig = model._cov_sigma # memoize dict for G matrix function self._g_memo = {}
[docs] def cov(self, orth=False): """ Compute asymptotic standard errors for impulse response coefficients Notes ----- L├╝tkepohl eq 3.7.5 Returns ------- """ if orth: return self._orth_cov() covs = self._empty_covm(self.periods + 1) covs[0] = np.zeros((self.neqs ** 2, self.neqs ** 2)) for i in range(1, self.periods + 1): Gi = self.G[i - 1] covs[i] = chain_dot(Gi, self.cov_a, Gi.T) return covs
[docs] def errband_mc(self, orth=False, svar=False, repl=1000, signif=0.05, seed=None, burn=100): """ IRF Monte Carlo integrated error bands """ model = self.model periods = self.periods if svar == True: return model.sirf_errband_mc(orth=orth, repl=repl, T=periods, signif=signif, seed=seed, burn=burn, cum=False) else: return model.irf_errband_mc(orth=orth, repl=repl, T=periods, signif=signif, seed=seed, burn=burn, cum=False)
[docs] def err_band_sz1(self, orth=False, svar=False, repl=1000, signif=0.05, seed=None, burn=100, component=None): """ IRF Sims-Zha error band method 1. Assumes symmetric error bands around mean. Parameters ---------- orth : bool, default False Compute orthogonalized impulse responses repl : int, default 1000 Number of MC replications signif : float (0 < signif < 1) Significance level for error bars, defaults to 95% CI seed : int, default None np.random seed burn : int, default 100 Number of initial simulated obs to discard component : neqs x neqs array, default to largest for each Index of column of eigenvector/value to use for each error band Note: period of impulse (t=0) is not included when computing principle component References ---------- Sims, Christopher A., and Tao Zha. 1999. "Error Bands for Impulse Response". Econometrica 67: 1113-1155. """ model = self.model periods = self.periods if orth: irfs = self.orth_irfs elif svar: irfs = self.svar_irfs else: irfs = self.irfs neqs = self.neqs irf_resim = model.irf_resim(orth=orth, repl=repl, T=periods, seed=seed, burn=100) q = util.norm_signif_level(signif) W, eigva, k =self._eigval_decomp_SZ(irf_resim) if component != None: if np.shape(component) != (neqs,neqs): raise ValueError("Component array must be " + str(neqs) + " x " + str(neqs)) if np.argmax(component) >= neqs*periods: raise ValueError("Atleast one of the components does not exist") else: k = component # here take the kth column of W, which we determine by finding the largest eigenvalue of the covaraince matrix lower = np.copy(irfs) upper = np.copy(irfs) for i in range(neqs): for j in range(neqs): lower[1:,i,j] = irfs[1:,i,j] + W[i,j,:,k[i,j]]*q*np.sqrt(eigva[i,j,k[i,j]]) upper[1:,i,j] = irfs[1:,i,j] - W[i,j,:,k[i,j]]*q*np.sqrt(eigva[i,j,k[i,j]]) return lower, upper
[docs] def err_band_sz2(self, orth=False, svar=False, repl=1000, signif=0.05, seed=None, burn=100, component=None): """ IRF Sims-Zha error band method 2. This method Does not assume symmetric error bands around mean. Parameters ---------- orth : bool, default False Compute orthogonalized impulse responses repl : int, default 1000 Number of MC replications signif : float (0 < signif < 1) Significance level for error bars, defaults to 95% CI seed : int, default None np.random seed burn : int, default 100 Number of initial simulated obs to discard component : neqs x neqs array, default to largest for each Index of column of eigenvector/value to use for each error band Note: period of impulse (t=0) is not included when computing principle component References ---------- Sims, Christopher A., and Tao Zha. 1999. "Error Bands for Impulse Response". Econometrica 67: 1113-1155. """ model = self.model periods = self.periods if orth: irfs = self.orth_irfs elif svar: irfs = self.svar_irfs else: irfs = self.irfs neqs = self.neqs irf_resim = model.irf_resim(orth=orth, repl=repl, T=periods, seed=seed, burn=100) W, eigva, k = self._eigval_decomp_SZ(irf_resim) if component != None: if np.shape(component) != (neqs,neqs): raise ValueError("Component array must be " + str(neqs) + " x " + str(neqs)) if np.argmax(component) >= neqs*periods: raise ValueError("Atleast one of the components does not exist") else: k = component gamma = np.zeros((repl, periods+1, neqs, neqs)) for p in range(repl): for i in range(neqs): for j in range(neqs): gamma[p,1:,i,j] = W[i,j,k[i,j],:] * irf_resim[p,1:,i,j] gamma_sort = np.sort(gamma, axis=0) #sort to get quantiles indx = round(signif/2*repl)-1,round((1-signif/2)*repl)-1 lower = np.copy(irfs) upper = np.copy(irfs) for i in range(neqs): for j in range(neqs): lower[:,i,j] = irfs[:,i,j] + gamma_sort[indx[0],:,i,j] upper[:,i,j] = irfs[:,i,j] + gamma_sort[indx[1],:,i,j] return lower, upper
[docs] def err_band_sz3(self, orth=False, svar=False, repl=1000, signif=0.05, seed=None, burn=100, component=None): """ IRF Sims-Zha error band method 3. Does not assume symmetric error bands around mean. Parameters ---------- orth : bool, default False Compute orthogonalized impulse responses repl : int, default 1000 Number of MC replications signif : float (0 < signif < 1) Significance level for error bars, defaults to 95% CI seed : int, default None np.random seed burn : int, default 100 Number of initial simulated obs to discard component : vector length neqs, default to largest for each Index of column of eigenvector/value to use for each error band Note: period of impulse (t=0) is not included when computing principle component References ---------- Sims, Christopher A., and Tao Zha. 1999. "Error Bands for Impulse Response". Econometrica 67: 1113-1155. """ model = self.model periods = self.periods if orth: irfs = self.orth_irfs elif svar: irfs = self.svar_irfs else: irfs = self.irfs neqs = self.neqs irf_resim = model.irf_resim(orth=orth, repl=repl, T=periods, seed=seed, burn=100) stack = np.zeros((neqs, repl, periods*neqs)) #stack left to right, up and down for p in range(repl): for i in range(neqs): stack[i, p,:] = np.ravel(irf_resim[p,1:,:,i].T) stack_cov=np.zeros((neqs, periods*neqs, periods*neqs)) W = np.zeros((neqs, periods*neqs, periods*neqs)) eigva = np.zeros((neqs, periods*neqs)) k = np.zeros((neqs)) if component != None: if np.size(component) != (neqs): raise ValueError("Component array must be of length " + str(neqs)) if np.argmax(component) >= neqs*periods: raise ValueError("Atleast one of the components does not exist") else: k = component #compute for eigen decomp for each stack for i in range(neqs): stack_cov[i] = np.cov(stack[i],rowvar=0) W[i], eigva[i], k[i] = util.eigval_decomp(stack_cov[i]) gamma = np.zeros((repl, periods+1, neqs, neqs)) for p in range(repl): c=0 for j in range(neqs): for i in range(neqs): gamma[p,1:,i,j] = W[j,k[j],i*periods:(i+1)*periods] * irf_resim[p,1:,i,j] if i == neqs-1: gamma[p,1:,i,j] = W[j,k[j],i*periods:] * irf_resim[p,1:,i,j] gamma_sort = np.sort(gamma, axis=0) #sort to get quantiles indx = round(signif/2*repl)-1,round((1-signif/2)*repl)-1 lower = np.copy(irfs) upper = np.copy(irfs) for i in range(neqs): for j in range(neqs): lower[:,i,j] = irfs[:,i,j] + gamma_sort[indx[0],:,i,j] upper[:,i,j] = irfs[:,i,j] + gamma_sort[indx[1],:,i,j] return lower, upper
def _eigval_decomp_SZ(self, irf_resim): """ Returns ------- W: array of eigenvectors eigva: list of eigenvalues k: matrix indicating column # of largest eigenvalue for each c_i,j """ neqs = self.neqs periods = self.periods cov_hold = np.zeros((neqs, neqs, periods, periods)) for i in range(neqs): for j in range(neqs): cov_hold[i,j,:,:] = np.cov(irf_resim[:,1:,i,j],rowvar=0) W = np.zeros((neqs, neqs, periods, periods)) eigva = np.zeros((neqs, neqs, periods, 1)) k = np.zeros((neqs, neqs)) for i in range(neqs): for j in range(neqs): W[i,j,:,:], eigva[i,j,:,0], k[i,j] = util.eigval_decomp(cov_hold[i,j,:,:]) return W, eigva, k
[docs] @cache_readonly def G(self): # Gi matrices as defined on p. 111 K = self.neqs # nlags = self.model.p # J = np.hstack((np.eye(K),) + (np.zeros((K, K)),) * (nlags - 1)) def _make_g(i): # p. 111 Lutkepohl G = 0. for m in range(i): # be a bit cute to go faster idx = i - 1 - m if idx in self._g_memo: apow = self._g_memo[idx] else: apow = la.matrix_power(self._A.T, idx) # apow = np.dot(J, apow) apow = apow[:K] self._g_memo[idx] = apow # take first K rows piece = np.kron(apow, self.irfs[m]) G = G + piece return G return [_make_g(i) for i in range(1, self.periods + 1)]
def _orth_cov(self): # Lutkepohl 3.7.8 Ik = np.eye(self.neqs) PIk = np.kron(self.P.T, Ik) H = self.H covs = self._empty_covm(self.periods + 1) for i in range(self.periods + 1): if i == 0: apiece = 0 else: Ci = np.dot(PIk, self.G[i-1]) apiece = chain_dot(Ci, self.cov_a, Ci.T) Cibar = np.dot(np.kron(Ik, self.irfs[i]), H) bpiece = chain_dot(Cibar, self.cov_sig, Cibar.T) / self.T # Lutkepohl typo, cov_sig correct covs[i] = apiece + bpiece return covs
[docs] def cum_effect_cov(self, orth=False): """ Compute asymptotic standard errors for cumulative impulse response coefficients Parameters ---------- orth : boolean Notes ----- eq. 3.7.7 (non-orth), 3.7.10 (orth) Returns ------- """ Ik = np.eye(self.neqs) PIk = np.kron(self.P.T, Ik) F = 0. covs = self._empty_covm(self.periods + 1) for i in range(self.periods + 1): if i > 0: F = F + self.G[i - 1] if orth: if i == 0: apiece = 0 else: Bn = np.dot(PIk, F) apiece = chain_dot(Bn, self.cov_a, Bn.T) Bnbar = np.dot(np.kron(Ik, self.cum_effects[i]), self.H) bpiece = chain_dot(Bnbar, self.cov_sig, Bnbar.T) / self.T covs[i] = apiece + bpiece else: if i == 0: covs[i] = np.zeros((self.neqs**2, self.neqs**2)) continue covs[i] = chain_dot(F, self.cov_a, F.T) return covs
[docs] def cum_errband_mc(self, orth=False, repl=1000, signif=0.05, seed=None, burn=100): """ IRF Monte Carlo integrated error bands of cumulative effect """ model = self.model periods = self.periods return model.irf_errband_mc(orth=orth, repl=repl, T=periods, signif=signif, seed=seed, burn=burn, cum=True)
[docs] def lr_effect_cov(self, orth=False): """ Returns ------- """ lre = self.lr_effects Finfty = np.kron(np.tile(lre.T, self.lags), lre) Ik = np.eye(self.neqs) if orth: Binf = np.dot(np.kron(self.P.T, np.eye(self.neqs)), Finfty) Binfbar = np.dot(np.kron(Ik, lre), self.H) return (chain_dot(Binf, self.cov_a, Binf.T) + chain_dot(Binfbar, self.cov_sig, Binfbar.T)) else: return chain_dot(Finfty, self.cov_a, Finfty.T)
[docs] def stderr(self, orth=False): return np.array([tsa.unvec(np.sqrt(np.diag(c))) for c in self.cov(orth=orth)])
[docs] def cum_effect_stderr(self, orth=False): return np.array([tsa.unvec(np.sqrt(np.diag(c))) for c in self.cum_effect_cov(orth=orth)])
[docs] def lr_effect_stderr(self, orth=False): cov = self.lr_effect_cov(orth=orth) return tsa.unvec(np.sqrt(np.diag(cov)))
def _empty_covm(self, periods): return np.zeros((periods, self.neqs ** 2, self.neqs ** 2), dtype=float)
[docs] @cache_readonly def H(self): k = self.neqs Lk = tsa.elimination_matrix(k) Kkk = tsa.commutation_matrix(k, k) Ik = np.eye(k) # B = chain_dot(Lk, np.eye(k**2) + commutation_matrix(k, k), # np.kron(self.P, np.eye(k)), Lk.T) # return np.dot(Lk.T, L.inv(B)) B = chain_dot(Lk, np.dot(np.kron(Ik, self.P), Kkk) + np.kron(self.P, Ik), Lk.T) return np.dot(Lk.T, L.inv(B))
[docs] def fevd_table(self): pass