"""
Vector Autoregression (VAR) processes

References
----------
Lutkepohl (2005) New Introduction to Multiple Time Series Analysis
"""

from __future__ import division, print_function
from statsmodels.compat.python import (range, lrange, string_types, StringIO, iteritems,
                                cStringIO)

from collections import defaultdict

import numpy as np
import numpy.linalg as npl
from numpy.linalg import cholesky as chol, solve
import scipy.stats as stats
import scipy.linalg as L

from statsmodels.tools.decorators import cache_readonly
from statsmodels.tools.tools import chain_dot
from statsmodels.tools.linalg import logdet_symm
from statsmodels.tsa.tsatools import vec, unvec

from statsmodels.tsa.vector_ar.irf import IRAnalysis
from statsmodels.tsa.vector_ar.output import VARSummary

import statsmodels.tsa.tsatools as tsa
import statsmodels.tsa.vector_ar.output as output
import statsmodels.tsa.vector_ar.plotting as plotting
import statsmodels.tsa.vector_ar.util as util
import statsmodels.tsa.base.tsa_model as tsbase
import statsmodels.base.wrapper as wrap

mat = np.array

#-------------------------------------------------------------------------------
# VAR process routines

def ma_rep(coefs, maxn=10):
    r"""
    MA(\infty) representation of VAR(p) process

    Parameters
    ----------
    coefs : ndarray (p x k x k)
    maxn : int
        Number of MA matrices to compute

    Notes
    -----
    VAR(p) process as

    .. math:: y_t = A_1 y_{t-1} + \ldots + A_p y_{t-p} + u_t

    can be equivalently represented as

    .. math:: y_t = \mu + \sum_{i=0}^\infty \Phi_i u_{t-i}

    e.g. can recursively compute the \Phi_i matrices with \Phi_0 = I_k

    Returns
    -------
    phis : ndarray (maxn + 1 x k x k)
    """
    p, k, k = coefs.shape
    phis = np.zeros((maxn+1, k, k))
    phis[0] = np.eye(k)

    # recursively compute Phi matrices
    for i in range(1, maxn + 1):
        for j in range(1, i+1):
            if j > p:
                break

            phis[i] += np.dot(phis[i-j], coefs[j-1])

    return phis

def is_stable(coefs, verbose=False):
    """
    Determine stability of VAR(p) system by examining the eigenvalues of the
    VAR(1) representation

    Parameters
    ----------
    coefs : ndarray (p x k x k)

    Returns
    -------
    is_stable : bool
    """
    A_var1 = util.comp_matrix(coefs)
    eigs = np.linalg.eigvals(A_var1)

    if verbose:
        print('Eigenvalues of VAR(1) rep')
        for val in np.abs(eigs):
            print(val)

    return (np.abs(eigs) <= 1).all()

def var_acf(coefs, sig_u, nlags=None):
    """
    Compute autocovariance function ACF_y(h) up to nlags of stable VAR(p)
    process

    Parameters
    ----------
    coefs : ndarray (p x k x k)
        Coefficient matrices A_i
    sig_u : ndarray (k x k)
        Covariance of white noise process u_t
    nlags : int, optional
        Defaults to order p of system

    Notes
    -----
    Ref: Lutkepohl p.28-29

    Returns
    -------
    acf : ndarray, (p, k, k)
    """
    p, k, _ = coefs.shape
    if nlags is None:
        nlags = p

    # p x k x k, ACF for lags 0, ..., p-1
    result = np.zeros((nlags + 1, k, k))
    result[:p] = _var_acf(coefs, sig_u)

    # yule-walker equations
    for h in range(p, nlags + 1):
        # compute ACF for lag=h
        # G(h) = A_1 G(h-1) + ... + A_p G(h-p)

        for j in range(p):
            result[h] += np.dot(coefs[j], result[h-j-1])

    return result

def _var_acf(coefs, sig_u):
    """
    Compute autocovariance function ACF_y(h) for h=1,...,p

    Notes
    -----
    Lutkepohl (2005) p.29
    """
    p, k, k2 = coefs.shape
    assert(k == k2)

    A = util.comp_matrix(coefs)
    # construct VAR(1) noise covariance
    SigU = np.zeros((k*p, k*p))
    SigU[:k,:k] = sig_u

    # vec(ACF) = (I_(kp)^2 - kron(A, A))^-1 vec(Sigma_U)
    vecACF = L.solve(np.eye((k*p)**2) - np.kron(A, A), vec(SigU))

    acf = unvec(vecACF)
    acf = acf[:k].T.reshape((p, k, k))

    return acf

def forecast(y, coefs, intercept, steps):
    """
    Produce linear MSE forecast

    Parameters
    ----------
    y :
    coefs :
    intercept :
    steps :

    Returns
    -------
    forecasts : ndarray (steps x neqs)

    Notes
    -----
    Lutkepohl p. 37

    Also used by DynamicVAR class
    """
    p = len(coefs)
    k = len(coefs[0])
    # initial value
    forcs = np.zeros((steps, k)) + intercept

    # h=0 forecast should be latest observation
    # forcs[0] = y[-1]

    # make indices easier to think about
    for h in range(1, steps + 1):
        # y_t(h) = intercept + sum_1^p A_i y_t_(h-i)
        f = forcs[h - 1]
        for i in range(1, p + 1):
            # slightly hackish
            if h - i <= 0:
                # e.g. when h=1, h-1 = 0, which is y[-1]
                prior_y = y[h - i - 1]
            else:
                # e.g. when h=2, h-1=1, which is forcs[0]
                prior_y = forcs[h - i - 1]

            # i=1 is coefs[0]
            f = f + np.dot(coefs[i - 1], prior_y)
        forcs[h - 1] = f

    return forcs

def forecast_cov(ma_coefs, sig_u, steps):
    """
    Compute theoretical forecast error variance matrices

    Parameters
    ----------

    Returns
    -------
    forc_covs : ndarray (steps x neqs x neqs)
    """
    k = len(sig_u)
    forc_covs = np.zeros((steps, k, k))

    prior = np.zeros((k, k))
    for h in range(steps):
        # Sigma(h) = Sigma(h-1) + Phi Sig_u Phi'
        phi = ma_coefs[h]
        var = chain_dot(phi, sig_u, phi.T)
        forc_covs[h] = prior = prior + var

    return forc_covs

def var_loglike(resid, omega, nobs):
    r"""
    Returns the value of the VAR(p) log-likelihood.

    Parameters
    ----------
    resid : ndarray (T x K)
    omega : ndarray
        Sigma hat matrix.  Each element i,j is the average product of the
        OLS residual for variable i and the OLS residual for variable j or
        np.dot(resid.T,resid)/nobs.  There should be no correction for the
        degrees of freedom.
    nobs : int

    Returns
    -------
    llf : float
        The value of the loglikelihood function for a VAR(p) model

    Notes
    -----
    The loglikelihood function for the VAR(p) is

    .. math::

        -\left(\frac{T}{2}\right)
        \left(\ln\left|\Omega\right|-K\ln\left(2\pi\right)-K\right)
    """
    logdet = logdet_symm(np.asarray(omega))
    neqs = len(omega)
    part1 = - (nobs * neqs / 2) * np.log(2 * np.pi)
    part2 = - (nobs / 2) * (logdet + neqs)
    return part1 + part2

def _reordered(self, order):
    #Create new arrays to hold rearranged results from .fit()
    endog = self.endog
    endog_lagged = self.endog_lagged
    params = self.params
    sigma_u = self.sigma_u
    names = self.names
    k_ar = self.k_ar
    endog_new = np.zeros([np.size(endog,0),np.size(endog,1)])
    endog_lagged_new = np.zeros([np.size(endog_lagged,0), np.size(endog_lagged,1)])
    params_new_inc, params_new = [np.zeros([np.size(params,0), np.size(params,1)])
                                  for i in range(2)]
    sigma_u_new_inc, sigma_u_new = [np.zeros([np.size(sigma_u,0), np.size(sigma_u,1)])
                                    for i in range(2)]
    num_end = len(self.params[0])
    names_new = []

    #Rearrange elements and fill in new arrays
    k = self.k_trend
    for i, c in enumerate(order):
        endog_new[:,i] = self.endog[:,c]
        if k > 0:
            params_new_inc[0,i] = params[0,i]
            endog_lagged_new[:,0] = endog_lagged[:,0]
        for j in range(k_ar):
            params_new_inc[i+j*num_end+k,:] = self.params[c+j*num_end+k,:]
            endog_lagged_new[:,i+j*num_end+k] = endog_lagged[:,c+j*num_end+k]
        sigma_u_new_inc[i,:] = sigma_u[c,:]
        names_new.append(names[c])
    for i, c in enumerate(order):
        params_new[:,i] = params_new_inc[:,c]
        sigma_u_new[:,i] = sigma_u_new_inc[:,c]

    return VARResults(endog=endog_new, endog_lagged=endog_lagged_new,
                      params=params_new, sigma_u=sigma_u_new,
                      lag_order=self.k_ar, model=self.model,
                      trend='c', names=names_new, dates=self.dates)

#-------------------------------------------------------------------------------
# VARProcess class: for known or unknown VAR process

class VAR(tsbase.TimeSeriesModel):
    r"""
    Fit VAR(p) process and do lag order selection

    .. math:: y_t = A_1 y_{t-1} + \ldots + A_p y_{t-p} + u_t

    Parameters
    ----------
    endog : array-like
        2-d endogenous response variable. The independent variable.
    dates : array-like
        must match number of rows of endog

    References
    ----------
    Lutkepohl (2005) New Introduction to Multiple Time Series Analysis
    """
    def __init__(self, endog, dates=None, freq=None, missing='none'):
        super(VAR, self).__init__(endog, None, dates, freq, missing=missing)
        if self.endog.ndim == 1:
            raise ValueError("Only gave one variable to VAR")
        self.y = self.endog #keep alias for now
        self.neqs = self.endog.shape[1]

    def _get_predict_start(self, start, k_ar):
        if start is None:
            start = k_ar
        return super(VAR, self)._get_predict_start(start)

    def predict(self, params, start=None, end=None, lags=1, trend='c'):
        """
        Returns in-sample predictions or forecasts
        """
        start = self._get_predict_start(start, lags)
        end, out_of_sample = self._get_predict_end(end)

        if end < start:
            raise ValueError("end is before start")
        if end == start + out_of_sample:
            return np.array([])

        k_trend = util.get_trendorder(trend)
        k = self.neqs
        k_ar = lags

        predictedvalues = np.zeros((end + 1 - start + out_of_sample, k))
        if k_trend != 0:
            intercept = params[:k_trend]
            predictedvalues += intercept

        y = self.y
        X = util.get_var_endog(y, lags, trend=trend, has_constant='raise')
        fittedvalues = np.dot(X, params)

        fv_start = start - k_ar
        pv_end = min(len(predictedvalues), len(fittedvalues) - fv_start)
        fv_end = min(len(fittedvalues), end-k_ar+1)
        predictedvalues[:pv_end] = fittedvalues[fv_start:fv_end]

        if not out_of_sample:
            return predictedvalues

        # fit out of sample
        y = y[-k_ar:]
        coefs = params[k_trend:].reshape((k_ar, k, k)).swapaxes(1,2)
        predictedvalues[pv_end:] = forecast(y, coefs, intercept, out_of_sample)
        return predictedvalues

    def fit(self, maxlags=None, method='ols', ic=None, trend='c',
            verbose=False):
        """
        Fit the VAR model

        Parameters
        ----------
        maxlags : int
            Maximum number of lags to check for order selection, defaults to
            12 * (nobs/100.)**(1./4), see select_order function
        method : {'ols'}
            Estimation method to use
        ic : {'aic', 'fpe', 'hqic', 'bic', None}
            Information criterion to use for VAR order selection.
            aic : Akaike
            fpe : Final prediction error
            hqic : Hannan-Quinn
            bic : Bayesian a.k.a. Schwarz
        verbose : bool, default False
            Print order selection output to the screen
        trend, str {"c", "ct", "ctt", "nc"}
            "c" - add constant
            "ct" - constant and trend
            "ctt" - constant, linear and quadratic trend
            "nc" - co constant, no trend
            Note that these are prepended to the columns of the dataset.

        Notes
        -----
        Lutkepohl pp. 146-153

        Returns
        -------
        est : VARResults
        """
        lags = maxlags

        if ic is not None:
            selections = self.select_order(maxlags=maxlags, verbose=verbose)
            if ic not in selections:
                raise Exception("%s not recognized, must be among %s"
                                % (ic, sorted(selections)))
            lags = selections[ic]
            if verbose:
                print('Using %d based on %s criterion' %  (lags, ic))
        else:
            if lags is None:
                lags = 1

        k_trend = util.get_trendorder(trend)
        self.exog_names = util.make_lag_names(self.endog_names, lags, k_trend)
        self.nobs = len(self.endog) - lags

        return self._estimate_var(lags, trend=trend)

    def _estimate_var(self, lags, offset=0, trend='c'):
        """
        lags : int
        offset : int
            Periods to drop from beginning-- for order selection so it's an
            apples-to-apples comparison
        trend : string or None
            As per above
        """
        # have to do this again because select_order doesn't call fit
        self.k_trend = k_trend = util.get_trendorder(trend)

        if offset < 0: # pragma: no cover
            raise ValueError('offset must be >= 0')

        y = self.y[offset:]

        z = util.get_var_endog(y, lags, trend=trend, has_constant='raise')
        y_sample = y[lags:]

        # Lutkepohl p75, about 5x faster than stated formula
        params = np.linalg.lstsq(z, y_sample)[0]
        resid = y_sample - np.dot(z, params)

        # Unbiased estimate of covariance matrix $\Sigma_u$ of the white noise
        # process $u$
        # equivalent definition
        # .. math:: \frac{1}{T - Kp - 1} Y^\prime (I_T - Z (Z^\prime Z)^{-1}
        # Z^\prime) Y
        # Ref: Lutkepohl p.75
        # df_resid right now is T - Kp - 1, which is a suggested correction

        avobs = len(y_sample)

        df_resid = avobs - (self.neqs * lags + k_trend)

        sse = np.dot(resid.T, resid)
        omega = sse / df_resid

        varfit = VARResults(y, z, params, omega, lags, names=self.endog_names,
                          trend=trend, dates=self.data.dates, model=self)
        return VARResultsWrapper(varfit)

    def select_order(self, maxlags=None, verbose=True):
        """
        Compute lag order selections based on each of the available information
        criteria

        Parameters
        ----------
        maxlags : int
            if None, defaults to 12 * (nobs/100.)**(1./4)
        verbose : bool, default True
            If True, print table of info criteria and selected orders

        Returns
        -------
        selections : dict {info_crit -> selected_order}
        """
        if maxlags is None:
            maxlags = int(round(12*(len(self.endog)/100.)**(1/4.)))

        ics = defaultdict(list)
        for p in range(maxlags + 1):
            # exclude some periods to same amount of data used for each lag
            # order
            result = self._estimate_var(p, offset=maxlags-p)

            for k, v in iteritems(result.info_criteria):
                ics[k].append(v)

        selected_orders = dict((k, mat(v).argmin())
                               for k, v in iteritems(ics))

        if verbose:
            output.print_ic_table(ics, selected_orders)

        return selected_orders

class VARProcess(object):
    """
    Class represents a known VAR(p) process

    Parameters
    ----------
    coefs : ndarray (p x k x k)
    intercept : ndarray (length k)
    sigma_u : ndarray (k x k)
    names : sequence (length k)

    Returns
    -------
    **Attributes**:
    """
    def __init__(self, coefs, intercept, sigma_u, names=None):
        self.k_ar = len(coefs)
        self.neqs = coefs.shape[1]
        self.coefs = coefs
        self.intercept = intercept
        self.sigma_u = sigma_u
        self.names = names

    def get_eq_index(self, name):
        "Return integer position of requested equation name"
        return util.get_index(self.names, name)

    def __str__(self):
        output = ('VAR(%d) process for %d-dimensional response y_t'
                  % (self.k_ar, self.neqs))
        output += '\nstable: %s' % self.is_stable()
        output += '\nmean: %s' % self.mean()

        return output

    def is_stable(self, verbose=False):
        """Determine stability based on model coefficients

        Parameters
        ----------
        verbose : bool
            Print eigenvalues of the VAR(1) companion

        Notes
        -----
        Checks if det(I - Az) = 0 for any mod(z) <= 1, so all the eigenvalues of
        the companion matrix must lie outside the unit circle
        """
        return is_stable(self.coefs, verbose=verbose)

    def plotsim(self, steps=1000):
        """
        Plot a simulation from the VAR(p) process for the desired number of
        steps
        """
        Y = util.varsim(self.coefs, self.intercept, self.sigma_u, steps=steps)
        plotting.plot_mts(Y)

    def mean(self):
        r"""Mean of stable process

        Lutkepohl eq. 2.1.23

        .. math:: \mu = (I - A_1 - \dots - A_p)^{-1} \alpha
        """
        return solve(self._char_mat, self.intercept)

    def ma_rep(self, maxn=10):
        r"""Compute MA(:math:`\infty`) coefficient matrices

        Parameters
        ----------
        maxn : int
            Number of coefficient matrices to compute

        Returns
        -------
        coefs : ndarray (maxn x k x k)
        """
        return ma_rep(self.coefs, maxn=maxn)

    def orth_ma_rep(self, maxn=10, P=None):
        r"""Compute Orthogonalized MA coefficient matrices using P matrix such
        that :math:`\Sigma_u = PP^\prime`. P defaults to the Cholesky
        decomposition of :math:`\Sigma_u`

        Parameters
        ----------
        maxn : int
            Number of coefficient matrices to compute
        P : ndarray (k x k), optional
            Matrix such that Sigma_u = PP', defaults to Cholesky descomp

        Returns
        -------
        coefs : ndarray (maxn x k x k)
        """
        if P is None:
            P = self._chol_sigma_u

        ma_mats = self.ma_rep(maxn=maxn)
        return mat([np.dot(coefs, P) for coefs in ma_mats])

    def long_run_effects(self):
        """Compute long-run effect of unit impulse

        .. math::

            \Psi_\infty = \sum_{i=0}^\infty \Phi_i

        """
        return L.inv(self._char_mat)

    @cache_readonly
    def _chol_sigma_u(self):
        return chol(self.sigma_u)

    @cache_readonly
    def _char_mat(self):
        return np.eye(self.neqs) - self.coefs.sum(0)

    def acf(self, nlags=None):
        """Compute theoretical autocovariance function

        Returns
        -------
        acf : ndarray (p x k x k)
        """
        return var_acf(self.coefs, self.sigma_u, nlags=nlags)

    def acorr(self, nlags=None):
        """Compute theoretical autocorrelation function

        Returns
        -------
        acorr : ndarray (p x k x k)
        """
        return util.acf_to_acorr(self.acf(nlags=nlags))

    def plot_acorr(self, nlags=10, linewidth=8):
        "Plot theoretical autocorrelation function"
        plotting.plot_full_acorr(self.acorr(nlags=nlags), linewidth=linewidth)

    def forecast(self, y, steps):
        """Produce linear minimum MSE forecasts for desired number of steps
        ahead, using prior values y

        Parameters
        ----------
        y : ndarray (p x k)
        steps : int

        Returns
        -------
        forecasts : ndarray (steps x neqs)

        Notes
        -----
        Lutkepohl pp 37-38
        """
        return forecast(y, self.coefs, self.intercept, steps)

    def mse(self, steps):
        """
        Compute theoretical forecast error variance matrices

        Parameters
        ----------
        steps : int
            Number of steps ahead

        Notes
        -----
        .. math:: \mathrm{MSE}(h) = \sum_{i=0}^{h-1} \Phi \Sigma_u \Phi^T

        Returns
        -------
        forc_covs : ndarray (steps x neqs x neqs)
        """
        ma_coefs = self.ma_rep(steps)

        k = len(self.sigma_u)
        forc_covs = np.zeros((steps, k, k))

        prior = np.zeros((k, k))
        for h in range(steps):
            # Sigma(h) = Sigma(h-1) + Phi Sig_u Phi'
            phi = ma_coefs[h]
            var = chain_dot(phi, self.sigma_u, phi.T)
            forc_covs[h] = prior = prior + var

        return forc_covs

    forecast_cov = mse

    def _forecast_vars(self, steps):
        covs = self.forecast_cov(steps)

        # Take diagonal for each cov
        inds = np.arange(self.neqs)
        return covs[:, inds, inds]

    def forecast_interval(self, y, steps, alpha=0.05):
        """Construct forecast interval estimates assuming the y are Gaussian

        Parameters
        ----------

        Notes
        -----
        Lutkepohl pp. 39-40

        Returns
        -------
        (lower, mid, upper) : (ndarray, ndarray, ndarray)
        """
        assert(0 < alpha < 1)
        q = util.norm_signif_level(alpha)

        point_forecast = self.forecast(y, steps)
        sigma = np.sqrt(self._forecast_vars(steps))

        forc_lower = point_forecast - q * sigma
        forc_upper = point_forecast + q * sigma

        return point_forecast, forc_lower, forc_upper


#-------------------------------------------------------------------------------
# VARResults class


class VARResults(VARProcess):
    """Estimate VAR(p) process with fixed number of lags

    Parameters
    ----------
    endog : array
    endog_lagged : array
    params : array
    sigma_u : array
    lag_order : int
    model : VAR model instance
    trend : str {'nc', 'c', 'ct'}
    names : array-like
        List of names of the endogenous variables in order of appearance in `endog`.
    dates


    Returns
    -------
    **Attributes**
    aic
    bic
    bse
    coefs : ndarray (p x K x K)
        Estimated A_i matrices, A_i = coefs[i-1]
    cov_params
    dates
    detomega
    df_model : int
    df_resid : int
    endog
    endog_lagged
    fittedvalues
    fpe
    intercept
    info_criteria
    k_ar : int
    k_trend : int
    llf
    model
    names
    neqs : int
        Number of variables (equations)
    nobs : int
    n_totobs : int
    params
    k_ar : int
        Order of VAR process
    params : ndarray (Kp + 1) x K
        A_i matrices and intercept in stacked form [int A_1 ... A_p]
    pvalues
    names : list
        variables names
    resid
    roots : array
        The roots of the VAR process are the solution to
        (I - coefs[0]*z - coefs[1]*z**2 ... - coefs[p-1]*z**k_ar) = 0.
        Note that the inverse roots are returned, and stability requires that
        the roots lie outside the unit circle.
    sigma_u : ndarray (K x K)
        Estimate of white noise process variance Var[u_t]
    sigma_u_mle
    stderr
    trenorder
    tvalues
    y :
    ys_lagged
    """
    _model_type = 'VAR'

    def __init__(self, endog, endog_lagged, params, sigma_u, lag_order,
                 model=None, trend='c', names=None, dates=None):

        self.model = model
        self.y = self.endog = endog  #keep alias for now
        self.ys_lagged = self.endog_lagged = endog_lagged #keep alias for now
        self.dates = dates

        self.n_totobs, neqs = self.y.shape
        self.nobs = self.n_totobs - lag_order
        k_trend = util.get_trendorder(trend)
        if k_trend > 0: # make this the polynomial trend order
            trendorder = k_trend - 1
        else:
            trendorder = None
        self.k_trend = k_trend
        self.trendorder = trendorder
        self.exog_names = util.make_lag_names(names, lag_order, k_trend)
        self.params = params

        # Initialize VARProcess parent class
        # construct coefficient matrices
        # Each matrix needs to be transposed
        reshaped = self.params[self.k_trend:]
        reshaped = reshaped.reshape((lag_order, neqs, neqs))

        # Need to transpose each coefficient matrix
        intercept = self.params[0]
        coefs = reshaped.swapaxes(1, 2).copy()

        super(VARResults, self).__init__(coefs, intercept, sigma_u, names=names)

    def plot(self):
        """Plot input time series
        """
        plotting.plot_mts(self.y, names=self.names, index=self.dates)

    @property
    def df_model(self):
        """Number of estimated parameters, including the intercept / trends
        """
        return self.neqs * self.k_ar + self.k_trend

    @property
    def df_resid(self):
        "Number of observations minus number of estimated parameters"
        return self.nobs - self.df_model

    @cache_readonly
    def fittedvalues(self):
        """The predicted insample values of the response variables of the model.
        """
        return np.dot(self.ys_lagged, self.params)

    @cache_readonly
    def resid(self):
        """Residuals of response variable resulting from estimated coefficients
        """
        return self.y[self.k_ar:] - self.fittedvalues

    def sample_acov(self, nlags=1):
        return _compute_acov(self.y[self.k_ar:], nlags=nlags)

    def sample_acorr(self, nlags=1):
        acovs = self.sample_acov(nlags=nlags)
        return _acovs_to_acorrs(acovs)

    def plot_sample_acorr(self, nlags=10, linewidth=8):
        "Plot theoretical autocorrelation function"
        plotting.plot_full_acorr(self.sample_acorr(nlags=nlags),
                                 linewidth=linewidth)

    def resid_acov(self, nlags=1):
        """
        Compute centered sample autocovariance (including lag 0)

        Parameters
        ----------
        nlags : int

        Returns
        -------
        """
        return _compute_acov(self.resid, nlags=nlags)

    def resid_acorr(self, nlags=1):
        """
        Compute sample autocorrelation (including lag 0)

        Parameters
        ----------
        nlags : int

        Returns
        -------
        """
        acovs = self.resid_acov(nlags=nlags)
        return _acovs_to_acorrs(acovs)

    @cache_readonly
    def resid_corr(self):
        "Centered residual correlation matrix"
        return self.resid_acorr(0)[0]

    @cache_readonly
    def sigma_u_mle(self):
        """(Biased) maximum likelihood estimate of noise process covariance
        """
        return self.sigma_u * self.df_resid / self.nobs

    @cache_readonly
    def cov_params(self):
        """Estimated variance-covariance of model coefficients

        Notes
        -----
        Covariance of vec(B), where B is the matrix
        [intercept, A_1, ..., A_p] (K x (Kp + 1))
        Adjusted to be an unbiased estimator
        Ref: Lutkepohl p.74-75
        """
        z = self.ys_lagged
        return np.kron(L.inv(np.dot(z.T, z)), self.sigma_u)

    def cov_ybar(self):
        r"""Asymptotically consistent estimate of covariance of the sample mean

        .. math::

            \sqrt(T) (\bar{y} - \mu) \rightarrow {\cal N}(0, \Sigma_{\bar{y}})\\

            \Sigma_{\bar{y}} = B \Sigma_u B^\prime, \text{where } B = (I_K - A_1
            - \cdots - A_p)^{-1}

        Notes
        -----
        Lutkepohl Proposition 3.3
        """

        Ainv = L.inv(np.eye(self.neqs) - self.coefs.sum(0))
        return chain_dot(Ainv, self.sigma_u, Ainv.T)

#------------------------------------------------------------
# Estimation-related things

    @cache_readonly
    def _zz(self):
        # Z'Z
        return np.dot(self.ys_lagged.T, self.ys_lagged)

    @property
    def _cov_alpha(self):
        """
        Estimated covariance matrix of model coefficients ex intercept
        """
        # drop intercept
        return self.cov_params[self.neqs:, self.neqs:]

    @cache_readonly
    def _cov_sigma(self):
        """
        Estimated covariance matrix of vech(sigma_u)
        """
        D_K = tsa.duplication_matrix(self.neqs)
        D_Kinv = npl.pinv(D_K)

        sigxsig = np.kron(self.sigma_u, self.sigma_u)
        return 2 * chain_dot(D_Kinv, sigxsig, D_Kinv.T)

    @cache_readonly
    def llf(self):
        "Compute VAR(p) loglikelihood"
        return var_loglike(self.resid, self.sigma_u_mle, self.nobs)

    @cache_readonly
    def stderr(self):
        """Standard errors of coefficients, reshaped to match in size
        """
        stderr = np.sqrt(np.diag(self.cov_params))
        return stderr.reshape((self.df_model, self.neqs), order='C')

    bse = stderr  # statsmodels interface?

    @cache_readonly
    def tvalues(self):
        """Compute t-statistics. Use Student-t(T - Kp - 1) = t(df_resid) to test
        significance.
        """
        return self.params / self.stderr

    @cache_readonly
    def pvalues(self):
        """Two-sided p-values for model coefficients from Student t-distribution
        """
        return stats.t.sf(np.abs(self.tvalues), self.df_resid)*2

    def plot_forecast(self, steps, alpha=0.05, plot_stderr=True):
        """
        Plot forecast
        """
        mid, lower, upper = self.forecast_interval(self.y[-self.k_ar:], steps,
                                                   alpha=alpha)
        plotting.plot_var_forc(self.y, mid, lower, upper, names=self.names,
                               plot_stderr=plot_stderr)

    # Forecast error covariance functions

    def forecast_cov(self, steps=1):
        r"""Compute forecast covariance matrices for desired number of steps

        Parameters
        ----------
        steps : int

        Notes
        -----
        .. math:: \Sigma_{\hat y}(h) = \Sigma_y(h) + \Omega(h) / T

        Ref: Lutkepohl pp. 96-97

        Returns
        -------
        covs : ndarray (steps x k x k)
        """
        mse = self.mse(steps)
        omegas = self._omega_forc_cov(steps)
        return mse + omegas / self.nobs

    #Monte Carlo irf standard errors
    def irf_errband_mc(self, orth=False, repl=1000, T=10,
                       signif=0.05, seed=None, burn=100, cum=False):
        """
        Compute Monte Carlo integrated error bands assuming normally
        distributed for impulse response functions

        Parameters
        ----------
        orth: bool, default False
            Compute orthoganalized impulse response error bands
        repl: int
            number of Monte Carlo replications to perform
        T: int, default 10
            number of impulse response periods
        signif: float (0 < signif <1)
            Significance level for error bars, defaults to 95% CI
        seed: int
            np.random.seed for replications
        burn: int
            number of initial observations to discard for simulation
        cum: bool, default False
            produce cumulative irf error bands

        Notes
        -----
        Lutkepohl (2005) Appendix D

        Returns
        -------
        Tuple of lower and upper arrays of ma_rep monte carlo standard errors

        """
        neqs = self.neqs
        mean = self.mean()
        k_ar = self.k_ar
        coefs = self.coefs
        sigma_u = self.sigma_u
        intercept = self.intercept
        df_model = self.df_model
        nobs = self.nobs

        ma_coll = np.zeros((repl, T+1, neqs, neqs))

        if (orth == True and cum == True):
            fill_coll = lambda sim : VAR(sim).fit(maxlags=k_ar).\
                              orth_ma_rep(maxn=T).cumsum(axis=0)
        elif (orth == True and cum == False):
            fill_coll = lambda sim : VAR(sim).fit(maxlags=k_ar).\
                              orth_ma_rep(maxn=T)
        elif (orth == False and cum == True):
            fill_coll = lambda sim : VAR(sim).fit(maxlags=k_ar).\
                              ma_rep(maxn=T).cumsum(axis=0)
        elif (orth == False and cum == False):
            fill_coll = lambda sim : VAR(sim).fit(maxlags=k_ar).\
                              ma_rep(maxn=T)

        for i in range(repl):
            #discard first hundred to eliminate correct for starting bias
            sim = util.varsim(coefs, intercept, sigma_u, steps=nobs+burn)
            sim = sim[burn:]
            ma_coll[i,:,:,:] = fill_coll(sim)

        ma_sort = np.sort(ma_coll, axis=0) #sort to get quantiles
        index = round(signif/2*repl)-1,round((1-signif/2)*repl)-1
        lower = ma_sort[index[0],:, :, :]
        upper = ma_sort[index[1],:, :, :]
        return lower, upper

    def irf_resim(self, orth=False, repl=1000, T=10,
                      seed=None, burn=100, cum=False):

        """
        Simulates impulse response function, returning an array of simulations.
        Used for Sims-Zha error band calculation.

        Parameters
        ----------
        orth: bool, default False
            Compute orthoganalized impulse response error bands
        repl: int
            number of Monte Carlo replications to perform
        T: int, default 10
            number of impulse response periods
        signif: float (0 < signif <1)
            Significance level for error bars, defaults to 95% CI
        seed: int
            np.random.seed for replications
        burn: int
            number of initial observations to discard for simulation
        cum: bool, default False
            produce cumulative irf error bands

        Notes
        -----
        Sims, Christoper A., and Tao Zha. 1999. "Error Bands for Impulse Response." Econometrica 67: 1113-1155.

        Returns
        -------
        Array of simulated impulse response functions

        """
        neqs = self.neqs
        mean = self.mean()
        k_ar = self.k_ar
        coefs = self.coefs
        sigma_u = self.sigma_u
        intercept = self.intercept
        df_model = self.df_model
        nobs = self.nobs
        if seed is not None:
            np.random.seed(seed=seed)

        ma_coll = np.zeros((repl, T+1, neqs, neqs))

        if (orth == True and cum == True):
            fill_coll = lambda sim : VAR(sim).fit(maxlags=k_ar).\
                              orth_ma_rep(maxn=T).cumsum(axis=0)
        elif (orth == True and cum == False):
            fill_coll = lambda sim : VAR(sim).fit(maxlags=k_ar).\
                              orth_ma_rep(maxn=T)
        elif (orth == False and cum == True):
            fill_coll = lambda sim : VAR(sim).fit(maxlags=k_ar).\
                              ma_rep(maxn=T).cumsum(axis=0)
        elif (orth == False and cum == False):
            fill_coll = lambda sim : VAR(sim).fit(maxlags=k_ar).\
                              ma_rep(maxn=T)

        for i in range(repl):
            #discard first hundred to eliminate correct for starting bias
            sim = util.varsim(coefs, intercept, sigma_u, steps=nobs+burn)
            sim = sim[burn:]
            ma_coll[i,:,:,:] = fill_coll(sim)

        return ma_coll


    def _omega_forc_cov(self, steps):
        # Approximate MSE matrix \Omega(h) as defined in Lut p97
        G = self._zz
        Ginv = L.inv(G)

        # memoize powers of B for speedup
        # TODO: see if can memoize better
        B = self._bmat_forc_cov()
        _B = {}
        def bpow(i):
            if i not in _B:
                _B[i] = np.linalg.matrix_power(B, i)

            return _B[i]

        phis = self.ma_rep(steps)
        sig_u = self.sigma_u

        omegas = np.zeros((steps, self.neqs, self.neqs))
        for h in range(1, steps + 1):
            if h == 1:
                omegas[h-1] = self.df_model * self.sigma_u
                continue

            om = omegas[h-1]
            for i in range(h):
                for j in range(h):
                    Bi = bpow(h - 1 - i)
                    Bj = bpow(h - 1 - j)
                    mult = np.trace(chain_dot(Bi.T, Ginv, Bj, G))
                    om += mult * chain_dot(phis[i], sig_u, phis[j].T)
            omegas[h-1] = om

        return omegas

    def _bmat_forc_cov(self):
        # B as defined on p. 96 of Lut
        upper = np.zeros((1, self.df_model))
        upper[0,0] = 1

        lower_dim = self.neqs * (self.k_ar - 1)
        I = np.eye(lower_dim)
        lower = np.column_stack((np.zeros((lower_dim, 1)), I,
                                 np.zeros((lower_dim, self.neqs))))

        return np.vstack((upper, self.params.T, lower))

    def summary(self):
        """Compute console output summary of estimates

        Returns
        -------
        summary : VARSummary
        """
        return VARSummary(self)

    def irf(self, periods=10, var_decomp=None, var_order=None):
        """Analyze impulse responses to shocks in system

        Parameters
        ----------
        periods : int
        var_decomp : ndarray (k x k), lower triangular
            Must satisfy Omega = P P', where P is the passed matrix. Defaults to
            Cholesky decomposition of Omega
        var_order : sequence
            Alternate variable order for Cholesky decomposition

        Returns
        -------
        irf : IRAnalysis
        """
        if var_order is not None:
            raise NotImplementedError('alternate variable order not implemented'
                                      ' (yet)')

        return IRAnalysis(self, P=var_decomp, periods=periods)

    def fevd(self, periods=10, var_decomp=None):
        """
        Compute forecast error variance decomposition ("fevd")

        Returns
        -------
        fevd : FEVD instance
        """
        return FEVD(self, P=var_decomp, periods=periods)

    def reorder(self, order):
        """Reorder variables for structural specification
        """
        if len(order) != len(self.params[0,:]):
            raise ValueError("Reorder specification length should match number of endogenous variables")
       #This convert order to list of integers if given as strings
        if isinstance(order[0], string_types):
            order_new = []
            for i, nam in enumerate(order):
                order_new.append(self.names.index(order[i]))
            order = order_new
        return _reordered(self, order)

#-------------------------------------------------------------------------------
# VAR Diagnostics: Granger-causality, whiteness of residuals, normality, etc.

    def test_causality(self, equation, variables, kind='f', signif=0.05,
                       verbose=True):
        """Compute test statistic for null hypothesis of Granger-noncausality,
        general function to test joint Granger-causality of multiple variables

        Parameters
        ----------
        equation : string or int
            Equation to test for causality
        variables : sequence (of strings or ints)
            List, tuple, etc. of variables to test for Granger-causality
        kind : {'f', 'wald'}
            Perform F-test or Wald (chi-sq) test
        signif : float, default 5%
            Significance level for computing critical values for test,
            defaulting to standard 0.95 level

        Notes
        -----
        Null hypothesis is that there is no Granger-causality for the indicated
        variables. The degrees of freedom in the F-test are based on the
        number of variables in the VAR system, that is, degrees of freedom
        are equal to the number of equations in the VAR times degree of freedom
        of a single equation.

        Returns
        -------
        results : dict
        """
        if isinstance(variables, (string_types, int, np.integer)):
            variables = [variables]

        k, p = self.neqs, self.k_ar

        # number of restrictions
        N = len(variables) * self.k_ar

        # Make restriction matrix
        C = np.zeros((N, k ** 2 * p + k), dtype=float)

        eq_index = self.get_eq_index(equation)
        vinds = mat([self.get_eq_index(v) for v in variables])

        # remember, vec is column order!
        offsets = np.concatenate([k + k ** 2 * j + k * vinds + eq_index
                                  for j in range(p)])
        C[np.arange(N), offsets] = 1

        # Lutkepohl 3.6.5
        Cb = np.dot(C, vec(self.params.T))
        middle = L.inv(chain_dot(C, self.cov_params, C.T))

        # wald statistic
        lam_wald = statistic = chain_dot(Cb, middle, Cb)

        if kind.lower() == 'wald':
            df = N
            dist = stats.chi2(df)
        elif kind.lower() == 'f':
            statistic = lam_wald / N
            df = (N, k * self.df_resid)
            dist = stats.f(*df)
        else:
            raise Exception('kind %s not recognized' % kind)

        pvalue = dist.sf(statistic)
        crit_value = dist.ppf(1 - signif)

        conclusion = 'fail to reject' if statistic < crit_value else 'reject'
        results = {
            'statistic' : statistic,
            'crit_value' : crit_value,
            'pvalue' : pvalue,
            'df' : df,
            'conclusion' : conclusion,
            'signif' :  signif
        }

        if verbose:
            summ = output.causality_summary(results, variables, equation, kind)

            print(summ)

        return results

    def test_whiteness(self, nlags=10, plot=True, linewidth=8):
        """
        Test white noise assumption. Sample (Y) autocorrelations are compared
        with the standard :math:`2 / \sqrt(T)` bounds.

        Parameters
        ----------
        plot : boolean, default True
            Plot autocorrelations with 2 / sqrt(T) bounds
        """
        acorrs = self.sample_acorr(nlags)
        bound = 2 / np.sqrt(self.nobs)

        # TODO: this probably needs some UI work

        if (np.abs(acorrs) > bound).any():
            print('FAIL: Some autocorrelations exceed %.4f bound. '
                   'See plot' % bound)
        else:
            print('PASS: No autocorrelations exceed %.4f bound' % bound)

        if plot:
            fig = plotting.plot_full_acorr(acorrs[1:],
                                           xlabel=np.arange(1, nlags+1),
                                           err_bound=bound,
                                           linewidth=linewidth)
            fig.suptitle(r"ACF plots with $2 / \sqrt{T}$ bounds "
                         "for testing whiteness assumption")

    def test_normality(self, signif=0.05, verbose=True):
        """
        Test assumption of normal-distributed errors using Jarque-Bera-style
        omnibus Chi^2 test

        Parameters
        ----------
        signif : float
            Test significance threshold

        Notes
        -----
        H0 (null) : data are generated by a Gaussian-distributed process
        """
        Pinv = npl.inv(self._chol_sigma_u)

        w = np.array([np.dot(Pinv, u) for u in self.resid])

        b1 = (w ** 3).sum(0) / self.nobs
        lam_skew = self.nobs * np.dot(b1, b1) / 6

        b2 = (w ** 4).sum(0) / self.nobs - 3
        lam_kurt = self.nobs * np.dot(b2, b2) / 24

        lam_omni = lam_skew + lam_kurt

        omni_dist = stats.chi2(self.neqs * 2)
        omni_pvalue = omni_dist.sf(lam_omni)
        crit_omni = omni_dist.ppf(1 - signif)

        conclusion = 'fail to reject' if lam_omni < crit_omni else 'reject'

        results = {
            'statistic' : lam_omni,
            'crit_value' : crit_omni,
            'pvalue' : omni_pvalue,
            'df' : self.neqs * 2,
            'conclusion' : conclusion,
            'signif' :  signif
        }

        if verbose:
            summ = output.normality_summary(results)
            print(summ)

        return results

    @cache_readonly
    def detomega(self):
        r"""
        Return determinant of white noise covariance with degrees of freedom
        correction:

        .. math::

            \hat \Omega = \frac{T}{T - Kp - 1} \hat \Omega_{\mathrm{MLE}}
        """
        return L.det(self.sigma_u)

    @cache_readonly
    def info_criteria(self):
        "information criteria for lagorder selection"
        nobs = self.nobs
        neqs = self.neqs
        lag_order = self.k_ar
        free_params = lag_order * neqs ** 2 + neqs * self.k_trend

        ld = logdet_symm(self.sigma_u_mle)

        # See Lutkepohl pp. 146-150

        aic = ld + (2. / nobs) * free_params
        bic = ld + (np.log(nobs) / nobs) * free_params
        hqic = ld + (2. * np.log(np.log(nobs)) / nobs) * free_params
        fpe = ((nobs + self.df_model) / self.df_resid) ** neqs * np.exp(ld)

        return {
            'aic' : aic,
            'bic' : bic,
            'hqic' : hqic,
            'fpe' : fpe
            }

    @property
    def aic(self):
        "Akaike information criterion"
        return self.info_criteria['aic']

    @property
    def fpe(self):
        """Final Prediction Error (FPE)

        Lutkepohl p. 147, see info_criteria
        """
        return self.info_criteria['fpe']

    @property
    def hqic(self):
        "Hannan-Quinn criterion"
        return self.info_criteria['hqic']

    @property
    def bic(self):
        "Bayesian a.k.a. Schwarz info criterion"
        return self.info_criteria['bic']

    @cache_readonly
    def roots(self):
        neqs = self.neqs
        k_ar = self.k_ar
        p = neqs * k_ar
        arr = np.zeros((p,p))
        arr[:neqs,:] = np.column_stack(self.coefs)
        arr[neqs:,:-neqs] = np.eye(p-neqs)
        roots = np.linalg.eig(arr)[0]**-1
        idx = np.argsort(np.abs(roots))[::-1] # sort by reverse modulus
        return roots[idx]

class VARResultsWrapper(wrap.ResultsWrapper):
    _attrs = {'bse' : 'columns_eq', 'cov_params' : 'cov',
              'params' : 'columns_eq', 'pvalues' : 'columns_eq',
              'tvalues' : 'columns_eq', 'sigma_u' : 'cov_eq',
              'sigma_u_mle' : 'cov_eq',
              'stderr' : 'columns_eq'}
    _wrap_attrs = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_attrs,
                                    _attrs)
    _methods = {}
    _wrap_methods = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_methods,
                                     _methods)
    _wrap_methods.pop('cov_params') # not yet a method in VARResults
wrap.populate_wrapper(VARResultsWrapper, VARResults)

class FEVD(object):
    """
    Compute and plot Forecast error variance decomposition and asymptotic
    standard errors
    """
    def __init__(self, model, P=None, periods=None):
        self.periods = periods

        self.model = model
        self.neqs = model.neqs
        self.names = model.model.endog_names

        self.irfobj = model.irf(var_decomp=P, periods=periods)
        self.orth_irfs = self.irfobj.orth_irfs

        # cumulative impulse responses
        irfs = (self.orth_irfs[:periods] ** 2).cumsum(axis=0)

        rng = lrange(self.neqs)
        mse = self.model.mse(periods)[:, rng, rng]

        # lag x equation x component
        fevd = np.empty_like(irfs)

        for i in range(periods):
            fevd[i] = (irfs[i].T / mse[i]).T

        # switch to equation x lag x component
        self.decomp = fevd.swapaxes(0, 1)

    def summary(self):
        buf = StringIO()

        rng = lrange(self.periods)
        for i in range(self.neqs):
            ppm = output.pprint_matrix(self.decomp[i], rng, self.names)

            buf.write('FEVD for %s\n' % self.names[i])
            buf.write(ppm + '\n')

        print(buf.getvalue())

    def cov(self):
        """Compute asymptotic standard errors

        Returns
        -------
        """
        raise NotImplementedError

    def plot(self, periods=None, figsize=(10,10), **plot_kwds):
        """Plot graphical display of FEVD

        Parameters
        ----------
        periods : int, default None
            Defaults to number originally specified. Can be at most that number
        """
        import matplotlib.pyplot as plt

        k = self.neqs
        periods = periods or self.periods

        fig, axes = plt.subplots(nrows=k, figsize=figsize)

        fig.suptitle('Forecast error variance decomposition (FEVD)')

        colors = [str(c) for c in np.arange(k, dtype=float) / k]
        ticks = np.arange(periods)

        limits = self.decomp.cumsum(2)

        for i in range(k):
            ax = axes[i]

            this_limits = limits[i].T

            handles = []

            for j in range(k):
                lower = this_limits[j - 1] if j > 0 else 0
                upper = this_limits[j]
                handle = ax.bar(ticks, upper - lower, bottom=lower,
                                color=colors[j], label=self.names[j],
                                **plot_kwds)

                handles.append(handle)

            ax.set_title(self.names[i])

        # just use the last axis to get handles for plotting
        handles, labels = ax.get_legend_handles_labels()
        fig.legend(handles, labels, loc='upper right')
        plotting.adjust_subplots(right=0.85)

#-------------------------------------------------------------------------------

def _compute_acov(x, nlags=1):
    x = x - x.mean(0)

    result = []
    for lag in range(nlags + 1):
        if lag > 0:
            r = np.dot(x[lag:].T, x[:-lag])
        else:
            r = np.dot(x.T, x)

        result.append(r)

    return np.array(result) / len(x)

def _acovs_to_acorrs(acovs):
    sd = np.sqrt(np.diag(acovs[0]))
    return acovs / np.outer(sd, sd)

if __name__ == '__main__':
    import statsmodels.api as sm
    from statsmodels.tsa.vector_ar.util import parse_lutkepohl_data
    import statsmodels.tools.data as data_util

    np.set_printoptions(linewidth=140, precision=5)

    sdata, dates = parse_lutkepohl_data('data/%s.dat' % 'e1')

    names = sdata.dtype.names
    data = data_util.struct_to_ndarray(sdata)
    adj_data = np.diff(np.log(data), axis=0)
    # est = VAR(adj_data, p=2, dates=dates[1:], names=names)
    model = VAR(adj_data[:-16], dates=dates[1:-16], names=names)
    # model = VAR(adj_data[:-16], dates=dates[1:-16], names=names)

    est = model.fit(maxlags=2)
    irf = est.irf()

    y = est.y[-2:]
    """
    # irf.plot_irf()

    # i = 2; j = 1
    # cv = irf.cum_effect_cov(orth=True)
    # print np.sqrt(cv[:, j * 3 + i, j * 3 + i]) / 1e-2

    # data = np.genfromtxt('Canada.csv', delimiter=',', names=True)
    # data = data.view((float, 4))
    """

    '''
    mdata = sm.datasets.macrodata.load().data
    mdata2 = mdata[['realgdp','realcons','realinv']]
    names = mdata2.dtype.names
    data = mdata2.view((float,3))
    data = np.diff(np.log(data), axis=0)

    import pandas as pn
    df = pn.DataFrame.fromRecords(mdata)
    df = np.log(df.reindex(columns=names))
    df = (df - df.shift(1)).dropna()

    model = VAR(df)
    est = model.fit(maxlags=2)
    irf = est.irf()
    '''

