============================================================================== Structural Vector Autoregression ============================================================================== Definition ============================================================================== A :math:`\smash{p}` th order structural vector autoregression generalizes has the form: .. math:: \smash{B_0\boldsymbol{Y}_{t} = \boldsymbol{k} + B_{1}\boldsymbol{Y}_{t-1} + B_{2}\boldsymbol{Y}_{t-2} + \ldots + + B_{p}\boldsymbol{Y}_{t-p} + \boldsymbol{u}_{t}}. .. raw:: - :math:`\smash{Y_{t} = (Y_{1t},Y_{2t},\ldots,Y_{nt})^{'}}` is an :math:`\smash{n \times 1}` vector of random variables. .. raw:: - :math:`\smash{\boldsymbol{k} = (k_1,k_2,\ldots,k_n)^{'}}` is an :math:`\smash{n \times 1}` vector of constants. .. raw:: - :math:`\smash{B_{j}}` is an :math:`\smash{n \times n}` matrix of autoregressive coefficients for :math:`\smash{j = 0,\ldots,p}`. .. raw:: - :math:`\smash{\boldsymbol{u}_{t} = (u_{1t},\ldots, u_{nt})^{'} }` is a vector white noise process: .. math:: \smash{E[\boldsymbol{u}_{t}] = \boldsymbol{0} \,\,\, \text{and} \,\,\, E[\boldsymbol{u}_{t}\boldsymbol{u}^{'}_{\tau}] = \bigg\{\begin{array}{c} U \hspace{10pt} t = \tau \\ 0 \hspace{10pt} \text{o/w} \\ \end{array}}. .. raw:: - :math:`\smash{\boldsymbol{Y}_{t}}` is referred to as a :math:`\smash{SVAR(p)}`. Structural Models ============================================================================== :math:`\smash{SVAR}` s .. raw:: - Allow for contemporaneous relationships among the variables in :math:`\smash{Y_{t}}`. - Not only are the elements of :math:`\smash{Y_{it}}` related to :math:`\smash{Y_{1\tau},Y_{2\tau},\ldots,Y_{n\tau}}` for :math:`\smash{\tau = t-1, \ldots,t-p}`, but they are also related to :math:`\smash{Y_{1t},Y_{2t},\ldots,Y_{nt}}`. .. raw:: - Such relationships allow for the expression of theoretical (structural) economic model relationships. Example ============================================================================== Consider the following bivariate model for asset returns, :math:`\smash{r_t}`, and order flow, :math:`\smash{x_t}`: .. math:: \begin{align*} r_t & = \alpha_{x0} x_t + \alpha_{r1} r_{t-1} + \alpha_{x1} x_{t-1} + \varepsilon_{rt} \\ x_t & = \beta_{r1} r_{t-1} + \beta_{x1} x_{t-1} + \varepsilon_{xt}. \end{align*} .. raw:: - :math:`\smash{r_t}` is typically measured as the differences in log prices. .. raw:: - :math:`\smash{x_t}` is signed trade volume: total number of shares traded at the offer (demand), less the total number of shares traded on the bid (supply). .. raw:: - That is, order flow is a measure of net demand or upward price preasure. .. raw:: - Notice that :math:`\smash{r_t}` depends on :math:`\smash{x_t}` and lags of both variables, whereas :math:`\smash{x_t}` only depends on lags. Example ============================================================================== The model can be expressed as .. math:: \begin{align} \left[\begin{array}{cc} 1 & -\alpha_{x0} \\ 0 & 1 \end{array} \right] \left[\begin{array}{c} r_t \\ x_t \end{array} \right] & = \left[\begin{array}{cc} \alpha_{r1} & \alpha_{x1} \\ \beta_{r1} & \beta_{x1} \end{array} \right] \left[\begin{array}{c} r_{t-1} \\ x_{t-1} \end{array} \right] + \left[\begin{array}{c} \varepsilon_{rt} \\ \varepsilon_{xt} \end{array} \right]. \end{align} :math:`\smash{SVAR}` as :math:`\smash{VAR}` ============================================================================== Under the assumption that :math:`\smash{B_0}` is invertible, a :math:`\smash{SVAR}` can be expressed as a traditional :math:`\smash{VAR}`: .. raw:: .. math:: \begin{align} \boldsymbol{Y}_{t} & = B_0^{-1} \boldsymbol{k} + B_0^{-1} B_{1}\boldsymbol{Y}_{t-1} + B_0^{-1} B_{2}\boldsymbol{Y}_{t-2} + \ldots + B_0^{-1} B_{p}\boldsymbol{Y}_{t-p} + B_0^{-1} \boldsymbol{u}_{t} \\ & = \boldsymbol{c} + \Phi_{1}\boldsymbol{Y}_{t-1} + \Phi_{2}\boldsymbol{Y}_{t-2} + \ldots + \Phi_{p}\boldsymbol{Y}_{t-p} + \boldsymbol{\varepsilon}_{t}. \end{align} .. raw:: - This is the reduced form representation of the :math:`\smash{SVAR}`. :math:`\smash{VAR}` as :math:`\smash{SVAR}` ============================================================================== Recall that a :math:`\smash{VAR(p)}` can be expressed in terms of orthoganalized errors via its :math:`\smash{VMA(\infty)}` representation: .. math:: \begin{align*} \boldsymbol{Y}_{t} & = \boldsymbol{\mu} + H^{-1}H\boldsymbol{\varepsilon}_{t} + \Psi_{1}(H^{-1}H)\boldsymbol{\varepsilon}_{t-1} + \ldots \\ & = \boldsymbol{\mu} + J_{0}\boldsymbol{u}_{t} + J_{1}\boldsymbol{u}_{t-1} + J_{2}\boldsymbol{u}_{t-2} + \ldots \end{align*} .. raw:: - :math:`\smash{H}` is the matrix such that :math:`\smash{\,\,\,\,H \Omega H^{'} = D}`, where :math:`\smash{\text{E}[\varepsilon_t \varepsilon_t^{\prime}] = \Omega}`. .. raw:: Consider the system .. math:: \smash{H \boldsymbol{Y}_{t} = H \boldsymbol{c} + H \Phi_{1}\boldsymbol{Y}_{t-1} + H \Phi_{2}\boldsymbol{Y}_{t-2} + \ldots + H \Phi_{p}\boldsymbol{Y}_{t-p} + H \boldsymbol{\varepsilon}_{t}}. .. raw:: - This is a :math:`\smash{SVAR}` with :math:`\smash{B_0 = H}`. :math:`\smash{B_i = H \Phi_i}` and orthogonal errors :math:`\smash{\boldsymbol{u}_t = H \boldsymbol{\varepsilon}_t}`. Identification ============================================================================== A :math:`\smash{SVAR}` is identified if the number of parameters in :math:`\smash{B_0}` and :math:`\smash{D}` are equal to those of :math:`\smash{\Omega}`. .. raw:: - Since :math:`\smash{D}` is diagonal (:math:`\smash{n}` parameters), :math:`\smash{B_0}` can have at most :math:`\smash{n(n-1)/2}` parameters. .. raw:: - For example, this condition is satisfied if :math:`\smash{B_0}` is lower triangular (with ones on the diagonal).