We know that Dynare uses another notation of state equations than the canonical notation used by Schmitt-Grohé and Uribe (2004) (SGU hereafter). I use the first-order linear equations as an example to explain the notation below.

Given the x_t as the vector of state variables determined at t and a_t as the TFP shock, SGU does not explicitly note the innovation of shocks u_t yet puts a_{t+1} into x_t and writes the production function in time t as Y_t=A_tL_t. Therefore the first-order linear equations work as

and the policy functions will be a_{t+1}=h_xa_t and y_t=g_xa_t (the true mapping should be on x_{t-1} but I use a_t for convenience). Therefore the innovation u_t is just an exogenous add-on to a_t.

Dynare uses another notation by putting a_t into x_t and the production function is still Y_t=A_{t}L_t. Therefore the first-order linear equations work as

and the policy functions will be a_{t}=h_xa_{t-1}+q_x\varepsilon_{t} and y_t=g_xa_{t-1}+w_x\varepsilon_{t} (dynare actually uses A\left[\begin{array}{c} E_{t}y_{t+1}\\ E_{t}a_{t+1} \end{array}\right]+B\left[\begin{array}{c} y_{t}\\ a_{t} \end{array}\right]+C\left[\begin{array}{c} y_{t-1}\\ a_{t-1} \end{array}\right]+D\varepsilon_{t}=0 but there is no fundamental difference)

The main difference between these two notations, 1 and 2, are the extra policy term q_x and w_x. The question now should be, how to solve them. For the first-order case, we have some papers such as Klein(2000) or Villemot(2011) who use Dynare’s notation and we can ignore SGU at all! However, for the second-order case, all the literature, such as SGU(2004), Kim et(2008) or Klein et(2011), use the SGU’s notation.

So **my question is how dynare separates or solves the q_x, w_x (and their higher order terms) out of SGU’s notation and solution method**?

My first conjecture to solve this problem is auxiliary variables. For instance, change the equation 1 to

where m_t is the auxiliary variable. It enters into the law of motion of TFP as a_t=\rho a_{t-1}+m_t and its own law of motion is m_{t+1}=0. So the column corresponds to m in the policy function h_x and g_x is what we want, q_x and w_x. In second-order case it is similar. Whereas this method is too tedious and I don’t believe Dynare have used it. The slides Computation first and second approximations verify my concern and Dynare, in fact, solves the response to contemporaneous innovation manually and directly instead of using SGU’s notation (to solve the problem first and then transfer it back to dynare’s notation).