Q2. What is the result of the second-order approximation of the steady state y?
As σ goes from 0 to 1 and the authors transition from the certain to uncertain model, the rest point of the solution transitions from the nonstochastic steady state y to the second-order approximation of the stochastic steady state y+ 12yσ2σ
Q3. What is the vech operator used to unfold a second partial derivative?
They use the matrix derivative structure and the associated chain rule of Magnus and Neudecker (2007, ch. 6) to unfold a three dimensional cube of second partial derivatives.
Q4. What is the idea of the Euler equation accuracy test in the neoclassical?
From Judd (1992), the idea of the Euler equation accuracy test in the neoclassical growth model is to find a unit-free measure that expresses the one-period optimization error in relation to current consumption.
Q5. What is the simplest way to solve the formyk derivative?
The unique stable solution of the forgoing, analogously to lower orders, takes the formyk, j,i = αyk−1, j−1,i−1 +β3Sk, j,i, with yk, j,i = 0, for k, j, i < 0(40) and β3 can be solved for by, again, formulating an appropriate Sylvester equation.
Q6. What is the significance of the nonlinear component in the baseline neoclassical?
Although there are a number of DSGE models and applications, i.e., welfare analysis, asset pricing and stochastic volatility for which the importance of nonlinear components and uncertainty in the policy function has been proved, the nonlinear components the authors analyzed in the baseline neoclassical growth model are quantitatively unimportant, this is not surprising as the model is known to be nearly linear.
Q7. What is the order in which derivatives with respect to appear?
Accordingly for higher-order derivatives, the order in which derivatives with respect to σ appear is inconsequential as it is a scalar and the authors choose to have the σ’s appear first.
Q8. What is the rewritten function u of the exogenous variable?
The known function u of the exogenous variable is rewritten similarlyut = u(σ,εt,εt−1, . . .) = ∞∑ i=0 Niεt−i(5)For notational ease in derivation, the authors will define vector xt , containing the complete set of variablesxt ≡ [ y′t−1 y ′ t y ′ t+1 u ′ t ]′ (6) xt is of dimension (nx×1) with (nx = 3ny+ne).
Q9. What is the central component of higher-order impulse responses?
Figure 5 highlights a central component of higher-order impulse responses: the break down of superposition or history dependence of the transfer function.
Q10. What is the average error of the second order approximations?
The second order approximations show an improvement as the horizon increases, whereas the third order approximations tend to be lower at first, rise and then fall again.
Q11. What is the difference between the two nonlinear moving average solutions?
Applying the techniques developed in Lan and Meyer-Gohde (2011), the existence and uniqueness16Thus, their nonlinear moving average solution parallels nonlinear state space solutions in a manner analogous to the linear case, where the recursion is in the coefficients as opposed to the variables themselves.
Q12. What is the shape of the kernels perpendicular to the diagonal?
The shapes of the kernels perpendicular to the diagonal have direct analogs in polynomials: on either side of the diagonal of figures 3a and 3b, the shape is reminiscent of the parabola of a quadratic26equation and the ‘s’ shape of the cubic equation can be found on either side of the diagonal of figure 4.
Q13. What is the potential for explosive behavior in the simulation of state-space perturbations?
The potential for explosive behavior in the simulation of state-space perturbations has lead to31the adaptation of ‘pruning’ algorithms, see Kim, Kim, Schaumburg, and Sims (2008), that appear ad-hoc relative to the perturbation solution itself.