RDP 2013-11: Issues in Estimating New-Keynesian Phillips Curves in the Presence of Unknown Structural Change 3. A Simple Variant of the Solution Algorithm

Kulish and Pagan (2012) set out an algorithm that can be used to compute solutions to a system in which there are changes in structural parameters that are potentially foreseen. It has more general application than the context we are working with here, so we provide a simplified discussion of its workings to highlight some of its significant features.

The system we consider has the format:

where z_t is a vector of n (=3) variables (π_t, x_t, r_t), c_t are the intercepts in the equations, and ε_t are identically and independently distributed (iid) (0,σ²) shocks. It is necessary to allow agents to have different beliefs about the timing of any breaks than is the case in reality. To study the effects of mean breaks we assume that agents always know the true shocks and the parameters A and B of the system, and that their beliefs may only be incorrect about c_t. Agents will be assumed to believe that the intercepts of the three equations at time t have the values rather than c_t. Then agents solve the system in Equation (4) (with replacing c_t) to form their expectations , where the ‘a’ indexes the agent's beliefs. Those expectations will then determine the actual outcomes of the system, i.e. the observed variables will be consistent with:

We adopt the Binder and Pesaran (1995) solution method to solve the system. Briefly, this involves converting Equation (4) into a purely forward-looking form, solving that, and then recovering the solution to the original system. In the case that agents believe that the economy is described by Equation (4), with the solution method comes down to solving

where Z_t = z_t − Pz_t−1, S = (I − BP)⁻¹B, Q = (I − BP)⁻¹T and P is chosen such that (A+BP²−P) = 0. Because shocks are iid, the solution will be

Now we can write the complete solution in one of two forms – either

or

In the first case, agents compute the weights to be applied to z_t−1 from what they believe the data-generating process (DGP) for the macroeconomic variables is, i.e. actual past outcomes are used in forming expectations. In the second, they solve for the complete path that they would have expected if the DGP was from the model compatible with their beliefs. The former seems more reasonable as the latter implies that agents persist in believing in a path even when they can observe persistent departures from it. Up until the break there will be no difference, but thereafter, unless the agents know exactly when the break occurs, there will be a systematic difference between z_t−1 and , since z_t will adjust to the structural changes when they happen. In ‘control’ jargon the path is often called the ‘open loop controller’ while that based on z_t−1 is the ‘closed loop controller’. Our algorithm will allow for expectations formation based on either z_t−1 or .

In the first case, expectations formation would follow

and the values of z_t that are generated by the model economy will then be

The solution for the second case is analogous to this.