Evaluating linear approximations in a two-country model with occasionally binding borrowing constraints

C.1 Static Euler errors

Euler errors, as suggested by Judd (1992) and described in more detail in den Haan (2010), are a widely used measure of the accuracy of numerical solutions of dynamic models. The policy iteration algorithm delivers policy functions which satisfy the Euler equations (up to the tolerance level used for algorithm convergence, in our case 10^–11) at the points in the state space chosen during the discretization procedure. The idea is to check the size of the approximation error at other points in the state space. We look at the midpoints lying in between discretization points and calculate the Euler equation errors at these points. Specifically, at each point we use our (linearly interpolated) policy functions to obtain “actual” values for current consumption, bonds and prices. We then compare these to the “implied” values obtained by directly solving the non-linear equations (3)–(8) for c_it, b_1t, p_t with the conditional expectations calculated using our policy functions.^[20] Santos (2000) shows that these Euler errors can be used to evaluate the accuracy of the solution since their size is closely related to the distance of the approximated policy functions to the true solution.^[21] For each variable x=c₁, c₂, b₁, p, we compute the absolute difference between actual values x^A and implied values x^I at each point considered. We express this difference as a percentage of x^I for x=c₁, c₂, p and as a fraction of mean income for bonds b, and denote them by Δx

(C7)Δx≡100|xA−xIxI| for x=c1, c2, pΔb≡100|bA−bI| (C7)

Tables 6 and 7 reports the Euler errors for the benchmark calibration.

The maximum and the average values of these are reported in Table 6. Euler errors are of similar order of magnitude for the three variables, so we focus on consumption. At most points in the state space Euler errors are very small, as indicated by the average being 1.8×10^–5%. Euler errors are larger close to the limits, with the maximum at 0.0077% of consumption. Overall, we find the errors to be reasonably small and, importantly, several orders of magnitude smaller than the differences between linear and policy iteration solutions reported in the previous section. This provides some confidence that the differences reported in section 4.3 of the main text are not simply due to large approximation errors in the policy iteration solution.

We have also computed Euler errors for the linear solution which we present in Table 7. Focusing on consumption, maximum Euler errors are almost 4% of consumption and average errors are 0.53%. These are several orders of magnitude higher than the policy iteration errors. The maximum error is of similar size to the maximum policy difference, but the average errors appear to be smaller than what the policy differences would suggest. Indeed, when we exclude the limit points, the maximum falls to 0.04% of consumption and the average is approximately 0.01% of consumption. Although still significantly larger than the PI solution errors, these could be argued to be reasonably small for a first approximation and they paint a very different picture than what policy differences would suggest. We interpret this as evidence of the importance of the expectations of future binding constraints, even when constraints do not currently bind. Whereas the global approximation method builds this in the conditional expectations, the linear solution does not. When computing standard Euler errors for the linear solution, we compute conditional expectations using the linear policy, i.e., ignoring the possibility of future binding constraints. Under this assumption, current actual decisions are not too far away from the implied ones. But the global solution is actually very different exactly because these expectations are very different. Thus, focusing on Euler errors can be misleading as a measure of the quality of approximation obtained with linear methods for a model with occasionally binding constraints, even if one focuses on points that are away from the limits.

The Euler errors of the solutions under the natural debt limit are presented in Table 8.

C.2 Dynamic Euler errors

den Haan (2010) suggests a second accuracy measure which he terms dynamic Euler errors, as opposed to the static Euler errors computed in the previous section. The idea is to evaluate whether small static Euler errors can accumulate to large errors in a simulation. This is especially relevant for our case, since we know that the linear solution yields non-stationary policy functions so one of the main concerns about its accuracy is exactly this accumulation of errors in a simulation.

To produce a simulated series, we first draw a sequence of realizations for the exogenous income shocks and initialize the series at the steady state values for the state variables. For the “actual” series, we use directly the policy functions in each period to compute values for bonds, consumptions and prices given the exogenous shock. The bond choice at t is then the state variable for t+1 and a simulation is produced recursively. For the “implied” series, we compute current values by evaluating expectations according to the policy function but directly solving the non-linear equilibrium conditions for current bonds, consumptions and prices. This implied value for the bond choice at t is then used as the state variable for t+1 and the simulation is thus produced recursively again. In order to focus on the possibility of future binding constraints we produce many short run simulations as opposed to one very long one. This is intended to give the linear solution the best chance to perform well. In a very long simulation, the unit root in the linear bond law of motion would imply bonds that drift arbitrarily far away from the steady state and, in particular, above the limits. We choose the simulation length to be 50 periods and compute average errors over 20,000 repetitions. The draws of the exogenous shocks are kept the same in the simulations using the linear and global solution to make those directly comparable.

We define Δx as in (C7) and report these dynamic Euler errors for the policy iteration method in the top panel of Table 9. It is important to note that the limits never bind in any of the simulations. This is partly due to the fact that the debt policy function is mean-reverting but also due to the fact that we are looking at short run simulations starting at zero bonds. The errors are thus small because the static Euler errors away from the limits are small and there does not seem to be substantial accumulation of errors in a 50 period simulation. The maximum consumption error occurs when bonds come close to the limit and is 0.0022%. The economy spends most of these simulations far from the limits so the average is very small at 3.74×10^–6%. The corresponding statistics for the linear solution are shown in the bottom panel of Table 9. Despite the short simulation length, in the linear equilibrium debt sometimes reaches the debt limits. In the implied series, this limit is imposed but in the actual series it is not, since the linear policies do not assume such limits. As a result, the maximum error in bonds can be extremely high, indeed it can be arbitrarily high if we increase the simulation length.^[22] For our short simulation experiment, this only happens for <2% of the periods. For consumption, the maximum error is more than 3% of consumption and the average is 0.033%. Consumption errors are significantly smaller than errors in bonds because of a moderating effect from prices. In the implied allocations, when bonds cannot adjust due to the limits, prices increase significantly, sometimes rising even above one. This provides significant consumption smoothing for the borrowing constrained country, keeping the consumption allocations from diverging even further away from the ones in the actual simulation.

To sum up, Euler errors can accumulate significantly for bonds as they drift above the limits, but this does not necessarily translate to large accumulation of errors for consumption, partly because prices adjust to provide some consumption smoothing. Before the limits are reached, actual and implied series stay very close even for the linear solution, indicating small dynamic Euler errors as long as the limits don’t bind in the simulation. Similarly to the previous section, we note that the use of the linear policies to evaluate expectations essentially shuts down any effects from the possibility of future binding constraints. As a result, small dynamic Euler errors for the linear solution does not necessarily imply that the dynamic properties of the linear economy match well with those of the non-linear economy with constraints. This point is illustrated by the cross-country consumption correlations reported in the main text.