Monte Carlo simulations

In this appendix, we conduct simulations to assess finite sample performance of the threshold and semiparametric models implemented in the paper under model misspecification. For simplicity, we restrict our attention to the one-period case and consider Equation 1 with p=h=1. We parametrize the model as follows: ϕ₁=0.4, αi=i−0.5(N+1)N, and x_i,t =4+ψ_ix_i,t–1+ν_i,t . In addition, the autoregressive roots of the process for x are specified as being local-to-unity, such that ψ_i =1+c_i /T, which reflects the near-integrated dynamics of the debt ratio. The local to unity parameters (c_i ) are drawn from a uniform distribution with support [–10, –2]. The innovations (u_i,t , ν_i,t ) are drawn from normal distribution with E[u_i,t ]=E[ν_i,t ]=0, σ_u =σ_ν =3, and Corr(u_i,t , ν_i,t )=–0.3, ∀i, t. For f(·) we consider piecewise-linear, quadratic, and cubic cases as follows:

DGP−1:f(xi,t)=0.0455xi,tI(xi,t≤τ)−0.0103xi,tI(xi,t>τ),

DGP−2:f(xi,t)=0.002(xi,t−40)2,

DGP−3:f(xi,t)=0.0001(xi,t−40)3.

For DGP-1, we consider τ∈{20, 40, 60} and set the predictive coefficients equal to the empirical estimates from our data set (see Table 4). The other DGPs are designed to generate smooth nonlinear functions in the support of x. Estimation is carried out as described in the main text for both the threshold and the semiparametric specifications. We set the number of Monte Carlo replications equal to 1000 and pick the sample size N=20, T=50 to closely resemble the data at hand. To evaluate model performance, we estimate f(·) under the assumption of both the threshold and semiparametric specifications regardless of the true DGP and assess the result of misspecification in each case. We evaluate the estimated function at x∈{10, 20, 30, 40, 50, 60, 70, 80} for each replication and report the median estimated value of f(x) across all replications for each model along with the value associated with the true DGP.

Panels A to C in Figure A.1 summarize results under the threshold DGPs. For the case of τ=20 shown in panel a the semiparametric model suggests a turning point around 25 and the point estimates remain closer to the true DGP than those from the threshold model for values of x above the threshold. For τ=40, the semiparametric model yields a largely accurate signal about the turning point, but its overall fit appears poor. The threshold model performs well with respect to goodness of fit below the threshold as well as estimating the threshold itself. Finally, in case of τ=60, the semiparametric model suggest two turning points, but the second one located closer to the true threshold is more significant in magnitude. In terms of goodness of fit, the semiparametric model performs close to the threshold model below the threshold, with the exception of very small values of x. Overall, the semiparametric model’s performance is satisfactory in terms of estimating the turning point in the underlying discontinuous function as well as goodness of fit in the regime with the larger number of observations despite violation of the smoothness assumption.

Figure A.1:

Fit of the Semiparametric and Threshold Models under the Threshold DGP. (A): τ=20. (B): τ=40. (C): τ=60.

Median fitted values across 1,000 Monte Carlo simulations are reported for the threshold and semiparametric models. True DGP is DGP-1 in the text.

Panels A and B in Figure A.2 illustrate the cases of quadratic (DGP-2) and cubic (DGP-3), respectively. In the quadratic case, shown in panel a, the goodness of fit of the semiparametric specification is remarkable as the underlying function fulfills the basic assumption of smoothness. The threshold model is useful in signaling the turning point in the underlying function, but performs poorly in goodness of fit. The relative flatness of the estimated function under the threshold assumption suggest that the accompanying confidence intervals would be relatively large. The differences between the two estimation strategies become even more evident when the true DGP is of the cubic form. In this case, the semiparametric model continues to perform well, but the threshold model signals only the second turning point and generally suffers from a positive bias in point estimates.

Figure A.2:

Fit of the Semiparametric and Threshold Models under Polynomial DGPs. (A): DGP-2. (B): DGP-3.

Median fitted values across 1,000 Monte Carlo simulations are reported for the threshold and semiparametric models. See the text for DGP-1 and DGP-2.

In general, this Monte Carlo exercise shows that the semiparametric model performs well in signaling a turning point even when the underlying assumption of smoothness is violated. It also appears that the threshold model can be useful in estimating turning points when the underlying function is a smooth low-degree polynomial. The semiparametric model has better goodness of fit properties under model misspecification. We obtain qualitatively similar results under different numerical parametrizations.