Dealing With Endogeneity in Threshold Models Using Copulas

Abstract

We suggest a new method dealing with the problem of endogeneity of the threshold variable in structural threshold regression models based on copula theory. This method enables us to relax the assumption that the threshold variable is normally distributed and to capture the dependence structure between the threshold regression error term and the threshold variable independently of the marginal distribution of the threshold variable. For Gaussian and Student’s t copulas, this dependent structure can be captured by copula-type transformations of the distribution of the threshold variable, for each regime of the model. Augmenting the threshold model under these transformations can control for the endogeneity problem of threshold variable. The single-factor correlation structure of the threshold regression error term with these transformations allows us to consistently estimate the threshold and the slope parameters of the model based on a least squares method. Based on a Monte Carlo study, we show that our method is robust to nonlinear dependence structures between the regression error term and the threshold variable implied by the Archimedean family of copulas. We illustrate the method by estimating a model of the foreign-trade multiplier for seven OECD economies.

Key Words:

• Bootstrap
• Copulas
• Foreign trade multiplier
• Structural threshold model

ACKNOWLEDGMENTS

We thank the editor (Todd Clark), an associate editor, two anonymous referees, Stelios Arvanitis, Yiannis Bilias, George Kapetanios, Dimitris Karlis, Andros Kourtelos, Andrew Patton, Peter Phillips, and Thanasis Stengos for comments on an earlier version of the article.

Notes

1 Threshold models have recently been gained growing interest (see Teräsvirta, Tjøstheim, and Granger 2010 for a survey). Applications include studies on business cycles (e.g., Potter 1995), the debt-growth relationship (e.g., Reinhart and Rogoff 2010; Kourtellos, Stengos, and Tan 2013), financial markets and volatility (see Tsay 2010 for a survey), monetary policy models (e.g., Davig and Leeper 2007; Kazanas and Tzavalis 2015).

2 To mitigate the effects of this assumption on the threshold parameter estimates, Kourtellos et al. (2017) suggested a semiparametric estimator.

3 For a concise overview of applications of copula theory to economic and financial data, see, for example, Patton (2006, 2012) and Fan and Patton (2014). Recently, this theory has been applied to capture contemporaneous dependence between the error terms of two time series modeled separately, by the self-exciting threshold autoregressive model using as threshold variables lagged values of these series (see, e.g., Wong et al. 2017).

4 Note that, if δ = 1, then $C_1(F_{u_1}(u_{1,t}),\frac{F_z(z)}{\delta })$ becomes $C_1(F_{u_1}(u_{1,t}),F_z(z)),$ $0\le F_z(z)\le 1$ , which gives the joint distribution of $(u_{1,t},z_t)$ for $z_t\in {\mathcal{Z}}_1\cup {\mathcal{Z}}_2$ , which is the set of real numbers $\mathcal{R}$. Analogously, we can obtain $C_2(F_{u_2}(u_{2,t}),F_z(z)),$ $0\le F_z(z)\le 1$ , when δ = 0.

5 Note that, as in Heckman’s approach (1979), the selectivity bias correction terms in Lee’s (1983) model are given by the inverse Mills ratios of the transformed distribution of the selection equation error term, truncated at the condition function values of the selection equation.

6 For instance, note that, apart from the Gaussian and Student’s t copulas, a closed form solution of $\mathbb{E}(u_{i,t}|{\mathcal{Z}}_i)$ , i = 1, 2, can also be derived for the case that C_i is the Farlie–Gumbel–Morgenstern (FGM) copula frequently used in practice, which assumes nonlinear dependence between $u_{i,t}$ and z_t. As shown in Crane and Van Der Hoek (2008), this can happen under the assumptions that $u_{i,t}\sim N(0,{\sigma }_i^2)$ and $z_t\sim N(0,1)$. Then, we have,

$\mathbb{E}(u_{i,t}|{\mathcal{Z}}_i)=-\frac{{\theta }_i}{\sqrt{\pi }}{\sigma }_i(1-\Phi (z_t|{\mathcal{Z}}_i)),\hspace{1em}{\theta }_i\in [-1,1],\mathrm{i}=1,2.$

7 A similar problem appears in binary choice selectivity models (see Vella 1998 for a survey). Note that, for nonnormal distributions of x_kt, the nonlinear pattern of the transformation ${\Phi }^{-1}(F_{x_k}(x_{\mathit{kt}}))$ , or $T_v^{-1}(F_{x_K}(x_{\mathit{kt}}))$ , reduces the correlation between $x_{\mathit{kt}}^{\ast }$ and x_kt. For instance, if $x_{\mathit{kt}}\sim U(0,1)$ , where U is the uniform distribution, we can easily see that

$x_{\mathit{kt}}^{\ast }={\Phi }^{-1}(F_{x_K}(x_{\mathit{kt}}))={\Phi }^{-1}(x_{\mathit{kt}})=\sqrt{2}{\text{erf}}^{-1}(2x_{\mathit{kt}}-1),$

where $\sqrt{2}{\text{erf}}^{-1}(2p-1),p\in (0,1)$ , is the inversion function of the standard normal distribution. Based on a Maclaurin series expansion of ${\text{erf}}^{-1}(2x_{\mathit{kt}}-1)$ around $2x_{\mathit{kt}}-1=0$ (i.e., $x_{\mathit{kt}}=\frac{1}{2}$ ), $x_{\mathit{kt}}^{\ast }$ can be written as

$x_{\mathit{kt}}^{\ast }=\sqrt{2}(\frac{1}{2}(2x_{\mathit{kt}}-1)+\frac{\pi }{24}{(2x_{\mathit{kt}}-1)}^3+\frac{7{\pi }^2}{24}{(2x_{\mathit{kt}}-1)}^5+\cdots ).$

The last relationship shows more clearly that the degree of correlation between $x_{\mathit{kt}}^{\ast }$ and x_kt is also determined by the higher order terms in the above expansion of $x_{\mathit{kt}}^{\ast }$ which mitigates the problem of multicollinearity.

8 The discontinuity structure of distribution of z_t, at $z_{\delta }$ , adds extra randomness which, in a nonparametric framework of estimating model (1), also helps to identify the slope coefficients of the model across its two regimes (see Yu and Phillips 2018).

9 Note that this approach is different from that of Hansen (2000)—see also Gonzalo and Pitarakis (2002) and Kourtelos et al. (2016)—assuming that the threshold effects vanish with T, that is, ${\beta }_2-{\beta }_1\to 0$ as $T\to \infty .$

10 Note that this approach is different from that of Hansen (2000)—see also Gonzalo and Pitarakis (2002) and Kourtelos et al. (2016)—assuming that the threshold effects vanish with T, that is, ${\beta }_2-{\beta }_1\to 0$ as $T\to \infty .$

11 Note that similar results are obtained if we set the value of $z_{\delta }$ at the 1st quantile of the distribution of $z_t.$

12 Similar performance to the nonparametric method of estimating $F_{x_2}(x_{2t})$ and $F_{z|{\mathcal{Z}}_i}(z_t|Z_i),i=1,2,$ has also found for the ECDF method.

13 Kraay (2012) used a similar relationship, without threshold effects, to obtain estimates of the government spending multiplier.

14 In particular, REER is calculated from the nominal effective exchange rate (NEER) and a measure of the relative price of country i and its trading partners j, for all $j\ne i$ (e.g., Darvas 2012): ${\text{REER}}_{\mathit{it}}=\frac{{\text{NEER}}_{\mathit{it}}\times {\text{CPI}}_{\mathit{it}}}{{\text{CPI}}_{\mathit{it}}^{\ast }}$ , where ${\text{NEER}}_{\mathit{it}}=\underset{j\ne i}{\prod ^N}{s}_{\mathit{it}}{(j)}^{w_{\mathit{ij}}}$ is the geometric mean of the nominal bilateral exchange rates of a country i with its trading partners j, w_ij is the weight of trading partner j, ${\text{CPI}}_{\mathit{it}}$ is the consumer price index of country i, and ${\text{CPI}}_{\mathit{it}}^{\ast }=\underset{j\ne i}{\prod ^N}{\text{CPI}}_j^{w_{\mathit{ij}}}$ is the geometric mean of the CPI’s of trading partners j.

15 Our earlier working paper version of this article (European Central Bank Working Paper No. 2136) shows graphs of the variables and lists their summary statistics. These are suppressed here for reasons of space.

16 Note that Canada can reject the linear relationship at a 15% level.

17 As shown by Roth (2013) and Ding (2016) for the multivariate case, $T_{2,v+1}$ is also a Student’s t distribution. See also Kotz and Nadarajah (2004).

18 Note that a column of this matrix also contains the values of the quantile ${\Phi }^{-1}$ , or $T_v^{-1}$ , based transformed variable of regressor ${\tilde{x}}_t$ dealing with its possible endogeneity.