Exact and asymptotic identification-robust inference for dynamic structural equations with an application to New Keynesian Phillips Curves

Abstract

Many models in econometrics involve endogeneity and lagged dependent variables. We start by observing that usual identification-robust (IR) tests are unreliable when model variables are nonstationary or nearly nonstationary. We propose IR methods which are also robust to nonstationarity: one Anderson-Rubin type procedure and two split-sample procedures. Our procedures are also robust to missing instruments. For distributional theory, three different sets of assumptions are considered. First, on assuming Gaussian structural errors, we show that two of the proposed statistics follow the standard F distribution. Second, for more general cases, we assume that the distribution of errors is completely specified up to an unknown scale factor, allowing the Monte Carlo test method to be applied. This assumption enables one to deal with non-Gaussian error distributions. For example, even when errors follow heavy-tailed distribution, such as the Cauchy distribution or more generally the family of stable distributions—which may not have moments and thus make inference difficult—our procedures provide simple and exact solutions. Third, we establish the asymptotic validity of our procedures under quite general distributional assumptions. We present simulation results showing that our procedures control their level correctly and have good power properties. The methods are applied to an empirical example, the New Keynesian Phillips curve, in which both weak identification and nonstationarity present challenges. The results of this empirical study suggest forward-looking behavior of U.S. inflation.

Keywords:

• Simultaneous equation model
• lagged dependent variable
• weak instruments
• identification-robust test
• exact inference
• Monte Carlo test
• New Keynesian Phillips curve

JEL CLASSIFICATION:

• C1
• C12
• C3
• C32
• C36
• E3
• E31

Acknowledgment

The authors thank the Editor, Esfandiar Maasoumi, and two anonymous referees for several useful comments.

Funding

This work was supported by the William Dow Chair in Political Economy (McGill University), the Bank of Canada (Research Fellowship), the Toulouse School of Economics (Pierre-de-Fermat Chair of excellence), the Universitad Carlos III de Madrid (Banco Santander de Madrid Chair of excellence), the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada, and the Fonds de recherche du Québec (Société et culture).

Notes

1 See, for example, the reviews of Stock et al. (2002), Andrews and Stock (2005), and Mikusheva (2013).

2 Recently, Chevillon et al. (2020) proposed the AR combining the IVX method of Magdalinos and Phillips (2009) in the SVAR to deal with near nonstationarity. However, their method cannot be a solution in our model because the filtered IV is a function of the lagged dependent variable, which is still correlated with equation errors. We show its failure in the simulation study.

3 While King (1980) and Dufour and King (1991) show the optimality of their tests, it is more complicated in this model to establish the optimality of the test based on (3.25)(3.25) $${\tilde{F}}^{\ast \ast }({\beta }_0, {\rho }_0)=\frac{y^{(2)}{({\beta }_0, {\rho }_0)}^{\prime }M[Z^{(2)}]y^{(2)}({\beta }_0, {\rho }_0)}{y^{(2)}{({\beta }_0, {\rho }_0)}^{\prime }{P}_0{({\widehat{\psi }}^{(1)})}^{\prime }M[P_0({\widehat{\psi }}^{(1)})Z^{(2)}⋮P_0({\widehat{\psi }}^{(1)}){\widehat{W}}^{(2)}({\widehat{\varphi }}^{(1)})]P_0({\widehat{\psi }}^{(1)})y^{(2)}({\beta }_0, {\rho }_0)}$$(3.25) because the null hypothesis imposes restrictions both on the mean and on the covariance matrix at the same time.

4 Note that x₁ in the IV set 1 is a time trend. By this case, we can check the performance of the tests with IVs including the trend, which is nonstationary.

5 But if ${\tau }^{\ast }$ is endogenously selected, the split sample procedures can fail to control the test level.

6 They present many confidence sets based on various specifications. We compare our results to the case with GDP deflator inflation, survey of professional forecasters (SPF), sample period 1984Q1-2011Q4 and labor share as a forcing variable because we take this specification.

7 Economic theory restricts the parameters in (44) as $0\le {\gamma }_f\le 1$ and $0\le {\gamma }_b\le 1.$ But we consider γ_f from $0$ to 1.2 and γ_b from –0.2 to 1.0 to describe the confidence sets more completely. The values of parameters vary with increments of 0.01. The same seed for the random number generator was used for all values on the parameter space.

8 For grid search, we consider the admissible range $[0,2]$ for $({\gamma }_f+{\gamma }_b).$ The values of $({\gamma }_f+{\gamma }_b)$ vary with increments of 0.02. For the other parameters, the same setup as the previous case is used.