Bias correction through filtering omitted variables and instruments

Abstract

This paper proposes a combination of the particle-filter-based method and the expectation-maximization algorithm (PFEM), in order to filter unobservable variables and hence, to reduce the omitted variables bias. Furthermore, I consider as an unobservable variable, an exogenous one that can be used as an instrument in the instrumental variable (IV) methodology. The aim is to show that the PFEM is able to eliminate or reduce both the omitted variable bias and the simultaneous equation bias by filtering the omitted variable and the unobserved instrument, respectively. In other words, the procedure provides (at least approximately) consistent estimates, without using additional information embedded in the omitted variable or in the instruments, since they are filtered by the observable variables. The validity of the procedure is shown both through simulations and through a comparison to an IV analysis which appeared in an important previous publication. As regards the latter point, I demonstrate that the procedure developed in this article yields similar results to those of the original IV analysis.

Keywords:

• omitted variables bias
• particle filter
• simultaneous equations bias

JEL Classification:

• C18
• C36
• E21

Notes

1 McCallum [9] was probably the first to highlight the OVB, and the method of IVs was first proposed by Reiersol [12,13] and rationalized by Sargan [16].

2 Regarding the SEB, to the best of the author's knowledge, there is nothing at all in the literature about general alternative solutions not based on the use of additional information. For example, in order to estimate the exogenous impact of the money supply on aggregate output, [14], following a narrative approach, considered subperiods such that, from an exogenous (but additional) information, the money supply is not influenced by output.

3 See, [4,15]; for an application to time series analysis, see [7].

4 See also [7] for an exposition of the EM algorithm related to the filtering of a latent polytomous variable.

5 The expected log-likelihood is parameterized by the vector of estimates at the $k+1$ iteration meanwhile the expectation is taken with respect to a second distribution parameterized by the vector of estimates of the kth iteration, see [7] for details.

6 See however, Section 3.2, for the inclusion of the variable z in the proposal and in the target distribution.

7 Note that these parameters can be estimated at each iteration (see also, [3]).

8 Of course, the reader may directly apply the PFEM described in the preceding section.

9 When the estimates approach the global maximum (where the log-likelihood, LL, is maximized), the parameter of $x_t^f$ of the latter regression approaches 1 and the constant becomes not significant.

10 The determination of the initial parameters is defined at the beginning of the section.

11 In the next section, it is shown that, under this assumption, PFEM and VPFEM are equivalent.

12 If $p=0$, this implies that no instrument (or a proxy of it) is available.

13 Note however, that the numerical optimization at the base of the PFEM requires cumbersome calculations and involves (numerical) approximations as well. Nevertheless, the PFEM provides consistent estimates (up to numerical approximations) as it is based on maximum-likelihood principles.

14 For ease of exposition, the expectation operator $E[.]$ is understood as conditional on the available information $y_t$, $z_t:E[.]\equiv E[.|y_t,z_t]$ unless specified differently.

15 Standard errors are in brackets.

16 Since one problem emerging with the IV method is that the available instruments may be poorly correlated with the endogenous variables.

17 Appendix 1 shows some details.

18 Note that since the relevant observation equation is linear, PFEM and VPFEM methods provide exactly the same results.

19 Some details are again provided in Appendix 1.

20 Although the best point estimate of their articles is $0.410$, according to the $R^2$.