Eliminating the omitted variable bias by a regime-switching approach

Abstract

This work shows a procedure that aims to eliminate or reduce the bias caused by omitted variables by means of the so-called regime-switching regressions. There is a bias estimation whenever the statistical (linear) model is under-specified, that is, when there are some omitted variables and they are correlated with the regressors. This work shows how an appropriate specification of a regime-switching model (independent or Markov-switching) can eliminate or reduce this correlation, hence the estimation bias. A demonstration is given, together with some Monte Carlo simulations. An empirical verification, based on Fisher's equation, is also provided.

Keywords:

• omitted variable bias
• regime-switching model
• EM algorithm
• Monte Carlo simulations
• Fisher's equation

Jel Classification: :

• C15
• C52
• E43

Notes

For ease of exposition, here the omitted variable is assumed to be polychotomous; however, it will be shown later that the procedure also works, if only approximately, when the omitted variable is defined in an open set of the real line.

T is the number of observations.

For general references see 11 14 15 20.

For references of methods using auxiliary information: 3 12.

Assumptions are labelled as A1, A2, … and definitions as D1, D2,…

For an overview of the theory of asymptotic distributions, see 9.

One may observe that the case wherein z is defined in the real line includes the case wherein z is polychotomous. However, for ease of exposition of the theorems in Section 3, it is preferable to consider these two cases as distinct.

The Ramsey RESET test or the Hausman test 6 may help detect missing regressors and nonlinearities. To detect serial correlation, the Durbin–Watson statistic and the Breush–Godfrey LM test may be used. The normality assumption may be tested by the Jarque–Bera test.

For references about Markov-switching models, see [5,7–10,19].

As in 19.

The weights should be the unconditional probabilities of each regime to occur.

For the sake of legibility, the parameters are not labelled by the value of q.

It can be shown that its second derivative is always negative.

The expression of the mean directly stems from the law of iterated expectations. The expression of the variance states that the total variance of z can be decomposed into two parts: the first part is the variance within regimes and the second part is the variance across regimes.

In fact, if one regime is assumed for z (k=1), then the conditional (on the unique regime) mean must be equal to the (constant) unconditional mean and its variance is also unconditional.

The subscript k stresses the fact that the decomposition of z depends on the number of the assumed regimes k.

For completeness of exposition, note that, by construction, it holds: E[ξ _k ]=0 and , , .

In fact, the number of the estimated parameters is the sum of the number of the columns of X (n) plus the constant and the standard deviation times the number of regimes (k) plus the number of the estimated transition probabilities minus 1 (k−1). Indeed, this is the case of the unconstrained RS regression. In the case of the constrained regression (Section 2) it must be .

The first-order autocovariance is 25 and the covariance with the regressor x is 10.

In this case, the unbiased parameter must be smaller than the biased parameter.

Assuming that in the short run prices are fixed.

According to the maximum-likelihood based criteria AIC and BIC, the autoregressive order of 1 is the most appropriate.

In particular, the fact that the 3-regime model seems to perform well for the purposes of this paper (and is to be preferred to the 10-regime model according to the SWC, Schwarz criterion, see Table 2) is also consistent with the results of 4.