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Empirical Economics

Erschienen in:

29.12.2017

Testing for the omission of relevant variables and regime-switching misspecification

verfasst von: Andrea Beccarini

Erschienen in: Empirical Economics | Ausgabe 3/2019

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Abstract

This article shows that the interpretation of statistical evidence of regime-switching is not unambiguous. The usual interpretation is that some parameters switch according to the values of a predefined latent variable. An alternative interpretation is that regime-switching, as a statistical evidence, is also possible when the linear model is underspecified and the omitted variable bias emerges. A formal test is proposed to verify a potentially spurious regression with regime-switching. Through this test, it is evident that regime-switching estimates presented in an academic paper, should be interpreted as a consequence of the misspecification considered here.

Nächster Artikel Lag length selection and p-hacking in Granger causality testing: prevalence and performance of meta-regression models

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Only regime-switching models with unobservable switching are considered.

The chapter is called: Macroeconomics, Non-linear Time Series in.

The identification problem arises when specifying the null hypothesis as a linear model given R-S estimates. This arises in particular from nuisance parameters which are the transitional probabilities that are unidentified under the null. This in turn makes the information matrix singular (under the null). Thus the standard inference tools cannot apply.

y, x and z are defined as stochastic variables such that the usual assumptions of the classical linear regression model apply. The model describes the causal relationship between the relevant variables.

The considered data-generating processes imply that the underlying parameters measure the causal effects between variables.

The usual definition of the plim operator applies. For references, see White (1999, chapter 2). Throughout the article, I intend the plim value of a matrix to constitute the plim value of each element of the matrix.

\(s_n\) is drawn from a discrete distribution whose support is \(1,2,\ldots r\), independently of X.

Where \({{\varvec{E}}}[.|{{\varvec{Y}}},{{\varvec{X}}},{\varvec{\rho }} _{\mathbf{{1}},{{\varvec{i}}}}]\) denotes the mathematical expectation operator conditional on realizations of Y and X and on the relevant parameters vector \(\rho _{1,i}\): \(\rho _{1,i} =[ \rho _{0,i} \quad \theta _i ]\) with \(\theta _i \equiv \left[ {\beta _i \sigma } \right] \) whose elements are defined in A8 and A9.

For concepts specific to the RSM, the reader may consider (Hamilton 1994, chapter 22).

This is due to exogeneity of \({{\varvec{X}}}\) assumed in A2 and in A9.

ID stays identically distributed.

Note that this is a sufficient (but not necessary) condition for applying the Central Limit Theorem in appendix.

The R-S estimators exist under the condition that \(\left( {{{\varvec{X}}}_\mathbf{{1}}^{{\prime }} {\widehat{{{\varvec{P}}}}}_{{\varvec{i}}} {{\varvec{X}}}_\mathbf{{1}}} \right) ^{-1}\) exists, \(i=1,2,\ldots ,r\). Note that the result of Lemma 2 presented later on is not necessary for its existence, implying that the econometric model does not need to be specified correctly.

The estimated smoothed probabilities of each regime and for each observation are organized in a diagonal matrix \(\widehat{{{\varvec{p}}}}_{{\varvec{i}}}\) consistently with A6 (where the main diagonal corresponds to the vector of the inferred smoothed probabilities conditional on regime i).

A similar analysis is provided in Timmerman (2000), but only for some R-S models.

\(\varvec{\rho }_{\mathbf{{1},{\varvec{i}}}}=[{\varvec{\rho }} _\mathbf{0 ,{{\varvec{i}}}}{\varvec{\theta }}_{{\varvec{i}}}]\, \hbox {with}\,{\varvec{\theta }}_{{\varvec{i}}} \equiv {\varvec{\beta }}_{{\varvec{i}}} \sigma \).

The reader can also easily verify this by considering the proofs of Lemma 5 and Proposition 1.

They are still organized in r matrices:\({{\varvec{S}}}_{{\varvec{i}}}\) is a \(N{\times }N\) diagonal matrix, such that the nth-element of the main diagonal is 1 if the nth-observation belongs to regime i\((i=1,2,\ldots ,r)\) and zero otherwise.

See Eq. (2.10).

Table 1 reports some parameters of Chen (2007) related to the regressions of real and nominal returns (S&P500) on the M2 growth rate (M2G), on changes in discount rates (DDR) and on changes on the Fed funds rates (DFF).

I.I.D.N. stays independently and identically normally distributed.

The sample is 1965m1–2004m11.

The reader should consider the absolute values, because the employed regressors may be negatively correlated with each other, although all measure the stance of the monetary policy. Thus, the covariances between each of them and the omitted variable may have different signs and so the bias.

The reader should compare \(\mathop \sum \nolimits _{i=0}^1 {\hat{\varvec{\pi }}}_i \widehat{\varvec{\gamma }}_{{\varvec{i}}}\, \hbox {with}\, {\varvec{\gamma }}_{{{\varvec{ols}}}}^{{\varvec{B}}}\).

Very similar estimates are obtained here to those of Chen (2007).

When recessions occur, the indicator equals \(-\,1\); when the expansions occur, the indicator equals \(+\,1\).

Table 3 (last row) reports these correlations.

Note that these estimates are equal to those reported in the first row of Table 3.

Note that Table 1 only reports the slope of the regressions: and , meanwhile, the test considers the entire vector of parameters.

There a constant was assumed in the linear model equal to zero. Here, the reader may verify that the average estimated constant of the R-S \(\mathop \sum \nolimits _{{i}=1}^1 \hat{\varvec{\pi }}_{{\varvec{i}}} {\widehat{{\alpha }}}_{i}\) regressions, is asymptotically equal to the parameter of the NBER indicator multiplied by its expected value, \(\mathop \sum \nolimits _{{i}=1}^r {\pi }_{i} \alpha z_{i}=\alpha \mathop \sum \nolimits _{i=1}^r \pi _i z^i=\alpha E[z]\), plus the constant of the linear model. Where \(z^0=-\,1\) and \(z^1 =+\,1\) as specified in Note 27.

Indeed, it cannot be concluded that this relationship is linear, as the monetary policy variables may still have nonlinear effects (although non R-S) on the stock market. The easiest way to verify this is to insert as regressor an interactive variable (which is the product of the NBER indicator and the measures of monetary policy). However, these interacting variables are never significant (results available upon request).

In fact, the NBER indicator cannot be employed for forecasting purposes, since it is delivered several months later with respect to its reference period. See also the conclusions of Morley et al. (2013) on this point.

To see this, note that the matrices \( i=1,\ldots r,\) provide a partition of the sample consistent with the postulated regimes.

This is equivalent to differentiating the first order conditions found in Hamilton (1990) in eq. (5.8). This is also equivalent to considering the transformation of the row data: \(\tilde{{{\varvec{y}}}}=(\hat{{\varvec{p}}}_{{\varvec{i}}})^{0.5} {{\varvec{y}}},\,{\tilde{{\varvec{x}}}}=(\hat{{{\varvec{p}}}}_{{\varvec{i}}})^{0.5}\)x, and then finding the OLS variance-covariance matrix of \((\tilde{{{\varvec{x}}}}^\prime \tilde{{{\varvec{x}}}})^{-1}(\tilde{{{\varvec{x}}}}^\prime \tilde{{{\varvec{y}}}})\).

Note first that the matrices \(\hat{{{\varvec{p}}}}_{{\varvec{i}}}\) are functions of \({{\hat{\varvec{{\rho }}}}}_{\mathbf{{1},{\varvec{i}}}}\)\((i=1,2,\ldots ,r)\) with \(\hat{{\varvec{\rho }}}_{\mathbf{{1},{\varvec{i}}}}\equiv [\hat{{\varvec{\rho }}}_\mathbf{0 , {{\varvec{i}}}} \quad \hat{{\varvec{\theta }}}_{{\varvec{i}}}]\) and \(\hat{{\varvec{\theta }}}_{{\varvec{i}}}\equiv [\hat{{\varvec{\beta }}_{{\varvec{i}}}}\, \hat{\sigma }]\) which are estimators of the parameters defined in A6, A8, and A9. Having then assumed that the relevant density and probability functions are continuous, I apply proposition 2.27 of White (1991), chapter 2.

See White (1991, chapter 2), Proposition 2.27.

Note that, Eqs. (A.7) and (A.8) continue to hold, if the true data-generating process entails switches in only one subset of parameters (say the constant or the variance).

Which is the result of the estimating procedure of Kim et al. (2008).

\({\varvec{\beta }} _{{1}} =\left[ {0.5 0.8} \right] ^{{{\prime }}},{\varvec{\beta }} _2 =\left[ {-0.5-0.8} \right] {{\prime }} \quad \pi _1 =0.6\), \(\Pr \left[ {s_t =1 {|}s_{t-1} =1} \right] =0.8\), \(\Pr \left[ {s_t =2 |s_{t-1} =2} \right] =0.7\), \(\sigma _1^2 =0.5, \quad \sigma _2^2 =2\).

\({\varvec{\beta }}_\mathbf{1} =\left[ {0.5 0.8} \right] ^{{{\prime }}}, {\varvec{\beta }}_2 =\left[ {-0.5-0.8} \right] {{\prime }} \quad \pi _1 =0.6\), \(\Pr \left[ {s_t =1{|}s_{t-1} =1} \right] =0.8\), \(\Pr \left[ {s_t =2{|}s_{t-1} =2} \right] =0.7\), \(\sigma _1^2 =0.5, \quad \sigma _2^2 =0.5\).

\({\varvec{\beta }}_\mathbf{1} =\left[ {0.5 0.8} \right] ^{{{\prime }}}, {\varvec{\beta }}_2 =\left[ {0.5-0.8} \right] {{\prime }} \quad \pi _1 =0.6\), \(\Pr \left[ {s_t =1 {|}s_{t-1} =1} \right] =0.8\), \(\Pr \left[ {s_t =2 {|}s_{t-1} =2} \right] =0.7\), \(\sigma _1^2 =0.5, \quad \sigma _2^2 =2\). The case of constant variance is also considered. This does not exhibits different behavior with respect to the simulations with R-S variance.

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Titel: Testing for the omission of relevant variables and regime-switching misspecification
verfasst von: Andrea Beccarini
Publikationsdatum: 29.12.2017
Verlag: Springer Berlin Heidelberg
Erschienen in: Empirical Economics / Ausgabe 3/2019
Print ISSN: 0377-7332
Elektronische ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-017-1373-8

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