Selecting the Lag Length for the *M*^{GLS} Unit Root Tests with Structural Change: A Warning Note for Practitioners Based on Simulations*Econometrics* **2017**, *5*(2), 17; doi:10.3390/econometrics5020017 - 16 April 2017**Abstract **

This is a simulation-based warning note for practitioners who use the ${M}^{GLS}$ unit root tests in the context of structural change using different selection lag length criteria. With $T=100$ , we find severe oversize problems when using some

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This is a simulation-based warning note for practitioners who use the ${M}^{GLS}$ unit root tests in the context of structural change using different selection lag length criteria. With $T=100$ , we find severe oversize problems when using some criteria, while other criteria produce an undersizing behavior. In view of this dilemma, we do not recommend using these tests. While such behavior tends to disappear when $T=250$ , it is important to note that most empirical applications use smaller sample sizes such as $T=100$ or $T=150$ . The $AD{F}^{GLS}$ test does not present an oversizing or undersizing problem. The only disadvantage of the $AD{F}^{GLS}$ test arises in the presence of $MA(1)$ negative correlation, in which case the ${M}^{GLS}$ tests are preferable, but in all other cases they are very undersized. When there is a break in the series, selecting the breakpoint using the Supremum method greatly improves the results relative to the Infimum method.
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Copula–Based vMEM Specifications versus Alternatives: The Case of Trading Activity*Econometrics* **2017**, *5*(2), 16; doi:10.3390/econometrics5020016 - 12 April 2017**Abstract **

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We discuss several multivariate extensions of the Multiplicative Error Model to take into account dynamic interdependence and contemporaneously correlated innovations (vector MEM or vMEM). We suggest copula functions to link Gamma marginals of the innovations, in a specification where past values and conditional

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We discuss several multivariate extensions of the Multiplicative Error Model to take into account dynamic interdependence and contemporaneously correlated innovations (vector MEM or vMEM). We suggest copula functions to link Gamma marginals of the innovations, in a specification where past values and conditional expectations of the variables can be simultaneously estimated. Results with realized volatility, volumes and number of trades of the JNJ stock show that significantly superior realized volatility forecasts are delivered with a fully interdependent vMEM relative to a single equation. Alternatives involving log–Normal or semiparametric formulations produce substantially equivalent results.
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Accuracy and Efficiency of Various GMM Inference Techniques in Dynamic Micro Panel Data Models*Econometrics* **2017**, *5*(1), 14; doi:10.3390/econometrics5010014 - 20 March 2017**Abstract **

Studies employing Arellano-Bond and Blundell-Bond generalized method of moments (GMM) estimation for linear dynamic panel data models are growing exponentially in number. However, for researchers it is hard to make a reasoned choice between many different possible implementations of these estimators and associated

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Studies employing Arellano-Bond and Blundell-Bond generalized method of moments (GMM) estimation for linear dynamic panel data models are growing exponentially in number. However, for researchers it is hard to make a reasoned choice between many different possible implementations of these estimators and associated tests. By simulation, the effects are examined in terms of many options regarding: (i) reducing, extending or modifying the set of instruments; (ii) specifying the weighting matrix in relation to the type of heteroskedasticity; (iii) using (robustified) 1-step or (corrected) 2-step variance estimators; (iv) employing 1-step or 2-step residuals in Sargan-Hansen overall or incremental overidentification restrictions tests. This is all done for models in which some regressors may be either strictly exogenous, predetermined or endogenous. Surprisingly, particular asymptotically optimal and relatively robust weighting matrices are found to be superior in finite samples to ostensibly more appropriate versions. Most of the variants of tests for overidentification and coefficient restrictions show serious deficiencies. The variance of the individual effects is shown to be a major determinant of the poor quality of most asymptotic approximations; therefore, the accurate estimation of this nuisance parameter is investigated. A modification of GMM is found to have some potential when the cross-sectional heteroskedasticity is pronounced and the time-series dimension of the sample is not too small. Finally, all techniques are employed to actual data and lead to insights which differ considerably from those published earlier.
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A Simple Test for Causality in Volatility*Econometrics* **2017**, *5*(1), 15; doi:10.3390/econometrics5010015 - 20 March 2017**Abstract **

An early development in testing for causality (technically, Granger non-causality) in the conditional variance (or volatility) associated with financial returns was the portmanteau statistic for non-causality in the variance of Cheng and Ng (1996). A subsequent development was the Lagrange Multiplier (LM) test

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An early development in testing for causality (technically, Granger non-causality) in the conditional variance (or volatility) associated with financial returns was the portmanteau statistic for non-causality in the variance of Cheng and Ng (1996). A subsequent development was the Lagrange Multiplier (LM) test of non-causality in the conditional variance by Hafner and Herwartz (2006), who provided simulation results to show that their LM test was more powerful than the portmanteau statistic for sample sizes of 1000 and 4000 observations. While the LM test for causality proposed by Hafner and Herwartz (2006) is an interesting and useful development, it is nonetheless arbitrary. In particular, the specification on which the LM test is based does not rely on an underlying stochastic process, so the alternative hypothesis is also arbitrary, which can affect the power of the test. The purpose of the paper is to derive a simple test for causality in volatility that provides regularity conditions arising from the underlying stochastic process, namely a random coefficient autoregressive process, and a test for which the (quasi-) maximum likelihood estimates have valid asymptotic properties under the null hypothesis of non-causality. The simple test is intuitively appealing as it is based on an underlying stochastic process, is sympathetic to Granger’s (1969, 1988) notion of time series predictability, is easy to implement, and has a regularity condition that is not available in the LM test.
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Goodness-of-Fit Tests for Copulas of Multivariate Time Series

*Econometrics* **2017**, *5*(1), 13; doi:10.3390/econometrics5010013 - 17 March 2017**Abstract **

In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations

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In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown that if the stochastic volatility matrices are diagonal, which is the case if the univariate time series are estimated separately instead of being jointly estimated, then the empirical copula process behaves as if the innovations were observed; a remarkable property. As a by-product, one also obtains the asymptotic behavior of rank-based measures of dependence applied to residuals of these time series models.

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Testing for a Structural Break in a Spatial Panel Model*Econometrics* **2017**, *5*(1), 12; doi:10.3390/econometrics5010012 - 6 March 2017**Abstract **

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We consider the problem of testing for a structural break in the spatial lag parameter in a panel model (spatial autoregressive). We propose a likelihood ratio test of the null hypothesis of no break against the alternative hypothesis of a single break. The

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We consider the problem of testing for a structural break in the spatial lag parameter in a panel model (spatial autoregressive). We propose a likelihood ratio test of the null hypothesis of no break against the alternative hypothesis of a single break. The limiting distribution of the test is derived under the null when both the number of individual units N and the number of time periods T is large or N is ﬁxed and T is large. The asymptotic critical values of the test statistic can be obtained analytically. We also propose a break-date estimator that can be employed to determine the location of the break point following evidence against the null hypothesis. We present Monte Carlo evidence to show that the proposed procedure performs well in ﬁnite samples. Finally, we consider an empirical application of the test on budget spillovers and interdependence in ﬁscal policy within the U.S. states.
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Structural Breaks, Inflation and Interest Rates: Evidence from the G7 Countries*Econometrics* **2017**, *5*(1), 11; doi:10.3390/econometrics5010011 - 17 February 2017**Abstract **

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This study reconsiders the common unit root/co-integration approach to test for the Fisher effect for the economies of the G7 countries. We first show that nominal interest and inflation rates are better represented as I(0) variables. Later, we use the Bai–Perron procedure to

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This study reconsiders the common unit root/co-integration approach to test for the Fisher effect for the economies of the G7 countries. We first show that nominal interest and inflation rates are better represented as I(0) variables. Later, we use the Bai–Perron procedure to show the existence of structural changes in the Fisher equation. After considering these breaks, we find very limited evidence of a total Fisher effect as the transmission coefficient of the expected inflation rates to nominal interest rates is very different than one.
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A Note on Identification of Bivariate Copulas for Discrete Count Data*Econometrics* **2017**, *5*(1), 10; doi:10.3390/econometrics5010010 - 15 February 2017**Abstract **

Copulas have enjoyed increased usage in many areas of econometrics, including applications with discrete outcomes. However, Genest and Nešlehová (2007) present evidence that copulas for discrete outcomes are not identified, particularly when those discrete outcomes follow count distributions. This paper confirms the Genest

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Copulas have enjoyed increased usage in many areas of econometrics, including applications with discrete outcomes. However, Genest and Nešlehová (2007) present evidence that copulas for discrete outcomes are not identified, particularly when those discrete outcomes follow count distributions. This paper confirms the Genest and Nešlehová result using a series of simulation exercises. The paper then proceeds to show that those identification concerns diminish if the model has a regression structure such that the exogenous variable(s) generates additional variation in the outcomes and thus more completely covers the outcome domain.
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Endogeneity, Time-Varying Coefficients, and Incorrect vs. Correct Ways of Specifying the Error Terms of Econometric Models*Econometrics* **2017**, *5*(1), 8; doi:10.3390/econometrics5010008 - 3 February 2017**Abstract **

Using the net effect of all relevant regressors omitted from a model to form its error term is incorrect because the coefficients and error term of such a model are non-unique. Non-unique coefficients cannot possess consistent estimators. Uniqueness can be achieved if; instead;

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Using the net effect of all relevant regressors omitted from a model to form its error term is incorrect because the coefficients and error term of such a model are non-unique. Non-unique coefficients cannot possess consistent estimators. Uniqueness can be achieved if; instead; one uses certain “sufficient sets” of (relevant) regressors omitted from each model to represent the error term. In this case; the unique coefficient on any non-constant regressor takes the form of the sum of a bias-free component and omitted-regressor biases. Measurement-error bias can also be incorporated into this sum. We show that if our procedures are followed; accurate estimation of bias-free components is possible.
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A Fast Algorithm for the Computation of HAC Covariance Matrix Estimators*Econometrics* **2017**, *5*(1), 9; doi:10.3390/econometrics5010009 - 25 January 2017**Abstract **

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This paper considers the algorithmic implementation of the heteroskedasticity and autocorrelation consistent (HAC) estimation problem for covariance matrices of parameter estimators. We introduce a new algorithm, mainly based on the fast Fourier transform, and show via computer simulation that our algorithm is up

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This paper considers the algorithmic implementation of the heteroskedasticity and autocorrelation consistent (HAC) estimation problem for covariance matrices of parameter estimators. We introduce a new algorithm, mainly based on the fast Fourier transform, and show via computer simulation that our algorithm is up to 20 times faster than well-established alternative algorithms. The cumulative effect is substantial if the HAC estimation problem has to be solved repeatedly. Moreover, the bandwidth parameter has no impact on this performance. We provide a general description of the new algorithm as well as code for a reference implementation in `R`.
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Between Institutions and Global Forces: Norwegian Wage Formation Since Industrialisation*Econometrics* **2017**, *5*(1), 6; doi:10.3390/econometrics5010006 - 12 January 2017**Abstract **

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This paper reviews the development of labour market institutions in Norway, shows how labour market regulation has been related to the macroeconomic development, and presents dynamic econometric models of nominal and real wages. Single equation and multi-equation models are reported. The econometric modelling

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This paper reviews the development of labour market institutions in Norway, shows how labour market regulation has been related to the macroeconomic development, and presents dynamic econometric models of nominal and real wages. Single equation and multi-equation models are reported. The econometric modelling uses a new data set with historical time series of wages and prices, unemployment and labour productivity. Impulse indicator saturation is used to achieve robust estimation of focus parameters, and the breaks are interpreted in the light of the historical overview. A relatively high degree of constancy of the key parameters of the wage setting equation is documented, over a considerably longer historical time period than earlier studies have done. The evidence is consistent with the view that the evolving system of collective labour market regulation over long periods has delivered a certain necessary level of coordination of wage and price setting. Nevertheless, there is also evidence that global forces have been at work for a long time, in a way that links real wages to productivity trends in the same way as in countries with very different institutions and macroeconomic development.
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Acknowledgement to Reviewers of *Econometrics* in 2016*Econometrics* **2017**, *5*(1), 7; doi:10.3390/econometrics5010007 - 11 January 2017**Abstract **
The editors of *Econometrics *would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2016.[...]
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Fractional Unit Root Tests Allowing for a Structural Change in Trend under Both the Null and Alternative Hypotheses*Econometrics* **2017**, *5*(1), 5; doi:10.3390/econometrics5010005 - 8 January 2017**Abstract **

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This paper considers testing procedures for the null hypothesis of a unit root process against the alternative of a fractional process, called a fractional unit root test. We extend the Lagrange Multiplier (LM) tests of Robinson (1994) and Tanaka (1999), which are locally

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This paper considers testing procedures for the null hypothesis of a unit root process against the alternative of a fractional process, called a fractional unit root test. We extend the Lagrange Multiplier (LM) tests of Robinson (1994) and Tanaka (1999), which are locally best invariant and uniformly most powerful, to allow for a slope change in trend with or without a concurrent level shift under both the null and alternative hypotheses. We show that the limit distribution of the proposed LM tests is standard normal. Finite sample simulation experiments show that the tests have good size and power. As an empirical analysis, we apply the tests to the Consumer Price Indices of the G7 countries.
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Consistency of Trend Break Point Estimator with Underspecified Break Number*Econometrics* **2017**, *5*(1), 4; doi:10.3390/econometrics5010004 - 5 January 2017**Abstract **

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This paper discusses the consistency of trend break point estimators when the number of breaks is underspecified. The consistency of break point estimators in a simple location model with level shifts has been well documented by researchers under various settings, including extensions such

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This paper discusses the consistency of trend break point estimators when the number of breaks is underspecified. The consistency of break point estimators in a simple location model with level shifts has been well documented by researchers under various settings, including extensions such as allowing a time trend in the model. Despite the consistency of break point estimators of level shifts, there are few papers on the consistency of trend shift break point estimators in the presence of an underspecified break number. The simulation study and asymptotic analysis in this paper show that the trend shift break point estimator does not converge to the true break points when the break number is underspecified. In the case of two trend shifts, the inconsistency problem worsens if the magnitudes of the breaks are similar and the breaks are either both positive or both negative. The limiting distribution for the trend break point estimator is developed and closely approximates the finite sample performance.
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Business Cycle Estimation with High-Pass and Band-Pass Local Polynomial Regression*Econometrics* **2017**, *5*(1), 1; doi:10.3390/econometrics5010001 - 5 January 2017**Abstract **

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Filters constructed on the basis of standard local polynomial regression (LPR) methods have been used in the literature to estimate the business cycle. We provide a frequency domain interpretation of the contrast filter obtained by the difference of a series and its long-run

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Filters constructed on the basis of standard local polynomial regression (LPR) methods have been used in the literature to estimate the business cycle. We provide a frequency domain interpretation of the contrast filter obtained by the difference of a series and its long-run LPR component and show that it operates as a kind of high-pass filter, so that it provides a noisy estimate of the cycle. We alternatively propose band-pass local polynomial regression methods aimed at isolating the cyclical component. Results are compared to standard high-pass and band-pass filters. Procedures are illustrated using the US GDP series.
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Regime Switching Vine Copula Models for Global Equity and Volatility Indices*Econometrics* **2017**, *5*(1), 3; doi:10.3390/econometrics5010003 - 4 January 2017**Abstract **

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For nearly every major stock market there exist equity and implied volatility indices. These play important roles within finance: be it as a benchmark, a measure of general uncertainty or a way of investing or hedging. It is well known in the academic

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For nearly every major stock market there exist equity and implied volatility indices. These play important roles within finance: be it as a benchmark, a measure of general uncertainty or a way of investing or hedging. It is well known in the academic literature that correlations and higher moments between different indices tend to vary in time. However, to the best of our knowledge, no one has yet considered a global setup including both equity and implied volatility indices of various continents, and allowing for a changing dependence structure. We aim to close this gap by applying Markov-switching *R*-vine models to investigate the existence of different, global dependence regimes. In particular, we identify times of “normal” and “abnormal” states within a data set consisting of North-American, European and Asian indices. Our results confirm the existence of joint points in a time at which global regime switching between two different *R*-vine structures takes place.
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Fixed-*b* Inference for Testing Structural Change in a Time Series Regression*Econometrics* **2017**, *5*(1), 2; doi:10.3390/econometrics5010002 - 30 December 2016**Abstract **

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This paper addresses tests for structural change in a weakly dependent time series regression. The cases of full structural change and partial structural change are considered. Heteroskedasticity-autocorrelation (HAC) robust Wald tests based on nonparametric covariance matrix estimators are explored. Fixed-*b* theory is

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This paper addresses tests for structural change in a weakly dependent time series regression. The cases of full structural change and partial structural change are considered. Heteroskedasticity-autocorrelation (HAC) robust Wald tests based on nonparametric covariance matrix estimators are explored. Fixed-*b* theory is developed for the HAC estimators which allows fixed-*b* approximations for the test statistics. For the case of the break date being known, the fixed-*b* limits of the statistics depend on the break fraction and the bandwidth tuning parameter as well as on the kernel. When the break date is unknown, supremum, mean and exponential Wald statistics are commonly used for testing the presence of the structural break. Fixed-*b* limits of these statistics are obtained and critical values are tabulated. A simulation study compares the finite sample properties of existing tests and proposed tests.
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The Status of Bridge Principles in Applied Econometrics*Econometrics* **2016**, *4*(4), 50; doi:10.3390/econometrics4040050 - 17 December 2016**Abstract **

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The paper begins with a figurative representation of the contrast between present-day and formal applied econometrics. An explication of the status of bridge principles in applied econometrics follows. To illustrate the concepts used in the explication, the paper presents a simultaneous-equation model of

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The paper begins with a figurative representation of the contrast between present-day and formal applied econometrics. An explication of the status of bridge principles in applied econometrics follows. To illustrate the concepts used in the explication, the paper presents a simultaneous-equation model of the equilibrium configurations of a perfectly competitive commodity market. With artificially generated data I carry out two empirical analyses of such a market that contrast the prescriptions of formal econometrics in the tradition of Ragnar Frisch with the commands of present-day econometrics in the tradition of Trygve Haavelmo. At the end I demonstrate that the bridge principles I use in the formal-econometric analysis are valid in the Real World—that is in the world in which my data reside.
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Testing for the Equality of Integration Orders of Multiple Series*Econometrics* **2016**, *4*(4), 49; doi:10.3390/econometrics4040049 - 15 December 2016**Abstract **

Testing for the equality of integration orders is an important topic in time series analysis because it constitutes an essential step in testing for (fractional) cointegration in the bivariate case. For the multivariate case, there are several versions of cointegration, and the version

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Testing for the equality of integration orders is an important topic in time series analysis because it constitutes an essential step in testing for (fractional) cointegration in the bivariate case. For the multivariate case, there are several versions of cointegration, and the version given in Robinson and Yajima (2002) has received much attention. In this definition, a time series vector is partitioned into several sub-vectors, and the elements in each sub-vector have the same integration order. Furthermore, this time series vector is said to be cointegrated if there exists a cointegration in any of the sub-vectors. Under such a circumstance, testing for the equality of integration orders constitutes an important problem. However, for multivariate fractionally integrated series, most tests focus on stationary and invertible series and become invalid under the presence of cointegration. Hualde (2013) overcomes these difficulties with a residual-based test for a bivariate time series. For the multivariate case, one possible extension of this test involves testing for an array of bivariate series, which becomes computationally challenging as the dimension of the time series increases. In this paper, a one-step residual-based test is proposed to deal with the multivariate case that overcomes the computational issue. Under certain regularity conditions, the test statistic has an asymptotic standard normal distribution under the null hypothesis of equal integration orders and diverges to infinity under the alternative. As reported in a Monte Carlo experiment, the proposed test possesses satisfactory sizes and powers.
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Higher Order Bias Correcting Moment Equation for M-Estimation and Its Higher Order Efficiency*Econometrics* **2016**, *4*(4), 48; doi:10.3390/econometrics4040048 - 8 December 2016**Abstract **

This paper studies an alternative bias correction for the M-estimator, which is obtained by correcting the moment equations in the spirit of Firth (1993). In particular, this paper compares the stochastic expansions of the analytically-bias-corrected estimator and the alternative estimator and finds that

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This paper studies an alternative bias correction for the M-estimator, which is obtained by correcting the moment equations in the spirit of Firth (1993). In particular, this paper compares the stochastic expansions of the analytically-bias-corrected estimator and the alternative estimator and finds that the third-order stochastic expansions of these two estimators are identical. This implies that at least in terms of the third-order stochastic expansion, we cannot improve on the simple one-step bias correction by using the bias correction of moment equations. This finding suggests that the comparison between the one-step bias correction and the method of correcting the moment equations or the fully-iterated bias correction should be based on the stochastic expansions higher than the third order.
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