Second-Order Covariance Matrix Formula for Heteroskedastic Generalized Linear Models

For the first time, we provide a matrix formula for second-order covariances of maximum likelihood estimates in heteroskedastic generalized linear models, thus generalizing the results of Cordeiro (2004 Cordeiro , G. M. ( 2004 ). Second-order covariance matrix of maximum likelihood estimates in generalized linear models . Statist. Probab. Lett. 66 : 153 – 160 .[Crossref], [Web of Science ®] , [Google Scholar]) and Cordeiro et al. (2006 Cordeiro , G. M. , Barroso , L. P. , Botter , D. A. (2006). Covariance matrix formula for generalized linear models with unknown dispersion. Commun. Statist. Theor. Meth. 35:113–120.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) related to the generalized linear models with known and unknown dispersion parameter, respectively. The covariance matrix formula does not involve cumulants of log-likelihood derivatives and can be easily obtained using simple matrix operations. We apply our main result to a simple model. Some simulations show that the second-order covariances can be quite pronounced in small to moderate samples. The usual covariances of the maximum likelihood estimates can be corrected by these second-order covariances.     

A Computer Program to Improve LR Tests for Generalized Linear Models

Generalized linear models enable the fitting of models to a wide range of data types. These models are based on exponential dispersion distributions. Improved likelihood ratio tests for these models were developed by Cordeiro (1983 Cordeiro , G. M. (1983). Improved likelihood ratio statistics for generalized linear models. Journal of the Royal Statistical Society, Series B: Methodological 45:404–413. [Google Scholar])Cordeiro (1987 Cordeiro , G. M. ( 1987 ). On the corrections to the likelihood ratio statistics . Biometrika 74 : 265 – 274 .[Crossref], [Web of Science ®] , [Google Scholar]). We present a simple R program source for calculating Bartlett corrections to improve likelihood ratio tests in these models. The program was tested on some special models, confirming all of the previously reported numerical results for the Bartlett corrections.     

Asymptotic Skewness in Exponential Family Nonlinear Models

In this article, we give an asymptotic formula of order n ^?1/2, where n is the sample size, for the skewness of the distributions of the maximum likelihood estimates of the pa-ra-meters in exponencial family nonlinear models. We generalize the result by Cordeiro and Cordeiro (2001 Cordeiro , H. H. , Cordeiro , G. M. ( 2001 ). Skewness for parameters in generalized linear models . Commun. Statist. Theor. Meth. 30 : 1317 – 1334 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The formula is given in matrix notation and is very suitable for computer implementation and to obtain closed form expressions for a great variety of models. Some special cases and two applications are discussed.     

Bias Correction in Generalized Nonlinear Models with Dispersion Covariates

We derive general formulae for the second-order biases of maximum likelihood estimates of the parameters in generalized nonlinear models with dispersion covariates. This result generalizes previous work by Botter and Cordeiro (1998 Botter , D. A. , Cordeiro , G. M. ( 1998 ). Improved estimates for generalized linear models with dispersion covariates . J. Statist. Comput. Simul. 62 : 91 – 104 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and Cordeiro and McCullagh (1991 Cordeiro , G. M. , McCullagh , P. ( 1991 ). Bias correction in generalized linear models . J. Roy Statist. Soc. B 53 : 629 – 643 . [Google Scholar]). The practical use of such bias corrections is illustrated in a simulation study.     

A Broad Class of Partially Specified Autoregressions on Multi-Casting Data

Extending the bifurcating autoregressive (BAR) process (cf. Cowan and Staudte, 1986 Cowan , R. , Staudte , R. G. ( 1986 ). The bifurcating autoregression model in cell lineage studies . Biometrics 42 : 769 – 783 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) to multi-casting (multi-splitting) data, Hwang and Choi (2009 Hwang , S. Y. , Choi , M. S. ( 2009 ). Modeling and large sample estimation for multi-casting autoregression . Statist. Prob. Lett. 79 : 1943 – 1950 .[Crossref], [Web of Science ®] , [Google Scholar]) introduced multi-casting autoregression (MCAR, for short) defined on multi-casting tree structured data. This article is concerned with the case when the MCAR model is partially specified only through conditional mean and variance without directly imposing autoregressive (AR) structure. The resulting class of models will be referred to as P-MCAR (partially specified MCAR). The P-MCAR considerably enlarges the class of multi-casting models including (as special cases) MCAR, random coefficient MCAR, conditionally heteroscedastic multi-casting models and binomial-thinning processes. Moment structures for this broad P-MCAR class are investigated. Least squares (LS) estimation method is discussed and asymptotic relative efficiency (ARE) of the generalized-LS over ordinary-LS is obtained in a closed form. A simulation study is conducted to illustrate results.     

Bias-Corrected Maximum Likelihood Estimators in Nonlinear Heteroscedastic Models

Nonlinear heteroscedastic models are widely used in econometrics and statistical applications. We derive matrix formulae for the second-order biases of the maximum likelihood estimators of the parameters in the mean and variance response which generalize previous results by Cook et al. (1986 Cook , D. R. , Tsai , C. L. , Wei , B. C. ( 1986 ). Bias in nonlinear regression . Biometrika 73 : 615 – 623 .[Crossref], [Web of Science ®] , [Google Scholar]) and Cordeiro (1993 Cordeiro , G. M. ( 1993 ). Bartlett corrections and bias correction for two heteroscedastic regression models . Commun. Statist. Theor. Meth. 22 : 169 – 188 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The biases of the estimators are easily obtained as vectors of regression coefficients from suitable weighted linear regressions. The practical use of such biases is illustrated in a simulation study and in an application to a real data set.     

Regression Analysis with Covariates Missing at Random: A Piece-wise Nonparametric Model for Missing Covariates

Statistical analysis for the regression model f _β(y | x, z) with missing values in the covariate vector X requires modeling of the covariate distribution g(x | z). Likelihood methods, including Ibrahim (1990 Ibrahim , J. G. ( 1990 ). Incomplete data in generalized linear models . J. Amer. Statist. Assoc. 85 : 765 – 769 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Chen (2004 Chen , H. Y. (2004). Nonparametric and semiparametric models for missing covariates in parametric regression. J. Amer. Statist. Assoc. 99:1176–1189.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), and Zhao (2005 Zhao , Y. ( 2005 ). Design and Efficient Estimation in Regression Analysis with Missing Data in Two-Phase Studies. Ph.D. thesis , University of Waterloo . [Google Scholar]), need either X or Z to be discrete. This article considers extending the likelihood methods to deal with cases where both X and Z may be continuous. We propose modeling the covariate distribution g(x | z) using a piece-wise nonparametric model, then a maximum likelihood estimate (MLE) of β can be computed following the maximum likelihood estimating procedure of Chen (2004 Chen , H. Y. (2004). Nonparametric and semiparametric models for missing covariates in parametric regression. J. Amer. Statist. Assoc. 99:1176–1189.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) or Zhao (2005 Zhao , Y. ( 2005 ). Design and Efficient Estimation in Regression Analysis with Missing Data in Two-Phase Studies. Ph.D. thesis , University of Waterloo . [Google Scholar]). The resulting estimation method is easy to implement and the asymptotic properties of the MLE follow under certain conditions. Extensive simulation studies for different models indicate that the proposed method is acceptable for practical implementation. A real data example is used to illustrate the method.     

Empirical Likelihood Ratio for Linear Transformation Models with Doubly Censored Data

Double censoring arises when T represents an outcome variable that can only be accurately measured within a certain range, [L, U], where L and U are the left- and right-censoring variables, respectively. When L is always observed, we consider the empirical likelihood inference for linear transformation models, based on the martingale-type estimating equation proposed by Chen et al. (2002 Chen , K. , Jin , Z. , Ying , Z. ( 2002 ). Semiparametric analysis of transformation models with censored data . Biometrika 89 : 659 – 668 .[Crossref], [Web of Science ®] , [Google Scholar]). It is demonstrated that both the approach of Lu and Liang (2006 Lu , W. , Liang , Y. ( 2006 ). Empirical likelihood inference for linear transformation models . Journal of Multivariate Analysis 97 : 1586 – 1599 .[Crossref], [Web of Science ®] , [Google Scholar]) and that of Yu et al. (2011 Yu , W. , Sun , Y. , Zheng , M. ( 2011 ). Empirical likelihood method for linear transformation models . Annals of the Institute of Statistical Mathematics 63 : 331 – 346 .[Crossref], [Web of Science ®] , [Google Scholar]) can be extended to doubly censored data. Simulation studies are conducted to investigate the performance of the empirical likelihood ratio methods.     

SECOND-ORDER ASYMPTOTICS FOR SCORE TESTS IN HETEROSKEDASTIC t REGRESSION MODELS

ABSTRACT

This paper develops corrected score tests for heteroskedastic t regression models, thus generalizing results by Cordeiro, Ferrari and Paula[1] Cordeiro, G.M., Ferrari, S.L.P. and Paula, G.A. 1993. Improved Score Tests for Generalized Linear Models. Journal of the Royal Statistical Society B, 55: 661–674.  [Google Scholar] and Cribari-Neto and Ferrari[2] Cribari-Neto, F. and Ferrari, S.L.P. 1995. Second-order Asymptotics for Score Tests in Generalised Linear Models. Biometrika, 82: 426–432. [Crossref], [Web of Science ®] , [Google Scholar] for normal regression models and by Ferrari and Arellano-Valle[3] Ferrari, S.L.P. and Arellano-Valle, R. 1996. Modified Likelihood Ratio and Score Tests in Linear Regression Models Using the t Distribution. Brazilian Journal of Probability and Statistics, 10: 15–33.  [Google Scholar] for homoskedastic t regression models. We present, in matrix notation, Bartlett-type correction formulae to improve score tests in this class of models. The corrected score statistics have a chi-squared distribution to order n ^?1, where n is the sample size. We apply our main result to a few special models and present simulation results comparing the performance of the usual score tests and their corrected versions.     

Semiparametric analysis of transformation models with doubly censored data

Double censoring arises when T represents an outcome variable that can only be accurately measured within a certain range, [L, U], where L and U are the left- and right-censoring variables, respectively. In this note, using Martingale arguments of Chen et al. [3 Chen, K., Jin, Z. and Ying, Z. 2002. Semiparametric analysis of transformation models with censored data. Biometrika, 89: 659–668. [Crossref], [Web of Science ®] [Google Scholar]], we propose an estimator (denoted by ?β) for estimating regression coefficients of transformation model when L is always observed. Under Cox proportional hazards model, the proposed estimator is equivalent to the partial likelihood estimator for left-truncated and right-censored data if the left-censoring variables L were regarded as left-truncated variables. In this case, the estimator ?β can be obtained by the standard software. A simulation study is conducted to investigate the performance of ?β. For the purpose of comparison, the simulation study also includes the estimator proposed by Cai and Cheng [2 Cai, T. and Cheng, S. 2004. Semiparametric regression analysis for doubly censored data. Biometrika, 91: 277–290. [Crossref], [Web of Science ®] [Google Scholar]] for the case when L and U are always observed.     

Second-Order Powers of a Class of Tests in the Presence of a Nuisance Parameter

The second-order local powers of a broad class of asymptotic chi-squared tests are considered in a composite case where both the parameter of interest and the nuisance parameter are possibly multidimensional for which no assumption has been made regarding global parametric orthogonality or curved exponentiality. The main result is that the second-order (point-by-point) local power identity holds if approximate third cumulants of a square-root version of the (modified) test statistic in the class vanish up to the second-order, which is an extension of Kakizawa (2010a Kakizawa , Y. ( 2010a ). Second-order power comparison of tests . Commun. Statist. Theor. Meth. 39 : 1424 – 1436 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) in the absence of the nuisance parameter. It is also shown that in the presence of the nuisance parameter, such a third cumulant condition does not always imply the second-order local unbiasedness of the resulting test. Then, the adjusted likelihood ratio test by Mukerjee (1993b Mukerjee , R. ( 1993b ). An extension of the conditional likelihood ratio test to the general multiparameter case . Ann. Inst. Statist. Math. 45 : 759 – 771 .[Crossref], [Web of Science ®] , [Google Scholar]) can be interpreted as the second-order local unbiased modification after applying the third cumulant condition.     

On Joint Determination of the Number of States and the Number of Variables in Markov-Switching Models: A Monte Carlo Study

In this article, we examine the performance of two newly developed procedures that jointly select the number of states and variables in Markov-switching models by means of Monte Carlo simulations. They are Smith et al. (2006 Smith , A. , Naik , P. A. , Tsai , C. ( 2006 ). Markov-switching model selection using Kullback–Leibler divergence . Journal of Econometrics 134 ( 2 ): 553 – 577 .[Crossref], [Web of Science ®] , [Google Scholar]) and Psaradakis and Spagnolo (2006 Psaradakis , Z. , Spagnolo , N. ( 2006 ). Joint determination of the state dimension and autoregressive order for models with Markov regime switching . Journal of Time Series Analysis 27 ( 2 ): 753 – 766 .[Crossref], [Web of Science ®] , [Google Scholar]), respectively. The former develops Markov switching criterion (MSC) designed specifically for Markov-switching models, while the latter recommends the use of standard complexity-penalised information criteria (BIC, HQC, and AIC) in joint determination of the state dimension and the autoregressive order of Markov-switching models. The Monte Carlo evidence shows that BIC outperforms MSC while MSC and HQC are preferable over AIC.     

The Identification of Potential Longitudinal Biomarkers and Measurements of Effectiveness for Biomarkers as Surrogates in Multivariate Survival Data

A proposed method based on frailty models is used to identify longitudinal biomarkers or surrogates for a multivariate survival. This method is an extention of earlier models by Wulfsohn and Tsiatis (1997 Wulfsohn , M. S. , Tsiatis , A. A. ( 1997 ). A joint model for survival and longitudinal data measured with error . Biometrics 53 ( 1 ): 330 – 339 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) and Song et al. (2002 Song , X. , Davidian , M. , Tsiatis , A. A. ( 2002 ). A Semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data . Biometrics 58 ( 4 ): 742 – 753 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). In this article, similar to Henderson et al. (2002 Henderson , R. , Diggle , P. J. , Dobson , A. ( 2002 ). Identification and efficacy of longitudinal markers for survival . Biostatistics 3 ( 1 ): 33 – 50 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), a joint likelihood function combines the likelihood functions of the longitudinal biomarkers and the multivariate survival times. We use simulations to explore how the number of individuals, the number of time points per individual and the functional form of the random effects from the longitudianl biomarkers influence the power to detect the association of a longitudinal biomarker and the multivariate survival time. The proposed method is illustrate by using the gastric cancer data.     

Using Frailty Models to Identify the Longitudinal Biomarkers in Survival Analysis

We introduce a score test to identify longitudinal biomarkers or surrogates for a time to event outcome. This method is an extension of Henderson et al. (2000 Henderson , R. , Diggle , P. , Dobson , A. ( 2000 ). Joint modelling of longitudinal measurements and event time data . Biostatistics 1 ( 4 ): 465 – 480 .[Crossref], [PubMed] , [Google Scholar], 2002 Henderson , R. , Diggle , P. , Dobson , A. ( 2002 ). Identification and efficacy of longitudinal markers for survival . Biostatistics 3 ( 1 ): 33 – 50 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). In this article, a score test is based on a joint likelihood function which combines the likelihood functions of the longitudinal biomarkers and the survival times. Henderson et al. (2000 Henderson , R. , Diggle , P. , Dobson , A. ( 2000 ). Joint modelling of longitudinal measurements and event time data . Biostatistics 1 ( 4 ): 465 – 480 .[Crossref], [PubMed] , [Google Scholar], 2002 Henderson , R. , Diggle , P. , Dobson , A. ( 2002 ). Identification and efficacy of longitudinal markers for survival . Biostatistics 3 ( 1 ): 33 – 50 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) assumed that the same random effect exists in the longitudinal component and in the Cox model and then they can derive a score test to determine if a longitudinal biomarker is associated with time to an event. We extend this work and our score test is based on a joint likelihood function which allows other random effects to be present in the survival function.

Considering heterogeneous baseline hazards in individuals, we use simulations to explore how the factors can influence the power of a score test to detect the association of a longitudinal biomarker and the survival time. These factors include the functional form of the random effects from the longitudinal biomarkers, in the different number of individuals, and time points per individual. We illustrate our method using a prothrombin index as a predictor of survival in liver cirrhosis patients.     

A finite-sample sensitivity analysis of the Dickey–Fuller test under local-to-unity detrending

In recent research, Elliott et al. (1996) Elliott, G. 1996. Efficient tests for an autoregressive unit root. Econometrica, 64: 813–836. [Crossref], [Web of Science ®] [Google Scholar] have shown the use of local-to-unity detrending via generalized least squares (GLS) to substantially increase the power of the Dickey–Fuller (1979) unit root test. In this paper the relationship between the extent of detrending undertaken, determined by the detrending parameter &art1;, and the power of the resulting GLS-based Dickey–Fuller (DF-GLS) test is examined. Using Monte Carlo simulation it is shown that the values of &art1; suggested by Elliott et al. (1996) Elliott, G. 1996. Efficient tests for an autoregressive unit root. Econometrica, 64: 813–836. [Crossref], [Web of Science ®] [Google Scholar] on the basis of a limiting power function seldom maximize the power of the DF-GLS test for the finite samples encountered in applied research. This result is found to hold for the DF-GLS test including either an intercept or an intercept and a trend term. An empirical examination of the order of integration of the UK household savings ratio illustrates these findings, with the unit root hypothesis rejected using values of &art1; other than that proposed by Elliott et al. (1996) Elliott, G. 1996. Efficient tests for an autoregressive unit root. Econometrica, 64: 813–836. [Crossref], [Web of Science ®] [Google Scholar].     

Additive Mixed Effect Model for Clustered Doubly Censored Data

Cai and Zeng (2011 Cai, J. and Zeng, D. 2011. Additive mixed effect model for clustered failure time data. Biometrics, 67(4): 1340–1351. [Crossref], [PubMed] , [Google Scholar]) proposed an additive mixed effect model to analyze clustered right-censored data. In this article, we demonstrate that the approach of Cai and Zeng (2011 Cai, J. and Zeng, D. 2011. Additive mixed effect model for clustered failure time data. Biometrics, 67(4): 1340–1351. [Crossref], [PubMed] , [Google Scholar]) can be extended to clustered doubly censored data. Furthermore, when both left- and right-censoring variables are always observed, we propose alternative estimators using the approach of Cai and Cheng (2004 Cai, T. and Cheng, S. C. 2004. Semiparametric regression analysis for doubly censored data. Biometrika, 91: 277–290. [Crossref], [Web of Science ®] , [Google Scholar]). A simulation study is conducted to investigate the performance of the proposed estimators.     

A Note on Importance Resampling for Multi-Dimensional Statistics

Johns (1988 Johns , M. V. (1988). Importance sampling for bootstrap confidence intervals. Journal of the American Statistical Association 83:709–714.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Davison (1988 Davison , A. C. ( 1988 ). Discussion of paper by D. V. Hinkley . Journal of the Royal Statistical Society Series B 50 : 356 – 357 . [Google Scholar]), and Do and Hall (1991 Do , K. A. , Hall , P. ( 1991 ). On importance sampling for the bootstrap . Biometrika 78 : 161 – 167 .[Crossref], [Web of Science ®] , [Google Scholar]) used importance sampling for calculating bootstrap distributions of one-dimensional statistics. Realizing that their methods can not be extended easily to multi-dimensional statistics, Fuh and Hu (2004 Fuh , C. D. , Hu , I. ( 2004 ). Efficient importance sampling for events of moderate deviations with applications . Biometrika 91 : 471 – 490 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed an exponential tilting formula for statistics of multi-dimension, which is optimal in the sense that the asymptotic variance is minimized for estimating tail probabilities of asymptotically normal statistics. For one-dimensional statistics, Hu and Su (2008 Hu , J. , Su , Z. ( 2008 ). Adaptive resampling algorithms for estimating bootstrap distributions . Journal of Statistical Planning and Inference 138 ( 6 ): 1763 – 1777 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed a multi-step variance minimization approach that can be viewed as a generalization of the two-step variance minimization approach proposed by Do and Hall (1991 Do , K. A. , Hall , P. ( 1991 ). On importance sampling for the bootstrap . Biometrika 78 : 161 – 167 .[Crossref], [Web of Science ®] , [Google Scholar]). In this article, we generalize the approach of Hu and Su (2008 Hu , J. , Su , Z. ( 2008 ). Adaptive resampling algorithms for estimating bootstrap distributions . Journal of Statistical Planning and Inference 138 ( 6 ): 1763 – 1777 .[Crossref], [Web of Science ®] , [Google Scholar]) to multi-dimensional statistics, which applies to general statistics and does not resort to asymptotics. Empirical results on a real survival data set show that the proposed algorithm provides significant computational efficiency gains.     

More Efficient Tests Robust to Heteroskedasticity of Unknown Form

ABSTRACT

In the presence of heteroskedasticity of unknown form, the Ordinary Least Squares parameter estimator becomes inefficient, and its covariance matrix estimator inconsistent. Eicker (1963 Eicker , B. ( 1963 ). Limit theorems for regression with unequal and dependant errors . Ann. Math. Statist. 34 : 447 – 456 .[Crossref] , [Google Scholar]) and White (1980 White , H. ( 1980 ). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity . Econometrica 48 : 817 – 838 .[Crossref], [Web of Science ®] , [Google Scholar]) were the first to propose a robust consistent covariance matrix estimator, that permits asymptotically correct inference. This estimator is widely used in practice. Cragg (1983 Cragg , J. G. ( 1983 ). More efficient estimation in the presence of heteroskedasticity of unknown form . Econometrica 51 : 751 – 63 .[Crossref], [Web of Science ®] , [Google Scholar]) proposed a more efficient estimator, but concluded that tests basd on it are unreliable. Thus, this last estimator has not been used in practice. This article is concerned with finite sample properties of tests robust to heteroskedasticity of unknown form. Our results suggest that reliable and more efficient tests can be obtained with the Cragg estimators in small samples.     

Standard Exponential Cure Rate Model with Informative Censoring

In this article, we consider the standard cure rate model proposed by Boag (1949 Boag , J. W. ( 1949 ). Maximum likelihood estimates of the proportion of patients cured by cancer therapy . J. Roy. Statist. Soc. B 11 : 15 – 53 . [Google Scholar]) and Berkson and Gage (1952 Berkson , J. , Gage , R. ( 1952 ). Survival curve for cancer patients following treatment . J. Amer. Statist. Assoc. 47 : 501 – 515 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). We present a new definition of informative censoring similar to Lawless (1982 Lawless , J. F. ( 1982 ). Statistical Models and Methods for Lifetime Data . New York : Wiley .[Crossref] , [Google Scholar]) and the corresponding likelihood function. Under informative censoring, we obtain the Fisher information matrix of the exponential standard cure rate model. We verify, with simulated data, the impact caused by informative censoring in the coverage probabilities and in the lengths of asymptotic confidence intervals of the parameters of interest. An example with real data is analyzed.     

DSGE Models with Student-t Errors

This paper deals with Dynamic Stochastic General Equilibrium (DSGE) models under a multivariate student-t distribution for the structural shocks. Based on the solution algorithm of Klein (2000) and the gamma-normal representation of the t-distribution, the TaRB-MH algorithm of Chib and Ramamurthy (2010 Chib , S. , Ramamurthy , S. ( 2010 ). Tailored randomized block MCMC methods with application to DSGE models . Journal of Econometrics 108 : 19 – 38 .[Crossref], [Web of Science ®] , [Google Scholar]) is used to estimate the model. A technique for estimating the marginal likelihood of the DSGE student-t model is also provided. The methodologies are illustrated first with simulated data and then with the DSGE model of Ireland (2004 Ireland , P. N. ( 2004 ). Technology shocks in the new keynesian model . Review of Economics and Statistics 86 ( 4 ): 923 – 936 .[Crossref], [Web of Science ®] , [Google Scholar]) where the results support the t-error model in relation to the Gaussian model.