Global identification of the dynamic shock-error model

The paper gives (necessary and sufficient) conditions for the global identifiability of dynamic regression models with errors in the variables. The conditions are simple counting rules combining the order parameters of a model. They are counterparts of conditions previously established for local identifiability in a series of papers.     

Bayesian estimation of the switching regression model with autocorrelated errors

In this paper, we make a Bayesian analysis of the switching (two-phase) regression model when the subset of the regression coefficients shifts and the error terms are generated by a first-order autoregressive process. The posterior distributions of the shift point and other parameters are derived, and some numerical studies are performed. From the numerical studies, we see that the shift point is accurately estimated when the shift of the regression coefficient is relatively large. Also, the conditional distributions of the autocorrelation and regression coefficients on the shift point are compared with the marginal ones.     

The behaviour of inconsistent instrumental variables estimators in dynamic systems with autocorrelated errors

To appropriately interpret time-series evidence when empirical relationships are incorrectly formulated, a general mis-specification framework is required. A linear, stationary, dynamic, simultaneous system with autoregressive errors is postulated to investigate instrumental variables ables estimators when the instruments are unknowingly correlated with the equation errors. The approach uses control variates (Hendry and Harrison, Journal of Econometrics, July 1974) to develop asymptotic distributions and exact moments for approximations to the econometric estimators. The accuracy of the asymptotic results for finite sample moments is corroborated by simulation. The analysis highlights the need for care in interpreting estimated equations and tests for predictive failure.     

z-Transform and identification of linear econometric models with autocorrelated errors

The paper consists of two main parts. In the first part we derive the solution of systems of linear stochastic difference equations by means of thez-transform. In the second part thisz-transform is used to treat the problem of identification of linear econometric systems (the term econometric is used to stress the special aspects of the identification problem dealt with in econometrics). It is shown, that under suitable restrictions observationally equivalent structures are related by unimodular matrices. Using this result, we state (rank-) conditions which ensure, that the unimodular matrices are constant, such that the classical econometric identification theorems can be applied. These conditions are given for stationary errors in the general case as well as in the MA, AR and ARMA case.     

Estimating the autocorrelated error model with trended data

A Monte Carlo study of the small sample properties of various estimators of the linear regression model with first-order autocorrelated errors. When independent variables are trended, estimators using Ttransformed observations (Prais-Winsten) are much more efficient than those using T–1 (Cochrane–Orcutt). The best of the feasible estimators isiterated Prais-Winsten using a sum-of-squared-error minimizing estimate of the autocorrelation coefficient ?. None of the feasible estimators performs well in hypothesis testing; all seriously underestimate standard errors, making estimated coefficients appear to be much more significant than they actually are.     

An examination of two-step estimators for models with lagged dependent variables and autocorrelated errors

The possible roles of the Durbin Equation and the first observation correction in improving the efficiency of parameter estimates in the lagged dependent variables-serial correlation model are examined. Unconstrained estimation of the Durbin Equation results in an estimate of ρ which is inefficient and its use in feasible generalized least squares does not provide asymptotically efficient estimates. Evidently, the first observation correction is a very important determinant of small sample properties in the present model. Asymptotically inefficient estimators which use a first observation correction frequently outperform Hatanaka's asymptotically efficient estimator in finite samples, essentially because it does not use the first observation.     

Identification in models with autoregressive errors

Consider the model

$\text{A(L)x}{}_{\mathrm{t}}\text{=B(L)y}{}_{\mathrm{t}}\text{+C(L)z}{}_{\mathrm{t}}\text{=u}{}_{\mathrm{t}}\text{, t=1,…,T}$

, where

$\text{A(L)=(B(L):C(L))}$

is a matrix of polynomials in the lag operator so that $\text{L}{}^{\mathrm{r}}\text{x}{}_{\mathrm{t}}\text{=x}{}_{\mathrm{t?r}}$, and $\text{y}{}_{\mathrm{t}}$ is a vector of n endogenous variables,

$\text{B(L)=}\text{∑}\text{s=0}\text{k}\text{B}{}_{\mathrm{s}}\text{L}{}^{\mathrm{s}}$

$\text{B}{}_0\text{I}{}_{\mathrm{n}}$, and the remaining $\text{B}{}_{\mathrm{s}}$ are $\text{n × n}$ square matrices,

$\text{C(L)=}\text{∑}\text{s=0}\text{k}\text{C}{}_{\mathrm{s}}\text{L}{}^{\mathrm{s}}$

, and $\text{C}{}_{\mathrm{s}}$ is $\text{n × m}$.Suppose that $\text{u}{}_{\mathrm{t}}$ satisfies

$\text{R(L)u}{}_{\mathrm{t}}\text{=e}{}_{\mathrm{t}}$

, where

$\text{R(L)=}\text{∑}\text{s=0}\text{r}\text{R}{}_{\mathrm{s}}\text{L}{}^{\mathrm{s}}$

, $\text{R}{}_0\text{=I}{}_{\mathrm{n}}$, and $\text{R}{}_{\mathrm{s}}$ is a $\text{n × n}$ square matrix. $\text{e}{}_{\mathrm{t}}$ may be white noise, or generated by a vector moving average stochastic process.Now writing

$\text{Ψ(L)=R(L)A(L)}$

, it is assumed that ignoring the implicit restrictions which follow from eq. (1), $\text{Ψ(L)}$ can be consistently estimated, so that if the equation

$\text{Ψ(L)x}{}_{\mathrm{t}}\text{=e}{}_{\mathrm{t}}$

has a moving average error stochastic process, suitable conditions [see E.J. Hannan] for the identification of the unconstrained model are satisfied, and that the appropriate conditions (lack of multicollinearity) on the data second moments matrices discussed by Hannan are also satisfied. Then the essential conditions for identification of the A(L) and R(L) can be considered by requiring that for the true $\text{Ψ(L)}$ eq. (1) has a unique solution for $\text{A(L)}$ and $\text{R(L)}$.There are three types of lack of identification to be distinguished. In the first there are a finite number of alternative factorisations. Apart from a factorisation condition which will be satisfied with probability one a necessary and sufficient condition for lack of identification is that $\text{A(L)}$ has a latent root λ in the sense that for some non-zero vector β,

$\text{β′A(λ)=0}$

.The second concept of lack of identification corresponds to the Fisher conditions for local identifiability on the derivatives of the constraints. It is shown that a necessary and sufficient condition that the model is locally unidentified in this sense is that R(L) and A(L) have a common latent root, i.e., that for some vectors δ and β,

$\text{R(λ)δ=0 and β′A(λ)=0}$

.Firstly it is shown that only if further conditions are satisfied will this lead to local unidentifiability in the sense that there are solutions of the equation

$\text{Ψ(z)=R(z)A(z)}$

in any neighbourhood of the true values.     

Small sample properties of maximum likelihood versus generalized method of moments based tests for spatially autocorrelated errors

Many applied researchers have to deal with spatially autocorrelated residuals (SAR). Available tests that identify spatial spillovers as captured by a significant SAR parameter, are either based on maximum likelihood (MLE) or generalized method of moments (GMM) estimates. This paper illustrates the properties of various tests for the null hypothesis of a zero SAR parameter in a comprehensive Monte Carlo study. The main finding is that Wald tests generally perform well regarding both size and power even in small samples. The GMM-based Wald test is correctly sized even for non-normally distributed disturbances and small samples, and it exhibits a similar power as its MLE-based counterpart. Hence, for the applied researcher the GMM Wald test can be recommended, because it is easy to implement.     

A note on a Bayesian estimator in an autocorrelated error model

For a simple autocorrelated error model studied by Fomby and Guilkey (1978) we demonstrate that there is a Bayesian counterpart to the class of sampling theory pre-test estimators.     

Copula-based multivariate GARCH model with uncorrelated dependent errors

Multivariate GARCH (MGARCH) models are usually estimated under multivariate normality. In this paper, for non-elliptically distributed financial returns, we propose copula-based multivariate GARCH (C-MGARCH) model with uncorrelated dependent errors, which are generated through a linear combination of dependent random variables. The dependence structure is controlled by a copula function. Our new C-MGARCH model nests a conventional MGARCH model as a special case. The aim of this paper is to model MGARCH for non-normal multivariate distributions using copulas. We model the conditional correlation (by MGARCH) and the remaining dependence (by a copula) separately and simultaneously. We apply this idea to three MGARCH models, namely, the dynamic conditional correlation (DCC) model of Engle [Engle, R.F., 2002. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics 20, 339–350], the varying correlation (VC) model of Tse and Tsui [Tse, Y.K., Tsui, A.K., 2002. A multivariate generalized autoregressive conditional heteroscedasticity model with time-varying correlations. Journal of Business and Economic Statistics 20, 351–362], and the BEKK model of Engle and Kroner [Engle, R.F., Kroner, K.F., 1995. Multivariate simultaneous generalized ARCH. Econometric Theory 11, 122–150]. Empirical analysis with three foreign exchange rates indicates that the C-MGARCH models outperform DCC, VC, and BEKK in terms of in-sample model selection and out-of-sample multivariate density forecast, and in terms of these criteria the choice of copula functions is more important than the choice of the volatility models.     

Robustness and the general dynamic factor model with infinite-dimensional space: Identification,estimation, and forecasting

General dynamic factor models have demonstrated their capacity to circumvent the curse of dimensionality in the analysis of high-dimensional time series and have been successfully considered in many economic and financial applications. As second-order models, however, they are sensitive to the presence of outliers—an issue that has not been analyzed so far in the general case of dynamic factors with possibly infinite-dimensional factor spaces (Forni et al. 2000, 2015, 2017). In this paper, we consider this robustness issue and study the impact of additive outliers on the identification, estimation, and forecasting performance of general dynamic factor models. Based on our findings, we propose robust versions of identification, estimation, and forecasting procedures. The finite-sample performance of our methods is evaluated via Monte Carlo experiments and successfully applied to a classical data set of 115 US macroeconomic and financial time series.     

R-estimation of the parameters of a multiple regression model with measurement errors

We consider the theory of R-estimation of the regression parameters of a multiple regression models with measurement errors. Using the standard linear rank statistics, R-estimators are defined and their asymptotic properties are studied as robust alternatives to the least squares estimator. This paper fills the gap of the rank theory for the estimation of regression parameters with measurement error models. Some simulation results are presented to show the effectiveness of the R-estimators.     

Estimation of parameters of parallelism model with elliptically distributed errors

In this paper we consider some improved estimators of the intercept and slope parameters in a parallelism model with errors belonging to a sub-class of elliptically contoured distributions. We derive the exact bias, MSE matrices and quadratic risk expressions for these estimators. It is shown that the dominance properties of these estimators are the same as under normal theory. Further, it is shown that the shrinkage factor of the Stein estimators is robust with respect to the regression parameters and unknown mixing distributions.     

The dynamic invariant multinomial probit model: Identification,pretesting and estimation

We present a new specification for the multinomial multiperiod probit model with autocorrelated errors. In sharp contrast with commonly used specifications, ours is invariant with respect to the choice of a baseline alternative for utility differencing. It also nests these standard models as special cases, allowing for data-based selection of the baseline alternatives for the latter. Likelihood evaluation is achieved under an Efficient Importance Sampling (EIS) version of the standard GHK algorithm. Several simulation experiments highlight identification, estimation and pretesting within the new class of multinomial multiperiod probit models.     

The adequacy of asymptotic approximations in the near-integrated autoregressive model with dependent errors

We consider the normalized least squares estimator of the parameter in a nearly integrated first-order autoregressive model with dependent errors. In a first step we consider its asymptotic distribution as well as asymptotic expansion up to order O_p(T⁻¹). We derive a limiting moment generating function which enables us to calculate various distributional quantities by numerical integration. A simulation study is performed to assess the adequacy of the asymptotic distribution when the errors are correlated. We focus our attention on two leading cases: MA(1) errors and AR(1) errors. The asymptotic approximations are shown to be inadequate as the MA root gets close to −1 and as the AR root approaches either −1 or 1. Our theoretical analysis helps to explain and understand the simulation results of Schwert (1989) and DeJong, Nankervis, Savin, and Whiteman (1992) concerning the size and power of Phillips and Perron's (1988) unit root test. A companion paper, Nabeya and Perron (1994), presents alternative asymptotic frameworks in the cases where the usual asymptotic distribution fails to provide an adequate approximation to the finite-sample distribution.     

The informative sample size for dynamic multiple equation systems with moving average errors

Abstract  The problem considered here, is that of finding suitable conditions for dynamic economic systems that exclude the existence of observationally equivalent structures. Here observational equivalence refers to equality of distributions or first and second moments of a small finite sample from the observable process. It is shown, that under these conditions we may act as if the lagged endogenous variables are nonrandom exogenous variables, when global identifiability is investigated.     

Expolynomial smoothing of autocorrelated time series

Summary In the following, causal pattern buried in autocorrelated noise is considered. The causal pattern may be described by models such as trends, polynomial trajectories, growing sines. Based on a new criterion — called expolynomial — estimators of coefficients of a polynomial model are obtained. Characteristic functions of the estimators are derived and the first two moments calculated. Continuous time series are briefly studied to show similarities between discrete and continuous observations. Popular exponential smoothing is a special case of the expolynomial smoothing.