ROBUST ESTIMATION FOR THE SPATIAL AUTOREGRESSIVE MODEL

A GMM approach for estimation of volatility and regression models when daily prices are subject to price limits

ABSTRACTUsing approximations of the score of the log-likelihood function, we derive moment conditions for estimating spatial regression models, starting with the spatial error model. Our approach results in computationally simple and robust estimators, such as a new moment estimator derived from the first-order approximation obtained by solving a quadratic moment equation, and performs similarly to existing generalized method of moments (GMM) estimators. Our estimator based on the second-order approximation resembles the GMM estimator proposed by Kelejian and Prucha in 1999. Hence, we provide an intuitive interpretation of their estimator. Additionally, we provide a convenient framework for computing the weighting matrix of the optimal GMM estimator. Heteroskedasticity robust versions of our estimators are also proposed. Furthermore, a first-order approximation for the spatial autoregressive model is considered, resulting in a computationally simple method of moment estimator. The performance of the considered estimators is compared in a Monte Carlo study.

In this article, we propose instrumental variables (IV) and generalized method of moments (GMM) estimators for panel data models with weakly exogenous variables. The model is allowed to include heterogeneous time trends besides the standard fixed effects (FE). The proposed IV and GMM estimators are obtained by applying a forward filter to the model and a backward filter to the instruments in order to remove FE, thereby called the double filter IV and GMM estimators. We derive the asymptotic properties of the proposed estimators under fixed T and large N, and large T and large N asymptotics where N and T denote the dimensions of cross section and time series, respectively. It is shown that the proposed IV estimator has the same asymptotic distribution as the bias corrected FE estimator when both N and T are large. Monte Carlo simulation results reveal that the proposed estimator performs well in finite samples and outperforms the conventional IV/GMM estimators using instruments in levels in many cases.

In this paper, we propose instrumental variables (IV) and generalized method of moments (GMM) estimators for panel data models with weakly exogenous variables. The model is allowed to include heterogeneous time trends besides the standard fixed effects. The proposed IV and GMM estimators are obtained by applying a forward filter to the model and a backward filter to the instruments in order to remove fixed effects, thereby called the double filter IV and GMM estimators. We derive the asymptotic properties of the proposed estimators under fixed T and large N, and large T and large N asymptotics where N and T denote the dimensions of cross section and time series, respectively. It is shown that the proposed IV estimator has the same asymptotic distribution as the bias corrected fixed effects estimator when both N and T are large. Monte Carlo simulation results reveal that the proposed estimator performs well in finite samples and outperforms the conventional IV/GMM estimators using instruments in levels in many cases.

Purpose The purpose of this study is to provide new empirical evidence on the impact of a variety of financial market forces on the ex post real cost of funds to corporations, namely, the ex post real interest rate yield on AAA-rated long-term corporate bonds in the USA. The study is couched within an open-economy loanable funds model, and it adopts annual data for the period 1973-2013, so that the results are current while being applicable only for the post-Bretton Woods era. The auto-regressive two-stage least squares (2SLS) and generalized method of moments (GMM) estimations reveal that the ex post real interest rate yield on AAA-rated long-term corporate bonds in the USA was an increasing function of the ex post real interest rate yields on six-month Treasury bills, seven-year Treasury notes, high-grade municipal bonds and the Moody’s BAA-rated corporate bonds, while being a decreasing function of the monetary base as a per cent of gross domestic product (GDP) and net financial capital inflows as a per cent of GDP. Finally, additional estimates reveal that the higher the budget deficit as a per cent of GDP, the higher the ex post real interest rate on AAA-rated long-term corporate bonds. Design/methodology/approach After developing an initial open-economy loanable funds model, the empirical dimension of the study involves auto-regressive, two-stage least squares and GMM estimates. The model is then expanded to include the federal budget deficit, and new AR/2SLS and GMM estimates are provided. Findings The AR/2SLS and GMM (generalized method of moments) estimations reveal that the ex post real interest rate yield on AAA-rated long-term corporate bonds in the USA was an increasing function of the ex post real interest rate yields on six-month Treasury bills, seven-year Treasury notes, high-grade municipal bonds and the Moody’s BAA-rated corporate bonds, while being a decreasing function of the monetary base as a per cent of GDP and net financial capital inflows as a per cent of GDP. Finally, additional estimates reveal that the higher the budget deficit as a per cent of GDP, the higher the ex post real interest rate on AAA-rated long -term corporate bonds. Originality/value The author is unaware of a study that adopts this particular set of real interest rates along with net capital inflows and the monetary base as a per cent of GDP and net capital inflows. Also, the data run through 2013. There have been only studies of deficits and real interest rates in the past few years.

We propose a generalized method of moments (GMM) estimator with optimal instruments for a probit model that includes a continuous endogenous regressor. This GMM estimator incorporates the probit error and the heteroscedasticity of the error term in the first-stage equation in order to construct the optimal instruments. The estimator estimates the structural equation and the first-stage equation jointly and, based on this joint moment condition, is efficient within the class of GMM estimators. To estimate the heteroscedasticity of the error term of the first-stage equation, we use the k-nearest neighbour (k-nn) non-parametric estimation procedure. Our Monte Carlo simulation shows that in the presence of heteroscedasticity and endogeneity, our GMM estimator outperforms the two-stage conditional maximum likelihood estimator. Our results suggest that in the presence of heteroscedasticity in the first-stage equation, the proposed GMM estimator with optimal instruments is a useful option for researchers.

Fully modified IV, GIVE and GMM estimation with possibly non-stationary regressors and instruments

This paper proposes a new generalized method of moments (GMM) estimator for spatial panel models with spatial moving average errors combined with a spatially autoregressive dependent variable. Monte Carlo results are given suggesting that the GMM estimator is consistent. The estimator is applied to English real estate price data.

Few estimation methods were discussed to handle the missing data problem in the panel data models. However, in the panel vector autoregressive (PVAR) model, there is no estimator to handle this problem. The traditional treatment in the case of incomplete data is to use the generalized method of moment (GMM) estimation based on only available data without imputation of the missing data. Therefore, this paper introduces a new GMM estimation for the PVAR model in case of incomplete data based on the mean imputation. Moreover, we make a Monte Carlo simulation study to study the efficiency of the proposed estimator. We compare between two GMM estimators based on the mean squared error (MSE) and relative bias (RB) criteria. The first is the GMM estimation based on the list-wise (LW) and the second is the GMM estimation using the mean imputation (MI) at multi-missing levels. The results showed that the MI estimator provides more efficiency than the LW estimator.

Recent works have proposed regression models which are invariant across data collection environments [24, 20, 11, 16, 8]. These estimators often have a causal interpretation under conditions on the environments and type of invariance imposed. One recent example, the Causal Dantzig (CD), is consistent under hidden confounding and represents an alternative to classical instrumental variable estimators such as Two Stage Least Squares (TSLS). In this work we derive the CD as a generalized method of moments (GMM) estimator. The GMM representation leads to several practical results, including 1) creation of the Generalized Causal Dantzig (GCD) estimator which can be applied to problems with continuous environments where the CD cannot be fit 2) a Hybrid (GCD-TSLS combination) estimator which has properties superior to GCD or TSLS alone 3) straightforward asymptotic results for all methods using GMM theory. We compare the CD, GCD, TSLS, and Hybrid estimators in simulations and an application to a Flow Cytometry data set. The newly proposed GCD and Hybrid estimators have superior performance to existing methods in many settings.

This paper considers a generalized method of moments(GMM) estimation for seasonal cointegration as the extension of Kleibergen (1999). We propose two iterative methods for the estimation according to whether parameters in the model are simultaneously estimated or not. It is shown that the GMM estimator coincides in form to a maximum likelihood estimator or a feasible two-step estimator. In addition, we derive its asymptotic distribution that takes the same form as that in Ahn and Reinsel (1994).

A limit theory for instrumental variables (IV) estimation that allows for possibly nonstationary processes was developed in Kitamura and Phillips (1992, Fully Modified IV, GIVE, and GMM Estimation with Possibly Non-stationary Regressors and Instruments, mimeo, Yale University). This theory covers a case that is important for practitioners, where the nonstationarity of the regressors may not be of full rank, and shows that the fully modified (FM) regression procedure of Phillips and Hansen (1990) is still applicable. FM. versions of the generalized method of moments (GMM) estimator and the generalized instrumental variables estimator (GIVE) were also developed, and these estimators (FM-GMM and FM-GIVE) were designed specifically to take advantage of potential stationarity in the regressors (or unknown linear combinations of them). These estimators were shown to deliver efficiency gains over FM-IV in the estimation of the stationary components of a model.This paper provides an overview of the FM-IV, FM-GMM, and FM-GIVE procedures and investigates the small sample properties of these estimation procedures by simulations. We compare the following five estimation methods: ordinary least squares, crude (conventional) IV, FM-IV, FM-GMM, and FM-GIVE. Our findings are as follows, (i) In terms of overall performance in both stationary and nonstationary cases, FM-IV is more concentrated and better centered than OLS and crude IV, though it has a higher root mean square error than crude IV due to occasional outliers, (ii) Among FM-IV, FM-GMM, and FM-GIVE, (a) when applied to the stationary coefficients, FM-GIVE generally outperforms FM-IV and FM-GMM by a wide margin, whereas the difference between the latter two is quite small when the AR roots of the stationary processes are rather large; and (b) when applied to the nonstationary coefficients, the three estimators are numerically very close. The performance of the FM-GIVE estimator is generally very encouraging.

This paper provides a first order asymptotic theory for generalized method of moments (GMM) estimators when the number of moment conditions is allowed to increase with the sample size and the moment conditions may be weak. Examples in which these asymptotics are relevant include instrumental variable (IV) estimation with many (possibly weak or uninformed) instruments and some panel data models that cover moderate time spans and have correspondingly large numbers of instruments. Under certain regularity conditions, the GMM estimators are shown to converge in probability but not necessarily to the true parameter, and conditions for consistent GMM estimation are given. A general framework for the GMM limit distribution theory is developed based on epiconvergence methods. Some illustrations are provided, including consistent GMM estimation of a panel model with time varying individual effects, consistent limited information maximum likelihood estimation as a continuously updated GMM estimator, and consistent IV structural estimation using large numbers of weak or irrelevant instruments. Some simulations are reported.

For panel data models with error components that are spatially correlated, the finite sample properties of alternative generalized method of moments (GMM) estimators are determined. We suggest using a continuously updated GMM estimator which is invariant to curvature altering transformations and which should improve small sample efficiency. A Monte Carlo study using a wide range of settings compares the small sample efficiency of various GMM approaches and the maximum likelihood estimator (MLE). The GMM estimators turn out to perform comparably to the MLE approach and even outperform the latter for complex weighting matrices and non-normally distributed errors.

GMM estimation of a maximum entropy distribution with interval data