Two-step Estimation Research Articles

Financial inclusion and stability in Ethiopia using bank-level data: A two-step system GMM estimation

Background This paper examines the impact of financial inclusion on bank stability within Ethiopian context, using panel data from 17 commercial banks over the period 2015-2023. Given the scarcity of research focused on the relationship between financial inclusion and bank stability in Ethiopia, this paper seeks to address a crucial gap by analyzing both conventional and digital aspects of financial inclusion in relation with bank stability. Methods A two-stage principal component analysis (PCA) was conducted to construct a composite financial inclusion index, integrating 10 conventional and 5 digital indicators. The study applied a two-step robust system generalized method of moments (GMM) to examine the effects of financial inclusion on bank stability, complemented by Granger causality testing to examine the directionality of this relationship. Results The result reveals a significant positive effect of financial inclusion on bank stability and Granger causality tests confirms a bi-directional relationship between financial inclusion and stability, indicating that improvements in financial inclusion foster greater stability and vice versa. Our results also highlight that while bank efficiency and GDP growth rate positively effect stability, total assets and income diversification exhibit detrimental effects. Conclusions It is essential to capitalize policy synergies to promote bank stability and to enhance financial inclusion through conventional and digital channels, while carefully managing the implications of risks associated with income diversification and asset distribution. Ensuring inclusive financial system is vital for maintaining bank stability, thus positioning it as a key priority for financial institutions.

A novel bounded ℤ-valued autoregressive model with its application on crime data

This paper tackles the challenge of autoregressive modelling for bounded Z -valued count time series, a topic that has been largely overlooked in the existing literature. To bridge this gap, we introduce the trinomial difference bounded Z -valued autoregressive (TDBZAR) model utilizing the trinomial difference thinning operator, and we explore its stochastic properties in detail. Two notable advantages of the TDBZAR model are that the incorporated trinomial difference thinning operator ensures the conditional expectation takes a linear form and allows its 1-step transition probability to be expressed as the convolution of two independent trinomial difference distributions. This significantly simplifies the computation of both the conditional least squares (CLS) and conditional maximum likelihood (CML) estimates, making them more practical and straightforward. Second, we examine the two-step CLS and CML estimates, establishing their asymptotic properties. Third, we compare the performance of these estimators through simulation studies. Finally, we apply the proposed model to a real dataset to investigate which type of crime (rape or sex offense) was more prevalent in police reports from January 1990 to December 1998.

Do Gender Norms Relax with Displacement?

Conflict-induced displacement is associated with loss of human and physical capital as well as psychological trauma. Households and social structures that produce and reproduce gender norms are disrupted. This paper investigates the extent to which gender norms relax in situations of displacement using predetermined indicators in the Colombia Demographic and Health Survey. Results of a two-step estimation involving kernel-based propensity score matching and multilevel linear regression models show that gender norms condoning violence against women relaxed with displacement, while those that limit women’s economic opportunities became more rigid. Findings also reveal a misalignment between attitudes and behaviors in specific domains of gender norms. Displaced women expressed lower support for patriarchy than non-displaced women, but they had less ability to make independent decisions over the use of contraceptives and over their own earnings. This implies that attitudes and behaviors do not always shift together, and, in displacement settings, do not always progress towards greater gender equality.

Does high volatility increase connectedness? A study of Asian equity markets

Financial market integration metrics based upon variance decompositions from vector autoregression (VAR) models have become quite popular, and prior literature shows how these metrics change over time. However, only a few studies explain what is driving connectedness to change. Using stock market volatilities from eight Asian equity markets, we postulate that volatility may be a driving force behind increased connectedness. We use a two-step VAR estimation procedure to analyze the dynamics between equity market volatility and connectedness. The first step estimates the VAR model over rolling windows; this yields a sequence of connectedness-measuring spillover indices. The second step re-estimates the VAR with the spillover index sequence as an endogenous variable. Impulse response analysis from the two-step procedure confirms that volatility shocks increase connectedness. The two-step procedure indicates that volatility shocks from Singapore, Hong Kong, and China affect connectedness the most, whereas Malaysia affects it the least. China’s direct effect on the volatilities of the other markets is small. However, since China has a considerable impact on connectedness, China may have an important indirect effect on the other markets by increasing the speed and ease with which volatility spreads from market to market.

A remaining useful life prediction method of aluminum electrolytic capacitor with adaptive degradation model selection

The degradation process of aluminum electrolytic capacitors(AECs) usually exhibits characteristics such as non-linearity and multi-stage. These degradation features lead to the difficulty to accurately predict the remaining useful life(RUL) of the whole degradation process of AECs. To address this, by dividing the capacitor degradation process into two stages, a two-stage RUL prediction method for AECs considering multiple degradation models is proposed in this paper.In the offline parameter estimation phase, the initial degradation model parameters of two stages are estimated using two-step maximum likelihood estimation combined with particle swarm optimization(PSO) algorithm. In the dynamic parameter updating phase, a sequential Bayesian method is used to update the model parameters. To select the optimal degradation model, an evaluation method based on historical RUL similarity is proposed to calculate the fitness of each model. Finally, the effectiveness of the method is verified on NASA’s accelerated degradation data set and several widely used methods are used for comparison. The experimental results show that the proposed method has higher accuracy, which proves the superiority of the method.

Determinants of Banks Profitability: Evidence from Nigeria Banking Industry

The study, investigated determinants of banks profitability in Nigeria. Other studies in this area concentrated on other profitability notions, this study incorporated Economic Value Added (EVA). Due to the presence of endogeneity problems inherent in most panel data sets of this nature we employed a two-step system GMM estimator to produce a robust estimation and account for the endogeneity problem in the data. The study revealed that bank size has a positive and significant impact on all measures of bank profitability with the exception of Net Interest Margin (NIM). This result remained consistent after macroeconomic factors were introduced in the model. The outcome also indicated that capital adequacy significantly explains bank profitability in Nigeria, with a sign change in the case of Economic Value Added (EVA) only. It was also evident that financial structure (FS) has a positive and significant impact on Return on Assets (ROA), NIM, and EVA but no evidence of significant relationship was found with ROE. The result indicated that liquidity risk (LR) has a negative and significant impact on ROA and EVA but failed to provide a significant impact on ROE and NIM. The result of the macroeconomic factors included in the model indicated that Gross Domestic Product (GDP) has a positive and significant impact on NIM and EVA while no significant relationship was found when related to ROA and ROE. Inflation as a macroeconomic variable showed a significant relationship with ROA and ROE with no evidence of significant relationship with NIM and EVA. It was concluded that the impact of firm specific factors and macroeconomic variables on the profitability of the Nigerian banking sector differ according to the profitability measure employed.

Female CEOs and Green Innovation: Evidence from Asian Firms

This study aims to examine how female CEOs influence green innovation and whether firm size moderates this connection. Our paper focuses on CEOs, who are considered the strategic leaders of corporations, because of their crucial role in making important decisions. This research paper examines how female CEOs influence green innovation (GI) in the Asian industrial sector. The primary goal is to address these research questions: Do Asian industrial firms with female and male CEOs differ in their GI efforts? Is there a positive moderating influence of Asian industrial enterprises’ size on the nexus between women in CEO positions and eco-innovation? Based on our research questions, firm size is likely a determining factor in the GI of female CEOs. This research employs rigorous econometric modeling to analyze a substantial dataset of listed Asian industrial companies from 2013 to 2022. We have found a significant positive correlation between female CEOs and GI in Asian industrial firms. It has been proven that female CEOs in the industrial sector are more inclined to promote environmentally friendly practices. Furthermore, the size of an industrial firm amplifies the beneficial influence of a female CEO on the firm’s chances of engaging in GI initiatives. Regarding the moderating effect of size, the size of companies significantly magnifies the impact of female CEOs on GI. The effectiveness of female CEOs on environmentally friendly practices is more prominent in large corporations than in smaller ones. Our outcomes remain robust with respect to endogeneity issues using two-step GMM estimators. This study proposes that stakeholders, particularly in Asian countries, should promote the increased representation of females in CEO roles, particularly within large corporations. This is because women-led companies demonstrate superior performance in GI endeavors. Hence, regulators must establish policies that facilitate the participation of women in CEO positions within large-scale enterprises. These policies may strengthen the private sector’s capacity to foster sustainable innovation.

The role of economic freedom in the development of international tourism in Asian countries

Purpose ― This paper aims to investigate the influence of economic freedom and its components, namely business freedom and trade freedom, on international tourist arrivals in Asian countries. Additionally, it examines the effect of important macroeconomic factors, such as foreign direct investment, exchange rates, political stability, GDP per capita, and inflation on international tourist arrivals in Asian countries.Methods ― The GMM two-step estimation system is used to analyze data from 25 Asian countries from 1995 to 2020.Findings ― The results show that economic and trade freedom positively influence tourism, while business freedom has a less distinct impact. Inflation positively contributes to tourist arrivals. Exchange rates and political stability show inconclusive effects.Implications ― The study recommends that governments prioritize expanding economic freedom to boost international tourism.Originality ― This is the first study on the impact of economic freedom on developing international tourism in Asian countries.

Enhancing Effort-Moderated Item Response Theory Models by Evaluating a Two-Step Estimation Method and Multidimensional Variations on the Model

Rapid-guessing behavior in data can compromise our ability to estimate item and person parameters accurately. Consequently, it is crucial to model data with rapid-guessing patterns in a way that can produce unbiased ability estimates. This study proposes and evaluates three alternative modeling approaches that follow the logic of the effort-moderated item response theory model (EM-IRT) to analyze response data with rapid-guessing responses. One is the two-step EM-IRT model, which utilizes the item parameters estimated by respondents without rapid-guessing behavior and was initially proposed by Rios and Soland without further investigation. The other two models are effort-moderated multidimensional models (EM-MIRT), which we introduce in this study and vary as both between-item and within-item structures. The advantage of the EM-MIRT model is to account for the underlying relationship between rapid-guessing propensity and ability. The three models were compared with the traditional EM-IRT model regarding the accuracy of parameter recovery in various simulated conditions. Results demonstrated that the two-step EM-IRT and between-item EM-MIRT model consistently outperformed the traditional EM-IRT model under various conditions, with the two-step EM-IRT estimation generally delivering the best performance, especially for ability and item difficulty parameters estimation. In addition, different rapid-guessing patterns (i.e., difficulty-based, changing state, and decreasing effort) did not affect the performance of the two-step EM-IRT model. Overall, the findings suggest that the EM-IRT model with the two-step parameter estimation method can be applied in practice for estimating ability in the presence of rapid-guessing responses due to its accuracy and efficiency. The between-item EM-MIRT model can be used as an alternative model when there is no significant mean difference in the ability estimates between examinees who exhibit rapid-guessing behavior and those who do not.

Nexus between natural resource endowments and economic growth in selected African countries

Africa began implementing the Sustainable Development Goals (SDGs) in 2015 emphasizing on sustainable management and effective use of natural capital to spur economic growth (Goals 12, 14, and 15). This study using World Bank data sets from 46 African countries selected for the years 2000 to 2022, examined the nexus between natural resource endowments and economic growth in Africa. We used the system generalized method of moments (sys-GMM) and dynamic panel threshold regression (DPTR) to analyze the data. The findings of the two-step sys-GMM estimation using 'xtabond2' revealed that when the institutional quality variable is added and excluded from the model, natural resource dependence negatively impacts economic growth, but the impact is greater when the institution is excluded. In the estimation of the interaction variable of natural resource dependence and institutional variable included in the model, natural resource dependence positively impacts economic growth. The results of the DPTR using "xthenreg" showed that when the threshold value of natural resource dependence is ≤ 1.73% of gross domestic product, natural resource dependence has a positive impact on economic growth and a negative impact when the threshold value is above 1.73%. Similarly, when the institutional quality threshold is ≤ 0.277, natural resources dependence impacts economic growth negatively; above the threshold (0.277), the impact is positive. In conclusion, natural resource endowment is a curse with no or low-quality institutions and a blessing with high-quality institutions. Thus, building strong institutions and proper utilization of natural resources helps to minimize the adverse impact of resource endowments on economic growth.

문화예술부문에 대한 재정지출이 온실가스배출에 미치는 영향분석

As the culture and arts sector develops alongside economic growth, the increase in greenhouse gas emissions highlights the importance of environmental considerations in the context of climate change. This study employs a two-step estimation method using a dynamic panel model to analyze the impact of fiscal expenditure for the culture and arts sector on greenhouse gas emissions, distinguishing between direct and indirect effects. The first-stage analysis reveals that increased fiscal expenditure in the culture and arts sector positively impacts regional economic growth. The second-stage analysis indicates the presence of an inverted U-shaped curve between per-capita GRDP and greenhouse gas emissions, supporting the Environmental Kuznets Curve hypothesis. While fiscal expenditure for the culture and arts sector directly increases greenhouse gas emissions, it indirectly reduces them through regional economic growth. However, the direct effect is greater and more significant than the indirect effect, resulting in a net increase in greenhouse gas emissions. These findings suggest that for effective greenhouse gas reduction policies at the local government level, it is necessary to improve energy efficiency in cultural and artistic activities and promote the use of eco-friendly energy sources. Additionally, there is a need for support that can raise environmental awareness among cultural and artistic professionals and induce changes in their energy consumption patterns.

Validation of ADNEX and IOTA two-step strategy and estimation of risk of complications during follow-up of adnexal masses in low-risk population.

To evaluate the ability of the Assessment of Different NEoplasias in the adneXa (ADNEX) model and the International Ovarian Tumour Analysis (IOTA) two-step strategy to predict malignancy in adnexal masses detected in an outpatient low-risk setting, and to estimate the risk of complications in masses with benign ultrasound morphology managed using clinical and ultrasound follow-up. This single-center study was performed at Hospital Universitari Dexeus, Barcelona, Spain, using interim data from the ongoing prospective observational IOTA Phase-5 (IOTA5) study. The primary aim of the IOTA5 study is to describe the cumulative incidence of complications during follow-up of adnexal masses classified as benign on ultrasound examination. Consecutive patients with an adnexal mass detected between June 2012 and September 2016 in a private center offering screening for gynecological cancer were included and followed up until February 2020. Tumors were classified as benign or malignant based on histology (if patients underwent surgery) or the outcome of clinical and ultrasound follow-up at 12 (range, 10-14) months. Multiple imputation was used when outcomes were uncertain. The ability of the ADNEX model without CA125 and of the IOTA two-step strategy to distinguish benign from malignant masses was evaluated retrospectively using the prospectively collected data. We assessed performance with regard to discrimination (area under the receiver-operating-characteristics curve (AUC)), calibration, classification (sensitivity and specificity) and clinical utility (Net Benefit). In the group of patients with a mass judged to be benign who were selected for conservative management, we evaluated the occurrence of spontaneous resolution or any mass complication during the first 5 years of follow-up by assessing the cumulative incidence of malignancy, torsion, cyst rupture and minor mass complications (inflammation, infection or adhesions) and the time to occurrence of an event. A total of 2654 patients were recruited to the study. After application of exclusion criteria, 2039 patients with a newly detected mass were included for the model validation. Of those, 1684 (83%) masses were benign, 49 (2%) masses were malignant and, for 306 (15%) masses, the outcome was uncertain and therefore imputed. The AUC was 0.95 (95% CI, 0.89-0.98) for ADNEX without CA125 and 0.94 (95% CI, 0.88-0.97) for the two-step strategy. Calibration performance could not be meaningfully interpreted because the small number of malignancies resulted in very wide confidence intervals. The two-step strategy had better clinical utility than did the ADNEX model at malignancy risk thresholds < 3%. There were 1472 (72%) patients whose mass was judged to be benign based on pattern recognition by an experienced ultrasound examiner and were managed with clinical and ultrasound follow-up. In this group, the 5-year cumulative incidence was 66% (95% CI, 63-69%) for spontaneous resolution of the mass, 0% (95% CI, 0-0.2%) for torsion, 0.1% (95% CI, < 0.1-0.4%) for cyst rupture, 0.2% (95% CI, 0.1-0.6%) for a borderline tumor and 0.2% (95% CI, 0.1-0.6%) for invasive malignancy. The ADNEX model and IOTA two-step strategy performed well to distinguish benign from malignant adnexal masses detected in a low-risk population. Conservative management is safe for masses with a benign ultrasound appearance in this population. © 2024 International Society of Ultrasound in Obstetrics and Gynecology.

Inferences for the distribution of the duration of response in a comparative clinical study.

Duration of response is an important endpoint used in drug development. Prolonged duration for response is often viewed as an early indication of treatment efficacy. However, there are numerous difficulties in studying the distribution of duration of response based on observed data subject to right censoring in practice. The most important obstacle is that the distribution of the duration of response is in general not identifiable in the presence of censoring due to the simple fact that there is no information on the joint distribution of time to response and time to progression beyond the largest follow-up time. In this article, we introduce the restricted duration of response as a replacement of the conventional duration of response. The distribution of restricted duration of response is estimable and we have proposed several nonparametric estimators in this article. The corresponding inference procedure and additional downstream analysis have been developed. Extensive numerical simulations have been conducted to examine the finite sample performance of the proposed estimators. It appears that a new regression-based two-step estimator for the survival function of the restricted duration of response tends to have a robust and superior performance, and we recommend its use in practice. A real data example from oncology has been used to illustrate the analysis for restricted duration of response.

A new two-step estimation approach for retrieving surface urban heat island intensity and footprint based on urban-rural temperature gradients

Past decades have seen substantial efforts devoted to observing, assessing, and documenting the urban heat island (UHI) phenomenon. However, the discrepant criteria of non-urban references and ambiguous distinctions between urban and rural landscapes pose great challenges in measuring UHI magnitudes and spatial extents. This study goes beyond the conventional urban-rural dichotomy and introduces a new two-step approach based on the continuous transition of thermal environments along urban-rural gradients. The approach is applied to quantify Surface UHI (SUHI) intensities and footprints across 283 Chinese cities from 2005 to 2018 using multiple satellite-derived data sources. The results include: 1) The two-step approach avoids the limitations in subjective rural reference selections and provides reliable quantification of SUHI characteristics in various cities over time. 2) The SUHI footprints extracted by our approach are more reasonable than those obtained by two existing methods, with footprint ratios generally ranging within 0 − 6 times the urban area. 3) The two-step approach provides more concentrated estimates of SUHI intensity. Typically, ignoring heat sources in non-built-up areas can cause an overestimation of SUHI effect and misidentification of remote rural areas with high temperatures. Overall, the two-step approach enables more accurate estimates of SUHI effect, thereby facilitating policy-making for SUHI mitigation.

Evaluation of enhanced external counterpulsation for diabetic foot based on a patient-specific 0D-1D cardiovascular system model

Background and objectiveDiabetic foot (DF) complications often lead to severe vascular issues. This study investigated the effectiveness of enhanced external counterpulsation (EECP) and its derived innovative compression strategies in addressing poor perfusion in DF. Although developing non-invasive and efficient treatment methods for DF is critical, the hemodynamic alterations during EECP remain underexplored despite promising outcomes in microcirculation. This research sought to address this gap by developing a patient-specific 0D-1D model based on clinical ultrasound data to identify potentially superior compression strategies that could substantially enhance blood flow in patients with DF complications. MethodsData were gathered from 10 patients with DF utilizing ultrasound for blood flow rate and computed tomography angiography (CTA) to identify lower limb conditions. Clinical measurements during standard EECP, with varying cuff pressures, facilitated the creation of a patient-specific 0D-1D model through a two-step parameter estimation process. The accuracy of this model was verified via comparison with the clinical measurements. Four compression strategies were proposed and rigorously evaluated using this model: EECP-Simp-I (removing hip cuffs), EECP-Simp-II (further removing the cuffs around the lower leg), EECP-Impr-I (removing all cuffs around the affected side), and EECP-Impr-II (building a loop circulation from the healthy side to the affected side). ResultsThe predicted results under the rest and standard EECP states were generally closely aligned with clinical measurements. The patient-specific 0D-1D model demonstrated that EECP-Simp-I and EECP-Impr-I contributed similar enhancement to perfusion in the dorsal artery (DA) and were comparable to standard EECP, while EECP-Simp-II had the least effect and EECP-Impr-II displayed the most significant enhancement. Pressure at the aortic root (AO) remained consistent across strategies. ConclusionsEECP-Simp-I is recommended for patients with DF, emphasizing device simplification. However, EECP-Simp-II is discouraged as it significantly diminished blood perfusion in this study, except in cases of limb fragility. EECP-Impr-II showed superior enhancement of blood perfusion in DA to all other strategies but required a more complex EECP device. Despite increased AO pressure in all the proposed compression strategies, safety could be guaranteed as the pressue remained within a safe range.

CFO co-option and dividend payments: the moderating role of CFO talent

Purpose This study aims to investigate the association between the co-option of the chief financial officer (CFO) and dividend payments, assessing whether the talent of the CFO affects this association. Design/methodology/approach The empirical analyses were based on hand-collected data for 922 firm-year observations from 157 European listed firms, during the period 2013–2019. Empirical models, based on a two-step estimation procedure, involved the use of instrumental variables and the generalised moment method. Findings The results show that CFO co-option is negatively associated with the level of dividend payments. It was also found that the degree of CFO talent moderates the negative association between CFO co-option and dividend payments. Research limitations/implications This investigation responds to the call for literature which examines how chief executive officer (CEO) – CFO relationships influence firms’ policies and outcomes. The study offers novel evidence for the individual-level characteristics of CFOs which are likely to reduce the effectiveness of CEO power and increase monitoring on corporate decisions on dividends. Practical implications The study sheds light on the effect of the interactions between CEOs and CFOs, which are important for investors’ expectations. In this regard, investors may be interested in the CFO profiles which may reduce CEO power over dividend policies. Originality/value Unlike previous research, which focused on CEOs, the authors are the first to shed light on the role of CFOs as key decision makers in influencing the dividend policies in modern corporations.

A novel skewed generalized normal distribution: properties, statistical inference, and its applications

In this paper, a novel skewed generalized normal distribution (NSGN), which encompasses several well-known distributions, is a two-component mixture model based on the generalized normal distribution. The NSGN distribution offers flexibility in modeling skewed data characterized by high kurtosis and strong skewness, making it highly applicable in finance, medicine, and engineering. The primary characteristics and properties of this new distribution are introduced, and the explicit expressions for the moments of order statistics are provided using recursive relations. The maximum likelihood estimation based on a profile likelihood approach, L-moments estimation, and a two-step estimation method combining Bayesian posterior maximum estimation with moment estimates are presented to estimate the four parameters of NSGN distribution. A simulation study compares the performance of these methods across different sample sizes and different parameters. The appropriateness of the proposed distribution has been tested by comparing it with several skewed distributions. Additionally, applications to real datasets showcase the beneficial properties of the NSGN for applied statistical research.

Oracle-efficient estimation and global inferences for variance function of functional data

A new two-step reconstruction-based moment estimator and an asymptotically correct smooth simultaneous confidence band as a global inference tool are proposed for the heteroscedastic variance function of dense functional data. Step one involves spline smoothing for individual trajectory reconstructions and step two employs kernel regression on the individual squared residuals to estimate each trajectory variability. Then by the method of moment an estimator for the variance function of functional data is constructed. The estimation procedure is innovative by synthesizing spline smoothing and kernel regression together, which allows one not only to apply the fast computing speed of spline regression but also to employ the flexible local estimation and the extreme value theory of kernel smoothing. The resulting estimator for the variance function is shown to be oracle-efficient in the sense that it is uniformly as efficient as the ideal estimator when all trajectories were known by “oracle”. As a result, an asymptotically correct simultaneous confidence band for the variance function is established. Simulation results support our asymptotic theory with fast computation. As an illustration, the proposed method is applied to the analyses of two real data sets leading to a number of discoveries.

Cross-border banking and bank stability: evidence from Sub-Saharan Africa

AbstractBased on bank-level data from 29 Sub-Saharan African countries between 2005 and 2019, we apply panel fixed effects (FE) and two-step system GMM estimators to investigate whether increased cross-border banking affects domestic banking sector stability. We find significant evidence that the stability of banks in host countries declines with an increased presence of foreign banks—and the impact is more pronounced on banks that are small and less efficient. The stability impact of foreign banks is also found to depend on the quality of governance institutional factors in the host country. The findings shed some important insights on the downside of financial liberalisation policy in developing countries and the need for increased cross-border collaboration between home and host supervisory authorities in the SSA region—especially in jurisdictions where the foreign bank affiliates are systemically important. The domestic supervisory authorities thus need to effectively manage the inherent trade-off between reaping the benefits from international financial integration while effectively safeguarding domestic banking systems against cross-border contagion and fragility.

Frisch–Waugh–Lovell theorem-type results for the k-Class and 2SGMM estimators

The Frisch–Waugh–Lovell (FWL) theorem shows that for the least squares estimator, parameter estimates from full and partial models are identically same. I show that in linear regression models with a mix of exogenous and endogenous regressors, FWL theorem-type results hold for the k-class estimators (including LIML) and the two-step optimal GMM estimator.