When A Factor Is Measured with Error: The Role of Conditional Heteroskedasticity in Identifying and Estimating Linear Factor Models

A new method is proposed for estimating linear triangular models, where identification results from the structural errors following a bivariate and diagonal GARCH(1,1) process. The associated estimator is a GMM estimator shown to have the usual √T-asymptotics. A Monte Carlo study of the estimator is provided as is an empirical application of estimating market betas from the CAPM. These market beta estimates are found to be statistically distinct from their OLS counterparts and to display expanded cross-sectional variation, the latter feature offering promise for their ability to provide improved pricing of cross-sectional expected returns.Measurement error; triangular models; factor models; heteroskedasticity; identification; many moments; GMM

Market Proxies, Correlation, and Relative Mean-Variance Efficiency: Still Living with the Roll Critique

A test of the CAPM is developed conditional on a prior belief about the correlation between the true market return and the proxy return used in the test. Consideration is given to the effect of the proxy's mismeasurement of the market return on the estimation of the market model. Failure to grant this consideration biases tests towards rejection by overstating the inefficiency of the proxy. An extension of the proposed test to a CAPM with conditioning information links mismeasurement of the market return to time-variation in beta.Asset pricing, CAPM, portfolio efficiency, multivariate testing, bootstrap hypothesis testing, triangular systems, endogeneity, identification, GMM, conditional heteroskedasticity, GARCH

Simple, Skewness-Based GMM Estimation of the Semi-Strong GARCH(1,1) Model

IV estimators with an instrument vector composed only of past squared residuals, while applicable to the semi-strong ARCH(1) model, do not extend to the semi-strong GARCH(1,1) case because of underidentification. Augmenting the instrument vector with past residuals, however, renders traditional IV estimation feasible, if the residuals are skewed. The proposed estimators are much simpler to implement than efficient IV estimators, yet they retain improved finite sample performance over QMLE. Jackknife versions of these estimators deal with the issues caused by many (potentially weak) instruments. A Monte Carlo study is included, as is an empirical application involving foreign currency spot returns

Simple GMM Estimation of the Semi-Strong GARCH(1,1) Model

IV estimators for the semi-strong ARCH(1) model that rely on past squared residuals alone as instruments do not extend to the GARCH case. Efficient IV estimators of the semi-strong GARCH(1,1) model require the derivative of the conditional variance as well as both the third and fourth conditional moments to be included within the instrument vector. This paper proposes IV estimators for the semi-strong GARCH(1,1) model that only rely on past residuals and past squared residuals as instruments. These estimators are based on the autocovariances of squared residuals, as in the ARCH(1) case described above, as well as on the covariances between squared residuals and past residuals. These latter covariances are nonzero if the residuals are skewed. Jackknife GMM estimators and jackknife continuous updating estimators (CUE) eliminate the bias caused by many (weak) instruments. The jackknife CUE is new and applies to cases where the optimal weighting matrix is unavailable out of a concern over the existence of higher moments. In these cases, a robust analog to the variance-covariance matrix determines the weighting matrix. A Monte Carlo study shows that a CUE based on the optimal weighting matrix as well as the jackknife CUE outperforms QMLE in finite samples. An empirical application involving Australian Dollar and Japanese Yen spot returns is also included

Simple, Skewness-Based GMM Estimation of the Semi-Strong GARCH(1,1) Model

IV estimators with an instrument vector composed only of past squared residuals, while applicable to the semi-strong ARCH(1) model, do not extend to the semi-strong GARCH(1,1) case because of underidentification. Augmenting the instrument vector with past residuals, however, renders traditional IV estimation feasible, if the residuals are skewed. The proposed estimators are much simpler to implement than efficient IV estimators, yet they retain improved finite sample performance over QMLE. Jackknife versions of these estimators deal with the issues caused by many (potentially weak) instruments. A Monte Carlo study is included, as is an empirical application involving foreign currency spot returns

Simple GMM Estimation of the Semi-Strong GARCH(1,1) Model

Efficient GMM estimation of the semi-strong GARCH(1,1) model requires simultaneous estimation of the conditional third and fourth moments. This paper proposes a simple alternative to efficient GMM based upon the unconditional skewness of residuals and the autocovariances of squared residuals. An advantage of this simple alternative is that neither the third nor the fourth conditional moment needs to be estimated. A second advantage is that linear estimators apply to all of the parameters in the model, making estimation straightforward in practice. The proposed estimators are IV-like with potentially many instruments. Sequential estimation involves TSLS in a first step followed by linear GMM. Simultaneous estimation involves either two-step GMM or CUE. A Monte Carlo study of the proposed estimators is included

Simple, Skewness-Based GMM Estimation of the Semi-Strong GARCH(1,1) Model

IV estimators with an instrument vector composed only of past squared residuals, while applicable to the semi-strong ARCH(1) model, do not extend to the semi-strong GARCH(1,1) case because of underidentification. Augmenting the instrument vector with past residuals, however, renders traditional IV estimation feasible, if the residuals are skewed. The proposed estimators are much simpler to implement than efficient IV estimators, yet they retain improved finite sample performance over QMLE. Jackknife versions of these estimators deal with the issues caused by many (potentially weak) instruments. A Monte Carlo study is included, as is an empirical application involving foreign currency spot returns

Simple GMM Estimation of the Semi-Strong GARCH(1,1) Model

IV estimators for the semi-strong ARCH(1) model that rely on past squared residuals alone as instruments do not extend to the GARCH case. Efficient IV estimators of the semi-strong GARCH(1,1) model require the derivative of the conditional variance as well as both the third and fourth conditional moments to be included within the instrument vector. This paper proposes IV estimators for the semi-strong GARCH(1,1) model that only rely on past residuals and past squared residuals as instruments. These estimators are based on the autocovariances of squared residuals, as in the ARCH(1) case described above, as well as on the covariances between squared residuals and past residuals. These latter covariances are nonzero if the residuals are skewed. Jackknife GMM estimators and jackknife continuous updating estimators (CUE) eliminate the bias caused by many (weak) instruments. The jackknife CUE is new and applies to cases where the optimal weighting matrix is unavailable out of a concern over the existence of higher moments. In these cases, a robust analog to the variance-covariance matrix determines the weighting matrix. A Monte Carlo study shows that a CUE based on the optimal weighting matrix as well as the jackknife CUE outperforms QMLE in finite samples. An empirical application involving Australian Dollar and Japanese Yen spot returns is also included

When A Factor Is Measured with Error: The Role of Conditional Heteroskedasticity in Identifying and Estimating Linear Factor Models

A new method is proposed for estimating linear triangular models, where identification results from the structural errors following a bivariate and diagonal GARCH(1,1) process. The associated estimator is a GMM estimator shown to have the usual √T-asymptotics. A Monte Carlo study of the estimator is provided as is an empirical application of estimating market betas from the CAPM. These market beta estimates are found to be statistically distinct from their OLS counterparts and to display expanded cross-sectional variation, the latter feature offering promise for their ability to provide improved pricing of cross-sectional expected returns

When A Factor Is Measured with Error: The Role of Conditional Heteroskedasticity in Identifying and Estimating Linear Factor Models

A new method is proposed for estimating linear triangular models, where identification results from the structural errors following a bivariate and diagonal GARCH(1,1) process. The associated estimator is a GMM estimator shown to have the usual √T-asymptotics. A Monte Carlo study of the estimator is provided as is an empirical application of estimating market betas from the CAPM. These market beta estimates are found to be statistically distinct from their OLS counterparts and to display expanded cross-sectional variation, the latter feature offering promise for their ability to provide improved pricing of cross-sectional expected returns