Contour projected dimension reduction

In regression analysis, we employ contour projection (CP) to develop a new dimension reduction theory. Accordingly, we introduce the notions of the central contour subspace and generalized contour subspace. We show that both of their structural dimensions are no larger than that of the central subspace Cook [Regression Graphics (1998b) Wiley]. Furthermore, we employ CP-sliced inverse regression, CP-sliced average variance estimation and CP-directional regression to estimate the generalized contour subspace, and we subsequently obtain their theoretical properties. Monte Carlo studies demonstrate that the three CP-based dimension reduction methods outperform their corresponding non-CP approaches when the predictors have heavy-tailed elliptical distributions. An empirical example is also presented to illustrate the usefulness of the CP method.Comment: Published in at http://dx.doi.org/10.1214/08-AOS679 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Quantile correlations and quantile autoregressive modeling

In this paper, we propose two important measures, quantile correlation (QCOR) and quantile partial correlation (QPCOR). We then apply them to quantile autoregressive (QAR) models, and introduce two valuable quantities, the quantile autocorrelation function (QACF) and the quantile partial autocorrelation function (QPACF). This allows us to extend the classical Box-Jenkins approach to quantile autoregressive models. Specifically, the QPACF of an observed time series can be employed to identify the autoregressive order, while the QACF of residuals obtained from the fitted model can be used to assess the model adequacy. We not only demonstrate the asymptotic properties of QCOR, QPCOR, QACF, and PQACF, but also show the large sample results of the QAR estimates and the quantile version of the Ljung-Box test. Simulation studies indicate that the proposed methods perform well in finite samples, and an empirical example is presented to illustrate usefulness

Markov-Switching Model Selection Using Kullback-Leibler Divergence

In Markov-switching regression models, we use Kullback-Leibler (KL) divergence between the true and candidate models to select the number of states and variables simultaneously. In applying Akaike information criterion (AIC), which is an estimate of KL divergence, we find that AIC retains too many states and variables in the model. Hence, we derive a new information criterion, Markov switching criterion (MSC), which yields a marked improvement in state determination and variable selection because it imposes an appropriate penalty to mitigate the over-retention of states in the Markov chain. MSC performs well in Monte Carlo studies with single and multiple states, small and large samples, and low and high noise. Furthermore, it not only applies to Markov-switching regression models, but also performs well in Markov- switching autoregression models. Finally, the usefulness of MSC is illustrated via applications to the U.S. business cycle and the effectiveness of media advertising.Research Methods/ Statistical Methods,

Semiparametric and Additive Model Selection Using an Improved Akaike Information Criterion

An improved AIC-based criterion is derived for model selection in general smoothing-based modeling, including semiparametric models and additive models. Examples are provided of applications to goodness-of-fit, smoothing parameter and variable selection in an additive model and semiparametric models, and variable selection in a model with a nonlinear function of linear terms.Statistics Working Papers Serie

Score Tests for the Single Index Model

The single index model is a generalization of the linear regression model with E(y|x) = g, where g is an unknown function. The model provides a flexible alternative to the linear regression model while providing more structure than a fully nonparametric approach. Although the fitting of single index models does not require distributional assumptions on the error term, the properties of the estimates depend on such assumptions, as does practical application of the model. In this article score tests are derived for three potential misspecifications of the single index model: heteroscedasticity in the errors, autocorrelation in the errors, and the omission of an important variable in the linear index. These tests have a similar structure to corresponding tests for nonlinear regression models. Monte Carlo simulations demonstrate that the first two tests hold their nominal size well and have good power properties in identifying model violations, often outperforming other tests. Testing for the need for additional covariates can be effective, but is more difficult. The score tests are applied to three real datasets, demonstrating that the tests can identify important model violations that affect inference, and that approaches that do not take model misspecifications into account can lead to very different results.Statistics Working Papers Serie

The examination of residual plots

This is the publisher's version, also available electronically from http://www3.stat.sinica.edu.tw/statistica/j8n2/j8n29/j8n29.htm.Linear and squared residual plots are proposed to assess nonlinearity and heteroscedasticity in regression diagnostics. It is shown that linear residual plots are useful for diagnosing nonlinearity and squared residual plots are powerful for detecting nonconstant variance. A paradigm for the graphical interpretation of residual plots is presented

Tobit Model Estimation and Sliced Inverse Regression

It is not unusual for the response variable in a regression model to be subject to censoring or truncation. Tobit regression models are a specific example of such a situation, where for some observations the observed response is not the actual response, but rather the censoring value (often zero), and an indicator that censoring (from below) has occurred. It is well-known that the maximum likelihood estimator for such a linear model (assuming Gaussian errors) is not consistent if the error term is not homoscedastic and normally distributed. In this paper we consider estimation in the Tobit regression context when those conditions do not hold, as well as when the true response is an unspecified nonlinear function of linear terms, using sliced inverse regression (SIR). The properties of SIR estimation for Tobit models are explored both theoretically and based on Monte Carlo simulations. It is shown that the SIR estimator has good properties when the usual linear model assumptions hold, and can be much more effective than other estimators when they do not. An example related to household charitable donations demonstrates the usefulness of the estimator.Statistics Working Papers Serie

Denoised least squares estimators: An application to estimating advertising effectiveness

This is the publisher's version, also available electronically from http://www3.stat.sinica.edu.tw/statistica/j10n4/j10n412/j10n412.htm.It is known in marketing science that an advertiser under- or overspends millions of dollars on advertising because the estimation of advertising effectiveness is biased. This bias is induced by measurement noise in advertising variables, such as awareness and television rating points, which are provided by commercial market research firms based on small-sample surveys of consumers. In this paper, we propose a denoised regression approach to deal with the problem of noisy variables. We show that denoised least squares estimators are consistent. Simulation results indicate that the denoised regression approach outperforms the classical regression approach. A marketing example is presented to illustrate the use of denoised least squares estimators