Testing Serial Correlation in Partially Linear Additive Models

The paper considers the problem of testing error serial correlation of partially linear additive measurement error model. We propose a test statistic and show that it converges to the standard chi-square distribution under the null hypothesis. Finally, a simulation study is conducted to illustrate the performance of the test approach.

ABSTRACTIn this paper, we introduce a partially linear single-index additive hazards model with current status data. Both the unknown link function of the single-index term and the cumulative baseline hazard function are approximated by B-splines under a monotonicity constraint on the latter. The sieve method is applied to estimate the nonparametric and parametric components simultaneously. We show that, when the nonparametric link function is an exact B-spline, the resultant estimator of regression parameter vector is asymptotically normal and achieves the semiparametric information bound and the rate of convergence of the estimator for the cumulative baseline hazard function is optimal. Simulation studies are presented to examine the finite sample performance of the proposed estimation method. For illustration, we apply the method to a clinical dataset with current status outcome.

The author considers serial correlation testing in seasonal time series models. He proposes a test statistic based on a spectral approach. Many tests of this type rely on kernel‐based spectral density estimators that assign larger weights to low order lags than to high ones. Under seasonality, however, large autocorrelations may occur at seasonal lags that classical kernel estimators cannot take into account. The author thus proposes a test statistic that relies on the spectral density estimator of Shin (2004), whose weighting scheme is more adapted to this context. The distribution of his test statistic is derived under the null hypothesis and he studies its behaviour under fixed and local alternatives. He establishes the consistency of the test under a general fixed alternative. He also makes recommendations for the choice of the smoothing parameters. His simulation results suggest that his test is more powerful against seasonality than alternative procedures based on classical weighting schemes. He illustrates his procedure with monthly statistics on employment among young Americans.

THIS NOTE REPORTS ON an empirical investigation into the relative importance of different explanations of market serial correlation. Market serial correlation taken as the first-order serial correlation in the logarithmic differences of a market index may apparently be explained by the serial correlation of individual stock returns. The other, less apparent, explanation advanced in the literature concerns nonsynchronous trading in individual stocks. Recently, Perry [9] and Atchinson, Butler, and Simonds [1]1 have shown that there seems to be yet an additional unexplained component present that tends to induce serial correlation in portfolio returns, at least in U.S. data. In this paper, the actual contribution of these explanations to the first-order autocorrelation of changes in the value-weighted market index on a markedly thin security market, the Helsinki Stock Exchange (HeSE)2 in Finland, will be examined. The HeSE is a specific case in the sense that its trading procedure may create additional first-order market serial correlation. We will show that the rules of the exchange produce additional intraday nonsynchroneity in stock returns and thus aggravate the problem of market serial correlation. An attempt to assess the importance of this additional cause of market serial correlation will also be made. In 1966, Lawrence Fisher [7] observed that an index return calculated on the basis of the latest recorded trading prices will tend to be serially correlated if share prices tend to react similarly to certain types of news. This will be the case when some share prices react almost immediately whereas others, simply because of an absence of trading, will experience a reaction delay. If nonsynchronous trading is taken to be the first cause of market serial correlation, the second one is the serial correlation of individual stock returns. Cohen, Hawawini, Maier, Schwartz, and Whitcomb [6] review the literature on why individual stock returns, especially in thin markets, may be serially corre-

Among several semiparametric models, the Cox proportional hazard model is widely used to assess the association between covariates and the time-to-event when the observed time-to-event is interval-censored. Often, covariates are measured with error. To handle this covariate uncertainty in the Cox proportional hazard model with the interval-censored data, flexible approaches have been proposed. To fill a gap and broaden the scope of statistical applications to analyze time-to-event data with different models, in this paper, a general approach is proposed for fitting the semiparametric linear transformation model to interval-censored data when a covariate is measured with error. The semiparametric linear transformation model is a broad class of models that includes the proportional hazard model and the proportional odds model as special cases. The proposed method relies on a set of estimating equations to estimate the regression parameters and the infinite-dimensional parameter. For handling interval censoring and covariate measurement error, a flexible imputation technique is used. Finite sample performance of the proposed method is judged via simulation studies. Finally, the suggested method is applied to analyze a real data set from an AIDS clinical trial.

Statistical inference for partially linear additive spatial autoregressive models

Partial compliance with assigned treatment regimes is common in drug trials and calls for a causal analysis of the effect of treatment actually received. As such observed exposure is no longer randomized, selection bias must be carefully accounted for. The framework of potential outcomes allows this by defining a subject-specific treatment-free reference outcome, which may be latent and is modelled in relation to the observed (treated) data. Causal parameters enter these structural models explicitly. In this paper we review recent progress in randomization-based inference for structural mean modelling, from the additive linear model to the structural generalized linear models. An arsenal of tools currently available for standard association regression has steadily been developed in the structural setting, providing many parallel features to help randomization-based inference. We argue that measurement error on exposure is an important practical complication that has, however, not yet been addressed. We show how standard additive linear structural mean models are robust against unbiased measurement error and how efficient, asymptotically unbiased inference can be drawn when the degree of measurement error bias is known. The impact of measurement error is illustrated in a blood pressure example and finite sample properties are verified by simulation. We end with a plea for more and careful use of this methodology and point to directions for further development.

We develop empirical models for the rate of spread and intensity of fires in grass fuels. The models are based on a well-known physical analogy for the rate of spread of a fire through a continuous fuelbed. Unlike other models based on this analogy, we do not attempt to directly estimate the model parameters. Rather, we use data on the rate of spread to indirectly estimate parameters that describe aggregate properties of the fire behaviour. The resulting models require information on the moisture content of the fuel and wind speed to predict the rate of spread of fires. To predict fire intensity, the models additionally use information on the heat yield of the fuel and the amount of fuel consumed. We evaluate the models by using them to predict the intensity of independent fires and by comparing them with linear and additive regression models. The additive model provides the best description of the training data but predicts independent data poorly and with high bias. Overall, the empirical models describe the data better than the linear model, and predict independent data with lower bias. Hence our physically motivated empirical models perform better than statistical models and are easier to parameterise than parameter-rich physical models. We conclude that our physically motivated empirical models provide an alternative to statistical models and parameter-rich physical models of fire behaviour.

We propose a method for fitting semiparametric models such as the proportional hazards (PH), additive risks (AR), and proportional odds (PO) models. Each of these semiparametric models implies that some transformation of the conditional cumulative hazard function (at each t) depends linearly on the covariates. The proposed method is based on nonparametric estimation of the conditional cumulative hazard function, forming a weighted average over a range of t-values, and subsequent use of least squares to estimate the parameters suggested by each model. An approximation to the optimal weight function is given. This allows semiparametric models to be fitted even in incomplete data cases where the partial likelihood fails (e.g., left censoring, right truncation). However, the main advantage of this method rests in the fact that neither the interpretation of the parameters nor the validity of the analysis depend on the appropriateness of the PH or any of the other semiparametric models. In fact, we propose an integrated method for data analysis where the role of the various semiparametric models is to suggest the best fitting transformation. A single continuous covariate and several categorical covariates (factors) are allowed. Simulation studies indicate that the test statistics and confidence intervals have good small-sample performance. A real data set is analyzed.

PM2.5 monitoring data of air pollutants was calibrated and predicted. Due to the lack of monitoring data limited by equipment, the mean and linear interpolation was used to fill in the missing data. ARIMA model (A) was established based on the fluctuation characteristics of PM2.5. The difference between the monitoring value and the standard value was taken as the dependent variable, and five Meteorological factors, namely wind, pressure, precipitation, temperature and humidity, were taken as the independent variables. Multiple regression model (B) was developed. Then, the additive model y = A + B was built. By comparing the average relative error, ARIMA and Multiple Linear Regression Additive Model based on linear interpolation was the best (0.1616), followed by the model with interaction (0.1828), and the third was ARIMA and Multiple Linear Regression Additive Model based on mean filling (0.3080). The three models reduced the average relative errors and improved the effects of forecast.

ABSTRACTIn practice, it is not uncommon to encounter the situation that a discrete response is related to both a functional random variable and multiple real-value random variables whose impact on the response is nonlinear. In this paper, we consider the generalized partial functional linear additive models (GPFLAM) and present the estimation procedure. In GPFLAM, the nonparametric functions are approximated by polynomial splines and the infinite slope function is estimated based on the principal component basis function approximations. We obtain the estimator by maximizing the quasi-likelihood function. We investigate the finite sample properties of the estimation procedure via Monte Carlo simulation studies and illustrate our proposed model by a real data analysis.

For tting penalized Fourier regression, Wu (2013) provide some R-functions in the settings of univariate nonparametric regression, partial linear models, additive models, and nonparametric logistic and Poisson regression. We improve these functions to follow the R-function format so that users could apply generic R-functions summary, plot, lines, fitted, predict, and residuals. In addition, we write some R-function to select penalty parameters by AICc or BIC. Some data examples are used to illustrate our R-functions. A small simulation study is conducted to compare the penalized Fourier to spline approaches.

Multi-scale tests for serial correlation

By counting the “peaks” (relative maxima) of a time series, it is possible to estimate the serial correlation in the series, under the assumption that the series is a first-order autoregressive process. The estimator is inefficient, but it is effective nonetheless, and since all computations can be performed mentally and no tables are required, it is suitable for the quick assessment of computer-generated diagnostic plots. It also enjoys certain robustness properties. As a test for serial correlation, the procedure is essentially the same as one proposed by Bienayme in 1874 and by Wallis and Moore in 1941, which is itself equivalent to tests based on the counts of runs up and down.

Standard serial correlation tests are derived assuming that the disturbances are homoscedastic, but this study shows that asympotic critical values are not accurate when this assumption is violated. Asymptotic critical values for the ARCH(2)-corrected LM, BP and BL tests are valid only when the underlying ARCH process is strictly stationary, whereas Wooldridge's robust LM test has good properties overall. These tests exhibit similar bahaviour even when the underlying process is GARCH (1,1). When the regressors include lagged dependent variables, the rejection frequencies under both the null and alternative hypotheses depend on the coefficientsof the lagged dependent variables and the other model parameters. They appear to be robust across various disturbance distributions under the null hypothesis.