This paper uses the estimation of the Self-Excited Multi Fractal (SEMF) model, which holds theoretical promise but has seen mixed results in practice, as a case study to explore the impact of distributional assumption...This paper uses the estimation of the Self-Excited Multi Fractal (SEMF) model, which holds theoretical promise but has seen mixed results in practice, as a case study to explore the impact of distributional assumptions on the model fitting process. In the case of the SEMF model, this examination shows that incorporating reasonable distributional assumptions including a non-zero mean and the leptokurtic Student’s t distribution can have a substantial impact on the estimation results and can mean the difference between parameter estimates that imply unstable and potentially explosive volatility dynamics versus ones that describe more reasonable and realistic dynamics for the returns. While the original SEMF model specification is found to yield unrealistic results for most of the series of financial returns to which it is applied, the results obtained after incorporating the Student’s t distribution and a mean component into the model specification suggest that the SEMF model is a reasonable model, implying realistic return behavior, for most, if not all, of the series of stock and index returns to which it is applied in this study. In addition, reflecting the sensitivity of the sample mean to the types of characteristics that the SEMF model is designed to capture, the results of this study also illustrate the value of incorporating the mean component directly into the model and fitting it in conjunction with the other model parameters rather than simply centering the returns beforehand by subtracting the sample mean from them.展开更多
Unlike height-diameter equations for standing trees commonly used in forest resources modelling,tree height models for cut-to-length(CTL)stems tend to produce prediction errors whose distributions are not conditionall...Unlike height-diameter equations for standing trees commonly used in forest resources modelling,tree height models for cut-to-length(CTL)stems tend to produce prediction errors whose distributions are not conditionally normal but are rather leptokurtic and heavy-tailed.This feature was merely noticed in previous studies but never thoroughly investigated.This study characterized the prediction error distribution of a newly developed such tree height model for Pin us radiata(D.Don)through the three-parameter Burr TypeⅫ(BⅫ)distribution.The model’s prediction errors(ε)exhibited heteroskedasticity conditional mainly on the small end relative diameter of the top log and also on DBH to a minor extent.Structured serial correlations were also present in the data.A total of 14 candidate weighting functions were compared to select the best two for weightingεin order to reduce its conditional heteroskedasticity.The weighted prediction errors(εw)were shifted by a constant to the positive range supported by the BXII distribution.Then the distribution of weighted and shifted prediction errors(εw+)was characterized by the BⅫdistribution using maximum likelihood estimation through 1000 times of repeated random sampling,fitting and goodness-of-fit testing,each time by randomly taking only one observation from each tree to circumvent the potential adverse impact of serial correlation in the data on parameter estimation and inferences.The nonparametric two sample Kolmogorov-Smirnov(KS)goodness-of-fit test and its closely related Kuiper’s(KU)test showed the fitted BⅫdistributions provided a good fit to the highly leptokurtic and heavy-tailed distribution ofε.Random samples generated from the fitted BⅫdistributions ofεw+derived from using the best two weighting functions,when back-shifted and unweighted,exhibited distributions that were,in about97 and 95%of the 1000 cases respectively,not statistically different from the distribution ofε.Our results for cut-tolength P.radiata stems represented the first case of any tree species where a non-normal error distribution in tree height prediction was described by an underlying probability distribution.The fitted BXII prediction error distribution will help to unlock the full potential of the new tree height model in forest resources modelling of P.radiata plantations,particularly when uncertainty assessments,statistical inferences and error propagations are needed in research and practical applications through harvester data analytics.展开更多
文摘This paper uses the estimation of the Self-Excited Multi Fractal (SEMF) model, which holds theoretical promise but has seen mixed results in practice, as a case study to explore the impact of distributional assumptions on the model fitting process. In the case of the SEMF model, this examination shows that incorporating reasonable distributional assumptions including a non-zero mean and the leptokurtic Student’s t distribution can have a substantial impact on the estimation results and can mean the difference between parameter estimates that imply unstable and potentially explosive volatility dynamics versus ones that describe more reasonable and realistic dynamics for the returns. While the original SEMF model specification is found to yield unrealistic results for most of the series of financial returns to which it is applied, the results obtained after incorporating the Student’s t distribution and a mean component into the model specification suggest that the SEMF model is a reasonable model, implying realistic return behavior, for most, if not all, of the series of stock and index returns to which it is applied in this study. In addition, reflecting the sensitivity of the sample mean to the types of characteristics that the SEMF model is designed to capture, the results of this study also illustrate the value of incorporating the mean component directly into the model and fitting it in conjunction with the other model parameters rather than simply centering the returns beforehand by subtracting the sample mean from them.
文摘Unlike height-diameter equations for standing trees commonly used in forest resources modelling,tree height models for cut-to-length(CTL)stems tend to produce prediction errors whose distributions are not conditionally normal but are rather leptokurtic and heavy-tailed.This feature was merely noticed in previous studies but never thoroughly investigated.This study characterized the prediction error distribution of a newly developed such tree height model for Pin us radiata(D.Don)through the three-parameter Burr TypeⅫ(BⅫ)distribution.The model’s prediction errors(ε)exhibited heteroskedasticity conditional mainly on the small end relative diameter of the top log and also on DBH to a minor extent.Structured serial correlations were also present in the data.A total of 14 candidate weighting functions were compared to select the best two for weightingεin order to reduce its conditional heteroskedasticity.The weighted prediction errors(εw)were shifted by a constant to the positive range supported by the BXII distribution.Then the distribution of weighted and shifted prediction errors(εw+)was characterized by the BⅫdistribution using maximum likelihood estimation through 1000 times of repeated random sampling,fitting and goodness-of-fit testing,each time by randomly taking only one observation from each tree to circumvent the potential adverse impact of serial correlation in the data on parameter estimation and inferences.The nonparametric two sample Kolmogorov-Smirnov(KS)goodness-of-fit test and its closely related Kuiper’s(KU)test showed the fitted BⅫdistributions provided a good fit to the highly leptokurtic and heavy-tailed distribution ofε.Random samples generated from the fitted BⅫdistributions ofεw+derived from using the best two weighting functions,when back-shifted and unweighted,exhibited distributions that were,in about97 and 95%of the 1000 cases respectively,not statistically different from the distribution ofε.Our results for cut-tolength P.radiata stems represented the first case of any tree species where a non-normal error distribution in tree height prediction was described by an underlying probability distribution.The fitted BXII prediction error distribution will help to unlock the full potential of the new tree height model in forest resources modelling of P.radiata plantations,particularly when uncertainty assessments,statistical inferences and error propagations are needed in research and practical applications through harvester data analytics.
