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Estimation and inference for variance functions
Published in Raymond J. Carroll, David Ruppert, Transfor mation and Weighting in Regression, 2017
Raymond J. Carroll, David Ruppert
Recall the basic asymptotic result for a fully parametric specification of the variance function as in (2.1). At least in the limit, generalized least-squares estimates are just as efficient as weighted least-squares estimates with known weights. Now suppose that a parametric specification of variance is not known, and that instead we are only sure that for an unknown function g0 either () Standarddeviationofyi=g0(xi)
Filtering and Smoothing
Published in Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala, Nonlinear Filtering, 2017
Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala
Wiener filtering which is an optimal filter that estimates a time-varying signal buried in noise using frequency domain analysis is presented followed by least squares (LS) filtering. The LS technique was invented by Gauss in 1809 and independently by Legendre in 1806 for planetary orbit studies and forms the basis for Kalman filtering [20]. LS estimation is used in polynomial curve fitting for obtaining the best fit for a set of measured data by minimizing the mean square error (MSE) between the measured and the estimated function/data. Weighted least squares or generalized least squares enables filtering by weighting the measurements depending on their accuracies. LS is a special case of maximum likelihood estimation method for linear systems with Gaussian noise. LS technique is applicable to linear, nonlinear, SISO (single-input single-output) and MISO (multi-input single-output) dynamic systems. The maximum likelihood estimation theory, and Wiener and Kolmogorov filtering theory for stationary processes also had some bearing (like the KF formulation) in using the mean square error (Appendix 1A) and innovation concepts [20,21].
Linear Regression
Published in Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos, Statistical and Econometric Methods for Transportation Data Analysis, 2020
Simon Washington, Matthew Karlaftis, Fred Mannering, Panagiotis Anastasopoulos
Equation (3.3) says that on average, model over-predictions are equal to model under-predictions—or that model disturbances sum to zero. Equation (3.4) shows that the variance of the disturbance term, σ2, is independent across observations. This property is referred to as the homoscedasticity assumption and implies that the net effect of model uncertainty, including unobserved effects, measurement errors, and true random variation, is not systematic across observations; instead it is random across observations and across covariates. When disturbances are heteroscedastic (vary systematically across observations), then alternative modeling approaches such as weighted least squares or generalized least squares may be required.
Can interdependency considerations enhance forecasts of bridge infrastructure condition? Evidence using a multivariate regression approach
Published in Structure and Infrastructure Engineering, 2020
Steven M. Lavrenz, Tariq Usman Saeed, Jackeline Murillo-Hoyos, Matthew Volovski, Samuel Labi
The general modeling structure for 3SLS in this paper is presented as: where ci are the condition ratings for the bridge deck (d), the superstructure (s), and the substructure (u), Zi are vectors of exogenous variables, βi are vectors of estimable parameters, λi and τi are the estimable scalars, and εi are the disturbance terms. The model estimation follows three stages: The endogenous variables in Equations (1)–(3), and are regressed against all exogenous variables (vector Z).Equations (1)–(3) are estimated, using the estimated values of and from step 1, and correlations between the disturbance terms and are calculated.Equations (1)–(3) are re-estimated, using generalized least squares (GLS), whereby the error term correlations estimated in step 2, represented using a variance-covariance matrix, are used to refine parameter estimates for β, λ, and τ.
Petroleum endowment and economic growth: examination of the resource curse phenomenon
Published in Energy Sources, Part B: Economics, Planning, and Policy, 2021
Then, we adopted a step-by-step methodology for choosing the appropriate exogenous variables. In this strategy, exogenous variables are introduced sequentially instead of simultaneously. Four models are to be considered. Model expresses the GDP growth rate in terms of the ratio of proved reserves of oil per inhabitants and oil rents. Next, we, respectively, introduce the variables « openness rate » and « investment rate » to form the models and . Finally, the only variable measuring the quality of institutions is « rule of law», while the «government effectiveness » is firstly excluded from the model to avoid multi-collinearity problems. This last model is denoted model . We also consider another model, which contains the same variables as , but rather we keep the «government effectiveness » instead of the « rule of law». This model is denoted model . Once appropriately identified, a number of goodness-of-fit tests are performed to the models , , , and . The Fisher significance test is used to test the overall significance of the parameters of each model. The results show that all models are overall significant. Then, the Hausman’s test proves that all models would be fitted with assumption of fixed effects. The Breusch-Pagan statistic is also calculated and shows the existence of heteroscedasticity problem in all models. To correct the heteroscedasticity problem, the Generalized Least Squares (GLS) regression is used. The approach of Swamy and Arora (1972) is chosen to correct the variance-covariance matrix. Finally, we used the Wooldridge test to check whether a serial correlation problem exists. The results of this test suggest the failing to reject the null hypothesis of absence of first-order autocorrelations (all Chi-Squared p-values > 5%). All estimation results are reported in Tables 4 and 5.