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Topic: Ordinary least squares

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	Least squares Summary
		Least squares is a mathematical optimization technique which, when given a series of measured data, attempts to find a function which closely approximates the data (a "best fit").
		+ bx + c, estimating a, b, and c by least squares, is an instance of linear regression because the vector of least-square estimates of a, b, and c is a linear transformation of the vector whose components are f(x
		Least squares estimation for linear models is notoriously non-robust to outliers.
	www.bookrags.com /Least_squares (1107 words)

	Partial Least Squares (PLS)
		In short, partial least squares regression is probably the least restrictive of the various multivariate extensions of the multiple linear regression model.
		For establishing the model, partial least squares regression produces a p by c weight matrix W for X such that T=XW, i.e., the columns of W are weight vectors for the X columns producing the corresponding n by c factor score matrix T.
		Ordinary least squares procedures for the regression of Y on T are then performed to produce Q, the loadings for Y (or weights for Y) such that Y=TQ+E.
	www.statsoft.com /textbook/stpls.html (3719 words)

	Partial Least Squares
		Ordinary least squares regression, as implemented in SAS/STAT procedures such as PROC GLM and PROC REG, has the single goal of minimizing sample response prediction error, seeking linear functions of the predictors that explain as much variation in each response as possible.
		Two different formulations for partial least squares are available: the original method of Wold (1966) and the SIMPLS method of de Jong (1993).
		The partial least squares method was originally developed in the 1960s by the econometrician Herman Wold (1966) for modeling "paths" of causal relation between any number of "blocks" of variables.
	support.sas.com /rnd/app/da/new/dapls.html (595 words)

	Least Squares Fitting (Regression)
		The counting of annual growth rings should be precise; of course the trees are probably a bit older than the count as it took a few years for the tree to grow to the height at which the core was taken.
		The line is adjusted until the sum of the squares of the y deviations from the line (shown above in blue) are as small as possible.
		You may be guided by the suggestions of known theory, by the requirements of a particular instructor, by standard practice (usually a least squares line), by knowledge of which points are most likely to be anomalous, or (unfortunately) by a desire to produce a particular answer.
	www.physics.csbsju.edu /stats/least_squares.html (1507 words)

	Least squares - Wikipedia, the free encyclopedia
		Most evaluations were useless; the only calculation precise enough to allow Zach, a German astronomer, to recover Ceres at the end of the year, was that of 24-year-old Carl Friedrich Gauss (the fundamentals of his approach had already been accomplished by him in 1795, when he was still 18 years old).
		In 1829 Gauss was able to state the reason for this procedure's outstanding success: The method of least squares is simply optimal in many respects.
		The precise argument is known as the Gauss-Markov theorem.
	en.wikipedia.org /wiki/Least_squares (866 words)

	Heteroscedasticity
		This is called weighted least squares because the ordinary least squares (OLS) expression is “weighted” (by the inverse of the variance).
		Therefore, weighted least squares estimates of the regression parameters can be obtained by performing an ordinary least squares regression on the transformed variables obtained by dividing the original variables by X
		That is, the ordinary least squares solution is weighted by the inverse of the variances.
	darkwing.uoregon.edu /~mauro/psy612/WLS.htm (553 words)

	Ordinary Least Squares Regression (OLS) in Blossom Statistics
		Estimation and permutation testing alternatives for the familiar ordinary least squares regression are available with the OLS command.
		LAD regression estimates and their permutation test are far less sensitive to the impacts of one or a few outliers (Cade and Richards 1996, Mielke and Berry 2001).
		The SAVETEST option allows the Monte Carlo generated random sample of test statistics to be saved to a single column in the specified file, where the first value is the observed test statistic.
	www.mesc.usgs.gov /products/publications/10002/ols.htm (1139 words)

	Overview
		In the nonlinear case, nonlinear ordinary least-squares estimation is performed iteratively using a linearization of the model with respect to the parameters.
		The instrumental solution to simultaneous equation bias in the nonlinear case is the same as the linear case except the linearization of the model with respect to the parameters is predicted by the instrumental regression.
		Nonlinear two-stage least squares is one of several instrumental variables methods available in the MODEL procedure to handle simultaneous equation bias.
	bama.ua.edu /sasv8/ets/chap14/sect1.htm (1124 words)

	PA 765: Two-Stage Least Squares
		Two-stage least squares regression (2SLS) is a method of extending regression to cover models which violate ordinary least squares (OLS) regression's assumption of recursivity, specifically models where the researcher must assume that the disturbance term of the dependent variable is correlated with the cause(s) of the independent variable(s).
		The researcher then proceeds to stage 2 of 2SLS, which is simply an ordinary least squares (OLS) regression, but using the newly created instrumental variables.
		Statistical properties of the two-stage least squares estimator under cointegration.
	www2.chass.ncsu.edu /garson/pa765/2sls.htm (1828 words)

	ORDINARY LEAST SQUARES
		rdinary Least Squares (commonly called Linear Regression) uses the method of least squares to estimate the "best fit" of a set of independent (X) variables against the dependent variable (Y) variable you wish to explain or predict.
		The method of Ordinary Least Squares is based upon a number of statistical assumptions such as...
		Problems such as multicollinearity (extreme correlation) among the explanatory variables cause difficulties in computing the least squares estimates.
	www.forecastingsolutions.com /ols.html (229 words)

	[No title]
		OLS Ordinary Least Squares (cont.)¡4'$$ª ¨ÕA. The Least Square Criteria (cont.) 1.
		OLS Ordinary Least Squares (cont.)¡&'$ª ¨ÌA. The Least Square Criteria (cont.) 1.
		OLS Ordinary Least Squares (cont.)¡&'$ª ¨A. The Least Square Criteria B.
	www.columbia.edu /itc/sipa/U4320y-003/overheads/Lecture08.ppt (2266 words)

	[No title]
		The colleague remarks that the data are obviously drawn from a generalized regression model, and that they should be using generalized least squares, not ordinary least squares.
		The researcher returns to the lab and computes a two step generalized least squares estimator using one of the methods we have discussed in class.
		There are (at least) three possible estimators for the asymptotic covariance matrix for this estimator: (1) the usual estimator that you compute when you compute ordinary least squares estimators, (2) the White estimator, and (3) the appropriate estimator based on equation (13-56) in your text in which the terms with unequal subscripts are zero.
	pages.stern.nyu.edu /~wgreene/Econometrics/Problem7.doc (1299 words)

	Weighted Least Squares
		Although the F statistic is the same in ALR as in Stata, the sums of squares differ.
		Note that the constant term is also adjusted here, so doing the ordinary least-squares fit for Z and M requires a regression through the origin.
		The weighted least squares calculation is based on the assumption that the variance of the observations is unknown, but that the relative variances are known.
	galton.uchicago.edu /~s343/Handouts/wls.html (1618 words)

	squares - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "squares" is defined.
		Example: "Gravity is inversely proportional to the square of the distance"
		Phrases that include squares: least squares, method of least squares, ordinary least squares, latin squares, least squares analysis, more...
	www.onelook.com /?w=squares (324 words)

	Annotated Mplus Output: Ordinary Least Squares Regression
		This page shows an example of ordinary least squares (OLS) regression with footnotes explaining the output.
		First an example is shown using Stata, and then an example is shown using Mplus, to help you relate the output you are likely to be familiar with (Stata) to output that may be new to you (Mplus).
		It is analogous to the mean square error (MSE) from the Stata output.
	www.ats.ucla.edu /stat/mplus/output/olsreg.htm (586 words)

	lscov :: Functions (MATLAB Function Reference)
		and, in effect, inverts that factor to transform the problem into ordinary least squares.
		The MATLAB backslash operator (\) enables you to perform linear regression by computing ordinary least-squares (OLS) estimates of the regression coefficients.
		The coefficient standard errors are equal to the square roots of the values on the diagonal of this covariance matrix:
	www.mathworks.com /access/helpdesk/help/techdoc/ref/lscov.shtml (497 words)

	Least Squares
		To fit a straight line by ordinary least squares, we use the lm() ("linear model") function:
		The residual standard error is the square root of the residual sum of squares divided by the degrees of freedom.
		Degrees of freedom measures how much data we have left to estimate the residual variance, and thus the residual standard error, after fitting the line.
	www.reed.edu /~jones/141/ls1.html (513 words)

	Chemometrics and Qualimetrics Group (Site not responding. Last check: 2007-11-04)
		bls.m: Function that calculates the coefficients of the regression line using the BLS (Bivariate Least Squares) method.
		jci.m: Joint confidence interval for the intercept and the slope of the regression straight line using the BLS (Bivariate Least Squares), WLS (Weighted Least Squares) and OLS (Ordinary Least Squares) methods.
		tcbeta.m: Estimation of the probabilities of beta error in the joint confidence interval for the intercept and the slope of the regression line using the BLS (Bivariate Least Squares), WLS (Weighted Least Squares) and OLS (Ordinary Least Squares) methods.
	www.quimica.urv.es /quimio/ang/mcat.html (292 words)

	Issues in estimating ARIMA models
		ARIMA models which include only AR terms are special cases of linear regression models, hence they can be fitted by ordinary least squares.
		Another way to look at the problem: you can't fit MA models using ordinary multiple regression because there's no way to specify ERRORS as an independent variable--the errors are not known until the model is fitted!
		This is just an ordinary multiple regression model in which "mu" is the constant term, "phi-1" is the coefficient of the first lag of y, and so on.
	www.duke.edu /~rnau/arimest.htm (849 words)

	Weighted least squares - Wikipedia, the free encyclopedia
		Weighted least squares is a method of regression, similar to least squares in that it uses the same minimization of the sum of the residuals:
		However, instead of weighting all points equally, they are weighted such that points with a greater weight contribute more to the fit:
		is the same as minimizing the ordinary least squares
	en.wikipedia.org /wiki/Weighted_least_squares (114 words)

	StRD Linear Least Squares Regression Glossary
		The general statistical model assumed for the linear least squares regression problems is
		The certified value for the residual sum of squares is defined by
		The certified values for the standard deviations of the estimates of the model parameters are the square roots of the diagonal elements of the variance-covariance matrix
	www.itl.nist.gov /div898/strd/lls/data/LINKS/c-Pontius.shtml (343 words)

	Ordinary Least Squares for Simple Regression
		This is the reason why we will never be able to calculate the true value of alpha and beta.
		estimate the values of alpha and beta as good as possible (using the least squares criterion).
		There are at least 32,000 Britons now living in India, according to new research - and its the economy that is sucking them in an, a twist of migration history between the two countries.
	www.xycoon.com /ols.htm (1269 words)

	Ordinary Least Squares (Site not responding. Last check: 2007-11-04)
		Let us now look into the sum of squares for common linear regression models.
		We recall that most of our linear regression experience is with the model
		The residual sum of squares of (6) is
	www.jstatsoft.org /v02/i05/pirls/node7.html (65 words)

	SSRN-On the Performance of the Ordinary Least Squares Method Under an Error Component Model by Parimal Mukhopadhyay, ...
		SSRN-On the Performance of the Ordinary Least Squares Method Under an Error Component Model by Parimal Mukhopadhyay, Rainer Schwabe
		Starting from the one-dimensional results by Wang et al (1994) we consider the performance of the ordinary least squares estimator in comparison to the best linear unbiased estimator under an error component model with random effects in units and tons.
		Upper bounds are derived for the first-order approximation to the difference between both estimators and for the spectral norm of the difference between their dispersion matrices.
	papers.ssrn.com /sol3/papers.cfm?abstract_id=110989 (210 words)

	Ordinary Least Squares
		command will estimate the parameters of a linear regression equation by the method of ordinary least squares.
		A property of ordinary least squares regression (when an intercept is included) is that the sum of the estimated residuals (and hence the mean of the estimated residuals) is 0.
		Different computers may have numerical differences in the reporting of this result.
	econometrics.com /intro/ols.htm (884 words)

	Summary of Ordinary Least Squares (Site not responding. Last check: 2007-11-04)
		Next: Geometry Up: Ordinary Least Squares Previous: Ordinary Least Squares
		in sense that we minimize the sum of squared deviations
		minimizes the sum of squared elements from the deviation vector
	emlab.berkeley.edu /pub/GMTheorem/node3.html (86 words)

	AST 443: Ordinary Least Squares (Site not responding. Last check: 2007-11-04)
		Given pairwise data (X,Y), where X is the independent variable, it is often useful to characterize Y as a function of X. The simplest way to do this is the ordinary least squares (OLS) fit, which minimizes the square of the deviations of the fit from the data points.
		Assume a functional form for Y, such as Y=aX+b
		This function is tabulated in Table C.3 of Bevington and Robinson.
	www.ess.sunysb.edu /fwalter/AST443/ols.html (197 words)

	Ordinary Least Squares
		Consider estimating the relationship between the dependent variable
		Ordinary least squares estimates of the parameters can be obtained with the next SHAZAM commands.
		SAMPLE 1 17 READ (THEIL.txt) YEAR CONSUME INCOME PRICE OLS CONSUME INCOME PRICE STOP
	shazam.econ.ubc.ca /intro/ols2.htm (315 words)

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