OLS is the abbreviation or designator of a method/algorithm to calculate estimates in statistical models. These models connect one ore more predictors (categorical or continuous) to a response (continuous). The influences/effects of the predictors in the model are simple factors (of the predictors; theses factors are called "parameters" of the model) that are themselves untransformed, i.e. "linear". Hence, the models are linear in the parameters. The values of these parameters are determined by fitting the model to the data. The goodness of the fit is measured by the sum of the squared differences (="residuals") between model predictions and data. The "best", that is, minimum sum of squared differences is calles the "least squares" (LS). When the "best" fit is determined under the assumption of normal distributed residuals, it is called "ordinary least squares" (OLS), in contrasts to GLS (generalized LS) where other probability distributions are used.

It is completely nonrelevant whether or not the values of the predictors are fixed at particular, defined values. The models just aim to give "best" estimates for the parameters given the data at hand. If the predictors are measured with error, there will be correspondingly more uncertainty in the estimates. There is only one case where uncertainty in the predictors has a deleterious effect on the prediction: when several predictors are correlated. But (strongly) correlated predictors cause a whole bunch of problems anyway...

Jochen Wilhelm . i totally agree practically or you can say in estimation stages , but when i think it mathematically it confuses me, that this randomness in predictor, effect the linearity of OLS and also its distribution, but books say it doesn't. plz explain more if you can...

I don't really get your problem. I don't understand wat you mean with "linearity of OLS" and "its [the OLS] distribution".

Raja, when the regressors (X) are stochastic you replace the hypothesis that E(e|x)=0, where e is the error of the regression, with the weaker assumption that cov(e,X)=0 and the general framework of OLS remains valid.

Normally, if xj are considered nonrandom variables, the distribution of u is not correlated with x and the following assumption of the error term holds : cov(xi,ui)=0. Thus what you have in this case, is violation of the assumption between independence of xj and ui, that is we measure x with an error. A good way to deal with this problem is to find an instrumental variable zi which is uncorrelated with ui, i.e cov(z,u)=0 but correlated with xi, i.e cov(z,x)≠0. Then you get a new estimator of β^ , usually denoted in textbooks by β^ IV, which is unbiased and consistent.

actully the randomness you both are considering is endogenity problem, i was on the point that if x (regressors) is by nature stochastic as y (response) then usual ols estimator become prodduct of two stochastic variables, then what will be its distribution or properties.

let we consider x and y two normal random varibles then their product may be chi square , but here what will happened when we have beta=(X'X)^-1X'Y

The properties of OLS for stochastic X depend on what you assume with regard to the X, and if you're interested in the finite-sample or the asymptotic distribution. Simplifying: If X is a matrix of r.v. with some finite higher order moments and full rank, and if X is independent from U, [n^(-1) X'X] converges in probability to a finite and invertible matrix, and [n^(-0.5)X'U] converges in distribution to a gaussian distribution; as a consequence, your beta is (under the previous assumptions) asymptotically normal, regardless of the exact distribution of X.... for further reading (so somewhat abstract) see the chapter by Poetscher/Prucha in Baltagi (2001) "Companion to ..."