assumption about the distribution of error term εi, it is impossible to infer the distribution of the estimators and . Nevertheless, we can apply the central limit theorem to derive their asymptotic properties as sample size n goes to infinity. While the sample size is necessarily finite, it is customary to assume that n is "large enough" so that the true distribution of the OLS estimator is close to its asymptotic limit. We can show that under the model assumptions, the least squares estimator for β is consistent (that is converges in probability to β) and asymptotically normal:[proof] where Intervals Using this asymptotic distribution, approximate two-sided confidence intervals for the j-th component of the vector can be constructed as at the confidence level, where q denotes the quantile function of standard normal distribution, and [·]jj is the j-th diagonal element of a matrix. Similarly, the least squares estimator for σ2 is also consistent and asymptotically normal (provided that the fourth moment of εi exists) with limiting distribution These asymptotic distributions can be used for prediction, testing hypotheses, constructing other estimators, etc.. As an example consider the problem of prediction. Suppose is some point within the domain of distribution of the regressors, and one wants to know what the response variable would have been at that point. The mean response is the quantity , whereas the predicted response is . Clearly the predicted response is a random variable, its distribution can be derived from that of : which allows construct confidence intervals for mean response to be constructed: at the confidence level. Hypothesis testing Two hypothesis tests are particularly widely used. First, one wants to know if the estimated regression equation is any better than simply predicting that all values of the response variable equal its sample mean (if not, it is said to have no explanatory power). The null hypothesis of no explanatory value of the estimated regression is tested using an F-test. If the calculated F-value is found to be large enough to exceed its critical value for the pre-chosen level of significance, the null hypothesis is rejected and the alternative hypothesis, that the regression has explanatory power, is accepted. Otherwise, the null hypothesis of no explanatory power is accepted. Second, for each explanatory variable of interest, one wants to know whether its estimated coefficient differs significantly from zero—that is, whether this particular explanatory variable in fact has explanatory