1. In a simple linear regression analysis, n independent paired data (yl, x1), ...., (yn, X\") are fitted to the model M1 yi: 180+l81(xia)+5i; i=1u-un. where the regressor x is a random variable with mean [1x and variance 0,? . The random error e is statistically independent of x and has mean zero and constant variance 02. a is a given real number (that is, the value of a is known). The values of [1,6,03,ch are all unknown. Before the n independent paired data for (y , x) are available, we need to construct estimators for the parameters. Construct the ordinary least squares (OLS) estimator of [31. Is the OLS estimator unbiased for ,81? why? Construct an estimator for 02 . Is the estimator unbiased for 02 ? why? Derive the variance ofthe OLS estimator of l , conditional on x's, in a) and construct an unbiased estimator of this conditional variance. Continue c), derive the variance of the OLS estimator of l in a) and construct an unbiased estimator of this variance. [Note: variance is unconditional variance, which differs from conditional variance] Construct an estimator for the expectation E(y), using model M1