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CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics. 2007 Instructor Longin Jan Latecki C22: The Method of Least Squares 22.1 – Least Squares Given is a bivariate dataset (x1, y1), …, (xn, yn), where x1, …, xn are nonrandom and Yi = α + βxi + Ui are random variables for i = 1, 2, . . ., n. The random variables U1, U2, …, Un have zero expectation and variance σ 2 Method of Least Squares: Choose a value for α and β such that n 2 S(α,β)=( ∑ ( yi− α− β x i) ) is minimal. 1 22.1 – Regression The observed value yi corresponding to xi and the value α+βxi on the regression line y = α + βx. n 2 ( y − α− β x ) ∑ i i 1 22.1– Estimation Method of Least Squares: Choose a value for α and β such that n S(α,β)=(∑ ( yi− α− β x i)2 ) is minimal. 1 To find the least squares estimates, we differentiate S(α, β) with respect to α and β, and we set the derivatives equal to 0: After some calculus magic, we get two equations to estimate α and β: 22.1– Estimation After some simple algebraic rearranging, we obtain: (slope) (intercept) Regression line y = 0.25 x –2.35 for points Var ( X ) E[ X 2 ] E[ X ]2 22.1– Least Square Estimators are Unbiased The estimators for α and β are unbiased. For the simple linear regression model, the random variable n ̂σ 2= 1 2 ̂ (Y − ̂α − β x ) ∑ i n− 2 i= 1 i is an unbiased estimator for σ2. 22.2– Residuals A way to explore whether the linear regression model is appropriate to model a given bivariate dataset is to inspect a scatter plot of the so-called residuals ri against the xi. The ith residual ri is defined as the vertical distance between the ith point and the estimated regression line: We always have 22.2– Heteroscedasticity Homoscedasticity: The assumption of equal variance of the Ui (and therefore Yi). In case the variance of Yi depends on the value of xi, we speak of heteroscedasticity. For instance, heteroscedasticity occurs when Yi with a large expected value have a larger variance than those with small expected values. This produces a “fanning out” effect, which can be observed in the figure: 22.3– Relation with Maximum Likelihood What are the maximum likelihood estimates for α and β? To apply the method of least squares no assumption is needed about the type of distribution of the Ui. In case the type of distribution of the Ui is known, the maximum likelihood principle can be applied. In particular, when the Ui are independent with an N(0, σ2) distribution. Then Yi has an N (α + βxi, σ2) distribution, making the probability density function When Yi are independent, and eachYi has an N(α+βxi, σ2) distribution, and assuming that the linear model is appropriate to model a given bivariate dataset, the residuals ri should look like the realization of a random sample from a normal distribution. An example is shown in the figure below: 22.3– Maximum Likelihood For fixed σ >0 the loglikelihood l (α, β, σ) obtains the maximum when n ∑1 ( yi− α− β x i)2 is minimal. Hence, when random variables independent with a N(0,σ 2) distribution, the maximum likelihood principle and the least squares method return the same estimators. The maximum likelihood estimator for σ 2 is: n 1 2 ̂ ̂σ = ∑ (Y i− ̂α − β xi ) n i= 1 2