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Text of the page (random words):
it β i 1 n x i x y i y i 1 n x i x 2 i 1 n x i x 2 y i y x i x i 1 n x i x 2 i 1 n x i x 2 j 1 n x j x 2 y i y x i x displaystyle begin aligned widehat beta frac sum _ i 1 n left x_ i bar x right left y_ i bar y right sum _ i 1 n left x_ i bar x right 2 1ex frac sum _ i 1 n left x_ i bar x right 2 frac y_ i bar y x_ i bar x sum _ i 1 n left x_ i bar x right 2 1ex sum _ i 1 n frac left x_ i bar x right 2 sum _ j 1 n left x_ j bar x right 2 frac y_ i bar y x_ i bar x 6pt end aligned we can see that the slope tangent of angle of the regression line is the weighted average of y i y x i x displaystyle frac y_ i bar y x_ i bar x that is the slope tangent of angle of the line that connects the i th point to the average of all points weighted by x i x 2 displaystyle x_ i bar x 2 because the further the point is the more important it is since small errors in its position will affect the slope connecting it to the center point more interpretation about the intercept edit the parameter α displaystyle widehat alpha is the intercept of the linear function y α β x displaystyle begin aligned y widehat alpha widehat beta x 5pt end aligned therefore the y displaystyle y intercept of the function found with simple linear regression is y i n t e r c e p t α y β x displaystyle y_ rm intercept widehat alpha bar y widehat beta bar x because β displaystyle widehat beta is the slope of the linear function β tan θ displaystyle widehat beta tan theta therefore the angle θ displaystyle theta the graph of the function makes with the x displaystyle x axis is equal to θ arctan β displaystyle theta arctan widehat beta interpretation about the correlation edit in the above formulation notice that each x i displaystyle x_ i is a constant known upfront value while the y i displaystyle y_ i are random variables that depend on the linear function of x i displaystyle x_ i and the random term ε i displaystyle varepsilon _ i this assumption is used when deriving the standard error of the slope and showing that it is unbiased in this framing when x i displaystyle x_ i is not actually a random variable what type of parameter does the empirical correlation r x y displaystyle r_ xy estimate the issue is that for each value i we ll have e x i x i displaystyle e x_ i x_ i and v a r x i 0 displaystyle var x_ i 0 a possible interpretation of r x y displaystyle r_ xy is to imagine that x i displaystyle x_ i defines a random variable drawn from the empirical distribution of the x values in our sample for example if x had 10 values from the natural numbers 1 2 3 10 then we can imagine x to be a discrete uniform distribution under this interpretation all x i displaystyle x_ i have the same expectation and some positive variance with this interpretation we can think of r x y displaystyle r_ xy as the estimator of the pearson s correlation between the random variable y and the random variable x as we just defined it numerical properties edit the regression line goes through the center of mass point x y displaystyle bar x bar y if the model includes an intercept term i e not forced through the origin the sum of the residuals is zero if the model includes an intercept term i 1 n ε i 0 displaystyle sum _ i 1 n widehat varepsilon _ i 0 the residuals and x values are uncorrelated whether or not there is an intercept term in the model meaning i 1 n x i ε i 0 displaystyle sum _ i 1 n x_ i widehat varepsilon _ i 0 the relationship between ρ x y displaystyle rho _ xy the correlation coefficient for the population and the population variances of y displaystyle y σ y 2 displaystyle sigma _ y 2 and the error term of ε displaystyle varepsilon σ ε 2 displaystyle sigma _ varepsilon 2 is 10 401 σ ε 2 1 ρ x y 2 σ y 2 displaystyle sigma _ varepsilon 2 1 rho _ xy 2 sigma _ y 2 for extreme values of ρ x y displaystyle rho _ xy this is self evident since when ρ x y 0 displaystyle rho _ xy 0 then σ ε 2 σ y 2 displaystyle sigma _ varepsilon 2 sigma _ y 2 and when ρ x y 1 displaystyle rho _ xy 1 then σ ε 2 0 displaystyle sigma _ varepsilon 2 0 statistical properties edit description of the statistical properties of estimators from the simple linear regression estimates requires the use of a statistical model the following is based on assuming the validity of a model under which the estimates are optimal it is also possible to evaluate the properties under other assumptions such as inhomogeneity but this is discussed elsewhere clarification needed unbiasedness edit the estimators α displaystyle widehat alpha and β displaystyle widehat beta are unbiased to formalize this assertion we must define a framework in which these estimators are random variables we consider the residuals ε i as random variables drawn independently from some distribution with mean zero in other words for each value of x the corresponding value of y is generated as a mean response α βx plus an additional random variable ε called the error term equal to zero on average under such interpretation the least squares estimators α displaystyle widehat alpha and β displaystyle widehat beta will themselves be random variables whose means will equal the true values α and β this is the definition of an unbiased estimator variance of the mean response edit since the data in this context is defined to be x y pairs for every observation the mean response at a given value of x say x d is an estimate of the mean of the y values in the population at the x value of x d that is e y x d y d displaystyle hat e y mid x_ d equiv hat y _ d the variance of the mean response is given by 11 var α β x d var α var β x d 2 2 x d cov α β displaystyle operatorname var left hat alpha hat beta x_ d right operatorname var left hat alpha right left operatorname var hat beta right x_ d 2 2x_ d operatorname cov left hat alpha hat beta right this expression can be simplified to var α β x d σ 2 1 m x d x 2 x i x 2 displaystyle operatorname var left hat alpha hat beta x_ d right sigma 2 left frac 1 m frac left x_ d bar x right 2 sum x_ i bar x 2 right where m is the number of data points to demonstrate this simplification one can make use of the identity i x i x 2 i x i 2 1 m i x i 2 displaystyle sum _ i x_ i bar x 2 sum _ i x_ i 2 frac 1 m left sum _ i x_ i right 2 variance of the predicted response edit further information prediction interval the predicted response distribution is the predicted distribution of the residuals at the given point x d so the variance is given by var y d α β x d var y d var α β x d 2 cov y d α β x d var y d var α β x d displaystyle begin aligned operatorname var left y_ d left hat alpha hat beta x_ d right right operatorname var y_ d operatorname var left hat alpha hat beta x_ d right 2 operatorname cov left y_ d left hat alpha hat beta x_ d right right operatorname var y_ d operatorname var left hat alpha hat beta x_ d right end aligned the second line follows from the fact that cov y d α β x d displaystyle operatorname cov left y_ d left hat alpha hat beta x_ d right right is zero because the new prediction point is independent of the data used to fit the model additionally the term var α β x d displaystyle operatorname var left hat alpha hat beta x_ d right was calculated earlier for the mean response since var y d σ 2 displaystyle operatorname var y_ d sigma 2 a fixed but unknown parameter that can be estimated the variance of the predicted response is given by var y d α β x d σ 2 σ 2 1 m x d x 2 x i x 2 σ 2 1 1 m x d x 2 x i x 2 displaystyle begin aligned operatorname var left y_ d left hat alpha hat beta x_ d right right sigma 2 sigma 2 left frac 1 m frac left x_ d bar x right 2 sum x_ i bar x 2 right 4pt sigma 2 left 1 frac 1 m frac x_ d bar x 2 sum x_ i bar x 2 right end aligned confidence intervals edit the formulas given in the previous section allow one to calculate the point estimates of α and β that is the coefficients of the regression line for the given set of data however those formulas do not tell us how precise the estimates are i e how much the estimators α displaystyle widehat alpha and β displaystyle widehat beta vary from sample to sample for the specified sample size confidence intervals were devised to give a plausible set of values to the estimates one might have if one repeated the experiment a very large number of times the standard method of constructing confidence intervals for linear regression coefficients relies on the normality assumption which is justified if either the errors in the regression are normally distributed the so called classic regression assumption or the number of observations n is sufficiently large in which case the estimator is approximately normally distributed the latter case is justified by the central limit theorem normality assumption edit under the first assumption above that of the normality of the error terms the estimator of the slope coefficient will itself be normally distributed with mean β and variance σ 2 i x i x 2 textstyle sigma 2 left sum _ i x_ i bar x 2 right where σ 2 is the variance of the error terms see proofs involving ordinary least squares at the same time the sum of squared residuals q is distributed proportionally to χ 2 with n 2 degrees of freedom and independently from β displaystyle widehat beta this allows us to construct a t value t β β s β t n 2 displaystyle t frac widehat beta beta s_ widehat beta sim t_ n 2 where s β 1 n 2 i 1 n ε i 2 i 1 n x i x 2 displaystyle s_ widehat beta sqrt frac frac 1 n 2 sum _ i 1 n widehat varepsilon _ i 2 sum _ i 1 n x_ i bar x 2 is the unbiased standard error estimator of the estimator β displaystyle widehat beta this t value has a student s t distribution with n 2 degrees of freedom using it we can construct a confidence interval for β β β s β t n 2 β s β t n 2 displaystyle beta in left widehat beta s_ widehat beta t_ n 2 widehat beta s_ widehat beta t_ n 2 right at confidence level 1 γ where t n 2 displaystyle t_ n 2 is the 1 γ 2 th displaystyle scriptstyle left 1 frac gamma 2 right text th quantile of the t n 2 distribution for example if γ 0 05 then the confidence level is 95 similarly the confidence interval for the intercept coefficient α is given by α α s α t n 2 α s α t n 2 displaystyle alpha in left widehat alpha s_ widehat alpha t_ n 2 widehat alpha s_ widehat alpha t_ n 2 right at confidence level 1 γ where s α s β 1 n i 1 n x i 2 1 n n 2 i 1 n ε i 2 i 1 n x i 2 i 1 n x i x 2 displaystyle s_ widehat alpha s_ widehat beta sqrt frac 1 n sum _ i 1 n x_ i 2 sqrt frac 1 n n 2 left sum _ i 1 n widehat varepsilon _ i 2 right frac sum _ i 1 n x_ i 2 sum _ i 1 n x_ i bar x 2 the us changes in unemployment gdp growth regression with the 95 confidence bands the confidence intervals for α and β give us the general idea where these regression coefficients are most likely to be for example in the okun s law regression shown here the point estimates are α 0 859 β 1 817 displaystyle widehat alpha 0 859 qquad widehat beta 1 817 the 95 confidence intervals for these estimates are α 0 76 0 96 β 2 06 1 58 displaystyle alpha in left 0 76 0 96 right qquad beta in left 2 06 1 58 right in order to represent this information graphically in the form of the confidence bands around the regression line one has to proceed carefully and account for the joint distribution of the estimators it can be shown 12 that at confidence level 1 γ the confidence band has hyperbolic form given by the equation α β ξ α β ξ t n 2 1 n 2 ε i 2 1 n ξ x 2 x i x 2 displaystyle alpha beta xi in left widehat alpha widehat beta xi pm t_ n 2 sqrt left frac 1 n 2 sum widehat varepsilon _ i 2 right cdot left frac 1 n frac xi bar x 2 sum x_ i bar x 2 right right when the model assumed the intercept is fixed and equal to 0 α 0 displaystyle alpha 0 the standard error of the slope turns into s β 1 n 1 i 1 n ε i 2 i 1 n x i 2 displaystyle s_ widehat beta sqrt frac 1 n 1 frac sum _ i 1 n widehat varepsilon _ i 2 sum _ i 1 n x_ i 2 with ε i y i y i displaystyle hat varepsilon _ i y_ i hat y _ i asymptotic assumption edit the alternative second assumption states that when the number of points in the dataset is large enough the law of large numbers and the central limit theorem become applicable and then the distribution of the estimators is approximately normal under this assumption all formulas derived in the previous section remain valid with the only exception that the quantile t n 2 of student s t distribution is replaced with the quantile q of the standard normal distribution occasionally the fraction 1 n 2 is replaced with 1 n when n is large such a change does not alter the results appreciably numerical example edit see also ordinary least squares example and linear least squares example this data set gives average masses for women as a function of their height in a sample of american women of age 30 39 although the ols article argues that it would be more appropriate to run a quadratic regression for this data the simple linear regression model is applied here instead height m x i 1 47 1 50 1 52 1 55 1 57 1 60 1 63 1 65 1 68 1 70 1 73 1 75 1 78 1 80 1 83 mass kg y i 52 21 53 12 54 48 55 84 57 20 58 57 59 93 61 29 63 11 64 47 66 28 68 10 69 92 72 19 74 46 i displaystyle i x i displaystyle x_ i y i displaystyle y_ i x i 2 displaystyle x_ i 2 x i y i displaystyle x_ i y_ i y i 2 displaystyle y_ i 2 1 1 47 52 21 2 1609 76 7487 2725 8841 2 1 50 53 12 2 2500 79 6800 2821 7344 3 1 52 54 48 2 3104 82 8096 2968 0704 4 1 55 55 84 2 4025 86 5520 3118 1056 5 1 57 57 20 2 4649 89 8040 3271 8400 6 1 60 58 57 2 5600 93 7120 3430 4449 7 1 63 59 93 2 6569 97 6859 3591 6049 8 1 65 61 29 2 7225 101 1285 3756 4641 9 1 68 63 11 2 8224 106 0248 3982 8721 10 1 70 64 47 2 8900 109 5990 4156 3809 11 1 73 66 28 2 9929 114 6644 4393 0384 12 1 75 68 10 3 0625 119 1750 4637 6100 13 1 78 69 92 3 1684 124 4576 4888 8064 14 1 80 72 19 3 2400 129 9420 5211 3961 15 1 83 74 46 3 3489 136 2618 5544 2916 σ displaystyle sigma 24 76 931 17 41 0532 1548 2453 58498 5439 there are n 15 points in this data set hand calculations would be started by finding the following five sums s x i x i 24 76 s y i y i 931 17 s x x i x i 2 41 0532 s y y i y i 2 58498 5439 s x y i x i y i 1548 2453 displaystyle begin aligned s_ x sum _ i x_ i 24 76 qquad s_ y sum _ i y_ i 931 17 5pt s_ xx sum _ i x_ i 2 41 0532 s_ yy sum _ i y_ i 2 58498 5439 5pt s_ xy sum _ i x_ i y_ i 1548 2453 end aligned these quantities would be used to calculate the estimates of the regression coefficients and their standard errors β n s x y s x s y n s x x s x 2 61 272 α 1 n s y β 1 n s x 39 062 s ε 2 1 n n 2 n s y y s y 2 β 2 n s x x s x 2 0 5762 s β 2 n s ε 2 n s x x s x 2 3 1539 s α 2 s β 2 1 n s x x 8 63185 displaystyle begin aligned widehat beta frac ns_ xy s_ x s_ y ns_ xx s_ x 2 61 272 8pt widehat alpha frac 1 n s_ y widehat beta frac 1 n s_ x 39 062 8pt s_ varepsilon 2 frac 1 n n 2 left ns_ yy s_ y 2 widehat beta 2 ns_ xx s_ x 2 right 0 5762 8pt s_ widehat beta 2 frac ns_ varepsilon 2 ns_ xx s_ x 2 3 1539 8pt s_ widehat alpha 2 s_ wi...
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