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The Simple Regression Model y = b0 + b1x + u Economics 20 - Prof. Anderson 1 Some Terminology In the simple linear regression model, where y = b0 + b1x + u, we typically refer to y as the Dependent Variable, or Left-Hand Side Variable, or Explained Variable, or Regressand Economics 20 - Prof. Anderson 2 Some Terminology, cont. In the simple linear regression of y on x, we typically refer to x as the Independent Variable, or Right-Hand Side Variable, or Explanatory Variable, or Regressor, or Covariate, or Control Variables Economics 20 - Prof. Anderson 3 A Simple Assumption The average value of u, the error term, in the population is 0. That is, E(u) = 0 This is not a restrictive assumption, since we can always use b0 to normalize E(u) to 0 Economics 20 - Prof. Anderson 4 Zero Conditional Mean We need to make a crucial assumption about how u and x are related We want it to be the case that knowing something about x does not give us any information about u, so that they are completely unrelated. That is, that E(u|x) = E(u) = 0, which implies E(y|x) = b0 + b1x Economics 20 - Prof. Anderson 5 E(y|x) as a linear function of x, where for any x the distribution of y is centered about E(y|x) y f(y) . x1 . E(y|x) = b + b x 0 1 x2 Economics 20 - Prof. Anderson 6 Ordinary Least Squares Basic idea of regression is to estimate the population parameters from a sample Let {(xi,yi): i=1, …,n} denote a random sample of size n from the population For each observation in this sample, it will be the case that yi = b0 + b1xi + ui Economics 20 - Prof. Anderson 7 Population regression line, sample data points and the associated error terms E(y|x) = b0 + b1x .{ u4 y y4 y3 y2 y1 u2 {. .} u3 } u1 . x1 x2 x3 Economics 20 - Prof. Anderson x4 x 8 Deriving OLS Estimates To derive the OLS estimates we need to realize that our main assumption of E(u|x) = E(u) = 0 also implies that Cov(x,u) = E(xu) = 0 Why? Remember from basic probability that Cov(X,Y) = E(XY) – E(X)E(Y) Economics 20 - Prof. Anderson 9 Deriving OLS continued We can write our 2 restrictions just in terms of x, y, b0 and b1 , since u = y – b0 – b1x E(y – b0 – b1x) = 0 E[x(y – b0 – b1x)] = 0 These are called moment restrictions Economics 20 - Prof. Anderson 10 Deriving OLS using M.O.M. The method of moments approach to estimation implies imposing the population moment restrictions on the sample moments What does this mean? Recall that for E(X), the mean of a population distribution, a sample estimator of E(X) is simply the arithmetic mean of the sample Economics 20 - Prof. Anderson 11 More Derivation of OLS We want to choose values of the parameters that will ensure that the sample versions of our moment restrictions are true The sample versions are as follows: n 1 y i 1 n 1 n n bˆ 0 bˆ1 xi 0 i ˆ bˆ x 0 x y b i i 0 1 i i 1 Economics 20 - Prof. Anderson 12 More Derivation of OLS Given the definition of a sample mean, and properties of summation, we can rewrite the first condition as follows y bˆ0 bˆ1 x , or bˆ0 y bˆ1 x Economics 20 - Prof. Anderson 13 More Derivation of OLS n ˆ x bˆ x 0 x y y b i i 1 1 i i 1 n n i 1 i 1 ˆ x y y b i i 1 xi xi x n n i 1 i 1 2 ˆ xi x yi y b1 xi x Economics 20 - Prof. Anderson 14 So the OLS estimated slope is n bˆ1 x x y y i i 1 i n x x i 1 2 i n providedthat xi x 0 2 i 1 Economics 20 - Prof. Anderson 15 Summary of OLS slope estimate The slope estimate is the sample covariance between x and y divided by the sample variance of x If x and y are positively correlated, the slope will be positive If x and y are negatively correlated, the slope will be negative Only need x to vary in our sample Economics 20 - Prof. Anderson 16 More OLS Intuitively, OLS is fitting a line through the sample points such that the sum of squared residuals is as small as possible, hence the term least squares The residual, û, is an estimate of the error term, u, and is the difference between the fitted line (sample regression function) and the sample point Economics 20 - Prof. Anderson 17 Sample regression line, sample data points and the associated estimated error terms y . y4 û4 { yˆ bˆ0 bˆ1 x y3 y2 y1 û2 { . .} û3 û1 } . x1 x2 x3 Economics 20 - Prof. Anderson x4 x 18 Alternate approach to derivation Given the intuitive idea of fitting a line, we can set up a formal minimization problem That is, we want to choose our parameters such that we minimize the following: n n ˆ ˆ ˆ ui yi b 0 b1xi i 1 2 i 1 Economics 20 - Prof. Anderson 2 19 Alternate approach, continued If one uses calculus to solve the minimization problem for the two parameters you obtain the following first order conditions, which are the same as we obtained before, multiplied by n ˆ y b n i 1 n i ˆ x 0 b 0 1 i ˆ bˆ x 0 x y b i i 0 1i i 1 Economics 20 - Prof. Anderson 20 Algebraic Properties of OLS The sum of the OLS residuals is zero Thus, the sample average of the OLS residuals is zero as well The sample covariance between the regressors and the OLS residuals is zero The OLS regression line always goes through the mean of the sample Economics 20 - Prof. Anderson 21 Algebraic Properties (precise) n uˆ n i 1 ˆ ui 0 and t hus, n i 1 n x uˆ i 1 i i i 0 0 y bˆ 0 bˆ1 x Economics 20 - Prof. Anderson 22 More terminology We can thinkof each observation as being made up of an explainedpart,and an unexplained part, yi yˆ i uˆi Wethendefine thefollowing: y y is the totalsum of squares (SST ) yˆ y is theexplainedsum of squares (SSE) uˆ is theresidual sum of squares (SSR) 2 i 2 i 2 i T henSST SSE SSR Economics 20 - Prof. Anderson 23 Proof that SST = SSE + SSR y y y yˆ yˆ y uˆ yˆ y uˆ 2 uˆ yˆ y yˆ y SSR 2 uˆ yˆ y SSE and we know that uˆ yˆ y 0 2 2 i i i i 2 i i 2 2 i i i i i i i Economics 20 - Prof. Anderson i 24 Goodness-of-Fit How do we think about how well our sample regression line fits our sample data? Can compute the fraction of the total sum of squares (SST) that is explained by the model, call this the R-squared of regression R2 = SSE/SST = 1 – SSR/SST Economics 20 - Prof. Anderson 25 Unbiasedness of OLS Assume the population model is linear in parameters as y = b0 + b1x + u Assume we can use a random sample of size n, {(xi, yi): i=1, 2, …, n}, from the population model. Thus we can write the sample model yi = b0 + b1xi + ui Assume E(u|x) = 0 and thus E(ui|xi) = 0 Assume there is variation in the xi Economics 20 - Prof. Anderson 26 Unbiasedness of OLS (cont) In order to think about unbiasedness, we need to rewrite our estimator in terms of the population parameter Start with a simple rewrite of the formula as bˆ 1 x x y i s i 2 x , where s xi x 2 x 2 Economics 20 - Prof. Anderson 27 Unbiasedness of OLS (cont) x x y x x b b x u x x b x x b x x x u b x x b x x x x x u i i i 0 1 i i 1 i i i i i 0 1 i 0 i i i i Economics 20 - Prof. Anderson 28 Unbiasedness of OLS (cont) x x 0, x x x x x i 2 i i i so, thenumeratorcan be rewrittenas b s xi x ui , and thus 2 1 x bˆ1 b1 x x u i s i 2 x Economics 20 - Prof. Anderson 29 Unbiasedness of OLS (cont) let d i xi x , so t hat 1 ˆ b i b1 2 d i ui , t hen sx 1 ˆ E b1 b1 2 d i E ui b1 sx Economics 20 - Prof. Anderson 30 Unbiasedness Summary The OLS estimates of b1 and b0 are unbiased Proof of unbiasedness depends on our 4 assumptions – if any assumption fails, then OLS is not necessarily unbiased Remember unbiasedness is a description of the estimator – in a given sample we may be “near” or “far” from the true parameter Economics 20 - Prof. Anderson 31 Variance of the OLS Estimators Now we know that the sampling distribution of our estimate is centered around the true parameter Want to think about how spread out this distribution is Much easier to think about this variance under an additional assumption, so Assume Var(u|x) = s2 (Homoskedasticity) Economics 20 - Prof. Anderson 32 Variance of OLS (cont) Var(u|x) = E(u2|x)-[E(u|x)]2 E(u|x) = 0, so s2 = E(u2|x) = E(u2) = Var(u) Thus s2 is also the unconditional variance, called the error variance s, the square root of the error variance is called the standard deviation of the error Can say: E(y|x)=b0 + b1x and Var(y|x) = s2 Economics 20 - Prof. Anderson 33 Homoskedastic Case y f(y|x) . x1 . E(y|x) = b + b x 0 1 x2 Economics 20 - Prof. Anderson 34 Heteroskedastic Case f(y|x) . . x1 x2 x3 Economics 20 - Prof. Anderson . E(y|x) = b0 + b1x x 35 Variance of OLS (cont) 1 ˆ Var b1 Var b1 2 d i ui sx 2 1 1 2 Var d i ui 2 sx sx 1 2 sx 2 2 2 d i Varui 1 d s s sx2 2 i 2 2 2 2 2 d i 1 2 s2 ˆ s 2 sx 2 Var b1 sx sx 2 Economics 20 - Prof. Anderson 36 Variance of OLS Summary The larger the error variance, s2, the larger the variance of the slope estimate The larger the variability in the xi, the smaller the variance of the slope estimate As a result, a larger sample size should decrease the variance of the slope estimate Problem that the error variance is unknown Economics 20 - Prof. Anderson 37 Estimating the Error Variance We don’t know what the error variance, s2, is, because we don’t observe the errors, ui What we observe are the residuals, ûi We can use the residuals to form an estimate of the error variance Economics 20 - Prof. Anderson 38 Error Variance Estimate (cont) uˆi yi bˆ0 bˆ1 xi b 0 b1 xi ui bˆ0 bˆ1 xi u bˆ b bˆ b i 0 0 1 1 T hen,an unbiased estimatorof s is 2 1 2 ˆ sˆ u i SSR / n 2 n 2 2 Economics 20 - Prof. Anderson 39 Error Variance Estimate (cont) sˆ sˆ 2 Standard errorof theregression recall thatsd bˆ s sx if we substitutesˆ for s then wehave thestandarderrorof bˆ1 , se bˆ1 sˆ / xi x 2 1 2 Economics 20 - Prof. Anderson 40