1、Economics 20 - Prof. Anderson,1,The Simple Regression Model,y = b0 + b1x + u,Economics 20 - Prof. Anderson,2,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 Regres

2、sand,Economics 20 - Prof. Anderson,3,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,4,A Simple

3、 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,5,Zero Conditional Mean,We need to make a crucial assumption about how u and x are relat

4、ed 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,6,.,.,x1,x2,E(y|x) as a linear function of x, where for any x th

5、e distribution of y is centered about E(y|x),E(y|x) = b0 + b1x,y,f(y),Economics 20 - Prof. Anderson,7,Ordinary Least Squares（OLS）,Basic idea of regression is to estimate the population總體 parameters from a sample Let (xi,yi): i=1, ,n denote a random sample of size n from the population For each obser

6、vation in this sample, it will be the case that yi = b0 + b1xi + ui,Economics 20 - Prof. Anderson,8,.,.,.,.,y4,y1,y2,y3,x1,x2,x3,x4,u1,u2,u3,u4,x,y,Population regression line, sample data points and the associated error terms,E(y|x) = b0 + b1x,Economics 20 - Prof. Anderson,9,Deriving OLS Estimates,T

7、o 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,10,Deriving OLS continued,We can write our 2 restrictions just in terms

8、 of x, y, b0 and b1 , since u = y b0 b1x E(y b0 b1x) = 0 Ex(y b0 b1x) = 0 These are called moment restrictions,Economics 20 - Prof. Anderson,11,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 t

9、his 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,12,More Derivation of OLS,We want to choose values of the parameters that will ensure that the sample versions of our moment res

10、trictions are true The sample versions are as follows:,Economics 20 - Prof. Anderson,13,More Derivation of OLS,Given the definition of a sample mean, and properties of summation, we can rewrite the first condition as follows,Economics 20 - Prof. Anderson,14,More Derivation of OLS,Economics 20 - Prof

11、. Anderson,15,So the OLS estimated slope is,Economics 20 - Prof. Anderson,16,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 correl

12、ated, the slope will be negative Only need x to vary in our sample,Economics 20 - Prof. Anderson,17,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, , is an estimate of the

13、 error term, u, and is the difference between the fitted line (sample regression function) and the sample point,Economics 20 - Prof. Anderson,18,.,.,.,.,y4,y1,y2,y3,x1,x2,x3,x4,1,2,3,4,x,y,Sample regression line, sample data points and the associated estimated error terms,Economics 20 - Prof. Anders

14、on,19,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:,Economics 20 - Prof. Anderson,20,Alternate approach, continued,If one uses calculus to solve t

15、he minimization problem for the two parameters you obtain the following first order conditions, which are the same as we obtained before, multiplied by n,Economics 20 - Prof. Anderson,21,Algebraic Properties of OLS,The sum of the OLS residuals is zero Thus, the sample average of the OLS residuals is

16、 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,22,Algebraic Properties (precise),Economics 20 - Prof. Anderson,23,More terminology,Economics 20 - Prof. Anderson,

17、24,Proof that SST = SSE + SSR,Economics 20 - Prof. Anderson,25,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

18、 = 1 SSR/SST,Economics 20 - Prof. Anderson,26,Using Stata for OLS regressions,Now that weve derived the formula for calculating the OLS estimates of our parameters, youll be happy to know you dont have to compute them by hand Regressions in Stata are very simple, to run the regression of y on x, jus

19、t type reg y x,Economics 20 - Prof. Anderson,27,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

20、|x) = 0 and thus E(ui|xi) = 0 Assume there is variation in the xi,Economics 20 - Prof. Anderson,28,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,Economics 20 - Prof.

21、 Anderson,29,Unbiasedness of OLS (cont),Economics 20 - Prof. Anderson,30,Unbiasedness of OLS (cont),Economics 20 - Prof. Anderson,31,Unbiasedness of OLS (cont),Economics 20 - Prof. Anderson,32,Unbiasedness Summary,The OLS estimates of b1 and b0 are unbiased Proof of unbiasedness depends on our 4 ass

22、umptions 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,33,Variance of the OLS Estimators,Now we know that the sampling distributio

23、n 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,34,Variance of OLS (cont),Var(u|x) = E(u2|

24、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,35,.,.,x1,x2,Homoskedastic Case,E(y|x) = b0 + b1x,y,f(y|x),Economics 20 - Prof. Anderson,36,.,x,x1,x2,y,f(y|x),Heteroskedastic Case