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Chapter 2

Chapter 2. Introductory Econometrics: A Modern Approach, 4 th ed. By Jeffrey M. Wooldridge. 2.1 Definition of the Simple Regression Model. In the simple linear regression model, where y = β 0 + β 1 x + u (2.1)

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Chapter 2

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  1. Chapter 2 Introductory Econometrics: A Modern Approach, 4th ed. By Jeffrey M. Wooldridge

  2. 2.1 Definition of the Simple Regression Model • In the simple linear regression model, where y = β0 + β1x + u (2.1) • It is also call the two-variable (bivariate) linear regression model • we typically refer to y as the • Dependent Variable, or • Explained Variable, or • Response Variable, or • Regressand

  3. 2.1 Definition of the Simple Regression Model • In the simple linear regression of y on x, we typically refer to x as the • Independent Variable, or • Explanatory Variable, or • Control Variables • Regressor

  4. 2.1 Definition of the Simple Regression Model • The variable u, called the error term or disturbance (innovation) in the relationship, represents factors other than x that affect y. • You can usefully think of u as standing for “unobserved” • Δy =β1Δx if Δu=0 (2.2) • The intercept parameterβ0 ,sometimes called the constant term.

  5. 2.1 Definition of the Simple Regression Model (Expample) • Soybean Yield and Fertilizer • A Simple Wage Equation

  6. A Simple Assumption • uand x arerandom variable, we need a concept grounded in probability. • Mathematically The average value of u, in the population is 0. That is, E(u) = 0 (2.5) • As long as the intercept β0 is included in the equation, nothing by assuming that the average value of u in population is zero

  7. Zero Conditional Mean • We need to make a crucial assumption about how u and x are related • If u and x are uncorrelated, then, as random variables, they are notlinearly related The crucial assumption is that the average value of u does not depend on the value of x • E(u|x) = E(u) (2.6) • zero conditional mean assumption E(u|x) = 0 • E(y|x) = β0 + β1x (2.8)

  8. Zero Conditional Mean Take the expected value of (2.1) conditional on x and using E(u|x) = 0 • E(y|x) = β0 + β1x (2.8) E(2.8) population regression function (PRF) • E(y|x), is a linear function of x. (systematic part) • The linarity means that a one-unit increase in x change the expectedvalue of y by the amount β1 • The u is call the unsystematic part.

  9. E(y|x) as a linear function of x, the mean one-unit increase in x, changetheE(y), byβ1amount.

  10. 2.2 Deriving the Ordinary Least Squares Estimates (OLS, OLSE) • 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 = β0 +β1xi + ui (2.9) y: the annual saving , x: the annual income

  11. E(savings=y l income=x ) linear model y E(y|x) = b0 + b1x . y4 { u4 . u3 y3 } . y2 u2 { u1 . } y1 x2 x1 x4 x3 x

  12. 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 • E(u) = 0 (2.10) • Cov(x,u) = E(xu) = 0 (2.11) • Why? Remember from basic probability that Cov(X,Y) = E(XY) – E(X)E(Y)

  13. Deriving OLS continued • We can write our 2 restrictions just in terms of x, y, β0 and β1 , since u = y –β0 –β1x • E(y –β0 –β1x) = 0 (2.12)~(2.10) • E[x(y –β0 –β1x)] = 0 (2.13)~(2.11) • These are called moment restrictions

  14. 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.

  15. 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: (2.14), (2.15)

  16. More Derivation of OLS • Given the definition of a sample mean, and properties of summation, we can rewrite the first condition as follows (2.16) (2.17)

  17. More Derivation of OLS

  18. So the OLS estimated slope is (2.18) (2.19)

  19. More OLS • ordinary least squares (OLS)

  20. 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;SRF) and the sample point (2.21)

  21. Sample regression line, sample data points and the associated estimated error terms . { . } } .

  22. 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: (2.22)

  23. Alternate approach, continued • If one uses calculus to solve the minimization problem for the two parameters you obtain the following first order conditions (F.O.C.), which are the same as we obtained before, multiplied by n

  24. OLS regression lins • Sample regression function (SRF)

  25. 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 (negatively) correlated, the slope will be positive (negative)

  26. explame • Wage and education N=526

  27. 2.3 Properties of OLS on Any Sample of Data • The sum, and the sample average of OLS residuals, is zero. • (2.30) (2) The sample covariance between the regressors and OLS residuals is zero. (2.31)

  28. 2.3 Properties of OLS on Any Sample of Data • The OLS regression line always goes through the mean (X,Y) of the sample

  29. More terminology

  30. Proof that SST = SSE + SSR (2.37)

  31. 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 (2.38)

  32. Explame 2.8In the CEO salary rgression

  33. Units of Measurement and Functional Form

  34. Units of Measurement and Functional Form (cont)

  35. Incoporating Nonlinearities in Simple Regression • Wage-education explame

  36. Units of Measurement and Functional Form (cont)

  37. 2.5 Expected Values and Variances of the OLS Estimators • Unbiasedness of OLS • Assumption SLR.1: the population model is linearin parameters as y = β0 + β1x + u (2.47) • Assumption SLR.2: 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 = β0+β1xi + ui (2.48)

  38. 2.5 Expected Values and Variances of the OLS Estimators • Assumption SLR.3: (sample variation in the Explanatory variable ) The sample outcomes on xi , namely, {xi, i=1,…n}, are not all the same value. • Assumption SLR.4:(zero condition Mean) The error u has expected value of zero given any value of the explanatory variable. E(u|x) = 0

  39. 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 (2.49)

  40. Unbiasedness of OLS (cont) From the numerator of bate1_hat (2.51)

  41. Unbiasedness of OLS (cont) (2.52)

  42. Unbiasedness of OLS (cont)

  43. Unbiasedness Summary • The OLS estimates of β1 and β0 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

  44. 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 • Assumption SLR.5: (Homoskedasticity) Var(u|x) = s2

  45. 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 • E(y|x)=β0 + β1x (2.55) • Var(y|x) = s2 (2.56)

  46. Homoskedastic Case y f(y|x) . E(y|x) = b0 + b1x . x1 x2

  47. Heteroskedastic Case f(y|x) y . . E(y|x) = b0 + b1x . x1 x2 x3 x

  48. Variance of OLS (cont)

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