《計量經(jīng)濟學導論》電子教案英文版(伍德里奇)

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1、Welcome to Economics 20 What is Econometrics? Economics 20 - Prof. Anderson Why study Econometrics? Rare in economics (and many other areas without labs!) to have experimental data Need to use nonexperimental, or observational, data to make inferences Important to be able to apply economic theor

2、y to real world data Economics 20 - Prof. Anderson Why study Econometrics? An empirical analysis uses data to test a theory or to estimate a relationship A formal economic model can be tested Theory may be ambiguous as to the effect of some policy change – can use econometrics to evaluate the p

3、rogram Economics 20 - Prof. Anderson Types of Data – Cross Sectional Cross-sectional data is a random sample Each observation is a new individual, firm, etc. with information at a point in time If the data is not a random sample, we have a sample-selection problem Economics 20 - Prof. Anderson

4、Types of Data – Panel Can pool random cross sections and treat similar to a normal cross section. Will just need to account for time differences. Can follow the same random individual observations over time – known as panel data or longitudinal data Economics 20 - Prof. Anderson Types of Data –

5、Time Series Time series data has a separate observation for each time period – e.g. stock prices Since not a random sample, different problems to consider Trends and seasonality will be important Economics 20 - Prof. Anderson The Question of Causality Simply establishing a relationship between

6、 variables is rarely sufficient Want to the effect to be considered causal If we’ve truly controlled for enough other variables, then the estimated ceteris paribus effect can often be considered to be causal Can be difficult to establish causality Economics 20 - Prof. Anderson Example: Returns t

7、o Education A model of human capital investment implies getting more education should lead to higher earnings In the simplest case, this implies an equation like Economics 20 - Prof. Anderson Example: (continued) The estimate of b1, is the return to education, but can it be considered causal? Wh

8、ile the error term, u, includes other factors affecting earnings, want to control for as much as possible Some things are still unobserved, which can be problematic Economics 20 - Prof. Anderson The Simple Regression Model y = b0 + b1x + u Economics 20 - Prof. Anderson Some Terminology In the si

9、mple 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 Some Terminology, cont. In the simple linear regression of y on x, we typically refer to x as the

10、Independent Variable, or Right-Hand Side Variable, or Explanatory Variable, or Regressor, or Covariate, or Control Variables Economics 20 - Prof. Anderson 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,

11、 since we can always use b0 to normalize E(u) to 0 Economics 20 - Prof. Anderson 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

12、 unrelated. That is, that E(u|x) = E(u) = 0, which implies E(y|x) = b0 + b1x Economics 20 - Prof. Anderson . . x1 x2 E(y|x) as a linear function of x, where for any x the distribution of y is centered about E(y|x) E(y|x) = b0 + b1x y f(y) Economics 20 - Prof. Anderson Ordinary Least Squares Bas

13、ic 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 . . . . y4 y1 y2 y3 x1 x2 x3 x4

14、 } } { { 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 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

15、(xu) = 0 Why? Remember from basic probability that Cov(X,Y) = E(XY) – E(X)E(Y) Economics 20 - Prof. Anderson 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 rest

16、rictions Economics 20 - Prof. Anderson 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

17、simply the arithmetic mean of the sample Economics 20 - Prof. Anderson 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: Economics 20 - Prof. Anderson More Derivation o

18、f OLS Given the definition of a sample mean, and properties of summation, we can rewrite the first condition as follows Economics 20 - Prof. Anderson More Derivation of OLS Economics 20 - Prof. Anderson So the OLS estimated slope is Economics 20 - Prof. Anderson Summary of OLS slope estimate The sl

19、ope 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 More OLS Int

20、uitively, 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 error term, u, and is the difference between the fitted line (sample regression function) and the sample poi

21、nt Economics 20 - Prof. Anderson . . . . 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. Anderson Alternate approach to derivation Given the intuitive idea of fitting a line, we can set up a

22、 formal minimization problem That is, we want to choose our parameters such that we minimize the following: Economics 20 - Prof. Anderson Alternate approach, continued If one uses calculus to solve the minimization problem for the two parameters you obtain the following first order conditions, whi

23、ch are the same as we obtained before, multiplied by n Economics 20 - Prof. Anderson 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

24、regression line always goes through the mean of the sample Economics 20 - Prof. Anderson Algebraic Properties (precise) Economics 20 - Prof. Anderson More terminology Economics 20 - Prof. Anderson Proof that SST = SSE + SSR Economics 20 - Prof. Anderson Goodness-of-Fit How do we think about how wel

25、l 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 Using Stata for OLS regressions Now that we’ve derived the

26、 formula for calculating the OLS estimates of our parameters, you’ll be happy to know you don’t have to compute them by hand Regressions in Stata are very simple, to run the regression of y on x, just type reg y x Economics 20 - Prof. Anderson Unbiasedness of OLS Assume the population model is li

27、near 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. Anderso

28、n 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. Anderson Unbiasedness of OLS (cont) Economics 20 - Prof. Anderson Unbiasedness of OLS (cont) E

29、conomics 20 - Prof. Anderson Unbiasedness of OLS (cont) Economics 20 - Prof. Anderson 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 desc

30、ription of the estimator – in a given sample we may be “near〞 or “far〞 from the true parameter Economics 20 - Prof. Anderson 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 distr

31、ibution is Much easier to think about this variance under an additional assumption, so Assume Var(u|x) = s2 (Homoskedasticity) Economics 20 - Prof. Anderson 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

32、, 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 . . x1 x2 Homoskedastic Case E(y|x) = b0 + b1x y f(y|x) Economics 20 - Prof. Anderson . x x1 x2 y f(y|x) Hete

33、roskedastic Case x3 . . E(y|x) = b0 + b1x Economics 20 - Prof. Anderson Variance of OLS (cont) Economics 20 - Prof. Anderson 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 th

34、e 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 Estimating the Error Variance We don’t know what the error variance, s2, is, because we don’t observe the errors, ui Wh

35、at we observe are the residuals, ??i We can use the residuals to form an estimate of the error variance Economics 20 - Prof. Anderson Error Variance Estimate (cont) Economics 20 - Prof. Anderson Error Variance Estimate (cont) Economics 20 - Prof. Anderson Multiple Regression Analysis y = b0 + b1x1

36、 + b2x2 + . . . bkxk + u 1. Estimation Economics 20 - Prof. Anderson Parallels with Simple Regression b0 is still the intercept b1 to bk all called slope parameters u is still the error term (or disturbance) Still need to make a zero conditional mean assumption, so now assume that E(u|x1,x2, …

37、,xk) = 0 Still minimizing the sum of squared residuals, so have k+1 first order conditions Economics 20 - Prof. Anderson Interpreting Multiple Regression Economics 20 - Prof. Anderson A “Partialling Out〞 Interpretation Economics 20 - Prof. Anderson “Partialling Out〞 continued Previous equation imp

38、lies that regressing y on x1 and x2 gives same effect of x1 as regressing y on residuals from a regression of x1 on x2 This means only the part of xi1 that is uncorrelated with xi2 are being related to yi so we’re estimating the effect of x1 on y after x2 has been “partialled out〞 Economics 20 - Pr

39、of. Anderson Simple vs Multiple Reg Estimate Economics 20 - Prof. Anderson Goodness-of-Fit Economics 20 - Prof. Anderson Goodness-of-Fit (continued) 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 e

40、xplained by the model, call this the R-squared of regression R2 = SSE/SST = 1 – SSR/SST Economics 20 - Prof. Anderson Goodness-of-Fit (continued) Economics 20 - Prof. Anderson More about R-squared R2 can never decrease when another independent variable is added to a regression, and usually will i

41、ncrease Because R2 will usually increase with the number of independent variables, it is not a good way to compare models Economics 20 - Prof. Anderson Assumptions for Unbiasedness Population model is linear in parameters: y = b0 + b1x1 + b2x2 +…+ bkxk + u We can use a random sample of size n,

42、 {(xi1, xi2,…, xik, yi): i=1, 2, …, n}, from the population model, so that the sample model is yi = b0 + b1xi1 + b2xi2 +…+ bkxik + ui E(u|x1, x2,… xk) = 0, implying that all of the explanatory variables are exogenous None of the x’s is constant, and there are no exact linear relationships among

43、them Economics 20 - Prof. Anderson Too Many or Too Few Variables What happens if we include variables in our specification that don’t belong? There is no effect on our parameter estimate, and OLS remains unbiased What if we exclude a variable from our specification that does belong? OLS will usu

44、ally be biased Economics 20 - Prof. Anderson Omitted Variable Bias Economics 20 - Prof. Anderson Omitted Variable Bias (cont) Economics 20 - Prof. Anderson Omitted Variable Bias (cont) Economics 20 - Prof. Anderson Omitted Variable Bias (cont) Economics 20 - Prof. Anderson Summary of Direction of B

45、ias Positive bias Negative bias b2 < 0 Negative bias Positive bias b2 > 0 Corr(x1, x2) < 0 Corr(x1, x2) > 0 Economics 20 - Prof. Anderson Omitted Variable Bias Summary Two cases where bias is equal to zero b2 = 0, that is x2 doesn’t really belong in model x1 and x2 are uncorrelated in t

46、he sample If correlation between x2 , x1 and x2 , y is the same direction, bias will be positive If correlation between x2 , x1 and x2 , y is the opposite direction, bias will be negative Economics 20 - Prof. Anderson The More General Case Technically, can only sign the bias for the more genera

47、l case if all of the included x’s are uncorrelated Typically, then, we work through the bias assuming the x’s are uncorrelated, as a useful guide even if this assumption is not strictly true Economics 20 - Prof. Anderson Variance of the OLS Estimators Now we know that the sampling distribution of

48、 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|x1, x2,…, xk) = s2 (Homoskedasticity) Economics 20 - Prof. Anderson Variance of OLS (cont) Let x sta

49、nd for (x1, x2,…xk) Assuming that Var(u|x) = s2 also implies that Var(y| x) = s2 The 4 assumptions for unbiasedness, plus this homoskedasticity assumption are known as the Gauss-Markov assumptions Economics 20 - Prof. Anderson Variance of OLS (cont) Economics 20 - Prof. Anderson Components of OL

50、S Variances The error variance: a larger s2 implies a larger variance for the OLS estimators The total sample variation: a larger SSTj implies a smaller variance for the estimators Linear relationships among the independent variables: a larger Rj2 implies a larger variance for the estimators Eco

51、nomics 20 - Prof. Anderson Misspecified Models Economics 20 - Prof. Anderson Misspecified Models (cont) While the variance of the estimator is smaller for the misspecified model, unless b2 = 0 the misspecified model is biased As the sample size grows, the variance of each estimator shrinks to zer

52、o, making the variance difference less important Economics 20 - Prof. Anderson 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, ??i We can use the residuals to form an estimate of the error va

53、riance Economics 20 - Prof. Anderson Error Variance Estimate (cont) df = n – (k + 1), or df = n – k – 1 df (i.e. degrees of freedom) is the (number of observations) – (number of estimated parameters) Economics 20 - Prof. Anderson The Gauss-Markov Theorem Given our 5 Gauss-Markov Assumptions it ca

54、n be shown that OLS is “BLUE〞 Best Linear Unbiased Estimator Thus, if the assumptions hold, use OLS Economics 20 - Prof. Anderson Multiple Regression Analysis y = b0 + b1x1 + b2x2 + . . . bkxk + u 2. Inference Economics 20 - Prof. Anderson Assumptions of the Classical Linear Model (CLM) So

55、far, we know that given the Gauss-Markov assumptions, OLS is BLUE, In order to do classical hypothesis testing, we need to add another assumption (beyond the Gauss-Markov assumptions) Assume that u is independent of x1, x2,…, xk and u is normally distributed with zero mean and variance s2: u ~ No

56、rmal(0,s2) Economics 20 - Prof. Anderson CLM Assumptions (cont) Under CLM, OLS is not only BLUE, but is the minimum variance unbiased estimator We can summarize the population assumptions of CLM as follows y|x ~ Normal(b0 + b1x1 +…+ bkxk, s2) While for now we just assume normality, clear that so

57、metimes not the case Large samples will let us drop normality Economics 20 - Prof. Anderson . . x1 x2 The homoskedastic normal distribution with a single explanatory variable E(y|x) = b0 + b1x y f(y|x) Normal distributions Economics 20 - Prof. Anderson Normal Sampling Distributions Economics 20 -

58、Prof. Anderson The t Test Economics 20 - Prof. Anderson The t Test (cont) Knowing the sampling distribution for the standardized estimator allows us to carry out hypothesis tests Start with a null hypothesis For example, H0: bj=0 If accept null, then accept that xj has no effect on y, controlli

59、ng for other x’s Economics 20 - Prof. Anderson The t Test (cont) Economics 20 - Prof. Anderson t Test: One-Sided Alternatives Besides our null, H0, we need an alternative hypothesis, H1, and a significance level H1 may be one-sided, or two-sided H1: bj > 0 and H1: bj < 0 are one-sided H1:

60、 bj ?? 0 is a two-sided alternative If we want to have only a 5% probability of rejecting H0 if it is really true, then we say our significance level is 5% Economics 20 - Prof. Anderson One-Sided Alternatives (cont) Having picked a significance level, a, we look up the (1 – a)th percentile in a t

61、distribution with n – k – 1 df and call this c, the critical value We can reject the null hypothesis if the t statistic is greater than the critical value If the t statistic is less than the critical value then we fail to reject the null Economics 20 - Prof. Anderson yi = b0 + b1xi1 + … + b

62、kxik + ui H0: bj = 0 H1: bj > 0 c 0 a (1 - a) One-Sided Alternatives (cont) Fail to reject reject Economics 20 - Prof. Anderson One-sided vs Two-sided Because the t distribution is symmetric, testing H1: bj < 0 is straightforward. The critical value is just the negativ

63、e of before We can reject the null if the t statistic < –c, and if the t statistic > than –c then we fail to reject the null For a two-sided test, we set the critical value based on a/2 and reject H1: bj ?? 0 if the absolute value of the t statistic > c Economics 20 - Prof. Anderson yi =

64、 b0 + b1Xi1 + … + bkXik + ui H0: bj = 0 H1: bj > 0 c 0 a/2 (1 - a) -c a/2 Two-Sided Alternatives reject reject fail to reject Economics 20 - Prof. Anderson Summary for H0: bj = 0 Unless otherwise stated, the alternative is assumed to be two-sided If we reject the nul

65、l, we typically say “xj is statistically significant at the a % level〞 If we fail to reject the null, we typically say “xj is statistically insignificant at the a % level〞 Economics 20 - Prof. Anderson Testing other hypotheses A more general form of the t statistic recognizes that we may want to te

66、st something like H0: bj = aj In this case, the appropriate t statistic is Economics 20 - Prof. Anderson Confidence Intervals Another way to use classical statistical testing is to construct a confidence interval using the same critical value as was used for a two-sided test A (1 - a) % confidence interval is defined as Economics 20 - Prof. Anderson Computing p-values for t tests An alternative to the classical approach is to ask, “what is the smallest significance level at which the null wo