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1、计量经济学导论电子教案英文版（伍德里奇）Welcome to Economics 20What is Econometrics?Economics 20 - Prof. AndersonWhy study Econometrics? Rare in economics (and many other areas without labs!) to have experimental data Need to use nonexperimental, or observational, data to make inferencesImportant to be able to apply ec
2、onomic theory to real world dataEconomics 20 - Prof. AndersonWhy 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 th
3、e programEconomics 20 - Prof. AndersonTypes 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 problemEconomics 20 - Prof. AndersonTypes of
4、 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 dataEconomics 20 - Prof. AndersonTypes of Data Time Series Time ser
5、ies 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 importantEconomics 20 - Prof. AndersonThe Question of Causality Simply establishing a relationship between variables is rarely sufficien
6、t Want to the effect to be considered causal If weve truly controlled for enough other variables, then the estimated ceteris paribus effect can often be considered to be causal Can be difficult to establish causalityEconomics 20 - Prof. AndersonExample: Returns to Education A model of human capital
7、investment implies getting more education should lead to higher earnings In the simplest case, this implies an equation likeEconomics 20 - Prof. AndersonExample: (continued) The estimate of b1, is the return to education, but can it be considered causal? While the error term, u, includes other facto
8、rs affecting earnings, want to control for as much as possible Some things are still unobserved, which can be problematicEconomics 20 - Prof. AndersonThe Simple Regression Modely = b0 + b1x + uEconomics 20 - Prof. AndersonSome Terminology In the simple linear regression model, where y = b0 + b1x + u
9、, we typically refer to y as theDependent Variable, orLeft-Hand Side Variable, orExplained Variable, orRegressandEconomics 20 - Prof. AndersonSome Terminology, cont. In the simple linear regression of y on x, we typically refer to x as theIndependent Variable, orRight-Hand Side Variable, orExplanato
10、ry Variable, orRegressor, orCovariate, orControl VariablesEconomics 20 - Prof. AndersonA 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 0Economics 20 - Prof. And
11、ersonZero 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 + b1xEcon
12、omics 20 - Prof. Anderson.x1x2E(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 + b1xyf(y)Economics 20 - Prof. AndersonOrdinary Least Squares Basic idea of regression is to estimate the population parameters from a sample Let (xi,yi): i=1, ,n
13、 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 + uiEconomics 20 - Prof. Anderson.y4y1y2y3x1x2x3x4u1u2u3u4xyPopulation regression line, sample data pointsand the associated error termsE(y|x) = b0 + b1xEconomics 20 - P
14、rof. AndersonDeriving 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. AndersonDeriving OLS continued We can write
15、our 2 restrictions just in terms of x, y, b0 and b1 , since u = y b0 b1x E(y b0 b1x) = 0 Ex(y b0 b1x) = 0These are called moment restrictionsEconomics 20 - Prof. AndersonDeriving OLS using M.O.M. The method of moments approach to estimation implies imposing the population moment restrictions on the
16、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 sampleEconomics 20 - Prof. AndersonMore Derivation of OLS We want to choose values of the parameters that will ensure that the sample versi
17、ons of our moment restrictions are true The sample versions are as follows:Economics 20 - Prof. AndersonMore Derivation of OLSGiven the definition of a sample mean, and properties of summation, we can rewrite the first condition as followsEconomics 20 - Prof. AndersonMore Derivation of OLSEconomics
18、20 - Prof. AndersonSo the OLS estimated slope isEconomics 20 - Prof. AndersonSummary 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
19、ated, the slope will be negative Only need x to vary in our sampleEconomics 20 - Prof. AndersonMore 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 err
20、or term, u, and is the difference between the fitted line (sample regression function) and the sample pointEconomics 20 - Prof. Anderson.y4y1y2y3x1x2x3x4?1?2?3?4xySample regression line, sample data pointsand the associated estimated error termsEconomics 20 - Prof. AndersonAlternate approach to deri
21、vation 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. AndersonAlternate approach, continued If one uses calculus to solve the minimization problem for the two p
22、arameters you obtain the following first order conditions, which are the same as we obtained before, multiplied by nEconomics 20 - Prof. AndersonAlgebraic 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 betwee
23、n the regressors and the OLS residuals is zero The OLS regression line always goes through the mean of the sampleEconomics 20 - Prof. AndersonAlgebraic Properties (precise)Economics 20 - Prof. AndersonMore terminologyEconomics 20 - Prof. AndersonProof that SST = SSE + SSREconomics 20 - Prof. Anderso
24、nGoodness-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/SSTEconomics 20 - Prof. AndersonUsing Stata for OLS r
25、egressions 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, just type reg y xEconomics 20 - Prof. AndersonUnbiasedness of OLS Assu
26、me 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 xiEconomi
27、cs 20 - Prof. AndersonUnbiasedness 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 asEconomics 20 - Prof. AndersonUnbiasedness of OLS (cont)Economics 20 - Prof. AndersonUnbiasedness o
28、f OLS (cont)Economics 20 - Prof. AndersonUnbiasedness of OLS (cont)Economics 20 - Prof. AndersonUnbiasedness 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
29、 description of the estimator in a given sample we may be “near” or “far” from the true parameterEconomics 20 - Prof. AndersonVariance 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 distri
30、bution is Much easier to think about this variance under an additional assumption, soAssume Var(u|x) = s2 (Homoskedasticity)Economics 20 - Prof. AndersonVariance 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 t
31、he 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) = s2Economics 20 - Prof. Anderson.x1x2Homoskedastic CaseE(y|x) = b0 + b1xyf(y|x)Economics 20 - Prof. Anderson.x x1x2yf(y|x)Heteroskedastic Casex3.E(y|x) = b0
32、 + b1xEconomics 20 - Prof. AndersonVariance of OLS (cont)Economics 20 - Prof. AndersonVariance 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 sa
33、mple size should decrease the variance of the slope estimate Problem that the error variance is unknown Economics 20 - Prof. AndersonEstimating the Error Variance We dont know what the error variance, s2, is, because we dont observe the errors, ui What we observe are the residuals, ?i We can use the
34、 residuals to form an estimate of the error varianceEconomics 20 - Prof. AndersonError Variance Estimate (cont)Economics 20 - Prof. AndersonError Variance Estimate (cont)Economics 20 - Prof. AndersonMultiple Regression Analysisy = b0 + b1x1 + b2x2 + . . . bkxk + u1. EstimationEconomics 20 - Prof. An
35、dersonParallels 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, ,xk) = 0 Still minimizing the sum of squared residuals, so have k+1 fi
36、rst order conditionsEconomics 20 - Prof. AndersonInterpreting Multiple RegressionEconomics 20 - Prof. AndersonA “Partialling Out” InterpretationEconomics 20 - Prof. Anderson“Partialling Out” continued Previous equation implies that regressing y on x1 and x2 gives same effect of x1 as regressing y on
37、 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 were estimating the effect of x1 on y after x2 has been “partialled out”Economics 20 - Prof. AndersonSimple vs Multiple Reg EstimateEconomics 20 - Prof. AndersonGoodness-
38、of-FitEconomics 20 - Prof. AndersonGoodness-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 explained by the model, call this the R-squared of regression R2 = SSE/SST = 1 SSR/SSTEcon
39、omics 20 - Prof. AndersonGoodness-of-Fit (continued)Economics 20 - Prof. AndersonMore about R-squared R2 can never decrease when another independent variable is added to a regression, and usually will increase Because R2 will usually increase with the number of independent variables, it is not a goo
40、d way to compare modelsEconomics 20 - Prof. AndersonAssumptions for Unbiasedness Population model is linear in parameters: y = b0 + b1x1 + b2x2 + bkxk + u We can use a random sample of size n, (xi1, xi2, xik, yi): i=1, 2, , n, from the population model, so that the sample model is yi = b0 + b1xi1 +
41、b2xi2 + bkxik + ui E(u|x1, x2, xk) = 0, implying that all of the explanatory variables are exogenous None of the xs is constant, and there are no exact linear relationships among themEconomics 20 - Prof. AndersonToo Many or Too Few Variables What happens if we include variables in our specification
42、that dont belong? There is no effect on our parameter estimate, and OLS remains unbiasedWhat if we exclude a variable from our specification that does belong? OLS will usually be biased Economics 20 - Prof. AndersonOmitted Variable BiasEconomics 20 - Prof. AndersonOmitted Variable Bias (cont)Economi
43、cs 20 - Prof. AndersonOmitted Variable Bias (cont)Economics 20 - Prof. AndersonOmitted Variable Bias (cont)Economics 20 - Prof. AndersonSummary of Direction of BiasPositive biasNegative biasb2 < 0Negative biasPositive biasb2 > 0Corr(x1, x2) < 0Corr(x1, x2) > 0Economics 20 - Prof. Anderso
44、nOmitted Variable Bias Summary Two cases where bias is equal to zerob2 = 0, that is x2 doesnt really belong in modelx1 and x2 are uncorrelated in the 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 opposit
45、e direction, bias will be negativeEconomics 20 - Prof. AndersonThe More General Case Technically, can only sign the bias for the more general case if all of the included xs are uncorrelated Typically, then, we work through the bias assuming the xs are uncorrelated, as a useful guide even if this ass
46、umption is not strictly trueEconomics 20 - Prof. AndersonVariance 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 additiona
47、l assumption, soAssume Var(u|x1, x2, xk) = s2 (Homoskedasticity)Economics 20 - Prof. AndersonVariance of OLS (cont) Let x stand 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-M