Econometrics計(jì)量經(jīng)濟(jì)學(xué)Topic5SRM3課件

1、econometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm31lstandard assumptions underlying ols or assumptions for classical linear regression model (clrm)lqualities of ols: lb.l.u.e. (gauss-markov theorem 高斯高斯-馬爾可夫定理馬爾可夫定理)lunbiasedness of ols:lmathematical proof & monte carlo experiment (蒙特卡羅實(shí)驗(yàn)蒙特卡羅實(shí)驗(yàn)) topic 5. simple linear

2、 regression model（3）econometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm32standard assumptionslassumption 1: linear regression model.the regression model is linear in the parameters and correctly specified. i.e.uxy21econometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm33lassumption 2: non-random regressors (or independent of errors)in other words, x i

3、s nonstochasticu is distributed independently of the explanatory variables.econometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm34lassumption 3: zero mean value of disturbance ut.the expected value of u, the disturbance term, in the population is 0. that is, e (ut) = 0econometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm35standard assumptionslassumptio

4、n 4: homoscedasticity or equal variance of ut.given the value of x, the variance of ut is the same for all observations. that is, the conditional variance of ut are identical.a discussion of the meaning of this condition and its implications will be deferred until we come to the topic of heterosceda

5、sticity (the violation of the condition).2)(uvareconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm36lassumption 5: no autocorrelation between the disturbances.given any two x values, xs and and xt (s t), the correlation between any two us and ut is zero. cov (us , ut) = 0the fifth condition is that value of the disturbance

6、 term in any observation should be independent of its value in any other observation.again, a discussion of the meaning and implications of this condition will be deferred until we come to the topic of autocorrelation (the violation of the condition).econometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm37normality assumptionu

7、 has a normal distribution in addition to the gauss-markov conditions, one usually assumes that the disturbance term is normally distributed. this is necessary for the validity of the usual tests.econometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm38qualities of ols: bluethe gauss-markov theoremlthe gauss-markov theorem:if t

8、he standard assumptions hold, then ols estimators are blue- “best linear unbiased estimator”.llinear (線性性線性性): the estimator is a linear function of the observations on y.lunbiased (無(wú)偏性無(wú)偏性): the expected value of the estimator equals the true parameter.lefficient (有效性有效性): if we consider only unbias

9、ed estimators of a parameter, the one with the smallest variance is called best, or efficient, estimator.l these are good qualities of ols, which is also why we choose to use ols.econometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm39linear ：lan estimator, say the ols estimator , is said to be a linear function of a random va

10、riable, such as the dependent variable y.xbbyuxy2121 :line fitted :model truequalities of ols: bluethe gauss-markov theorem2econometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm310unbiased ：xbbyuxy2121 :line fitted :model truelthe average or expected values of parameters are equal to the true value 1，2.2211)(,)(bebequalities

11、of ols: bluethe gauss-markov theoremeconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm311efficient :xbbyuxy2121 :line fitted :model truelit has minimum variance in the class of all such linear unbiased estimators.lan unbiased estimator with the least variance is known as an efficient estimator.lminimum variance makes the f

12、itted values b1 ，b2 more valuable.qualities of ols: bluethe gauss-markov theoremeconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm312given the assumptions of the classical linear regression model, the ordinary least-squares estimators(普通最小二乘估計(jì)量), in the class of unbiased linear estimators, have minimum variance, that is, t

13、hey are best linear unbiased estimators(最佳線性無(wú)偏估計(jì)量), blue.econometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm313unbiasedness of olsuxy 21 xbby21 the regression coefficients are special types of random variable. we will demonstrate this using the simple regression model in which y depends on x. the two equations show the true

14、 model and the fitted regression.econometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm314uxy 21 xbby21 )(var),(cov)(var),(cov),(cov0)(var),(cov),(cov),(cov)(var) ,(cov)(var),(cov2221212xuxxuxxxxuxxxxxuxxxyxb we will investigate the behaviour of the ordinary least squares (ols) estimator of the slope coefficient, shown above.u

15、nbiasedness of olseconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm315)(var),(cov)(var),(cov),(cov0)(var),(cov),(cov),(cov)(var) ,(cov)(var),(cov2221212xuxxuxxxxuxxxxxuxxxyxb uxy 21 xbby21 using the second covariance rule, we can bring 2 outside the middle term.unbiasedness of olseconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm316uxy 21 xbby21

16、 cov (x, x) is the same as var (x). thus we have decomposed b2 into two components: the true value, 2, and an error term. )(var),(cov)(var),(cov),(cov0)(var),(cov),(cov),(cov)(var) ,(cov)(var),(cov2221212xuxxuxxxxuxxxxxuxxxyxb unbiasedness of olseconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm317uxy 21 xbby21 the error t

17、erm depends on the value of the disturbance term in every observation in the sample, and thus it is a special type of random variable.)(var),(cov)(var),(cov),(cov0)(var),(cov),(cov),(cov)(var) ,(cov)(var),(cov2221212xuxxuxxxxuxxxxxuxxxyxb unbiasedness of olseconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm318we will inves

18、tigate its effect on b2 in two ways: firstly, mathematical proof; and secondly, using monte carlo experiment.uxy 21 xbby21 )(var),(cov)(var),(cov),(cov0)(var),(cov),(cov),(cov)(var) ,(cov)(var),(cov2221212xuxxuxxxxuxxxxxuxxxyxb unbiasedness of olseconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm319uxy 21 xbby21 )var(),(co

19、v22xuxb 22222),(cov)(var1)var(),(cov)()var(),(cov)( uxexxuxeexuxebeby virtue of the assumption that x is non-stochastic, var (x) is non-stochastic. thus, using the second expected value rule, we may take it out of the second term.unbiasedness of ols: mathematical proofeconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm320ux

20、y 21 xbby21 )var(),(cov22xuxb 22222),(cov)(var1)var(),(cov)()var(),(cov)( uxexxuxeexuxebethe expected value of cov(x, u) is zero. we will prove this.unbiasedness of ols: mathematical proofeconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm321uxy 21 xbby21 )var(),(cov22xuxb 00)(1)()(1)(1)(1),(cov1111 niiniiiniiiniiixxnuuexxn

21、uuxxenuuxxneuxewe have used the second expected value rule to bring the factor (1/n) out of the expected value expression, and the first expected value rule to rewrite the expectation as the sum of the expectations of the individual terms.unbiasedness of ols: mathematical proofeconometrics計(jì)量經(jīng)濟(jì)學(xué)topic

22、5srm322uxy 21 xbby21 )var(),(cov22xuxb 00)(1)()(1)(1)(1),(cov1111 niiniiiniiiniiixxnuuexxnuuxxenuuxxneuxesince x is non-stochastic, the term involving it and its mean can be taken out of each expectation as a factor.unbiasedness of ols: mathematical proofeconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm323uxy 21 xbby21 )v

23、ar(),(cov22xuxb 00)(1)()(1)(1)(1),(cov1111 niiniiiniiiniiixxnuuexxnuuxxenuuxxneuxethe expected value of u in each observation is zero, and hence so is the expected value of its sample mean. thus the expectation of cov (x, u) is zero.unbiasedness of ols: mathematical proofeconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm32

24、4uxy 21 xbby21 xbuxxbyb22121 12212212212211)(0)()()()()()( xxbexxxbeuexeexbuxebewe have just demonstrated that e(b2) is equal to 2. hence b1 is an unbiased estimator of 1. .unbiasedness of ols: mathematical proofeconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm325choose model in which y is determined by x, parameter value

25、s, and uchoosedata for xchooseparametervalueschoosedistribution for umodelgenerate the values of ya monte carlo experiment is a laboratory-style exercise usually undertaken with the objective of evaluating the properties of regression estimators under controlled conditions.unbiasedness of ols: monte

26、 carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm326choose model in which y is determined by x, parameter values, and uchoosedata for xchooseparametervalueschoosedistribution for umodelgenerate the values of ywe will use it to investigate the behaviour of ols regression coefficients when applied to a simp

27、le regression model. we will also generate values for the disturbance term randomly from a known distribution.unbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm327choose model in which y is determined by x, parameter values, and uchoosedata for xchooseparametervalueschoosedistribu

28、tion for umodelgenerate the values of ythe values of y in the sample will be determined by the values of x, the parameters and the values of the disturbance term.unbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm328choose model in which y is determined by x, parameter values, and

29、uchoosedata for xchooseparametervalueschoosedistribution for umodelgenerate the values of yestimatorsestimate the values of the parameterswe will then use the regression technique to obtain estimates of the parameters using only the data on y and x.unbiasedness of ols: monte carlo experimenteconomet

30、rics計(jì)量經(jīng)濟(jì)學(xué)topic5srm329choose model in which y is determined by x, parameter values, and uchoosedata for xchooseparametervalueschoosedistribution for umodelgenerate the values of yestimatorsestimate the values of the parametersy = 1 + 2x + ux =1, 2, . , 20 1 = 2.0 2 = 0.5u is independent n(0,1)y = 2.0

31、 + 0.5x + ugenerate the values of yin this experiment we have 20 observations in the sample. x takes the values 1, 2, ., 20. 1 is equal to 2.0 and 2 is equal to 0.5.unbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm330we will then regress y on x using the ols estimation technique

32、and see how well our estimates b1 and b2 correspond to the true values 1 and 2.choose model in which y is determined by x, parameter values, and uchoosedata for xchooseparametervalueschoosedistribution for umodelgenerate the values of yestimatorsestimate the values of the parametersy = 1 + 2x + ux =

33、1, 2, . , 20 1 = 2.0 2 = 0.5u is independent n(0,1)y = 2.0 + 0.5x + ugenerate the values of yb2 = cov(x, y)/var(x); estimate the values of the parametersxbyb21 unbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm331x2.0+0.5x u yx 2.0+0.5x u y1112123134145156167178189191020y = 2.0 +

34、0.5x + uhere are the values of x, chosen quite arbitrarily.unbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm332x2.0+0.5x u yx 2.0+0.5x u y12.5117.523.0128.033.5138.544.0149.054.5159.565.01610.075.51710.586.01811.096.51911.5107.02012.0y = 2.0 + 0.5x + ugiven our choice of numbers

35、for 1 and 2, we can derive the non-stochastic component of y.unbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm333the non-stochastic component is displayed graphically.0.002.004.006.008.0010.0012.0014.000.005.0010.0015.0020.00xy5.00.2 yxunbiasedness of ols: monte carlo experimente

36、conometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm334x2.0+0.5x u yx 2.0+0.5x u y12.5-0.59117.51.5923.0-0.24128.0-0.9233.5-0.83138.5-0.7144.00.03149.0-0.2554.5-0.38159.51.6965.0-2.191610.00.1575.51.031710.50.0286.00.241811.0-0.1196.52.531911.5-0.91107.0-0.132012.01.42y = 2.0 + 0.5x + unext, we generate randomly a value of th

37、e disturbance term for each observation using a n(0,1) distribution (normal with zero mean and unit variance).unbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm335x2.0+0.5x u yx 2.0+0.5x u y12.5-0.591.91117.51.599.0923.0-0.242.76128.0-0.927.0833.5-0.832.67138.5-0.717.7944.00.034.0

38、3149.0-0.258.7554.5-0.384.12159.51.6911.1965.0-2.192.811610.00.1510.1575.51.036.531710.50.0210.5286.00.246.241811.0-0.1110.8996.52.539.031911.5-0.9110.59107.0-0.136.872012.01.4213.42we generate values of y.y = 2.0 + 0.5x + uunbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm336the

39、20 observations are displayed graphically.0.002.004.006.008.0010.0012.0014.000.005.0010.0015.0020.00yxunbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm337choose model in which y is determined by x, parameter values, and uchoosedata for xchooseparametervalueschoosedistribution for

40、 umodelgenerate the values of yestimatorsestimate the values of the parametersy = 1 + 2x + ux =1, 2, . , 20 1 = 2.0 2 = 0.5u is independent n(0,1)y = 2.0 + 0.5x + ugenerate the values of ywe have reached this point in the monte carlo experiment.unbiasedness of ols: monte carlo experimenteconometrics

41、計(jì)量經(jīng)濟(jì)學(xué)topic5srm338choose model in which y is determined by x, parameter values, and uchoosedata for xchooseparametervalueschoosedistribution for umodelgenerate the values of yestimatorsestimate the values of the parametersy = 1 + 2x + ux =1, 2, . , 20 1 = 2.0 2 = 0.5u is independent n(0,1)y = 2.0 + 0

42、.5x + ugenerate the values of yb2 = cov(x, y)/var(x); estimate the values of the parametersxbyb21 we will now apply the ols estimators for b1 and b2 to the data for x and y, and see how well the estimates correspond to the true values.unbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic

43、5srm339the regression estimators use only the observed data for x and y.0.002.004.006.008.0010.0012.0014.000.005.0010.0015.0020.00yxunbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm340here is the regression equation fitted to the data.0.002.004.006.008.0010.0012.0014.000.005.0010

44、.0015.0020.00yxxy54.063.1 unbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm341for comparison, the non-stochastic component of the true relationship is also displayed. 2 (true value 0.50) has been overestimated and 1 (true value 2.00) has been underestimated.0.002.004.006.008.0010

45、.0012.0014.000.005.0010.0015.0020.00yxxy54.063.1 unbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm342we will repeat the process, starting with the same non-stochastic components of y.0.002.004.006.008.0010.0012.0014.000.005.0010.0015.0020.00yxxy5.00.2 unbiasedness of ols: monte c

46、arlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm343as before, the values of y are modified by adding randomly-generated values of the disturbance term.0.002.004.006.008.0010.0012.0014.000.005.0010.0015.0020.00yxunbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm344the new values of the di

47、sturbance term are drawn from the same n(0,1) distribution as the previous ones but, except by coincidence, will be different from them.0.002.004.006.008.0010.0012.0014.000.005.0010.0015.0020.00yxunbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm345this time the slope coefficient

48、has been overestimated and the intercept underestimated.0.002.004.006.008.0010.0012.0014.000.005.0010.0015.0020.00yxxy48.052.2 unbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm346we will repeat the process once more.0.002.004.006.008.0010.0012.0014.000.005.0010.0015.0020.00yxxy5.00.2 unbiasedness of ols: monte carlo experimenteconometrics計(jì)量經(jīng)濟(jì)學(xué)topic5srm347a new set of random