计量经济学英文课件：Chapter 4 Multiple Regression Analysis Inference

1、 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Chapter 4 Multiple RegressionAnalysis: Inference 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a

2、 publicly accessible website, in whole or in part.Chapter 4 Multiple RegressionAnalysis: Inference4.2 Testing Hypotheses about a Single Population Parameter: The t Test4.3 Confidence Intervals4.4 Testing Hypotheses about a Single Linear Combination of the Parameters4.5 Testing Multiple Linear Restri

3、ctions: The F Test4.1 Sampling Distributions of the OLS Estimators4.6 An application estimation of the weights of CPI components in ChinaAssignments: Promblems 1, 2, 4, 5, 7, 8, 10 Computer Exercises C1, C2, C3, C8, C9 C8: smpl if marr=1 and fsize=2 (401ksubs.wf1)The End 2013 Cengage Learning. All R

4、ights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Statistical inference in the regression modelHypothesis tests about population parametersConstruction of confidence intervals Sampling distributions of the OLS estimatorsThe OLS

5、estimators are random variablesWe already know their expected values and their variancesHowever, for hypothesis tests we need to know their distributionIn order to derive their distribution we need additional assumptionsAssumption about distribution of errors: normal distributionChapter 4 Multiple R

6、egressionAnalysis: Inference4.1 Sampling Distributions of the OLS Estimators (1/5)ChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Assumption MLR.6 (Normality of error terms)independently

7、ofIt is assumed that the unobservedfactors are normally distributed around the population regression function.The form and the variance of the distribution does not depend onany of the explanatory variables.It follows that:Chapter 4 Multiple RegressionAnalysis: Inference4.1 Sampling Distributions of

8、 the OLS Estimators (2/5)ChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Discussion of the normality assumptionThe error term is the sum of many“ different unobserved factorsSums of indep

9、endent factors are normally distributed (CLT)Problems: How many different factors? Number large enough? Possibly very heterogenuous distributions of individual factors How independent are the different factors?The normality of the error term is an empirical questionAt least the error distribution sh

10、ould be close“ to normalChapter 4 Multiple RegressionAnalysis: Inference4.1 Sampling Distributions of the OLS Estimators (3/5)ChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Discussion of

11、 the normality assumption (cont.)Examples where normality cannot hold: Wages (nonnegative; also: minimum wage) Number of arrests (takes on a small number of integer values) Unemployment (indicator variable, takes on only 1 or 0)In some cases, normality can be achieved through transformations of the

12、dependent variable (e.g. use log(wage) instead of wage)Important: For the purposes of statistical inference, the assumption of normality can be replaced by a large sample sizeChapter 4 Multiple RegressionAnalysis: Inference4.1 Sampling Distributions of the OLS Estimators (4/5)ChapterEnd 2013 Cengage

13、 Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.TerminologyTheorem 4.1 (Normal sampling distributions)Under assumptions MLR.1 MLR.6:The estimators are normally distributed around the true parameters with the va

14、riance that was derived earlierThe standardized estimators follow a standard normal distributionGauss-Markov assumptions“Classical linear model (CLM) assumptions“Chapter 4 Multiple RegressionAnalysis: Inference4.1 Sampling Distributions of the OLS Estimators (5/5)ChapterEnd 2013 Cengage Learning. Al

15、l Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.4.2.1 Theorem 4.2 t Distribution for the Standardized EstimatorsChapter 4 Multiple RegressionAnalysis: Inference4.2 Testing Hypotheses about a Single Population Parameter: The

16、 t Test4.2.3 Two-Sided Alternatives4.2.4 Testing Other Hypotheses about b bj4.2.2 Testing against One-Sided Alternatives4.2.5 Computing p-Values for t Tests4.2.6 A Reminder on the Language of Classical Hypothesis Testing4.2.7 Economic, or Practical, versus Statistical SignificanceChapterEnd 2013 Cen

17、gage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Under assumptions MLR.1 MLR.6:If the standardization is done using the estimated standard deviation (= standard error), the normal distribution is replaced by

18、 a t-distributionNote: The t-distribution is close to the standard normal distribution if n-k-1 is large.Chapter 4 Multiple RegressionAnalysis: Inference4.2.1 Theorem 4.2 t Distribution for the Standardized Estimators (1/3)2222210,11 #jjjjjjjjjn ksdseNsdnkbbbbbbbbb Proof:SectionChapterEnd 2013 Cenga

19、ge Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Null hypothesis (for more general hypotheses, see below)t-statistic (or t-ratio)Distribution of the t-statistic if the null hypothesis is trueThe t-statistic wi

20、ll be used to test the above null hypothesis. The farther the estimated coefficient is away from zero, the less likely it is that the null hypothesis holds true. But what does far“ away from zero mean? This depends on the variability of the estimated coefficient, i.e. its standard deviation. The t-s

21、tatistic measures how many estimated standard deviations the estimated coefficient is away from zero. The population parameter is equal to zero, i.e. after controlling for the other independent variables, there is no effect of xj on y Chapter 4 Multiple RegressionAnalysis: Inference4.2.1 Theorem 4.2

22、 t Distribution for the Standardized Estimators (2/3)SectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Goal: Define a rejection rule so that, if it is true, H0 is rejected only with

23、a small probability (= significance level, e.g. 5%)The precise rejection rule depends on the alternative hypothesis and the chosen significance level of the test.A significance level: the probability of rejecting H0 when it is true.Chapter 4 Multiple RegressionAnalysis: Inference4.2.1 Theorem 4.2 t

24、Distribution for the Standardized Estimators (3/3)SectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Test against .Testing against one-sided alternatives (greater than zero)4.2.2 Test

25、ing against One-Sided Alternatives (1/8)Reject the null hypothesis in favour of the alternative hypothesis if the estimated coefficient is too large“ (i.e. larger than a critical value).Construct the critical value so that, if the null hypothesis is true, it is rejected in, for example, 5% of the ca

26、ses.In the given example, this is the point of the t-distribution with 28 degrees of freedom that is exceeded in 5% of the cases.! Reject if t-statistic greater than 1.701Chapter 4 Multiple RegressionAnalysis: InferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned,

27、 copied or duplicated, or posted to a publicly accessible website, in whole or in part.Example: Wage equationTest whether, after controlling for education and tenure, higher work experience leads to higher hourly wages(1) Test against .One would either expect a positive effect of experience on hourl

28、y wage or no effect at all.Standard errors4.2.2 Testing against One-Sided Alternatives (2/8)Chapter 4 Multiple RegressionAnalysis: InferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or

29、in part.Example: Wage equation (cont.)The effect of experience on hourly wage is statistically greater than zero at the 5% (and even at the 1%) significance level.“Thought the estimated return for another year of experience, holding tenure and education fixed, is not especially large, we have persua

30、sively shown that the partial effect of experience is positive in the population.t-statisticCritical values for the 5% and the 1% significance level (these are conventional significance levels). The null hypothesis is rejected because the t-statistic exceeds the critical value.(2)Degrees of freedom;

31、here the standard normal approximation applies(3)(4)4.2.2 Testing against One-Sided Alternatives (3/8)Chapter 4 Multiple RegressionAnalysis: InferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in

32、 whole or in part.Test against .Testing against one-sided alternatives (less than zero)Reject the null hypothesis in favour of the alternative hypothesis if the estimated coefficient is too small“ (i.e. smaller than a critical value).Construct the critical value so that, if the null hypothesis is tr

33、ue, it is rejected in, for example, 5% of the cases.In the given example, this is the point of the t-distribution with 18 degrees of freedom so that 5% of the cases are below the point.! Reject if t-statistic less than -1.7344.2.2 Testing against One-Sided Alternatives (4/8)Chapter 4 Multiple Regres

34、sionAnalysis: InferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Example: Student performance and school sizeTest whether smaller school size leads to better student performa

35、nceTest against .Do larger schools hamper student performance or is there no such effect?Percentage of studentspassing maths testAverage annual tea-cher compensationSchool enrollment(= school size)Staff per one thousand students4.2.2 Testing against One-Sided Alternatives (5/8)Chapter 4 Multiple Reg

36、ressionAnalysis: InferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Example: Student performance and school size (cont.)One cannot reject the hypothesis that there is no effe

37、ct of school size on student performance (not even for a lax significance level of 15%).t-statisticCritical values for the 5% and the 15% significance level.The null hypothesis is not rejected because the t-statistic is not smaller than the critical value.Degrees of freedom;here the standard normal

38、approximation applies4.2.2 Testing against One-Sided Alternatives (6/8)Chapter 4 Multiple RegressionAnalysis: InferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Example: Stud

39、ent performance and school size (cont.)Alternative specification of functional form:Test against .R-squared slightly higher 4.2.2 Testing against One-Sided Alternatives (7/8)Chapter 4 Multiple RegressionAnalysis: InferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scann

40、ed, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Example: Student performance and school size (cont.)The hypothesis that there is no effect of school size on student performance can be rejected in favor of the hypothesis that the effect is negative.t-statisti

41、cCritical value for the 5% significance level ! reject null hypothesisHow large is the effect?(small effect)+ 10% enrollment ! -0.129 percentage points students pass test4.2.2 Testing against One-Sided Alternatives (8/8)Chapter 4 Multiple RegressionAnalysis: InferenceSectionChapterEnd 2013 Cengage L

42、earning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Testing against two-sided alternativesTest against .Reject the null hypothesis in favour of the alternative hypothesis if the absolute value of the estimated coeffi

43、cient is too large.Construct the critical value so that, if the null hypothesis is true, it is rejected in,for example, 5% of the cases.In the given example, these are the points of the t-distribution so that 5% of the cases lie in the two tails.! Reject if absolute value of t-statistic is less than

44、 -2.06 or greater than 2.064.2.3 Two-Sided Alternatives (1/3)Chapter 4 Multiple RegressionAnalysis: InferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Example: Determinants o

45、f college GPALectures missed per weekThe effects of hsGPA and skipped are significantly different from zero at the 1% significance level. The effect of ACT is not significantly different from zero, not even at the 10% significance level. For critical values, use standard normal distribution4.2.3 Two

46、-Sided Alternatives (2/3)Chapter 4 Multiple RegressionAnalysis: InferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Statistically significant“ variables in a regressionIf a re

47、gression coefficient is different from zero in a two-sided test, the corresponding variable is said to be statistically significant“If the number of degrees of freedom is large enough so that the normal approximation applies, the following rules of thumb apply:statistically significant at 10 % level

48、“statistically significant at 5 % level“statistically significant at 1 % level“4.2.3 Two-Sided Alternatives (3/3)Chapter 4 Multiple RegressionAnalysis: InferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible

49、website, in whole or in part.Testing more general hypotheses about a regression coefficientNull hypothesist-statisticThe test works exactly as before, except that the hypothesized value is substracted from the estimate when forming the statisticHypothesized value of the coefficient4.2.4 Testing Othe

50、r Hypotheses about b bj (1/3)Chapter 4 Multiple RegressionAnalysis: InferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.Example: Campus crime and enrollmentAn interesting hypo

51、thesis is whether crime increases by one percent if enrollment is increased by one percentThe hypothesis is rejected at the 5% levelEstimate is different from one but is this difference statistically significant?4.2.4 Testing Other Hypotheses about b bj (2/3)Chapter 4 Multiple RegressionAnalysis: In

52、ferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.4.2.4 Testing Other Hypotheses about b bj (3/3)Chapter 4 Multiple RegressionAnalysis: InferenceSectionChapterEnd 2013 Cengage

53、 Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.4.2.5 Computing p-Values for t Tests (1/2)Computing p-values for t-testsIf the significance level is made smaller and smaller, there will be a point where the nul

54、l hypothesis cannot be rejected anymoreThe reason is that, by lowering the significance level, one wants to avoid more and more to make the error of rejecting a correct H0The smallest significance level at which the null hypothesis is still rejected, is called the p-value of the hypothesis testA sma

55、ll p-value is evidence against the null hypothesis because one would reject the null hypothesis even at small significance levelsA large p-value is evidence in favor of the null hypothesisP-values are more informative than tests at fixed significance levelsChapter 4 Multiple RegressionAnalysis: Infe

56、renceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.How the p-value is computed (here: two-sided test)The p-value is the significance level at which one is indifferent between rej

57、ecting and not rejecting the null hypothesis. In the two-sided case, the p-value is thus the probability that the t-distributed variable takes on a larger absolute value than the realized value of the test statistic, e.g.:From this, it is clear that a null hypothesis is rejected if and only if the c

58、orresponding p-value is smaller than the significance level.For example, for a significance level of 5% the t-statistic would not lie in the rejection region.value of test statisticThese would be the critical values for a 5% significance level4.2.5 Computing p-Values for t Tests (2/2)Chapter 4 Multi

59、ple RegressionAnalysis: InferenceSectionChapterEnd 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.4.2.6 A Reminder on the Language of Classical Hypothesis TestingExample 4.5 Housing Prices and Air

60、PollutionWe do not want to test that bnox=0. Instead,H0: bnox=1t=(.954+1)/.117=.393There is little evidence that the elasticity is different from 1.we fail to reject H0 at the x% level.H0 is accepted at the x% level.H0: bnox=.9t=(.954+.9)/.117=.462Chapter 4 Multiple RegressionAnalysis: InferenceSect