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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: InferenceWooldridge: Introductory Econometrics: A Modern Approach, 5eInstructed by professor Yuan, Huiping 20

2、13 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: Inference4.2 Testing Hypotheses about a Single Population Parameter: The t Test4.3 Confidence Intervals4.4 Testi

3、ng Hypotheses about a Single Linear Combination of the Parameters4.5 Testing Multiple Linear Restrictions: 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 C

4、1, C2, C3, C8, C9 C8: smpl if marr=1 and fsize=2 (401ksubs.wf1)The End 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. Statistical inference in the regression modelHypothesis tests about population

5、 parametersConstruction of confidence intervals Sampling distributions of the OLS estimatorsThe OLS 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 ad

6、ditional assumptionsAssumption about distribution of errors: normal distributionChapter 4 Multiple RegressionAnalysis: Inference4.1 Sampling Distributions of the OLS Estimators (1/5) Chapter End 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publ

7、icly accessible website, in whole or in part. Assumption MLR.6 (Normality of error terms)independently of It 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

8、 variables.It follows that:Chapter 4 Multiple RegressionAnalysis: Inference4.1 Sampling Distributions of the OLS Estimators (2/5) Chapter End 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. Discuss

9、ion of the normality assumptionThe error term is the sum of many“ different unobserved factorsSums of independent factors are normally distributed (CLT)Problems: How many different factors? Number large enough? Possibly very heterogenuous distributions of individual factors How independent are the d

10、ifferent factors?The normality of the error term is an empirical questionAt least the error distribution should be close“ to normalChapter 4 Multiple RegressionAnalysis: Inference4.1 Sampling Distributions of the OLS Estimators (3/5) Chapter End 2013 Cengage Learning. All Rights Reserved. May not be

11、 scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Discussion of 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 (ind

12、icator variable, takes on only 1 or 0)In some cases, normality can be achieved through transformations of the 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 Multi

13、ple RegressionAnalysis: Inference4.1 Sampling Distributions of the OLS Estimators (4/5) Chapter End 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. TerminologyTheorem 4.1 (Normal sampling distribut

14、ions)Under assumptions MLR.1 MLR.6:The estimators are normally distributed around the true parameters with the variance that was derived earlier The standardized estimators follow a standard normal distributionGauss-Markov assumptions“ Classical linear model (CLM) assumptions“ Chapter 4 Multiple Reg

15、ressionAnalysis: Inference4.1 Sampling Distributions of the OLS Estimators (5/5) Chapter End 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. 4.2.1 Theorem 4.2 t Distribution for the Standardized Es

16、timatorsChapter 4 Multiple RegressionAnalysis: Inference4.2 Testing Hypotheses about a Single Population Parameter: The t Test4.2.3 Two-Sided Alternatives4.2.4 Testing Other Hypotheses about bj4.2.2 Testing against One-Sided Alternatives4.2.5 Computing p-Values for t Tests4.2.6 A Reminder on the Lan

17、guage of Classical Hypothesis Testing4.2.7 Economic, or Practical, versus Statistical Significance Chapter End 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. Under assumptions MLR.1 MLR.6:If the s

18、tandardization is done using the estimated standard deviation (= standard error), the normal distribution is replaced by 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 Distrib

19、ution for the Standardized Estimators (1/3) 2 2 2 2 2 1 0,1 1 #j j jj jjj jj n ksdse Nsdn k b b bb bb b bb Proof: S ctionChapt r End 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. Null hypothesis

20、(for more general hypotheses, see below)t-statistic (or t-ratio)Distribution of the t-statistic if the null hypothesis is trueThe t-statistic will 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

21、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-statistic measures how many estimated standard deviations the estimated coefficient is away from zero. The population parameter is equal to zero, i.e. after

22、controlling for the other independent variables, there is no effect of xj on y Chapter 4 Multiple RegressionAnalysis: Inference4.2.1 Theorem 4.2 t Distribution for the Standardized Estimators (2/3) S ctionChapt r End 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicate

23、d, 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 a small probability (= significance level, e.g. 5%)The precise rejection rule depends on the alternative hypothesis and the chosen significance level of

24、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 Distribution for the Standardized Estimators (3/3) S ctionChapt r End 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicate

25、d, 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 Testing against One-Sided Alternatives (1/8)Reject the null hypothesis in favour of the alternative hypothesis if the estimated coefficient is too large“

26、(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 cases.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-statist

27、ic greater than 1.701Chapter 4 Multiple RegressionAnalysis: InferenceS ctionChapt r End 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: Wage equationTest whether, after controlling for edu

28、cation and tenure, higher work experience leads to higher hourly wages(1) Test against . One would either expect a positive effect of experience on hourly wage or no effect at all.Standard errors 4.2.2 Testing against One-Sided Alternatives (2/8)Chapter 4 Multiple RegressionAnalysis: Inference S cti

29、onChapt r End 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: Wage equation (cont.)The effect of experience on hourly wage is statistically greater than zero at the 5% (and even at the 1%)

30、 significance level.“Thought the estimated return for another year of experience, holding tenure and education fixed, is not especially large, we have persuasively shown that the partial effect of experience is positive in the population.t-statisticCritical values for the 5% and the 1% significance

31、level (these are conventional significance levels). The null hypothesis is rejected because the t-statistic exceeds the critical value.(2) Degrees of freedom;here the standard normal approximation applies(3)(4) 4.2.2 Testing against One-Sided Alternatives (3/8)Chapter 4 Multiple RegressionAnalysis:

32、Inference S ctionChapt r End 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 (less than zero)Reject the null hypothesis in favour of the alterna

33、tive 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 true, 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 freed

34、om 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 RegressionAnalysis: InferenceS ctionChapt r End 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted

35、to a publicly accessible website, in whole or in part. Example: Student performance and school sizeTest whether smaller school size leads to better student performanceTest against . Do larger schools hamper student performance or is there no such effect?Percentage of studentspassing maths test Avera

36、ge annual tea-cher compensation School enrollment(= school size)Staff per one thousand students 4.2.2 Testing against One-Sided Alternatives (5/8)Chapter 4 Multiple RegressionAnalysis: Inference S ctionChapt r End 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated,

37、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 effect of school size on student performance (not even for a lax significance level of 15%).t-statisticCritical values for the 5% and

38、 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 approximation applies4.2.2 Testing against One-Sided Alternatives (6/8)Chapter 4 Multiple RegressionAnalysis: Inference S ctionCh

39、apt r End 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.)Alternative specification of functional form:Test against .R-squared slightly higher 4.

40、2.2 Testing against One-Sided Alternatives (7/8)Chapter 4 Multiple RegressionAnalysis: Inference S ctionChapt r End 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

41、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-statisticCritical value for the 5% significance level ! reject null hypothesisHow large is the effect? (small effect)+ 10% enrollme

42、nt ! -0.129 percentage points students pass test 4.2.2 Testing against One-Sided Alternatives (8/8)Chapter 4 Multiple RegressionAnalysis: Inference S ctionChapt r End 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in

43、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 coefficient is too large.Construct the critical value so that, if the null hypothesis is true, it is rejected in,for example

44、, 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 -2.06 or greater than 2.06 4.2.3 Two-Sided Alternatives (1/3)Chapter 4 Multiple RegressionAnalysis: Inference S ctio

45、nChapt r End 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 of college GPA Lectures missed per weekThe effects of hsGPA and skipped are significantly different from zero at t

46、he 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-Sided Alternatives (2/3)Chapter 4 Multiple RegressionAnalysis: Inference S ctionChapt r End 2013 Cengage Learni

47、ng. 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 regression coefficient is different from zero in a two-sided test, the corresponding variable is said to be sta

48、tistically 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“statistically significant at 5 % level“statistically significant at 1 % level“4.2.3 Two-Sided Alternatives (

49、3/3)Chapter 4 Multiple RegressionAnalysis: Inference S ctionChapt r End 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. Testing more general hypotheses about a regression coefficientNull hypothesis

50、t-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 Other Hypotheses about bj (1/3)Chapter 4 Multiple RegressionAnalysis: Inference S ctionChapt r End 2013 Cengag

51、e 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 hypothesis is whether crime increases by one percent if enrollment is increased by one percentThe hypothesis

52、is rejected at the 5% levelEstimate is different from one but is this difference statistically significant?4.2.4 Testing Other Hypotheses about bj (2/3)Chapter 4 Multiple RegressionAnalysis: Inference S ctionChapt r End 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplic

53、ated, or posted to a publicly accessible website, in whole or in part. 4.2.4 Testing Other Hypotheses about bj (3/3)Chapter 4 Multiple RegressionAnalysis: Inference S ctionChapt r End 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly access

54、ible 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 null hypothesis cannot be rejected anymoreThe reason is that, by lowering the significance level, one want

55、s 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 small p-value is evidence against the null hypothesis because one would reject the null hypothesis even at

56、 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: Inference S ctionChapt r End 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or dupl

57、icated, 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 rejecting and not rejecting the null hypothesis. In the two-sided case, the p-value is thus the probabi

58、lity 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 corresponding p-value is smaller than the significance level. For example, for a significance level o

59、f 5% the t-statistic would not lie in the rejection region.value of test statisticThese would be the critical values for a 5% significance level 4.2.5 Computing p-Values for t Tests (2/2)Chapter 4 Multiple RegressionAnalysis: Inference S ctionChapt r End 2013 Cengage Learning. All Rights Reserved. M

60、ay 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 PollutionWe do not want to test that bnox=0. Instead,H0: bnox=1t=(.954+1)/.117=.393There is lit

61、tle evidence that the elasticity is different from 1.we fail to reject H 0 at the x% level.H0 is accepted at the x% level.H0: bnox=.9t=(.954+.9)/.117=.462 Chapter 4 Multiple RegressionAnalysis: Inference S ctionChapt r End 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or dup

62、licated, or posted to a publicly accessible website, in whole or in part. 4.2.7 Economic, or Practical, versus Statistical Significance (1/2)economic significance: statistical significance:Example 4.6 Participation Rates in 401(k) Plans Consider btotemp.Example 4.7 Effect of Job Training on Firm Scr

63、ap Rates Consider bhrsemp. Some researchers insist on using smaller significance levels as the sample size increases. Most researchers are also willing to entertain larger significance levels in applications with small sample sizes . ,jb j j jt seb b b Chapter 4 Multiple RegressionAnalysis: Inferenc

64、e S ctionChapt r End 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. Guidelines: If the variable is statistically significant at the usual levels, discuss the magnitude of the coefficient to get an

65、 idea of its economic importance. The fact that a coefficient is statistically significant does not necessarily mean it is economically or practically significant! If a variable is statistically and economically important but has the wrong“ sign, the regression model might be misspecified. If a vari

66、able is statistically insignificant at the usual levels (10%, 5%, 1%), one may think of dropping it from the regression. If the sample size is small, effects might be imprecisely estimated so that the case for dropping insignificant variables is less strong. variables with small t statistics that have the “wrong” sign. (multicollinearity)4.2.7 Economic, or Practical, versus Statistical Significance (2/2)Chapter 4 Multiple RegressionAnalysis: Inference S ctionChapt r End 2013 Cengage Learning. Al