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1、Chapter 4,Multiple RegressionAnalysis:Inference,Wooldridge:Introductory Econometrics:A Modern Approach,5eInstructed by professor Yuan,Huiping,Chapter 4 Multiple RegressionAnalysis:Inference,4.2 Testing Hypotheses about a Single Population Parameter:The t Test,4.3 Confidence Intervals,4.4 Testing Hyp
2、otheses about a Single Linear Combination of the Parameters,4.5 Testing Multiple Linear Restrictions:The F Test,4.1 Sampling Distributions of the OLS Estimators,4.6 An application estimation of the weights of CPI components in China,Assignments:Promblems 1,2,4,5,7,8,10 Computer Exercises C1,C2,C3,C8
3、,C9 C8:smpl if marr=1 and fsize=2(401ksubs.wf1),The End,Statistical inference in the regression modelHypothesis tests about population parametersConstruction of confidence intervals Sampling distributions of the OLS estimatorsThe OLS estimators are random variablesWe already know their expected valu
4、es 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 distribution,Chapter 4 Multiple RegressionAnalysis:Inference,4.1 Sampling Distributions of the OLS E
5、stimators(1/5),Chapter,End,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 variables.It follo
6、ws that:,Chapter 4 Multiple RegressionAnalysis:Inference,4.1 Sampling Distributions of the OLS Estimators(2/5),Chapter,End,Discussion of the normality assumptionThe error term is the sum of many“different unobserved factorsSums of independent factors are normally distributed(CLT)Problems:How many di
7、fferent factors?Number large enough?Possibly very heterogenuous distributions of individual factorsHow independent are the different factors?The normality of the error term is an empirical questionAt least the error distribution should be close“to normal,Chapter 4 Multiple RegressionAnalysis:Inferen
8、ce,4.1 Sampling Distributions of the OLS Estimators(3/5),Chapter,End,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(indicator variable,takes on only 1 or 0)
9、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 size,Chapter 4 Multiple RegressionAnalysis:Inference,4.1 Sampli
10、ng Distributions of the OLS Estimators(4/5),Chapter,End,TerminologyTheorem 4.1(Normal sampling distributions),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 nor
11、mal distribution,Gauss-Markov assumptions“,Classical linear model(CLM)assumptions“,Chapter 4 Multiple RegressionAnalysis:Inference,4.1 Sampling Distributions of the OLS Estimators(5/5),Chapter,End,4.2.1 Theorem 4.2 t Distribution for the Standardized Estimators,Chapter 4 Multiple RegressionAnalysis:
12、Inference,4.2 Testing Hypotheses about a Single Population Parameter:The t Test,4.2.3 Two-Sided Alternatives,4.2.4 Testing Other Hypotheses about bj,4.2.2 Testing against One-Sided Alternatives,4.2.5 Computing p-Values for t Tests,A Reminder on the Language of Classical Hypothesis Testing,4.2.7 Econ
13、omic,or Practical,versus Statistical Significance,Chapter,End,Under assumptions MLR.1 MLR.6:,If the standardization is done using the estimated standard deviation(=standard error),the normal distribution is replaced by a t-distribution,Note:The t-distribution is close to the standard normal distribu
14、tion if n-k-1 is large.,Chapter 4 Multiple RegressionAnalysis:Inference,4.2.1 Theorem 4.2 t Distribution for the Standardized Estimators(1/3),Proof:,Section,Chapter,End,Null hypothesis(for more general hypotheses,see below)t-statistic(or t-ratio)Distribution of the t-statistic if the null hypothesis
15、 is true,The 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 true.But what does far“away from zero mean?,This depends on the variability of the estimated coefficient,i.e.its standa
16、rd 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 controlling for the other independent variables,there is no effect of xj on y,Chapter 4 Multiple RegressionAnalysis:Inference,
17、4.2.1 Theorem 4.2 t Distribution for the Standardized Estimators(2/3),Section,Chapter,End,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 signif
18、icance level of the test.A significance level:the probability of rejecting H0 when it is true.,Chapter 4 Multiple RegressionAnalysis:Inference,4.2.1 Theorem 4.2 t Distribution for the Standardized Estimators(3/3),Section,Chapter,End,Test against.,Testing against one-sided alternatives(greater than z
19、ero),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“(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%o
20、f 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-statistic greater than 1.701,Chapter 4 Multiple RegressionAnalysis:Inference,Section,Chapter,End,Example:Wage equationTest whether,after controlling for
21、 education 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,Sec
22、tion,Chapter,End,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 per
23、suasively shown that the partial effect of experience is positive in the population.,t-statistic,Critical 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 freed
24、om;here the standard normal approximation applies,(3),(4),4.2.2 Testing against One-Sided Alternatives(3/8),Chapter 4 Multiple RegressionAnalysis:Inference,Section,Chapter,End,Test against.,Testing against one-sided alternatives(less than zero),Reject the null hypothesis in favour of the alternative
25、 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 freedom so that 5
26、%of the cases are below the point.!Reject if t-statistic less than-1.734,4.2.2 Testing against One-Sided Alternatives(4/8),Chapter 4 Multiple RegressionAnalysis:Inference,Section,Chapter,End,Example:Student performance and school sizeTest whether smaller school size leads to better student performan
27、ce,Test against.,Do larger schools hamper student performance or is there no such effect?,Percentage of studentspassing maths test,Average 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
28、 RegressionAnalysis:Inference,Section,Chapter,End,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-statistic,Critical values for the 5%and the 15%signific
29、ance 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 applies,4.2.2 Testing against One-Sided Alternatives(6/8),Chapter 4 Multiple RegressionAnalysis:Inference,Section,Chapter,End,Examp
30、le:Student 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:Inference,Section,Chapter,End,Example:Student performance and school size(cont.),
31、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-statistic,Critical value for the 5%significance level!reject null hypothesis,How large is the effect?,(small effect),+10%enrollment!-0.129 percentage
32、 points students pass test,4.2.2 Testing against One-Sided Alternatives(8/8),Chapter 4 Multiple RegressionAnalysis:Inference,Section,Chapter,End,Testing against two-sided alternatives,Test against.,Reject the null hypothesis in favour of the alternative hypothesis if the absolute value of the estima
33、ted coefficient 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 t
34、han-2.06 or greater than 2.06,4.2.3 Two-Sided Alternatives(1/3),Chapter 4 Multiple RegressionAnalysis:Inference,Section,Chapter,End,Example:Determinants of college GPA,Lectures missed per week,The effects of hsGPA and skipped are significantly different from zero at the 1%significance level.The effe
35、ct of ACT is not significantly different from zero,not even at the 10%significance level.,For critical values,use standard normal distribution,4.2.3 Two-Sided Alternatives(2/3),Chapter 4 Multiple RegressionAnalysis:Inference,Section,Chapter,End,Statistically significant“variables in a regressionIf a
36、 regression 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
37、“,statistically significant at 5%level“,statistically significant at 1%level“,4.2.3 Two-Sided Alternatives(3/3),Chapter 4 Multiple RegressionAnalysis:Inference,Section,Chapter,End,Testing more general hypotheses about a regression coefficientNull hypothesist-statisticThe test works exactly as before
38、,except that the hypothesized value is substracted from the estimate when forming the statistic,Hypothesized value of the coefficient,4.2.4 Testing Other Hypotheses about bj(1/3),Chapter 4 Multiple RegressionAnalysis:Inference,Section,Chapter,End,Example:Campus crime and enrollmentAn interesting hyp
39、othesis is whether crime increases by one percent if enrollment is increased by one percent,The hypothesis is rejected at the 5%level,Estimate is different from one but is this difference statistically significant?,4.2.4 Testing Other Hypotheses about bj(2/3),Chapter 4 Multiple RegressionAnalysis:In
40、ference,Section,Chapter,End,4.2.4 Testing Other Hypotheses about bj(3/3),Chapter 4 Multiple RegressionAnalysis:Inference,Section,Chapter,End,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 n
41、ull 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 smal
42、l 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 levels,Chapter 4 Multiple RegressionAnalysis:Infer
43、ence,Section,Chapter,End,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 probability that the t-distributed variable takes on a lar
44、ger 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 of 5%the t-statistic would not lie in the rejection regi
45、on.,value of test statistic,These 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,Section,Chapter,End,4.2.6 A Reminder on the Language of Classical Hypothesis Testing,Example 4.5 Housing Prices and Air
46、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=-.462,Chapter 4 Multiple RegressionAnalysis:Inference,Sec
47、tion,Chapter,End,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 Scrap Rates Consider bhrsemp.Some researchers insist on using sm
48、aller significance levels as the sample size increases.Most researchers are also willing to entertain larger significance levels in applications with small sample sizes.,Chapter 4 Multiple RegressionAnalysis:Inference,Section,Chapter,End,Guidelines:If the variable is statistically significant at the
49、 usual levels,discuss the magnitude of the coefficient to get an 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 wron
50、g“sign,the regression model might be misspecified.If a variable 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 l
