第二章一元線性回歸的檢驗和預(yù)測2016

1、CHAPTER TWOHYPOTHESIS TESTING AND PREDITION OF TWO-VARIABLE LINEAR REGRESSION MODELn1. t test and confidence intervaln2. prediction: point predictorn interval predictorn3. report and analysisn4. caseHypothesis test: Is a given observation or finding sufficiently closed to hypothesized value so that

2、we cant reject the stated hypothesis? Therere two approaches to proceed the hypothesis test: test of significance and confidence interval 2.1 test of significance and confidence interval )(,()(,(111000VNVN2212220)()1()(iuiuxVxXnV2.1.1 test of significance1100)()(EE Constituting standardized normal d

3、istribution But to obtain , we need Usually , true is unknown, now we substitute for . ) 1 , 0()() 1 , 0()(111000NVNV)(),(10VV2u2u2u222netu2uFurthermore we substitute for and get a t-statistic2212220)()1()(iuiuxVxXnV)(),(10VV)2()(000ntVT)2()(111ntVT)(),(10VV1、Put forward hypothesis:Null hypothesis:H

4、0：Alternative hypothesis:H1： 2、Constitute t-statistic and compute the value of T:0)2()()(ntVTVT03、Decision rule:If |T| ta/2(n-2)，reject the H0, , it means that the explanatory variable indeed infects dependent variable. The variable X is statistically significant .Otherwise, accept H0，it indicates t

5、hat there isnt significant correlation between explanatory and dependent variablesupplement：decision rule of t-testH0:H1:reject H0雙尾1= 1*1 1*|T|t/2右尾1 1*1 1*Tt左尾1 1*1 1*T -t: level of significance,fixed(1- )%: confidencedf: degree of freedomCritical valueP-value:probabilityactual probability of obta

6、ining a value of T-TEST as much as or greater than obtained in OLS estimation If fixed is greater than P, we may reject the null hypothesis.2.1.2 confidence intervalFor statistic T follows t distribution：Given level of significance：)2()(ntVT1)2()2(22ntTntPaa)2()2(22ntTntaa)2()()2(22ntVntaaIs called

7、the confidence interval of the true given confidence or the level of significanceIf zero falls within the confidence interval, we accept the null hypothesis. Otherwise, reject null hypothesis.Correspondingly,) ()2(2Vnta )()2(, )()2(22VntVntaa)()2()()2(22VntVntaa2.2 the analysis of variance SS D.F.ES

8、S 1RSSN-2TSSN-1iixy12iy2ieRSSESSTSS2221iiyeTSSESSR2.3 predictionvMean prediction vIndividual predictionWe have sample regression functionA future value of X is known as X0，then the future conditional means of Y can be predicted as following:ttXY100Y)(0YEIs estimator of the future conditional means o

9、f YIt is called point predictorAnother question is interval predictor: which is the confidence interval of given confidence 0100XY)(0YE2.3.1 MEAN PREDICTIONPrediction of conditional means Y on given X2u222netu2u),/(22012000）（tnuxxXYENY222netu01002201)2(200)(XYxxtYYEtnun)2(2000)()(ntYYEYTis the level

10、 of significance2.3.2 individual predictionPrediction of an individual Y value on given X222netu），（)(0020eNe2u222netu2u）（22012021)(tnuxxe1、The distribution of residual000YYe）（22012021)(tnuxxe2. Constitute the T-statistic2201)2(2001tnunxxtYY)2(200000)()( 0nteYYeeTis the level of significancettXY0XXYT

11、he predictive The predictive ability falls ability falls markedly as Xmarkedly as X0 0 departs departs progressively progressively from the sample from the sample mean!mean!Confidence interval for meansConfidence interval for individualsConclusion:The greater n, the more accurate prediction The clos

12、er for X to sample means X0, the more accurate prediction The width of confidence bands is smallest when The wider the sample range of X, the more accurate prediction XX 02.4 normality testsThe hypothesis test is based on the assumption that the disturbance follows the normal distribution. But ,in f

13、act, is this assumption satisfied? We use the residual to check it Histogram of Residuals Normal Probability Plot self-studyJB testdf. 2on with distributi square-chi thefollow 24)3(63)K and 0(S ondistributi normal thefollow variblerandom the:on.distributi normal of test formal sample-largeor symptot

14、ic,an 220JBkSnJBH2.5 report and analysis of the result of regression )10(8,96. 000004. 002. 02 .148 . 303. 04 . 682. 01202nDFRpTSEXYttAnalysis:1、Are the signs of estimated regression coefficient accordance with our expectation or economic theory?2、Is the slope coefficient statistical significant? (s

15、tatistically significantly different from Zero.)3、How well does the regression model explain the change or variation of Y?If R2 is great enough ,we can say that the regression model seems quite good.4、Is our model satisfied the normality assumption ？案例1 1977-1991， 美國， 討論股票和黃金的保值作用，貨幣戰(zhàn)爭中的觀點是否得到支持？ 每盎司紐約黃金美元價格消費價格指數(shù)（CPI）紐約股票交易所指數(shù)(NYSE)147.9860.653.69193.4465.253.7307.627