第二章 一元線性回歸的檢驗(yàn)和預(yù)測(cè)2011

1、CHAPTER TWOHYPOTHESIS TESTING OF TWO-VARIABLE LINEAR REGRESSION MODELHypothesis test: Is a given observation or finding sufficiently closed to hypothesized value so that we cant reject the stated hypothesis? Therere two approaches to proceed the hypothesis test: test of significance and confidence i

2、nterval 2.1 test of significance and confidence interval )(,()(,(111000VNVN2212220)()1()(iuiuxVxXnV2.1.1 test of significance1100)()(EE Constituting standardized normal distribution But to obtain , we need Usually , true is unknown, now we substitute for . ) 1 , 0()() 1 , 0()(111000NVNV)(),(10VV2u2u

3、2u222netu2uFurthermore we substitute for and get a t-statistic2212220)()1()(iuiuxVxXnV)(),(10VV)2()(000ntVT)2()(111ntVT)(),(10VV1、Put forward hypothesis:Null hypothesis:H0：Alternative hypothesis:H1： 2、Constitute t-statistic and compute the value of T:0)2()()(ntVTVT03、Decision rule:If |T| ta/2(n-2)，r

4、eject the H0, , it means that the explanatory variable indeed infects dependent variable. The variable X is statistically significant .Otherwise, accept H0，it indicates that there isnt significant correlation between explanatory and dependent variablesupplement：decision rule of t-testH0:H1:reject H0

5、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 obtaining 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 h

6、ypothesis.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 the confidence interval of the true given confidence or the level of significanceIf the true given by the null hypothesis falls wi

7、thin 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.ESS 1RSSN-2TSSN-1iixy12iy2ieRSSESSTSS2221iiyeTSSESSR2.3 predictionvMean prediction vIndividual pred

8、ictionWe 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 of YIt is called point predictorAnother question is interval predictor: which is the confidence in

9、terval 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 of significance2.3.2 individual predictionPrediction of an individual Y value on given X222netu）

10、，（)(0020eNe2u222netu2u）（22012021)(tnuxxe1、The distribution of residual000YYe）（22012021)(tnuxxe2. Constitute the T-statistic2201)2(2001tnunxxtYY)2(200000)()( 0nteYYeeTis the level of significancettXY0XXYThe predictive The predictive ability falls ability falls markedly as Xmarkedly as X0 0 departs de

11、parts 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 closer for X to sample means X0, the more accurate prediction The width of confidence bands is smalle

12、st 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 fact, is this assumption satisfied? We use the residual to check it Histogram of Residuals Normal

13、Probability Plot2.5 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? (statistically significantly different from Z