Interval EstimationHypothesis TestingRejection Regions for Specific HypothesisExamples of Hypothesis TestsThe p-ValueLinear Combinations of ParametersKey WordsLecture Note 4: Interval Estimation & HypothesisTestingMoshe BuchinskyUCLAFall, 2014B uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2 014 1 / 80Topics to be Covered1Interval Estimation2Hypothesis Tests3Rejection Regions for Speci…c Alternatives4Examples of Hypothesis Tests5The p-value6Linear Combinations of ParametersB uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2 014 2 / 80Interval Estimat ionInterval EstimationThere are two types of estimates: point estimates and intervalestimatesPoint estimates:The estimate b2is a point estimate of the unknown populationparameter in the regression model.Interval estimates:Interval estimation proposes a range of values in which the trueparameter is likely to fallProviding a range of values gives a sense of what the parameter valuemight be, and the precision with which we have estimated it.Such intervals are often called con…dence intervals.We prefer to call them interval estimates because the term“con…dence” is widely misunderstood and misusedB uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2 014 3 / 80Interval Estimat ionThe normal distribution of b2, the least squares estimator for β2, isb2 N β2,σ2∑Ni =1(xix)2!A standardized normal random variable is obtained from b2bysubtracting its mean and dividing by its standard deviation:Z =b2 β2rσ2.∑Ni =1(xix)2 N(0, 1)(3.1)B uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2 014 4 / 80Interval Estimat ionWe know that:Pr(1.96 Z 1.96)= .95Substituting:Pr0BB@1.96 b2 β2rσ2.∑Ni =1(xix)2 1.961CCA= .95Rearranging:Pr b21.96sσ2∑Ni =1(xix)2 β2 b2+ 1.96sσ2∑Ni =1(xix)2!= .95B uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2 014 5 / 80Interval Estimat ionThe two end-points b21.96rσ2.∑Ni =1(xix)2provide aninterval estimator.In repeated sampling 95% of the intervals constructed this way willcontain the true value of the parameter β2.This easy derivation of an interval estimator is based on theassumption SR6 and that we know the variance of the error term σ2.B uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2 014 6 / 80Interval Estimat ionReplacing σ2withbσ2creates a random variable t:t =b2 β2rbσ2.∑Ni =1(xix)2=b2 β2qbVar(b2)=b2 β2bSe(b2) t(N 2)(3.2)The ratio t =(b2 β2)/bSe(b2)has a t-distribution with (N 2)degrees of freedom, which we denote as:t t(N 2)In general we can say, if assumptions SR1–SR6 hold in the simplelinear regression model, thent =bk βkbSe(bk) t(N 2), for k = 1, 2. (3.3)B uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2 014 7 / 80Interval Estimat ionThe t-distribution is a bell shaped curve centered at zeroIt looks like the standard normal distribution, except it is more spreadout, with a larger variance and thicker tailsThe shape of the t-distribution is controlled by a single parametercalled the degrees of freedom, often abbreviated as df.B uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2 014 8 / 80Interval Estimat ionFigure 3.1: Critical values from a t-distributionB uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2 014 9 / 80Interval Estimat ionWe can …nd a “critical value” from a t-distribution such thatPr(t tc)= Pr(t tc)=α2,where α is a probability often taken to be α = 0.01, or α = 0.05.The critical value tcfor degrees of freedom m is the percentile valuet(1α/2,m)Each shaded “tail” area contains α/2 of the probability, so that 1 αof the probability is contained in the center portion.B uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2014 10 / 80Interval Estimat ionConsequently, we can make the probability statementPr(tc t tc)= 1 α, (3.4)orPr tcbk βkbSe(bk) tc!= 1 α,orPrbktcbSe(bk) βk bk+ tcbSe(bk)= 1 α, (3.5)B uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2014 11 / 80Interval Estimat ionWhen bkandbSe(bk)are estimated values (numbers), based on agiven sample of data, then bktcbSe(bk)is called a 100(1 α)%interval estimate of βk.Equivalently it is called a 100(1 α)% con…dence interval.Usually α = 0.01 or α = 0.05, so that we obtain a 99% con…denceinterval or a 95% con…dence interval.B uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2014 12 / 80Interval Estimat ionThe interpretation of con…dence intervals requires a great deal of careThe properties of the interval estimation procedure are based on thenotion of repeated samplingAny one interval estimate, based on one sample of data, may or maynot contain the true parameter βk, and because βkis unknown, wewill never know whether it does or does notWhen “con…dence intervals” are discussed, remember that ourcon…dence is in the procedure used to construct the interval estimate;it is not in any one interval estimate calculated from a sample of dataB uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2014 13 / 80Interval Estimat ionFor the food expenditure dataPrb22.024bSe(b2) β2 b2+ 2.024bSe(b2)= 0.95 (3.6)The critical value tc= 2.024, which is appropriate for α = 0.05 and38 degrees of freedomTo construct an interval estimate for β2we use the least squaresestimate b2= 10.21 and its estimated standard errorbSe(b2)=qbVar(b2)=p4.38 = 2.09B uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2014 14 / 80Interval Estimat ionA “95% con…dence interval estimate” for β2:b2tcbSe(b2)= 10.21 2.024 2.09 =[5.97, 14.45]When the procedure we used is applied to many random samples ofdata from the same population, then 95% of all the interval estimatesconstructed using this procedure will contain the true parameterB uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2014 15 / 80Interval Estimat ionIs β2actually in the interval [5.97, 14.45]?We do not know, and we will never knowWhat we do know is that when the procedure we used is applied tomany random samples of data from the same population, then 95%of all the interval estimates constructed using this procedure willcontain the true parameterThe interval estimation procedure “works” 95% of the timeWhat we can say about the interval estimate based on our one sampleis that, given the reliability of the procedure, we would be “surprised”if β2is not in the interval [5.97, 14.45].B uchinsky (UCLA ) Econ 103, Lecture 4 Fall, 2014 16 / 80Interval Estimat ionWhat is the usefulness of an interval estimate of β2?When reporting
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