b) The mean of the fitted (predicted) values of Y is equal to the mean of the Y values: c) The residuals of the regression line sum up to zero: = ei


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1 Mathematcal Propertes of the Least Squares Regresson The least squares regresson lne obeys certan mathematcal propertes whch are useful to know n practce. The followng propertes can be establshed algebracally: a The least squares regresson lne passes through the pont of sample means of Y and X. Ths can be easly seen from (4.9 whch can be rewrtten as follows, Y = b +b.x (4. b The mean of the ftted (predcted values of Y s equal to the mean of the Y values: Let Yˆ = b+b. X then we have  Y ˆ = ( b+b.x = ( Y  b.x +b. X =Y  b.x +b.x = Y n n (4.3 c The resduals of the regresson lne sum up to zero: e = ( Y  Yˆ =Y  Y ˆ = (4.4 n n d The resduals e are uncorrelated wth the X values: e X = e X  e X snce e = = e ( X  X = ( Y  Y.( X  X  b. ( X  X snce b =Y  b.x = snce b = ( Y  Y.( X  X / ( X  X e The resduals e are uncorrelated wth the ftted values Y. Ths property follows logcally from the prevous one snce each ftted value of Y s lnear functon of the correspondng X value. f The least squares regresson splts the varaton n the Y varable nto two components  the explaned varaton due to the varaton n X and the resdual varaton: where, TSS = RSS + ESS (4.5
2 TSS = (Y  Y ESS = (Yˆ  Yˆ RSS = e = (Y  Yˆ (4.6 TSS s the total varaton observed n the dependent varable Y. It s called the total sum of squares. ESS, the Explaned Sum of Squares, s the varaton of the predcted values (b +b.x. Ths s the varaton n Y accounted for by the varaton n the explanatory varable X. What s left s the RSS, the Resdual Sum of Squares. The reason why the ESS and RSS neatly add up to the TSS s that the resduals are uncorrelated wth the ftted Y values and, hence, there s no term wth the sum of covarances. Ths last property suggests a useful way to measure the goodness of ft of the estmated sample regresson. Ths s done as follows, R = ESS/TSS (4.7 where R, called Rsquare, s the coeffcent of determnaton. It gves us the proporton of the total sum of squares of the dependent varable explaned by the varaton n the explanatory varable. In fact, the R equals the square of the lnear correlaton coeffcent between the observed and the predcted values of the dependent varable Y, computed as follows, r = Cov (Y,Y ˆ V( Y.V( Yˆ = (Y  Y.( Yˆ  Yˆ (Y  Y. (Yˆ  Yˆ (4.7a A correlaton coeffcent measures the degree of lnear assocaton between two varables. Note, however, that f the underlyng relaton between the varables s nonlnear, the correlaton coeffcent may perform poorly, notwthstandng that fact that a strong nonlnear assocaton exsts between two varables.
3 Statstcal Propertes of LS Lnear Regresson We brefly revew the man ponts wthout much further elaboraton, apart from a few specfc ponts whch concern regresson only. We shall merely remnd you of the results of formal dervatons wthout botherng about proofs whch can be found n most ntroductory texts on statstcs or econometrcs. Standard Errors Gven the assumptons of the classcal lnear regresson model, the varances of the least squares estmators are gven by, var( b = σ (4.9 n σ var( b = ( X  X (4. Furthermore, an unbased estmator of σ s gven by s as follows: s = (Y  b  b X n  (4. where s s called the standard error of regresson snce σ s the varance of the error term whch measures the devaton of ndvduals ponts from the regresson lne. Replacng σ by s n (4.9 and (4., we get unbased estmates of the varances of b and b. Obvously, the estmated standard errors are the square roots of these varances. The total sum of squares of X, ( X  X whch features n the denomnator of the varances of the ntercept and slope coeffcents s a measure of the total varaton n the X values. Thus, other thngs beng equal, the hgher the varaton n the X values, the lower wll be the varances of the estmators, whch mples that hgher wll be the precson n estmaton. In other words, the range of observed X plays a crucal role n the relablty of
4 the estmates. Thnk about ths. It would ndeed be dffcult to measure the response of Y on X f X hardly vares at all. The greater the range over whch X vares, the easer t s to capture ts mpact on the varaton n Y. Samplng Dstrbutons To construct the confdence ntervals and to perform tests of hypotheses we need the probablty dstrbuton of the errors whch mples that we use the normalty assumpton of the error terms. Under ths assumpton, the least squares estmators b and b each follow a normal dstrbuton. However, snce we generally do not know the varance of the error term, we cannot make use of the normal dstrbuton drectly. Instead, we use the tdstrbuton defned as follows n the case of b, b  β = _ t se( b t ( n  (4. where se(b, the standard error of b, s gven by, se( b = s [ ( X  X ] (4.3 usng (4. and (4.. The statstc, t (n, denotes the Student's tdstrbuton wth (n degrees of freedom. The reason why we now have only (n degrees of freedom s that, n smple regresson, we use the sample data to estmate coeffcents: the slope and the ntercept of the lne. In the case of the sample mean, n contrast, we only estmated one parameter (the mean tself from the sample. Smlarly, for b, we get, b  β = _ t se( b t ( n  (4.4 where se(b, the standard error of b, s gven by, se( X b = s + (4.5 n ( X  X
5 usng (4.9 and (4.. Confdence Intervals for the Parameters ß and ß The confdence lmts for ß and ß wth (α per cent confdence coeffcent (say, 95 per cent, n whch case α=.5 are gven by, + _ t n ,.se( b (4.6 b + _ t n ,.se( b (4.7 b respectvely, where t(n,α/ s the (α/ percentle of a tdstrbuton wth (n degrees of freedom, and se(b and se(b are gven by (4.3 and (4.5 respectvely. Confdence Interval for the Condtonal Mean of Y At tmes, we may be nterested to construct a confdence nterval for the condtonal mean. For example, after fttng a regresson of household savngs on ncome, we may want to construct a confdence nterval for average savngs gven the level of ncome n order to assess the savngs potental of a certan type of households. Suppose, µ = β +.X (4.8 β.e. µ s the condtonal mean of Y gven X=X. The pont estmate of µ s gven by, +b. X b whle ts (α per cent confdence nterval can be obtaned as follows, µ + _ t n ,.se( µ (4.9 where, ( X  X se( µ = s + (4.3 n ( X  X
6 Confdence Interval for the Predcted Y Values There are other occasons where we mght be nterested n the uncertanty n predcton on the bass of the estmated regresson. For example, when estmatng a regresson of paddy yeld (physcal output per unt area on annual ranfall, we may want to predct next year's yeld gven the antcpated ranfall. In ths case, our nterest s not to obtan a confdence nterval of the condtonal mean of the yeld.e. the mean yeld at a gven level of ranfall. Rather, we want to fnd a confdence nterval for the yeld (Y tself, gven the ranfall (X? Obvously, n ths case, = β + β. X + ε = µ + ε Y where µ s gven by (4.8. The (α per cent confdence nterval for the Y gven X=X s then obtaned as follows, where, + _ t n ,.se(y (4.3 Y ( X  X se( Y = s + + (4.3 n ( X  X In ths case, therefore, the standard error of Y s larger than that of µ snce the latter corresponds to the condtonal mean of the yeld for a gven level of ranfall, whle the former corresponds to the predcted value of the yeld. In both cases, (4.3 and (4.3, the confdence ntervals wll be larger, the farther the X value s away from ts mean n the sample. Standard Error of a Resdual Fnally, the resduals e are the estmators of errors ε (see (4.7 and (4.8. The standard error of e s obtaned as follows, ( X  X se ( e = s  h where h = + n ( X  X (4.33
7 where s s gven by (4.. Note that whle the standard devaton of the error term s assumed to be homoscedastc, equaton 4.33 shows that the resduals of the regresson lne are heteroscedastc n nature. The standard error of each resdual depends on the value of h. The statstc h s called the hat statstc: h wll be larger, the greater the dstance of X from ts mean. A value of X whch s far away from ts mean (for example, an outler n the unvarate analyss of X wll produce a large hat statstc whch, as we shall see n secton 4.7, can exert undue nfluence on the locaton of a regresson lne. A data pont wth a large hat statstc s sad to exert leverage on the least squares regresson lne, the mportance of whch wll be shown n secton 4.7.
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