Interval Estimation in the Classical Normal Linear Regression Model. 1. Introduction

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1 ECONOMICS 35* -- NOTE 7 ECON 35* -- NOTE 7 Interval Estmaton n the Classcal Normal Lnear Regresson Model Ths note outlnes the basc elements of nterval estmaton n the Classcal Normal Lnear Regresson Model (the CNLRM). Interval estmaton --.e., the constructon of confdence ntervals for unknown populaton parameters -- s one of the two alternatve approaches to statstcal nference; the other s hypothess testng.. Introducton We have prevously derved pont estmators of all the unknown populaton parameters n the Classcal Normal Lnear Regresson Model (CNLRM) for whch the populaton regresson equaton, or PRE, s Y = β + β X + u where u s d as N(0, σ ) ( =,...,N). () The unknown parameters of the PRE are and () the regresson coeffcents β and β () the error varance σ. The pont estmators of these unknown populaton parameters are and () the unbased OLS regresson coeffcent estmators β ˆ and β ˆ () the unbased error varance estmator Assume that we have computed the pont estmates β $, β $, and σ $ of the unknown parameters for a gven set of sample data (Y, X ), =,..., N. ˆσ. ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page of 6 pages

2 ECONOMICS 35* -- NOTE 7 We therefore begn wth the followng OLS sample regresson equaton (or OLS-SRE): Y = β $ + β $ X + u $ ( =,...,N). () where $ ( )( ) = = xy X X Y Y β = x ( X X) β$ = Y β$ X = OLS estmate of β ; σ$ $ = u N = unbased OLS estmate of σ σˆ σˆ Vâr(ˆ β ) = = ; x ( X X) σˆ σˆ sê(ˆ β ) = Vâr(ˆ β ) = = x ; x Vâr(ˆ σˆ X β ) = = ; N ( ) x N X X σˆ X ) = Vâr(ˆ β) = N x σˆ X sê(ˆ β. OLS estmate of β ; Under the assumptons of the Classcal Normal Lnear Regresson Model (CNLRM) -- ncludng n partcular the normalty assumpton A9 -- the sample t-statstcs for ˆβ and β ˆ each have the t-dstrbuton wth (N ) degrees of freedom:.e., t( β$ $ β β β β ) Var $ ( β $ ) se$( β $ = = ) ~ t[n ]; t ( $ β$ $ β β β β ) Var $ ( β $ ) se$( β $ = = ) ~ t[n ]. ; ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page of 6 pages

3 ECONOMICS 35* -- NOTE 7. Interval Estmaton: Some Basc Ideas. General Form of a Confdence Interval A two-sded confdence nterval for the slope coeffcent β takes the general form Pr(β $ L β β $ ) = Pr(β $ U $δ β β $ + $δ ) = (3) where = the sgnfcance level (0 < < ), = the confdence level (or confdence coeffcent), δ $ = a postvely-valued sample statstc, β $ = β $ $δ L = the lower confdence lmt, β $ = β $ + $δ = the upper confdence lmt. U The nterval [β $, β $ ] = [β $ L U $δ, β $ + $δ ] s called the two-sded ( )-level confdence nterval, or two-sded 00( ) percent confdence nterval, for the slope coeffcent β. ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page 3 of 6 pages

4 ECONOMICS 35* -- NOTE 7. Interpretaton of Confdence Intervals Pr(β $ L β β $ ) = Pr(β $ U $δ β β $ + $δ ) = (3). The confdence nterval [β $, β $ ] s a random nterval. L U The confdence lmts β $ = $β L δ $ and β $ = β $ U + $δ are random varables (or sample statstcs) that vary n value from one sample to another because the values of β $ and δ $ vary from sample to sample. But for any one sample of data and the correspondng estmates of β $ and $δ, the confdence lmts β $ = β $ $δ and β $ = β $ + $δ are smply fxed L U numbers,.e., they take fxed values. Therefore, any one confdence nterval calculated for a partcular sample of data s a fxed -- meanng nonrandom -- nterval.. The correct nterpretaton of the confdence nterval [β $, β $ L U ] s based on the concept of repeated samplng. In repeated samples of the same sze from the same populaton, 00( ) percent of the confdence ntervals constructed usng the formulas for ˆβ L and ˆβ wll contan the true value of the populaton parameter β. U For example, f the confdence level = 0.95, 95 percent of the confdence ntervals computed usng repeated samples of the same sze from the same populaton wll contan the true value of β. But any one confdence nterval for β, based on one sample of data, may or may not contan the true value of β. Snce the true value of β s unknown, we do not know whether that value does or does not le nsde any one confdence nterval. ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page 4 of 6 pages

5 ECONOMICS 35* -- NOTE 7 3. Confdence Intervals for the Regresson Coeffcents β and β 3. Confdence Interval for β : Dervaton A two-step dervaton: Step : Start wth a probablty statement formulated n terms of t( ˆβ ), the t-statstc for β ˆ. Ths probablty statement mplctly defnes the two-sded ( )- level confdence nterval for β. Step : Re-arrange ths probablty statement to obtan an equvalent probablty statement formulated n terms of β rather than t(β ˆ ). The resultant probablty statement explctly defnes the two-sded ( )-level confdence nterval for β. Step : The two-sded ( )-level confdence nterval for β s mplctly defned by the probablty statement where ( t [N ] t(ˆ β ) t [N ] ) = Pr / (4) / = the confdence level attached to the confdence nterval; = the sgnfcance level, where 0 < < ; t / [N ] = the crtcal value of the t-dstrbuton wth N degrees of freedom at the / (or 00/ percent) sgnfcance level; and t(β $ ) s the t-statstc for β $ gven by ˆ β β β t(ˆ β ) = =. (5) Vâr(ˆ β ) sê(ˆ β ) ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page 5 of 6 pages

6 ECONOMICS 35* -- NOTE 7 The t(β ˆ ) statstc has the t[n ] dstrbuton area = / area = area = / t / t / Pr(t < t / ) = / Pr( t / t t / ) = Pr(t > t / ) = / ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page 6 of 6 pages

7 ECONOMICS 35* -- NOTE 7 Step : Express the double nequalty nsde the brackets n probablty statement (4) n terms of β rather than t(β $ ). ( t [N ] t(ˆ β ) t [N ] ) = Pr / (4) / () Substtute n the double nequalty t [N ] t(ˆ β ) t / [N / the expresson for t(β $ ) gven n (5) above: ] β$ β t [ N ] t N / se$( β $ [ ]. (6.) / ) () Multply the double nequalty (6.) by the postve number se $( β $ ) > 0: ˆ / sê(ˆ β ) β β t / sê(ˆ β ). (6.) t (3) Subtract β $ from both sdes of nequalty (6.): β $ t se $( β $ ) β β $ + t se $( $ ). (6.3) β / / (4) Multply all terms n nequalty (6.3) by, rememberng to reverse the drecton of the nequaltes: β$ t se$( β $ ) β β$ t se $( $ ). (6.4) + β / / ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page 7 of 6 pages

8 ECONOMICS 35* -- NOTE 7 RESULT: The probablty statement (4) can be wrtten as ( t [N ]sê(ˆ β ) β + t [N ]sê(ˆ β )) = Pr. (7) The two-sded ( )-level confdence nterval for β can therefore be wrtten as β$ [ ] $( $ ) $ [ ] $( $ t/ N se β β β + t N se β ) or more compactly as β$ ± t [ N ] se$( β $ ) or [ β ˆ t [N ] sê(ˆ ), β ˆ t [N ]sê(ˆ )] β + β where at the ( ) confdence level, or 00( ) percent confdence level, β$ $ [ ] $( $ L = β t N se β ) = the lower 00( ) percent confdence lmt for β and β$ $ [ ] $( $ U = β + t N se β ) = the upper 00( ) percent confdence lmt for β ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page 8 of 6 pages

9 ECONOMICS 35* -- NOTE 7 Two-sded ( )-level confdence nterval for β s centered around β ˆ tal area = / area = tal area = / β $ β ˆ β $ L U = t [N ]sê(ˆ ) = the lower ( )-level confdence lmt for β L β = + t [N ]sê(ˆ ) = the upper ( )-level confdence lmt for β U β ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page 9 of 6 pages

10 ECONOMICS 35* -- NOTE 7 3. Confdence Interval for β : Dervaton The confdence nterval (or nterval estmator) for the ntercept coeffcent β s derved, nterpreted, and constructed n exactly the same way as the confdence nterval for the slope coeffcent β.. The two-sded ( )-level confdence nterval for β s mplctly defned by the probablty statement where ( [N ] t(ˆ β ) t [N ] ) = Pr (8) t / / = the confdence level attached to the confdence nterval; = the sgnfcance level, where 0 < < ; t / [N ] = the crtcal value of the t-dstrbuton wth (N ) degrees of freedom at the / (or 00(/) percent) sgnfcance level; and t(β $ ) s the t-statstc for β $ gven by t( β $ ) = β$ β Var $ ( β $ ) = β$ β. (9) se$( β $ ). The double nequalty nsde the brackets n probablty statement (8) can be expressed n terms of β rather than t(β $ ), usng a dervaton analogous to that used n dervng the confdence nterval for β. ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page 0 of 6 pages

11 ECONOMICS 35* -- NOTE 7 RESULT: The probablty statement (8) can be wrtten as ( t [N ]sê(ˆ β ) β + t [N ]sê(ˆ β )) = Pr. (0) The two-sded ( )-level confdence nterval for β can therefore be wrtten as β$ [ ] $( $ ) $ [ ] $( $ t / N se β β β + t N se β ) or more compactly as β$ ± t [ N ] se$( β $ ) or [ β ˆ t [N ] sê(ˆ ), β ˆ t [N ]sê(ˆ )] β + β where at the ( ) confdence level, or 00( ) percent confdence level, β$ $ [ ] $( $ L = β t N se β ) = the lower 00( ) percent confdence lmt for β and β$ $ [ ] $( $ U = β + t N se β ) = the upper 00( ) percent confdence lmt for β ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page of 6 pages

12 ECONOMICS 35* -- NOTE Procedure for Computng Confdence Intervals Consder the problem of computng a confdence nterval for the slope coeffcent β. Recall that the two-sded ( )-level confdence nterval for β s gven by the double nequalty β$ t [ N ] se$( β $ ) β β$ + t [ N ] se$( β $ ). Step : After estmatng the PRE () by OLS, retreve from the estmaton results the OLS estmate β $ of β and the estmated standard error se $( β $ ). Step : Select the value of the confdence level ( ), whch amounts to selectng the value of. Although the choce of confdence level s essentally arbtrary, the values most commonly used n practce are: = 0.0 ( ) = 0.99,.e., the 00( ) = 00(0.99) = 99 percent confdence level; = 0.05 ( ) = 0.95,.e., the 00( ) = 00(0.95) = 95 percent confdence level; = 0.0 ( ) = 0.90,.e., the 00( ) = 00(0.90) = 90 percent confdence level. Step 3: Obtan the value of t [ N ], the / crtcal value of the t-dstrbuton wth N degrees of freedom, ether from statstcal tables of the t- dstrbuton or from a computer software program. Step 4: Use the values of β $, se$( β $ ), and t [ N ] to compute the upper and lower 00( ) percent confdence lmts for β : = + t [N ]sê( ˆ ) = upper 00( )% confdence lmt for β ; U β = t [N ]sê( ˆ ) = lower 00( )% confdence lmt for β. L β ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page of 6 pages

13 ECONOMICS 35* -- NOTE 7 4. Determnants of the Confdence Intervals for β and β Consder for example the two-sded 00( )% confdence nterval for β : or β$ [ ] $( $ ) $ [ ] $( $ t N se β β β + t N se β ) [ β $ [ ] $( β$ ), β $ [ ] $( β$ t N se + t N se ) ] The two-sded confdence nterval for β s wder () the greater the value of se$( β $ ), the estmated standard error of β $,.e., the less precse s the estmate of β; () the greater the crtcal value [N ],.e., the greater the chosen value of t the confdence level ( ) for the gven sample sze N. Explanaton: Gven sample sze N, the value of t [N ] s negatvely related to the value of, and so s postvely related to the value of ( ). Example: Suppose sample sze N = 30, so that the degrees-of-freedom N- = 8. Then from a table of percentage ponts for the t-dstrbuton, we obtan the followng values of t [ N ] = t [ 8] for dfferent values of : = 0.0 ( ) = 0.99: / = and t [8] =.763; = 0.0 ( ) = 0.98: / = 0.0 and t 0.0 [8] =.467; = 0.05 ( ) = 0.95: / = 0.05 and t 0.05 [8] =.048; = 0.0 ( ) = 0.90: / = 0.05 and t 0.05 [8] =.70. Note that hgher values of ( ) --.e., hgher confdence levels -- correspond to hgher crtcal values of t [ 8]. ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page 3 of 6 pages

14 ECONOMICS 35* -- NOTE 7 5. Two-Sded 00( )% Confdence Intervals for β : Examples Two-Sded 00( ) Percent Confdence Interval for β j : Formulas In general, the two-sded 00(-) percent confdence nterval for regresson coeffcent β j s: where [ t [N k]sê( ), + t [N k]sê( ) ] j j j = + t [N k]sê( ) = upper 00( ) % confdence lmt for β ju j j = t [N k]sê(ˆ β ) = lower 00( ) % confdence lmt for β jl j j j DATA: auto.dta A sample of 74 cars sold n North Amerca n 978. MODEL: prce = β + β weght + u ( =,..., N) N = 74 ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page 4 of 6 pages

15 ECONOMICS 35* -- NOTE 7 Compute the two-sded 95% confdence nterval for β. regress prce weght Source SS df MS Number of obs = F(, 7) = 9.4 Model Prob > F = Resdual R-squared = Adj R-squared = 0.80 Total Root MSE = prce Coef. Std. Err. t P> t [95% Conf. Interval] weght _cons dsplay nvttal(7, 0.05) $β =.044 se$( β $ ) = N = 74 k = N k = 74 = 7 = 0.95 = 0.95 = 0.05 / = 0.05/ = 0.05 t / [N ] = t 0.05 [7] =.9935 t / [N ]s (ˆ β ) = t0.05[n ] ê(ˆ β ) =.9935( ) = 0.75 ê s = + t [N ]sê( ˆ ) = =.7953 =.795 U β = t [N ]sê( ˆ ) = =.989 =.93 L β Result: The two-sded 95% confdence nterval for β s [.93,.795]. ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page 5 of 6 pages

16 ECONOMICS 35* -- NOTE 7 Compute the two-sded 99% confdence nterval for β. regress prce weght, level(99) Source SS df MS Number of obs = F(, 7) = 9.4 Model Prob > F = Resdual R-squared = Adj R-squared = 0.80 Total Root MSE = prce Coef. Std. Err. t P> t [99% Conf. Interval] weght _cons dsplay nvttal(7, 0.005) scalar bu99 = _b[weght] *_se[weght]. scalar bl99 = _b[weght] *_se[weght]. scalar lst bu99 bl99 bu99 = bl99 = $β =.044 se$( β $ ) = N = 74 k = N k = 74 = 7 = 0.99 = 0.99 = 0.0 / = 0.0/ = t / [N ] = t [7] =.6459 t / [N ]s (ˆ β ) = t0.005[n ] ê(ˆ β ) =.6459( ) = ê s β ˆ = + t [N ]sê( ˆ ) = = = 3.04 U β β ˆ = t [N ]sê( ˆ ) = = =.047 L β Result: The two-sded 99% confdence nterval for β s [.047, 3.04]. ECON 35* -- Note 7: Interval Estmaton n the CNLRM Page 6 of 6 pages

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