# THE TWO-VARIABLE LINEAR REGRESSION MODEL

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1 THE TWO-VARIABLE LINEAR REGRESSION MODEL Herma J. Bieres Pesylvaia State Uiversity April 30, 202. Itroductio Suppose you are a ecoomics or busiess maor i a college close to the beach i the souther part of the US, for example souther Califoria, where the weather is almost always ice the whole year aroud. I order to support yourself through college, you have started your ow (weeked) busiess: a ice cream parlor o the beach. You have experieced that o hot weekeds you usually sell more ice cream tha o cold weekeds. Also, you have recorded the average temperature ad the sales of ice cream durig eight weekeds. Let Y be the sales of ice cream o weeked, measured i \$00, ad let X be the average temperature o weeked, measured i uits of 0 degrees Fahreheit: Table : Ice cream data Sales (uit = \$00) Temperature (uit = 0 degrees) Y = 8 X = 5 Y 2 = 0 X 2 = 7 Y 3 = 8 X 3 = 6 Y 4 = 3 X 4 = 8 Y 5 = 5 X 5 = 0 Y 6 = 4 X 6 = 9 Y 7 = X 7 = 7 Y 8 = 9 X 8 = 8 You wat to use this iformatio to forecast ext weeked's sales of ice cream, give a good forecast of ext weeked's temperature. Such a forecast of the sales will eable you to These lecture otes are based o lecture otes that I wrote while teachig at the Uiversity of Califoria, Sa Diego, i the witer of 987.

3 scheme, because large positive errors ca be offset by large egative errors. Therefore, use the sum of squared errors as your measure of the accuracy of your forecasts: Q(ˆα,ˆβ) ' (Y & Yˆ ) 2 ' (Y & ˆα & ˆβX ) 2, where is the sample size ( = 8 i our example), ad miimize Q(α^,β^) to show (see the Appedix) that Q(α^,β^) is miimal for \$\$ ' ' (X & X )(Y & Ȳ ) ' (X & X ) 2 ' ' (X & X )Y ' (X & X ) 2 ˆα ad ˆβ. It ca be \$" ' Ȳ & \$\$ X, where X ' (/)' X ad Ȳ ' (/)' Y. () I the ice cream parlor case we have ' 8, X ' 7.5, Ȳ ', ' X 2 ' 468, ' X Y ' 687, ' (X & X )(Y & Ȳ ) ' ' X Y &. X.Ȳ ' 27, ' (X & X ) 2 ' ' X 2 &. X 2 ' 8, so that ˆβ '.5, ˆα ' &0.25. Thus, our best forecastig scheme is Y ˆ ' &0.25 %.5X. This is the straight lie i Figure. hece ˆ Y x \$00 Now suppose that the forecast of ext weeked's temperature is 75 degrees. The X = 7.5, ˆ Y ' &0.25 %.5 (7.5) = \$,00. =. Therefore, the best forecast of ext weeked's sales is: 2. The two-variable liear regressio model. I order to aswer the questio how good this forecast is, we have to make assumptios about the true relatioship betwee the depedet variable Y ad the idepedet variable X, (also called the explaatory variable). The true relatioship we are goig to assume is the twovariable liear regressio model: Y ' " % \$.X % U, ',2,...,. (2) 3

8 replace α ad β i (7) by their OLS estimators: where #F 2 ' (Y & \$" & \$\$.X ) 2 ' \$U 2, (8) \$U ' Y & \$" & \$\$.X (9) is called the regressio residual. However, the estimator (8) is biased, due to the fact that Propositio 5. Uder the assumptios I - V, E[' Û 2 ] ' ( & 2)σ2. Proof: See the Appedix. This result suggests to use \$F 2 ' &2 \$U 2 (0) as a estimator of σ 2 istead of (8), because the by Propositio 5, ˆσ 2 is a ubiased estimator of σ 2 : The sum ' Û 2 Residual Sum of Squares (RSS), ad shortly SER. Thus, E[\$F 2 ] ' F 2. () is called the Sum of Squares Residuals, shortly SSR, or also called the ˆσ ' ˆσ 2 is called the Stadard Error of the Residuals, SSR ' ' \$ U 2, SER ' ' \$ U 2 &2 ' SSR &2 (' \$F). (2) Fially, ote that the sum of squared residuals ca be computed as follows: See the Appedix. SSR ' (Y & Ȳ ) 2 & \$\$ 2 (X & X ) 2. (3) 8

10 The results i Propositio 6 ow eable us to test hypotheses about α ad β. I particular the ull hypothesis that β = 0 is of importace, because this hypothesis implies that X has o effect o Y. The test statistic for testig this hypothesis is the t-value (or t-statistic) of ˆβ: def. \$t \$ (' t&value of \$\$ \$\$) ' \$F \$ ' \$\$ ' (X & X ) 2 \$F - t &2 if \$ ' 0. (7) If β > 0 ad 6 4 the the t-value of ˆβ coverges i probability to +4, ad if β < 0 ad 6 4 the the t-value of ˆβ coverges i probability to!4. Moreover, if the sample size is large the by Propositio 7 we may use the stadard ormal distributio istead of the t distributio to fid critical values of the test. Similarly, def. t \$ \$" (' t&value of \$") ' \$" \$F \$" - t &2 if " ' 0. (8) However, the hypothesis α = 0 is ofte of o iterest. I the ice cream example, ' (X & X ) 2 ' 8 Y ' (X & X ) 2 ' , ad by (3), ' (Y & Ȳ ) 2 ' ' Y 2 &.Ȳ 2 ' 020 & 8 2 ' 52 ˆσ 2 ' &2 Û 2 ' &2 (Y & Ȳ ) 2 & ˆβ 2 &2 (X & X ) 2 Hece, ' 52 & (.5)2.8 8&2 ' Y ˆσ \$\$ t \$ \$ ' ' (X & X ) 2 \$F ' (9) Assumig that the coditios of Propositio 6 hold, the ull hypothesis H 0 : β ' 0 ca be tested 0

11 agaist the alterative hypothesis H : β 0 usig the two-sided t-test at say the 5% sigificace level, as follows. Uder the ull hypothesis, (9) is a radom drawig from the t distributio with!2 = 6 degrees of freedom. Look up i the table of the t distributio the value t ( such that for T - t 6, P[ T > t ( ] ' This value is t ( ' The accept the ull hypothesis if &t ( ' &2.447 # ˆtˆβ # ' t (, ad reect the ull hypothesis i favor of the alterative hypothesis if hypothesis ˆt β > t ( ' Thus, i the ice cream example we reect the ull H 0 : β ' 0 because ˆt β ' > ' t (. This test is illustrated i Figure 2 below. The curved lie i Figure 2 is the desity of the t distributio with 6 degrees of freedom. The grey areas are each 0.025, so that the total grey area is Figure 2 Two-sided t-test of H 0 : β ' 0 agaist the alterative hypothesis H : β 0. The ull hypothesis H 0 : β ' 0 ca be tested agaist the alterative hypothesis H : β > 0 at the 5% sigificace level by the right-sided t-test. Now look up i the table of the t distributio the value t ( such that for T - t 6, P[T > t ( ] ' This value correspods to the critical value of the two-sided t-test at the 0% sigificace level: ull hypothesis if hypothesis if t ( '.943. The accept the ˆtˆβ # t ( '.943, ad reect the ull hypothesis i favor of the alterative ˆt β > t ( '.943. Thus, i the ice cream case we reect the ull hypothesis

12 H 0 : β ' 0 i favor of the alterative hypothesis H : β > 0. This right-sided t-test is illustrated i Figure 3 below. Agai, the curved lie i Figure 3 is the desity of the t distributio with 6 degrees of freedom, ad the grey area is Figure 3 Right-sided t-test of H 0 : β ' 0 agaist the alterative hypothesis H : β > 0. If the sample size is large, so that ˆtˆβ - N(0,) if β ' 0, the a alterative way of testig the ull hypothesis β = 0 agaist the alterative hypothesis β 0 is to use the (two-sided) p-value: For example, if def. \$p \$ (' p&value of \$\$) ' P[ U > \$ t \$ ], where U - N(0,). (20) ˆpˆβ < 0.05 we reect the ull hypothesis β = 0 i favor of the alterative hypothesis β 0 at the 5% sigificace level, ad if = 0. The p-value for ˆα is defied ad used similarly. ˆpˆβ \$ 0.05 we accept the ull hypothesis β Although a t-value is a test statistics of the ull hypothesis that the correspodig coefficiet i the regressio model is zero, it is quite easy to rebuild the t-value for testig other ull hypotheses, as follows. Suppose you wat to test the ull hypothesis that is a give umber, for example β 0 '. The β ' β 0, where β 0 2

13 \$\$&\$ 0 \$F \$ \$ ' \$ \$ \$F \$ \$ & \$ 0 \$F \$ \$ ' \$ \$ \$F \$ \$ & \$ 0 \$ \$ \$\$\$F \$ \$ ' \$ \$ \$F \$ \$ & \$ 0 \$\$ ' \$\$&\$ 0 \$\$. \$ t \$ \$, (2) so that by Propositio 5, \$t \$ \$,\$'\$ 0 ' \$\$&\$ 0 \$\$. \$ t \$ \$ - t &2. (22) For example, suppose that i the ice cream case we wat to test the ull hypothesis H 0 : β '. The t \$ \$,\$' \$ ' \$& \$.\$t \$ \$\$ '.5& , (23) which uder the ull hypothesis H 0 : β ' is a radom drawig from the t distributio with 6 degrees of freedom. Note that the value of this test statistic is i the acceptace regios i Figures 2 ad 3. This trick is useful if the ecoometric software you are usig oly reports the t-values but ot the stadard errors. If the stadard errors are reported, you ca compute ˆtˆβ,β'β directly as 0 ˆtˆβ,β'β ' (ˆβ&β Of course, if oly the stadard errors are reported ad ot the t-values you 0 0 )/ˆσˆβ. ca compute the t-value of ˆβ as ˆtˆβ ' ˆβ/ ˆσˆβ. 6. The R 2 The R 2 of a regressio model compares the sum of squared residuals (SSR) of the model with the SSR of a regressio model without regressors: Y ' " % U, ',2,...,. (24) It is easy to verify that the OLS estimator α of α is ust the sample mea of the Y s: #" ' Ȳ ' Y. 3

14 Therefore, the SSR of regressio model (24) is Squares (TSS), is ' (Y &Ȳ )2, which is called the Total Sum of The R 2 is ow defied as: TSS ' (Y & Ȳ ) 2. (26) R 2 def. ' & SSR TSS. (27) The R 2 is always betwee zero ad oe, because SSR # TSS. (Exercise: Why?) If SSR = TSS, so that R 2 = 0, the model (24) explais the depedet variable other words, the explaatory variables s equally well as model (2). I Y i (2) do ot matter: β = 0. The other extreme case is where R 2 =, which correspods to SSR = 0. The the depedet variable X i model (2) is completely explaied by X, without error: / Thus, the R 2 Y α % βx. measures how well the explaatory variables X are able to explai the correspodig depedet variables Y. For example, i the ice cream case, SSR =.5 ad TSS = 52, hece R 2 = Loosely speakig, this meas that about 78% of the variatio of ice cream sales ca be explaied by the variatio i temperature. Y 7. Presetig regressio results Whe you eed to report regressio results you should iclude, ext to the OLS estimates of course, either the correspodig t-values or the stadard errors, the sample size, the stadard error of the residuals (SER), ad the R 2, because this iformatio will eable the reader to udge your results. For example, our ice cream estimatio results should be displayed as either Sales ' &0.25 %.5Temp., ' 8, SER ' , R 2 ' (&0.00) (4.597) or (t&values betwee brackets) 4

15 Sales ' &0.25 %.5Temp., ' 8, SER ' , R 2 ' ( ) ( ) (stadard errors betwee brackets) It is helpful to the reader if you would idicate whether you have displayed the t-values betwee brackets or the stadard errors, but you oly eed to metio this oce. 8. Out-of-sample forecastig The liear regressio model was itroduced as a forecastig scheme. The questio we ow address is: How reliable is a out-of-sample forecast? Cosider the liear regressio model (2), ad suppose we observe X %. The the forecast of is Yˆ % ' ˆα % ˆβ.X %, where the OLS estimators ˆα ad ˆβ are computed o the basis of Y % the observatios for =,2,...,. The actual but ukow value of so that the forecast error is: Y % = α + β.x % % U %, Y % is Y % & \$ Y % ' U % & (\$"&") & (\$\$&\$).X % ' U % & % (X % & X )(X & X ).U ' i' (X i &. (28) X ) 2 See the Appedix for the latter equality. It follows ow from Lemma 3 that uder Assumptios I through V, Y % & ˆ Y % - N[0,σ 2 Y % &Ŷ % ], where F 2 Y % & \$Y % ' F 2 % % (X % & X ) 2 ' (X & X ) 2. (29) See the Appedix. Deotig, \$F 2 Y % & \$Y % ' \$F 2 % % (X % & X ) 2 ' (X & X ) 2, (30) it follows ow similar to Propositio 6 that 5

16 Propositio 8. Uder assumptios I - V, (Y % & ˆ Y % )/ ˆσ Y% &Ŷ % - t &2. This result ca be used to costruct a 95% cofidece iterval, say, of Y %. Look up i the table of the t distributio the critical value t ( of the two-sided t-test with!2 degrees of freedom. The it follows from Propositio 7 that 0.95 ' P[&t ( # (Y % & \$ Y % )/\$F Y% & \$Y % # t ( ] ' P[&t ( \$F Y% & \$Y % # Y % & \$ Y % # t ( \$F Y% & \$Y % ] (3) ' P[ \$ Y % & t ( \$F Y% & \$Y % # Y % # \$ Y % % t ( \$F Y% & \$Y % ] Thus, the 95% cofidece iterval of Y % is [ ˆ Y % & t (ˆσ Y% &Ŷ %, ˆ Y % % t (ˆσ Y% &Ŷ % ]. Observe from (30) that ˆσ Y% &Ŷ icreases with (X ad so does the width of the % % & X ) 2, cofidece iterval. Thus, the father X % is away from X, the more ureliable the forecast Yˆ % of Y % becomes. Also observe from (30) that ˆσ Y% &Ŷ \$ ˆσ, ad that ˆσ gets close to % Y% &Ŷ ˆσ % if is large because lim 64 ' (X & X ) 2 ' Relaxig the o-radom regressor assumptio As said before, the assumptio that the regressors X are o-radom is too strog a assumptio i ecoomics. Therefore, we ow assume that the X s are radom variables. This requires the followig modificatios of the Assumptios I-V: Assumptio I * : The pairs (X,Y ), ',2,3,...,, are idepedet ad idetically distributed. Assumptio II * : The coditioal expectatios E[U X ] are equal to zero: E[U X ] / 0. Assumptio III * : The coditioal expectatios fiite, costat ad equal: assumptio.) E[U 2 X ] / σ 2 < 4. E[U 2 X ] do ot deped o the X 's ad are (This is called the homoscedasticity 6

18 def. Kurtosis ' def. Skewess ' E[U 4 ]/F 4 & 3 ' 0, E[U 3 ] ' 0 (34) Therefore, the ormality coditio ca be tested by testig whether the kurtosis ad the skewess of the model errors are zero, usig the residuals. This is the idea behid the Jarque-Bera 3 ad Kiefer-Salmo 4 tests. Uder the ull hypothesis (34) the test statistic ivolved has a χ 2 2 distributio. Heteroscedasticity 5 does ot hold: We say that the errors U of regressio model (2) are heteroskedastic if assumptio III * E[U 2 X ] ' R(X ) for some fuctio R(.). (35) Heteroscedasticity ofte occurs i practice. It is actually the rule rather tha the exceptio. The mai cosequece of heteroscedasticity is that the coditioal variace formulas i Propositios 2 ad 3 do o loger hold, although the ubiasedess result i Propositio is ot affected by heteroscedasticity. Therefore, the Propositios 4-8 are o loger valid as well. I particular, the coditioal variace of ˆβ [see (60)] uder heteroscedasticity takes the form var(\$\$ X,...,X ) ' E[(\$\$&\$) 2 X,...,X ] ' ' (X & X ) 2 R(X ) ' i' (X i & X ) 2 2. (36) A cure for the heteroscedasticity problem is to replace the stadard error of ˆβ by 3 Jarque, C.M.ad A.K. Bera, (980), "Efficiet Tests for Normality, Homoscedasticity ad Serial Idepedece of Regressio Residuals". Ecoomics Letters 6, 255BB Kiefer, N. ad M. Salmo (983), "Testig Normality i Ecoometric Models", Ecoomic Letters, Also spelled as "Heteroskedasticity." 8

19 #F \$ \$ ' &2 ' (X & X ) 2 \$U 2 ' i' (X i & X ) 2 2. (37) This is kow as the Heteroscedasticity Cosistet (H.C.) stadard error. The H.C. t-value the becomes tˆβ ' ˆβ/ σˆβ. Uder the ull hypothesis β = 0 this t-value is o loger t distributed, but the stadard ormal approximatio remais valid if the sample size is large. A popular test for heteroscedasticity is the Breusch-Paga 6 test. Give that E[U 2 X ] ' g(( 0 % ( X ) for some ukow fuctio g(.). (38) the Breusch-Paga test tests the ull hypothesis agaist the alterative hypothesis H 0 : ( ' 0 ] E[U 2 X ] ' g(( 0 ) ' F 2, say (39) H 0 : ( 0 ] E[U 2 X ] ' g(( 0 %( X ) ' R(X ), say. (40) Uder the ull hypothesis (39) of homoskedasticity the test statistic of the Breusch-Paga test has a χ 2 distributio 7, ad the test is coducted right-sided. 2. How close are OLS estimators? The ice cream data i Table is ot based o ay actual observatios o sales ad temperature; I have picked the umbers for X ad Y quite arbitrarily. Therefore, there is o way to fid out how close the OLS estimates ˆα ' &0.25, ˆβ '.5 are to the ukow parameters α ad β. Actually, we do ot kow either whether the liear regressio model (2) ad its assumptios are applicable to this artificial data. I order to show how well OLS estimators approximate the correspodig parameters I 6 Breusch, T. ad A. Paga (979), "A Simple Test for Heteroscedasticity ad Radom Coefficiet Variatio", Ecoometrica 47, I the multiple regressio case the degrees of freedom is equal to the umber of parameters mius for the itercept. 9

21 Table 2: Artificial regressio estimatio results ˆβ ˆα SER (' ˆσ) R 2 estimate: (t&value): (7.87) (.675) estimate: (t&value): (2.753) (8.237) estimate: (t&value): (47.24) (26.037) Eve for a sample size of = 0 the OLS estimator ˆβ is already pretty close to its true value, ad the same applies to ˆσ, but ˆα is too far away from the true value α =. However, for = 00 the OLS estimators ˆβ ad ˆα deviate oly about ±4% from their true values α = β =, ad deviates about -% from its true value. I the case = 000 these deviatios reduce to about ±2%. The R 2 's are too high, ad oly for = 000 is the R 2 reasoably close to its true value. However, the R 2 is oly a descriptive statistic; it does ot play a role i hypotheses testig, so that the ureliability of the R 2 i small samples is harmless. Notice the quite dramatic icrease of the t-values. Recall that these t-values are the test statistics of the ull hypotheses that the correspodig parameters are zero. Because the true parameters are equal to, what you see i Table 2 is the icrease of the power of the t-test with the sample size. ˆσ 2

22 APPENDIX Proof of (): The first-order coditios for a miimum of Q(ˆα,ˆβ) ' ' (Y & ˆα & ˆβX ) 2 are: dq(\$",\$\$)/d\$" ' 0 ] 2(Y & \$" & \$\$X )(&) ' 0 ] (Y & \$" & \$\$X ) ' 0 ] Y & \$" & (\$\$X ) ' 0 ] Y ' \$" % \$\$ X ' 0 ] Ȳ ' \$" % \$\$. X, (42) ad dq(\$",\$\$)/d\$\$ ' 0 ] ] ] ] ] 2(Y & \$" & \$\$X )(&X ) ' 0 (Y X & \$"X & \$\$X 2 ) ' 0 X Y & \$" X Y ' \$" X & \$\$ X % \$\$ X Y ' \$" X % \$\$ X 2 ' 0 X 2 X 2 (43) where X ' (/)' X ad Ȳ ' (/)' Y are the sample meas of the X 's ad Y 's, respectively. The last equatios i (42) ad (43) are called the ormal equatios: Ȳ ' \$" % \$\$. X, (44) X Y ' \$". X % \$\$ X 2. (45) To solve these ormal equatios, substitute ˆα ' Ȳ & ˆβ. X i (45). The we get 22

23 hece X Y ' (Ȳ & ˆβ X) X % ˆβ X 2 ' Ȳ. X & ˆβ X 2 % ˆβ ' X.Ȳ % ˆβ X 2 X 2 & X 2 X Y & X.Ȳ ' \$\$ X 2 & X 2. (46) Equatio (46) ca also be writte as (X & X)(Y & Ȳ ) ' \$\$ (X & X ) 2, (47) because ad similarly (X & X)(Y & Ȳ ) ' ' ' X Y & X.Y & X.Ȳ % X.Ȳ X Y & X. X Y & X.Ȳ (X & X ) 2 ' X 2 & X 2. Y & Ȳ. X % X.Ȳ (48) (49) Moreover, (X & X)(Y & Ȳ ) ' (X & X)Y & (X & X)Ȳ ' (X & X)Y & ( X & X)Ȳ ' (X & X)Y (50) 23

24 The result () ow follows from (44) ad (46) through (50). Proof of Propositio. Recall from () that \$\$ ' ' (X & X )Y ' (X & X ) 2. (5) Substitute model (2) i (5). The \$\$ ' ' (X & X )("%\$X %U ) ' (X & X ) 2 ' "' (X & X ) % \$' (X & X )X % ' (X & X )U ' (X & X ) 2 ' \$. ' (X & X )X ' (X & X ) 2 % ' (X & X )U ' (X & X ) 2 (52) ' \$ % ' (X & X )U ' (X & X ) 2, where the last step follows from the fact that similar to (50), (X & X) 2 ' (X & X)(X & X) ' (X & X )X. (53) Now take the mathematical expectatio at both sides of (52). The, E[\$\$] ' \$ % E ' (X & X)U ' (X & X) 2 ' \$ % ' (X & X)E(U ) ' \$, ' (X & X) 2 (54) because takig the mathematical expectatio of a costat (β) does ot effect that costat, ad takig the mathematical expectatio of a liear fuctio of radom variables is equal to takig the liear fuctio of the mathematical expectatio of these radom variables. The last coclusio i (54) follows from assumptio II, ad the secod step i (54) ca be take because 24

25 we have assumed that the X 's are o-radom (assumptio IV). Next cosider ˆα. We have already established that ˆα ' Ȳ & ˆβ. X. Substitutig the right- had side of (52) for ˆβ i this equatio yields \$" ' Ȳ & \$ % ' (X & X )U ' (X & X ) 2. X ' Ȳ & \$. X & ' X(X & X )U. ' (X & X ) 2 (55) Substitutig i (55) yields Ȳ ' Y ' (α%βx %U ) ' α % β. X % U \$" ' " % U & ' X(X & X )U ' i' (X i & X ) 2 ' " % X(X & & X ).U ' i' (X i &. X ) 2 (56) Similar as for ˆβ we therefore have: E[\$"] ' " % X(X & & X ) E[U ' i' (X i & ] ' ". X ) 2 (57) This completes the proof of Propositio. Proof of Lemma : We have E ' v U ' w U ' E' i' ' v i w U i U where the last equality i (58) follows from ' ' i' v w F 2, v i w E(U i U ) E(U i U ) ' E(U i )E(U ) ' 0 if i, (58) ' E(U 2 ) ' F2 if i '. (59) 25

26 Proof of Propositio 2: It follows from formula (52) ad Lemma 2 that var(\$\$) ' E[(\$\$&\$) 2 ] ' E X & X ' i' (X i & X ) 2 U 2 ' F 2 X & X ' i' (X i & X ) 2 ' F 2 ' (X & X ) 2 ' i' (X i & X ' ' ) 2 2 F2 (X & X ) 2 ' (X & X ' F2. ) 2 2 ' (X & X ) 2 2 (60) Similarly, it follows from formula (56) ad Lemma 2 that var(\$") ' E[(\$"&") 2 ] ' E & X(X & X) ' i' (X i & X) 2 U 2 ' F 2 & X(X & X) ' i' (X i & X) 2 2 ' F 2 2 & 2 X(X & X) % ' i' (X i & X) 2 X 2 (X & X ) 2 ' i' (X i & X ) 2 2 ' F 2 & 2 X(/)' (X & X) % ' i' (X i & X) 2 X 2 ' (X & X ) 2 ' i' (X i & X ) 2 2 (6) ' F 2 % X 2 ' (X & X ) 2 ' F 2 (/)' (X & X ) 2 % X 2 ' (X & X ) 2 ' F 2 ' X 2 ' (X & X) 2, where the last equality follows from the fact that (/)' (X & X ) 2 ' (/)' X 2 & X 2. Fially, it follows from Lemma ad the formulas (52) ad (56) that 26

27 cov(\$",\$\$) ' E[(\$"&")(\$\$&\$)] ' E ' F 2 & X(X & X) X(X & & X ) (X & X ) ' i' (X i & X ) 2 ' i' (X i & X ) 2 U ' i' (X i & X) 2 X & X ' i' (X i & X ) 2 U (62) which ca be rewritte as (/)' cov(\$",\$\$) ' F 2 (X & X ) & X' (X & X ) 2 ' i' (X i & X ) 2 2 Proof of Propositio 5. Observe first from (44) ad (9) that ' &F 2. X ' (X & X ) 2. (63) so that we ca write \$U ' Ȳ & \$" & \$\$. X ' 0 (64) \$U ' \$U & i' \$U i ' (Y & Ȳ ) & \$\$.(X & X ). (65) Next, observe from (2) that Substitutig the former equatio i (65) yields hece Y & Ȳ ' U & Ū % β.(x & X ), where Ū ' (/)' U. \$U ' (U & Ū ) & (\$\$&\$)(X & X ), (66) \$U 2 ' (U &Ū ) & (\$\$&\$)(X & X ) 2 ' (U &Ū ) 2 & 2(\$\$&\$) (X & X )(U &Ū ) % (\$\$&\$) 2 (X & X ) 2 (67) ' (U &Ū ) 2 & 2(\$\$&\$) (X & X )U % (\$\$&\$) 2 (X & X ) 2, 27

28 where the last equality follows from the fact that ' (X & X )Ū ' 0. It follows from (52), (67) ad the equality ' (U &Ū )2 ' ' U 2 & Ū 2 that \$U 2 ' ' (U &Ū ) 2 & (\$\$&\$) 2 (X & X ) 2 ' U 2 U 2 & ' i' U i 2 & ( \$\$&\$) 2 (X & X ) 2. & Ū 2 & (\$\$&\$) 2 (X & X ) 2. (68) Takig expectatios ad usig Lemma 2 ad Propositio 2 it follows ow from (68) that E[' \$ U 2 ] ' ' E[U 2 ] & E ' i' U 2 i & E( \$\$&\$) 2 ' (X & X ) 2 ' F 2 & F 2 & F 2 ' (&2)F 2. (69) Proof of (3): SSR ' ' ' ' \$U 2 ' (Y & \$" & \$\$.X ) 2 ' (Y & Ȳ ) & \$\$.(X & X ) 2 (Y & Ȳ ) 2 & 2\$\$ (Y & Ȳ ) 2 & \$\$ 2 (X & X ) 2. (Y & (Ȳ&\$\$. X ) & \$\$.X ) 2 (Y & Ȳ )(X & X ) % \$\$ 2 (X & X ) 2 (70) Proof of (28): It follows from (3) that Y % & \$ Y % ' U % & (\$"&") & (\$\$&\$).X % ' U % & X(X & & X ) ' i' (X i & X ) 2.U & X % (X & X ) ' i' (X i & X ) 2 U (7) ' U % & % (X % & X )(X & X ).U ' i' (X i &. X ) 2 28

29 Proof of (29): It follows from (28) ad Lemma 3 that F 2 Y % & \$Y % ' F 2 % % (X % & X )(X & X ) ' i' (X i & X ) 2 2.F 2 ' F 2 % % 2. (X % & X )' (X & X ) ' i' (X i & X ) 2 % (X % & X )2 ' (X & X ) 2 (' i' (X i & X ) 2 ) 2 (72) ' F 2 % % (X % & X ) 2 ' (X & X ) 2. 29

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