The Multiple Regression Model: Hypothesis Tests and the Use of Nonsample Information

Transcription

1 Chaper 8 The Muliple Regression Model: Hypohesis Tess and he Use of Nonsample Informaion An imporan new developmen ha we encouner in his chaper is using he F- disribuion o simulaneously es a null hypohesis consising of wo or more hypoheses abou he parameers in he muliple regression model. The heories ha economiss develop also someimes provide nonsample informaion ha can be used along wih he informaion in a sample of daa o esimae he parameers of a regression model. A procedure ha combines hese wo ypes of informaion is called resriced leas squares. Slide 8.1

2 I can be a useful echnique when he daa are no informaion-rich, a condiion called collineariy, and he heoreical informaion is good. The resriced leas squares procedure also plays a useful pracical role when esing hypoheses. In his chaper we adop assumpions MR1-MR6, including normaliy. If he errors are no normal, hen he resuls presened in his chaper will hold approximaely if he sample is large. Wha we discover in his chaper is ha a single null hypohesis ha may involve one or more parameers can be esed via a -es or an F-es. Boh are equivalen. A join null hypohesis, ha involves a se of hypoheses, is esed via an F-es. Slide 8.

3 8.1 The F-Tes The F-es for a se of hypoheses is based on a comparison of he sum of squared errors from he original, unresriced muliple regression model o he sum of squared errors from a regression model in which he null hypohesis is assumed o be rue. Consider he Bay Area Rapid Food hamburger chain example where weekly oal revenue of he chain (r) is a funcion of a price index of all producs sold (p) and weekly expendiure on adverising (a). r = β+β p +β a + e (8.1.1) 1 3 Suppose ha we wish o es he hypohesis ha changes in price have no effec on oal revenue agains he alernaive ha price does have an effec. Slide 8.3

4 The null and alernaive hypoheses are: H 0: β = 0 and H1: β 0. The resriced model, ha assumes he null hypohesis is rue, is r = β+β a + e (8.1.) 1 3 The sum of squared errors from equaion (8.1.) will be larger han ha from equaion (8.1.1). The idea of he F-es is ha if hese sums of squared errors are subsanially differen, hen he assumpion ha he null hypohesis is rue has significanly reduced he abiliy of he model o fi he daa, and hus he daa do no suppor he null hypohesis. If he null hypohesis is rue, we expec ha he daa are compaible wih he condiions placed on he parameers. Thus, we expec lile change in he sum of squared errors when he null hypohesis is rue. We call he sum of squared errors in he model ha assumes a null hypohesis o be rue he resriced sum of squared errors, or SSE R Slide 8.4

5 The sum of squared errors from he original model is he unresriced sum of squared errors, or SSE U. I is always rue ha SSE SSE 0. Le J be he number of hypoheses. R U The general F-saisic is given by F = ( R U) SSE ( T K ) SSE SSE J U (8.1.3) If he null hypohesis is rue, hen he saisic F has an F-disribuion wih J numeraor degrees of freedom and T K denominaor degrees of freedom. We rejec he null hypohesis if he value of he F-es saisic becomes oo large. We compare he value of F o a criical value F c, which leaves a probabiliy α in he upper ail of he F-disribuion wih J and T K degrees of freedom. For he unresriced and resriced models in equaions (8.1.1) and (8.1.), respecively, we find Slide 8.5

6 SSEU = SSE = R The F-es saisic is: ( R U) SSE ( T K ) SSE SSE J F = = = 4.33 U ( ) ( 5 3) For he F(1,49 ) disribuion he α =.05 criical value is F c = Since F = 4.33 F c we rejec he null hypohesis and conclude ha price does have a significan effec on oal revenue. F (1,49) The p-value for his es is p = P[ 4.33] =.047, which is less han α =.05, and hus we rejec he null hypohesis on his basis as well. Slide 8.6

7 The p-value can also be obained using modern sofware. Table 8.1 EViews Oupu for Tesing Price Coefficien Null Hypohesis: C()=0 F-saisic Probabiliy When esing one "equaliy" null hypohesis agains a "no equal o" alernaive hypohesis, eiher a -es or an F-es can be used and he oucomes will be idenical. The reason for his is ha here is an exac relaionship beween he - and F- disribuions. The square of a random variable wih df degrees of freedom is an F random variable wih disribuion F( 1,df ). Slide 8.7

8 8.1.1 The F-Disribuion: Theory An F random variable is formed by he raio of wo independen chi-square random variables ha have been divided by heir degrees of freedom. V If 1 ~ χ( m ) and ( ) 1 V ~ χ m and if V1 and V are independen, hen F = V 1 m 1 V m ~ m m F (8.1.4) (. ) 1 The F-disribuion is said o have numeraor degrees of freedom and m1 m denominaor degrees of freedom. The values of and m deermine he shape of he m1 disribuion, which in general looks like Figure 8.1. The range of he random variable is (0, ) and i has a long ail o he righ. Slide 8.8

9 When you ake courses in economeric heory, you prove ha V V SSE SSE σ R U 1 = ~ SSE = χ σ U ~ ( T K) χ (8.1.5) ( J ) (8.1.6) and ha and V are independen. V1 The resul for requires he relevan null hypohesis o be rue; ha for V does no. V1 σ V1 Noe ha cancels when we ake he raio of o V, yielding V J ( R U) ( ) SSE SSE J 1 F = = V ( T K ) SSE U T K (8.1.7) Slide 8.9

10 8. Tesing he Significance of a Model An imporan applicaion of he F-es is for wha is called "esing he overall significance of a model". Consider again he general muliple regression model wih (K 1) explanaory variables and K unknown coefficiens y = β+ x β+ x β+ + x β + e (8..1) K K To examine wheher we have a viable explanaory model, we se up he following null and alernaive hypoheses H H : β = 0, β = 0,, β = 0 : a leas one of he β is nonzero K k (8..) The null hypohesis has K 1 pars, and i is called a join hypohesis. If his null hypohesis is rue, none of he explanaory variables influence y, and hus our model is of lile or no value. Slide 8.10

11 If he alernaive hypohesis H 1 is rue, hen a leas one of he parameers is no zero. The alernaive hypohesis does no indicae, however, which variables hose migh be. Since we are esing wheher or no we have a viable explanaory model, he es for (8.4.) is someimes referred o as a es of he overall significance of he regression model. To es he join null hypohesis H :β =β = =β = 0, which acually is K hypoheses, we will use a es based on he F-disribuion. If he join null hypohesis H : β = 0, β = 0,, β = 0 is rue, hen he resriced model is 0 3 The leas squares esimaor of β 1 in his resriced model is sample mean of he observaions on he dependen variable. y 1 K K = β+ e (8..3) The resriced sum of squared errors from he hypohesis (8..) is b * 1 y = = T y, which is he Slide 8.11

12 SSE = ( y b ) = ( y y) = SST * R 1 In his one case, in which we are esing he null hypohesis ha all he model parameers are zero excep he inercep, he resriced sum of squared errors is he oal sum of squares (SST) from he full unconsrained model. The unresriced sum of squared errors is he sum of squared errors from he unconsrained model, or SSE U = SSE. The number of hypoheses is J = K 1. Thus o es he overall significance of a model he F-es saisic can be modified as ( SST SSE)/( K 1) F = (8..4) SSE /( T K) The calculaed value of his es saisic is compared o a criical value from he F (K-1,T-K) disribuion. To illusrae, we es he overall significance of he regression used o explain he Bay Area Burger s oal revenue. We wan o es wheher he coefficiens of price and of Slide 8.1

13 adverising expendiure are boh zero, agains he alernaive ha a leas one of he coefficiens is no zero. Thus, in he model we wan o es agains he alernaive r = β+β p +β a + e 1 3 H0: β = 0, β 3 = 0 H1: β 0, or β3 0, or boh are nonzero The ingrediens for his es, and he es saisic value iself, are repored in he Analysis of Variance Table repored by mos regression sofware. The SHAZAM oupu for he Bay Area Rapid Food daa appears in Table 8.. From his able, we see ha SSE = SST = and SSE = SSE = R U Slide 8.13

14 Table 8. ANOVA Table obained using SHAZAM ANALYSIS OF VARIANCE - FROM MEAN SS DF MS F REGRESSION ERROR P-VALUE TOTAL The values of Mean Square are he raios of he Sums of Squares values o he degrees of freedom, DF. In urn, he raio of he Mean Squares is he F-value for he es of overall significance of he model. For he Bay Area Burger daa his calculaion is F ( SST SSE) /( K 1) ( ) / = = = = SSE /( T K) /(5 3) Slide 8.14

15 The 5% criical value for he F saisic wih (, 49) degrees of freedom is F c = Since > 3.187, we rejec H 0 and conclude ha he esimaed relaionship is a significan one. Insead of looking up he criical value, we could have made our conclusion based on he p-value 8.3 An Exended Model We have hypohesized r = β+β p +β a + e (8.3.1) 1 3 As he level of adverising expendiure increases, we would expec diminishing reurns o se in. Slide 8.15

16 One way of allowing for diminishing reurns o adverising is o include he squared value of adverising, a, ino he model as anoher explanaory variable, so r = β+β p +β a +β a + e (8.3.) The response of E(r) o a is ΔEr ( ) Er ( ) = =β 3+ β4a Δa a ( p held consan) (8.3.3) We expec ha β 3 > 0. Also, o achieve diminishing reurns he response mus decline as a increases. Tha is, we expec β 4 < 0. The leas squares esimaes, using he daa in Table 7.1, are r ˆ = p +.948a a (6.58) (3.459) (0.786) (0.0361) (s.e.) (R8.4) Anoher 6 weeks of daa were colleced. Slide 8.16

17 Combining all he daa we obain he following leas squares esimaed equaion r ˆ = p a 0.068a (3.74) (1.58) (0.4) (0.0159) (s.e.) (R8.5) The esimaed coefficien of now has he expeced sign. Is -value of = 1.68 a implies ha b4 is significanly differen from zero, using a one-ailed es and α = Tesing Some Economic Hypoheses The Significance of Adverising Our expanded model is r = β+β p +β a +β a + e (8.4.1) How would we es wheher adverising has an effec upon oal revenue? If eiher β 3 or β 4 are no zero hen adverising has an effec upon revenue. Slide 8.17

18 Based on one-ailed -ess we can conclude ha individually, β 3 and β 4, are no zero, and of he correc sign. The join es will use he F-saisic in (8.1.3) o es H : β = 0, β = Compare he unresriced model in equaion o he resriced model, which assumes he null hypohesis is rue. The resriced model is r = β+β p + e (8.4.) 1 The elemens of he es are: 1. The join null hypohesis H0: β 3 = 0, β 4 = 0. The alernaive hypohesis H1: β3 0, or β4 0, or boh are nonzero Slide 8.18

19 ( SSER SSEU)/ J 3. The es saisic is F = SSE /( T K) U where J=, T=78 and K= 4. SSE U = is he sum of squared errors from (8.4.1). SSE R = is he sum of squared errors from (8.4.) ~ ( JT, K ) 4. If he join null hypohesis is rue, hen F F. The criical value Fc comes from he F (,74) disribuion, and for he α =.05 level of significance i is The value of he F-saisic is F = > F c and we rejec he null hypohesis ha boh β 3 = 0 and β 4 = 0 and conclude ha a leas one of hem is no zero, implying ha adverising has a significan effec upon oal revenue The Opimal Level of Adverising From (8.3.3) he marginal benefi from anoher uni of adverising is he increase in oal revenue: ΔEr ( ) Δa ( p held consan) = β + β a 3 4 Slide 8.19

20 Adverising expendiures should be increased o he poin where he marginal benefi of $1 of adverising falls o $1, or where β + β = 1 3 4a Using he leas squares esimaes for β 3 and β 4 in (R8.5) we can esimae he opimal level of adverising from (.068) ˆ = 1 Solving, we obain a ˆ = , which implies ha he opimal weekly adverising expendiure is $44, Suppose ha he franchise managemen, based on experience in oher ciies, hinks ha $44, is oo high, and ha he opimal level of adverising is acually abou $40,000. a Slide 8.0

21 The null hypohesis we wish o es is H : β + β (40) = 1 agains he alernaive ha H1: β 3+ β4(40) 1. The es saisic is ( b3+ 80 b4) 1 = se( b + 80 b ) 3 4 var( 垐 b + 80 b ) = var( b ) + 80 var( 垐 b ) + (80)cov( b, b ) = Then, he calculaed value of he -saisic is = = The criical value for his wo-ailed es comes from he (74) disribuion. A he α=.05 level of significance c = and hus we canno rejec he null hypohesis ha he opimal level of adverising is $40,000 per week. Slide 8.1

22 ( SSER SSEU)/ J Alernaively, using an F-es, he es saisic is F = SSE /( T K) U where J=1, T=78 and K= 4. SSE U = is he sum of squared errors from he full unresriced model in (8.4.1). SSE R is he sum of squared errors from he resriced model in which i is assumed ha he null hypohesis is rue. The resriced model is Rearranging we have ( r a ) =β +β p + (1 80 β ) a +β a + e ( r a) =β 1+β p +β4( a 80 a) + e Slide 8.

23 Esimaing his model by leas squares yields he resriced sum of squared errors SSE R = The calculaed value of he F-saisic is F = ( ) / / 74 =.0637 The criical value F c comes from he F (1,74) disribuion. For α =.05 he criical value is F c = Slide 8.3

24 8.4.3 The Opimal Level of Adverising and Price Weekly oal revenue is expeced o be $175,000 if adverising is $40,000, and p=$. In he conex of our model, Er ( ) =β +β p+β a+β a =β +β () +β (40) +β (40) = We now formulae he wo join hypoheses H0: β 3 + β 4(40) = 1, β 1+ β + 40β β 4 = 175 Because here are J= hypoheses o es joinly we will use an F-es. Slide 8.4

25 ( SSER SSEU)/ J The es saisic is F =, where J=. The compued value of he F- SSEU /( T K) saisic is F=1.75. The criical value for he es comes from he F (,74) disribuion and is F c = Since F < F c we do no rejec he null hypohesis 8.5 The Use of Nonsample Informaion From he heory of consumer choice in microeconomics, we know ha he demand for a good will depend on he price of ha good, on he prices of oher goods, paricularly subsiues and complemens, and on income. In he case of beer, i is reasonable o relae he quaniy demanded (q) o he price of beer (p B ), he price of oher liquor (p L ), he price of all oher remaining goods and services (p R ), and income (m). We wrie his relaionship as Slide 8.5

26 (,,, ) q= f p p p m (8.5.1) B L R We assume he log-log funcional form is appropriae for his demand relaionship ln q= β +β ln p +β ln p +β ln p +β ln m (8.5.) 1 B 3 L 4 R 5 A relevan piece of nonsample informaion is ha economic agens do no suffer from money illusion. Having all prices and income change by he same proporion is equivalen o muliplying each price and income by a consan λ ( B) ( L) ( R) ( ) p p p m ( ) ln q=β 1+βln λ p +β3ln λ p +β4ln λ p +β5ln λm =β +β ln +β ln +β ln +β ln + β +β +β +β lnλ (8.5.3) 1 B 3 L 4 R Slide 8.6

27 For here o be no change in ln(q) when all prices and income go up by he same proporion, i mus be rue ha β +β 3 +β 4 +β 5 = 0 (8.5.4) To inroduce he nonsample informaion, we solve he parameer resricion β +β 3+β 4 +β 5 = 0 for one of he βk s. β4 = β β3 β 5 (8.5.6) Subsiuing his expression ino he original model gives Slide 8.7

28 ( ) ( ln ln ) ( ln ln ) ( ln ln ) ln q =β +β ln p +β ln p + β β β ln p +β ln m + e 1 B 3 L 3 5 R 5 =β +β p p +β p p +β m p + e (8.5.7) 1 B R 3 L R 5 R p B p L m =β 1+β ln +β 3ln +β 5ln + e pr pr pr To ge resriced leas squares esimaes, we apply he leas squares esimaion o he resriced model p B p L m ln qˆ = ln ln ln pr pr pr (3.714) (0.166) (0.84) (0.47) (R8.8) * Le he resriced leas squares esimaes in equaion be denoed as b k. Slide 8.8

29 b = b b b * * * * = ( 1.994) = The resriced leas squares esimaor is biased, and we impose are exacly rue. * ( k) E b β, unless he consrains The second propery of he resriced leas squares esimaor is ha is variance is smaller han he variance of he leas squares esimaor, wheher he consrains imposed are rue or no. By incorporaing he addiional informaion wih he daa, we usually give up unbiasedness in reurn for reduced variances. k Slide 8.9

30 8.6 Model Specificaion Wha are he imporan consideraions when choosing a model? Wha are he consequences of choosing he wrong model? Are here ways of assessing wheher a model is adequae? Omied and Irrelevan Variables Suppose ha, in a paricular indusry, he wage rae of employees, depends on heir experience E and heir moivaion M, W W = β+β E +β M e (8.6.1) Daa on moivaion are no available. So, insead, we esimae he model W = β+β 1 E + v (8.6.) Slide 8.30

31 By esimaing (8.6.) we are imposing he resricion = 0 when i is no rue. β 3 The leas squares esimaor for β 1 and β will generally be biased, alhough i will have lower variance. The consequences of omiing relevan variables may lead you o hink ha a good sraegy is o include as many variables as possible in your model. However i may inflae he variances of your esimaes because of he presence of irrelevan variables. Suppose ha he correc specificaion is Bu we esimae he model W = β+β E +β M + e (8.6.3) 1 3 W = β+β E +β M +β C + e where C is he number of children of he -h employee, and where, in realiy, β 4= 0. Then, is an irrelevan variable. Including i does no make he leas squares esimaor C biased, bu i does mean he variances of, and b will be greaer han hose obained by esimaing he correc model in (8.6.3). b1 b 3 Slide 8.31

32 8.6.1a Omied Variable Bias: A Proof Suppose he rue model is y = β+β 1 x+β 3 h+ e, y =β +β x+ e omiing h from he model. 1, bu we esimae he model Then we use he esimaor So, b wh ( x x)( y y) ( x x) y = = * ( x x) ( x x) where w = =β +β + x x ( x x) we 3 Eb ( ) =β +β wh β * 3 Taking a closer look, we find ha Slide 8.3

33 wh Consequenly, ( x x) h ( x x)( h h) = = ( x x) ( x x) ( x x)( h h) ( T 1) cov( ˆ x, h) = = cov( ˆ x, h) Eb ( ) =β +β β * 3 var( ˆ x ) ( x ) ( 1) var( ˆ x T x) Knowing he sign of β and he sign of he covariance beween x and h ells us he direcion of he bias. While omiing a variable from he regression usually biases he leas squares esimaor, if he sample covariance, or sample correlaion, beween and he omied variable is zero, hen he leas squares esimaor in he misspecified model is sill unbiased. h x Slide 8.33

34 8.6. Tesing for Model Misspecificaion: The RESET Tes The RESET es (Regression Specificaion Error Tes) is designed o deec omied variables and incorrec funcional form. I proceeds as follows. Suppose ha we have specified and esimaed he regression model Le he prediced values of he y be y = β+β x +β x + e (8.6.4) y = b + b x + b x ˆ (8.6.5) Consider he following wo arificial models y = β+β x +β x +γ y ˆ 3 y = β+β x +β x +γ y垐 +γ y + e e (8.6.6) (8.6.7) In (8.6.6) a es for misspecificaion is a es of H : γ H : γ = 0 agains he alernaive 0 1 Slide 8.34

35 In (8.6.7), esing H0: γ 1 =γ = 0 agains H1: γ1 0 or γ 0 is a es for misspecificaion. Rejecion of implies he original model is inadequae and can be improved. A H 0 failure o rejec says he es has no been able o deec any misspecificaion. H 0 Overall, he general philosophy of he es is: If we can significanly improve he model by arificially including powers of he predicions of he model, hen he original model mus have been inadequae. As an example of he es, consider he beer demand example used in Secion 8.5 o illusrae he inclusion of non-sample informaion. ln( q ) =β +β ln( p ) +β ln( p ) +β ln( p ) +β ln( m ) + e (8.6.8) 1 B 3 L 4 R 5 Esimaing his model, and hen augmening i wih squares of he predicions, and squares and cubes of he predicions, yields he RESET es resuls in he op half of Table 8.5. Slide 8.35

36 The F-values are quie small and heir corresponding p-values of 0.93 and 0.70 are well above he convenional significance level of There is no evidence from he RESET es o sugges he log-log model is inadequae. Table 8.5 RESET Tes Resuls for Beer Demand Example Ramsey RESET Tes: LOGLOG Model F-saisic (1 erm) Probabiliy F-saisic ( erms) Probabiliy Ramsey RESET Tes: LINEAR Model F-saisic (1 erm) Probabiliy F-saisic ( erms) Probabiliy Now, suppose ha we had specified a linear model insead of a log-log model. Slide 8.36

37 q = β 1+β pb +β 3pL +β 4pR +β 5m + e (8.6.9) Augmening his model wih he squares and hen he squares and cubes of he predicions q yields he RESET es resuls in he boom half of Table 8.5. The p- ˆ values of and are below 0.05 suggesing he linear model is inadequae. 8.7 Collinear Economic Variables When daa are he resul of an unconrolled experimen many of he economic variables may move ogeher in sysemaic ways. Such variables are said o be collinear, and he problem is labeled collineariy, or mulicollineariy when several variables are involved. Consider he problem faced by Bay Area Rapid Food. Suppose i has been common pracice o coordinae hese wo adverising devices, so ha a he same ime Slide 8.37

38 adverising appears in he newspapers here are flyers disribued conaining coupons for price reducions on hamburgers. I will be difficul for such daa o reveal he separae effecs of he wo ypes of ads. Because he wo ypes of adverising expendiure move ogeher, i may be difficul o sor ou heir separae effecs on oal revenue. Consider a producion relaionship. There are cerain facors of producion (inpus), such as labor and capial, ha are used in relaively fixed proporions. Proporionae relaionships beween variables are he very sor of sysemaic relaionships ha epiomize collineariy. A relaed problem exiss when he values of an explanaory variable do no vary or change much wihin he sample of daa. When an explanaory variable exhibis lile variaion, hen i is difficul o isolae is impac. Slide 8.38

39 8.7.1 The Saisical Consequences of Collineariy 1. Whenever here are one or more exac linear relaionships among he explanaory variables, hen he condiion of exac collineariy, or exac mulicollineariy, exiss. In his case he leas squares esimaor is no defined.. When nearly exac linear dependencies among he explanaory variables exis, some of he variances, sandard errors and covariances of he leas squares esimaors may be large. 3. When esimaor sandard errors are large, i is likely ha he usual -ess will lead o he conclusion ha parameer esimaes are no significanly differen from zero. This oucome occurs despie possibly high R or F-values indicaing significan explanaory power of he model as a whole. 4. Esimaors may be very sensiive o he addiion or deleion of a few observaions, or he deleion of an apparenly insignifican variable. Slide 8.39

40 5. Despie he difficulies in isolaing he effecs of individual variables from such a sample, accurae forecass may sill be possible Idenifying and Miigaing Collineariy One simple way o deec collinear relaionships is o use sample correlaion coefficiens beween pairs of explanaory variables. A rule of humb is ha a correlaion coefficien beween wo explanaory variables greaer han 0.8 or 0.9 indicaes a srong linear associaion and a poenially harmful collinear relaionship. A second simple and effecive procedure for idenifying he presence of collineariy is o esimae so-called auxiliary regressions. In hese leas squares regressions he lef-hand-side variable is one of he explanaory variables, and he righ-hand-side variables are all he remaining explanaory variables. For example, he auxiliary regression for x is x = a x + a x + L + a x + error K K Slide 8.40

41 If he R from his arificial model is high, above.80, he implicaion is ha a large porion of he variaion in x is explained by variaion in he oher explanaory variables. One soluion is o obain more informaion and include i in he analysis. One form he new informaion can ake is more, and beer, sample daa. We may add srucure o he problem by inroducing nonsample informaion in he form of resricions on he parameers. Slide 8.41

42 8.8 Predicion Consider y = β+ x β+ x β+ e (8.8.1) where he e are uncorrelaed random variables wih mean 0 and variance Given a se of values for he explanaory variables, ( 1 x0 x 03 ), he predicion problem is o predic he value of he dependen variable y 0, which is given by σ. y = β+ x β+ x β+ e (8.8.) The random error e 0 we assume o be uncorrelaed wih each of he sample errors e and o have he same mean, 0, and variance, Under hese assumpions, he bes linear unbiased predicor of y 0 is given by σ. Slide 8.4

43 ŷ = b + x b + x b (8.8.3) This predicor is unbiased in he sense ha he average value of he forecas error is zero. Tha is, if f = ( y yˆ ) 0 is he forecas error hen E( f ) = 0. 0 The predicor is bes in ha for any oher linear and unbiased predicor of y 0, he variance of he forecas error is larger han var( f ) var ( y yˆ ) =. 0 0 var( f ) = var[( β +β x +β x + e ) ( b + b x + b x )] = var( e b b x b x ) = var( e ) + var( b) + x var( b ) + x var( b ) + x cov( b, b ) + x cov( b, b ) x x cov( b, b ) (8.8.4) Slide 8.43

44 Each of hese erms involves σ which we replace wih is esimaor ˆσ o obain he esimaed variance of he forecas error var( ˆ f ). The square roo of his quaniy is he sandard error of he forecas, se( f ) = var( ˆ f ). If he random errors e and e 0 are normally disribued, or if he sample is large, hen f y ˆ 0 y0 = se( f ) var ˆ y yˆ ( ) 0 0 ~ ( T K) (8.8.5) A 100(1 α)% inerval predicor for y 0 is yˆ 0 ± cse( f), where c is a criical value from he (T-K) disribuion. Slide 8.44