# Chapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t )

Save this PDF as:

Size: px
Start display at page:

Download "Chapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t )"

## Transcription

1 Chaper 4 Properies of he Leas Squares Esimaors Assumpions of he Simple Linear Regression Model SR1. SR. y = β 1 + β x + e E(e ) = 0 E[y ] = β 1 + β x SR3. var(e ) = σ = var(y ) SR4. cov(e i, e j ) = cov(y i, y j ) = 0 SR5. SR6. x is no random and akes a leas wo values e ~ N(0, σ ) y ~ N[(β 1 + β x ), σ ] (opional) Undergraduae Economerics, nd Ediion Chaper 4 1

2 4.1 The Leas Squares Esimaors as Random Variables To repea an imporan passage from Chaper 3, when he formulas for b 1 and b, given in Equaion (3.3.8), are aken o be rules ha are used whaever he sample daa urn ou o be, hen b 1 and b are random variables since heir values depend on he random variable y whose values are no known unil he sample is colleced. In his conex we call b 1 and b he leas squares esimaors. When acual sample values, numbers, are subsiued ino he formulas, we obain numbers ha are values of random variables. In his conex we call b 1 and b he leas squares esimaes. Undergraduae Economerics, nd Ediion Chaper 4

3 4. The Sampling Properies of he Leas Squares Esimaors The means (expeced values) and variances of random variables provide informaion abou he locaion and spread of heir probabiliy disribuions (see Chaper.3). As such, he means and variances of b 1 and b provide informaion abou he range of values ha b 1 and b are likely o ake. Knowing his range is imporan, because our objecive is o obain esimaes ha are close o he rue parameer values. Since b 1 and b are random variables, hey may have covariance, and his we will deermine as well. These predaa characerisics of b 1 and b are called sampling properies, because he randomness of he esimaors is brough on by sampling from a populaion. Undergraduae Economerics, nd Ediion Chaper 4 3

4 4..1 The Expeced Values of b 1 and b The leas squares esimaor b of he slope parameer β, based on a sample of T observaions, is b ( ) T x y x y = T x x (3.3.8a) The leas squares esimaor b 1 of he inercep parameer β 1 is b1 = y bx (3.3.8b) are he sample means of he observaions on y where y = y / T and x = x / T and x, respecively. Undergraduae Economerics, nd Ediion Chaper 4 4

5 We begin by rewriing he formula in Equaion (3.3.8a) ino he following one ha is more convenien for heoreical purposes: b = β + we (4..1) where w is a consan (non-random) given by w = x x ( x x) (4..) Since w is a consan, depending only on he values of x, we can find he expeced value of b using he fac ha he expeced value of a sum is he sum of he expeced values (see Chaper.5.1): Undergraduae Economerics, nd Ediion Chaper 4 5

6 ( ) Eb ( ) = Eβ + we = E( β ) + Ewe ( ) = β + we( e) =β [since E( e) = 0] (4..3) When he expeced value of any esimaor of a parameer equals he rue parameer value, hen ha esimaor is unbiased. Since E(b ) = β, he leas squares esimaor b is an unbiased esimaor of β. If many samples of size T are colleced, and he formula (3.3.8a) for b is used o esimae β, hen he average value of he esimaes b obained from all hose samples will be β, if he saisical model assumpions are correc. However, if he assumpions we have made are no correc, hen he leas squares esimaor may no be unbiased. In Equaion (4..3) noe in paricular he role of he assumpions SR1 and SR. The assumpion ha E(e ) = 0, for each and every, makes Undergraduae Economerics, nd Ediion Chaper 4 6

7 we ( e ) = 0 and E(b ) = β. If E(e ) 0, hen E(b ) β. Recall ha e conains, among oher hings, facors affecing y ha are omied from he economic model. If we have omied anyhing ha is imporan, hen we would expec ha E(e ) 0 and E(b ) β. Thus, having an economeric model ha is correcly specified, in he sense ha i includes all relevan explanaory variables, is a mus in order for he leas squares esimaors o be unbiased. The unbiasedness of he esimaor b is an imporan sampling propery. When sampling repeaedly from a populaion, he leas squares esimaor is correc, on average, and his is one desirable propery of an esimaor. This saisical propery by iself does no mean ha b is a good esimaor of β, bu i is par of he sory. The unbiasedness propery depends on having many samples of daa from he same populaion. The fac ha b is unbiased does no imply anyhing abou wha migh happen in jus one sample. An individual esimae (number) b may be near o, or far from β. Since β is never known, we will never know, given one sample, wheher our 7 Undergraduae Economerics, nd Ediion Chaper 4

8 esimae is close o β or no. The leas squares esimaor b 1 of β 1 is also an unbiased esimaor, and E(b 1 ) = β a The Repeaed Sampling Conex To illusrae unbiased esimaion in a slighly differen way, we presen in Table 4.1 leas squares esimaes of he food expendiure model from 10 random samples of size T = 40 from he same populaion. Noe he variabiliy of he leas squares parameer esimaes from sample o sample. This sampling variaion is due o he simple fac ha we obained 40 differen households in each sample, and heir weekly food expendiure varies randomly. Undergraduae Economerics, nd Ediion Chaper 4 8

9 Table 4.1 Leas Squares Esimaes from 10 Random Samples of size T=40 n b 1 b The propery of unbiasedness is abou he average values of b 1 and b if many samples of he same size are drawn from he same populaion. The average value of b 1 in hese 10 samples is b 1 = The average value of b is b = If we ook he averages of esimaes from many samples, hese averages would approach he rue Undergraduae Economerics, nd Ediion Chaper 4 9

10 parameer values β 1 and β. Unbiasedness does no say ha an esimae from any one sample is close o he rue parameer value, and hus we can no say ha an esimae is unbiased. We can say ha he leas squares esimaion procedure (or he leas squares esimaor) is unbiased. 4..1b Derivaion of Equaion 4..1 In his secion we show ha Equaion (4..1) is correc. The firs sep in he conversion of he formula for b ino Equaion (4..1) is o use some ricks involving summaion signs. The firs useful fac is ha 1 ( x x) = x x x + T x = x x T x + T x T = x T x + T x = x T x (4..4a) Undergraduae Economerics, nd Ediion Chaper 4 10

11 Then, saring from Equaion(4..4a), ( x ) ( x x) = x T x = x x x = x (4..4b) T To obain his resul we have used he fac ha x = x / T, so x = T x. The second useful fac is x y ( x x)( y y) xy Tx y xy (4..5) T = = This resul is proven in a similar manner by using Equaion (4..4b). Undergraduae Economerics, nd Ediion Chaper 4 11

12 If he numeraor and denominaor of b in Equaion (3.3.8a) are divided by T, hen using Equaions (4..4) and (4..5) we can rewrie b in deviaion from he mean form as b = ( x x)( y y) ( x x) (4..6) This formula for b is one ha you should remember, as we will use i ime and ime again in he nex few chapers. Is primary advanage is is heoreical usefulness. The sum of any variable abou is average is zero, ha is, ( x x) = 0 (4..7) Then, he formula for b becomes Undergraduae Economerics, nd Ediion Chaper 4 1

13 b ( x x)( y y) ( x x) y y ( x x) = = ( x x) ( x x) ( x x) y ( x x) = = y = ( x x) ( x x) w y (4..8) where w is he consan given in Equaion (4..). To obain Equaion (4..1), replace y by y = β 1 + β x + e and simplify: (4..9a) b = w y = w( β +β x + e ) =β w +β wx + we 1 1 Undergraduae Economerics, nd Ediion Chaper 4 13

14 Firs, w = 0, his eliminaes he erm β 1 w. Secondly, wx = 1 (by using Equaion (4..4b)), so β wx =β, and (4..9a) simplifies o Equaion (4..1), which is wha we waned o show. b = β + we (4..9b) The erm w = 0, because ( x x) 1 w = ( ) 0 = x x = ( using ( x x) = 0) ( x x) ( x x) To show ha wx = 1 we again use ( x x) = 0 ( x x) is Undergraduae Economerics, nd Ediion Chaper 4. Anoher expression for 14

15 ( x x) = ( x x)( x x) = ( x x) x x ( x x) = ( x x) x Consequenly, wx ( x x) x ( x x) x = = = 1 ( x x) ( x x) x 4.. The Variances and Covariance of b 1 and b The variance of he random variable b is he average of he squared disances beween he values of he random variable and is mean, which we now know is E(b ) = β. The variance (Chaper.3.4) of b is defined as Undergraduae Economerics, nd Ediion Chaper 4 15

16 var(b ) = E[b E(b )] I measures he spread of he probabiliy disribuion of b. In Figure 4.1 are graphs of wo possible probabiliy disribuion of b, f 1 (b ) and f (b ), ha have he same mean value bu differen variances. The probabiliy densiy funcion f (b ) has a smaller variance han he probabiliy densiy funcion f 1 (b ). Given a choice, we are ineresed in esimaor precision and would prefer ha b have he probabiliy disribuion f (b ) raher han f 1 (b ). Wih he disribuion f (b ) he probabiliy is more concenraed around he rue parameer value β, giving, relaive o f 1 (b ), a higher probabiliy of geing an esimae ha is close o β. Remember, geing an esimae close o β is our objecive. The variance of an esimaor measures he precision of he esimaor in he sense ha i ells us how much he esimaes produced by ha esimaor can vary from sample o sample as illusraed in Table 4.1. Consequenly, we ofen refer o he sampling Undergraduae Economerics, nd Ediion Chaper 4 16

17 variance or sampling precision of an esimaor. The lower he variance of an esimaor, he greaer he sampling precision of ha esimaor. One esimaor is more precise han anoher esimaor if is sampling variance is less han ha of he oher esimaor. If he regression model assumpions SR1-SR5 are correc (SR6 is no required), hen he variances and covariance of b 1 and b are: Undergraduae Economerics, nd Ediion Chaper 4 17

18 x var( b1 ) =σ T( x x) var( b ) = σ ( x x) (4..10) x cov( b1, b) =σ ( x x) Le us consider he facors ha affec he variances and covariance in Equaion (4..10). 1. The variance of he random error erm, σ, appears in each of he expressions. I reflecs he dispersion of he values y abou heir mean E(y). The greaer he variance σ, he greaer is he dispersion, and he greaer he uncerainy abou Undergraduae Economerics, nd Ediion Chaper 4 18

19 where he values of y fall relaive o heir mean E(y). The informaion we have abou β 1 and β is less precise he larger is σ. The larger he variance erm σ, he greaer he uncerainy here is in he saisical model, and he larger he variances and covariance of he leas squares esimaors.. The sum of squares of he values of x abou heir sample mean, ( x x), appears in each of he variances and in he covariance. This expression measures how spread ou abou heir mean are he sample values of he independen or explanaory variable x. The more hey are spread ou, he larger he sum of squares. The less hey are spread ou he smaller he sum of squares. The larger he sum of squares, ( x x), he smaller he variance of leas squares esimaors and he more precisely we can esimae he unknown parameers. The inuiion behind his is demonsraed in Figure 4.. On he righ, in panel (b), is a daa scaer in which he values of x are widely spread ou along he x-axis. In panel (a) he daa are Undergraduae Economerics, nd Ediion Chaper 4 19

20 bunched. The daa in panel (b) do a beer job of deermining where he leas squares line mus fall, because hey are more spread ou along he x-axis. 3. The larger he sample size T he smaller he variances and covariance of he leas squares esimaors; i is beer o have more sample daa han less. The sample size T appears in each of he variances and covariance because each of he sums consiss of T erms. Also, T appears explicily in var(b 1 ). The sum of squares erm ( x x) ges larger and larger as T increases because each of he erms in he sum is posiive or zero (being zero if x happens o equal is sample mean value for an observaion). Consequenly, as T ges larger, boh var(b ) and cov(b 1, b ) ge smaller, since he sum of squares appears in heir denominaor. The sums in he numeraor and denominaor of var(b 1 ) boh ge larger as T ges larger and offse one anoher, leaving he T in he denominaor as he dominan erm, ensuring ha var(b 1 ) also ges smaller as T ges larger. Undergraduae Economerics, nd Ediion Chaper 4 0

21 4. The erm Σx appears in var(b 1 ). The larger his erm is, he larger he variance of he leas squares esimaor b 1. Why is his so? Recall ha he inercep parameer β 1 is he expeced value of y, given ha x = 0. The farher our daa from x = 0 he more difficul i is o inerpre β 1, and he more difficul i is o accuraely esimae β 1. The erm Σx measures he disance of he daa from he origin, x = 0. If he values of x are near zero, hen Σx will be small and his will reduce var(b 1 ). Bu if he values of x are large in magniude, eiher posiive or negaive, he erm Σx will be large and var(b 1 ) will be larger. 5. The sample mean of he x-values appears in cov(b 1, b ). The covariance increases he larger in magniude is he sample mean x, and he covariance has he sign ha is opposie ha of x. The reasoning here can be seen from Figure 4.. In panel (b) he leas squares fied line mus pass hrough he poin of he means. Given a fied line hrough he daa, imagine he effec of increasing he esimaed slope, b. Since he line mus pass hrough he poin of he means, he effec mus be o lower he 1 Undergraduae Economerics, nd Ediion Chaper 4

22 poin where he line his he verical axis, implying a reduced inercep esimae b 1. Thus, when he sample mean is posiive, as shown in Figure 4., here is a negaive covariance beween he leas squares esimaors of he slope and inercep. Deriving he variance of b : The saring poin is Equaion (4..1). var( b ) = var β + we = var we [since β is a consan] = w var( e) [using cov( ei, ej) = 0] = σ w [using var( e) =σ ] = ( ) ( ) σ ( x x) (4..11) Undergraduae Economerics, nd Ediion Chaper 4

23 The very las sep uses he fac ha w = = { ( x x) } ( x x) 1 ( x x) (4..1) Deriving he variance of b 1 : From Equaion (3.3.8b) b= y bx= 1 (β β x ) 1 1 e bx T + + = β + β x+ e b x= β ( b β ) x+ e 1 1 b β ( b β ) x+ e = 1 1 Undergraduae Economerics, nd Ediion Chaper 4 3

24 Since E(b ) = β and E() e = 0, i follows ha Eb ( ) = β (β β ) x+ 0= β Then var( b) = E[( b β )] = E[(( b β ) x+ e) ] = xe[( b β )] + Ee [ ] xe[( b β )] e = x var( b ) + E[ e ] xe[( b β )] e Now Undergraduae Economerics, nd Ediion Chaper 4 4

25 var( b ) = σ ( x ) x and Ee ( ) = E[( 1 e) ] 1 [( ) E e ] T = T = 1 E e + ee T [ cross-produc erms in i j] 1 { E [ e ] E [cross-produc erms in ee i j ]} = + T = 1 ( Tσ ) = 1 σ T T = = (Noe: var( e) [( ( )) ] [ E e E e E e ]) Undergraduae Economerics, nd Ediion Chaper 4 5

26 Finally, E[( b β ) e] = E[( )( 1 )] (from Equaion (4..9b)) we e T = E 1 we + ee T = 1 T = 1 σ T [ ( cross-produc erms in i j)] ( ) we e w = 0 ( w = 0) Therefore, Undergraduae Economerics, nd Ediion Chaper 4 6

27 var( b) = x E[( b β )] + E[ e ] 1 = σ x + σ ( x x) T = = = Tx + ( x x) σ [ ] T ( x ) x Tx + x Tx σ [ ] T ( x ) x x σ [ ] T ( x ) x Undergraduae Economerics, nd Ediion Chaper 4 7

28 Deriving he covariance of b 1 and b : cov( b, b ) = E[( b β )( b β )] = E[( e ( b β ))( x b β )] = Eeb xe b [ ( β )] [( β )] = σ x (Noe: Eb ( β ) = 0) ( x x) 4..3 Linear Esimaors The leas squares esimaor b is a weighed sum of he observaions y, b = w y [see Equaion (4..8)]. In mahemaics weighed sums like his are called linear Undergraduae Economerics, nd Ediion Chaper 4 8

29 combinaions of he y ; consequenly, saisicians call esimaors like b, ha are linear combinaions of an observable random variable, linear esimaors. Puing ogeher wha we know so far, we can describe b as a linear, unbiased esimaor of β, wih a variance given in Equaion (4..10). Similarly, b 1 can be described as a linear, unbiased esimaor of β 1, wih a variance given in Equaion (4..10). Undergraduae Economerics, nd Ediion Chaper 4 9

30 4.3 The Gauss-Markov Theorem Gauss-Markov Theorem: Under he assumpions SR1-SR5 of he linear regression model he esimaors b 1 and b have he smalles variance of all linear and unbiased esimaors of β 1 and β. They are he Bes Linear Unbiased Esimaors (BLUE) of β 1 and β. Le us clarify wha he Gauss-Markov heorem does, and does no, say. 1. The esimaors b 1 and b are bes when compared o similar esimaors, hose ha are linear and unbiased. The Theorem does no say ha b 1 and b are he bes of all possible esimaors.. The esimaors b 1 and b are bes wihin heir class because hey have he minimum variance. Undergraduae Economerics, nd Ediion Chaper 4 30

31 3. In order for he Gauss-Markov Theorem o hold, he assumpions (SR1-SR5) mus be rue. If any of he assumpions 1-5 are no rue, hen b 1 and b are no he bes linear unbiased esimaors of β 1 and β. 4. The Gauss-Markov Theorem does no depend on he assumpion of normaliy (assumpion SR6). 5. In he simple linear regression model, if we wan o use a linear and unbiased esimaor, hen we have o do no more searching. The esimaors b 1 and b are he ones o use. 6. The Gauss-Markov heorem applies o he leas squares esimaors. I does no apply o he leas squares esimaes from a single sample. Proof of he Gauss-Markov Theorem: Le b = k y (where he k are consans) be any oher linear esimaor of β. * Suppose ha k = w + c, where c is anoher consan and w is given in Equaion Undergraduae Economerics, nd Ediion Chaper 4 31

32 (4..). While his is ricky, i is legal, since for any k ha someone migh choose we can find c. Ino his new esimaor subsiue y and simplify, using he properies of w, w = 0 and w x = 1. Then i can be shown ha b = k y = ( w + c ) y = ( w + c )( β +β x + e ) * 1 = ( w + c ) β + ( w + c ) β x + ( w + c ) e 1 =β w +β c +β wx +β c x + ( w + c ) e 1 1 (4.3.1) =β c +β +β c x + ( w + c ) e 1 Undergraduae Economerics, nd Ediion Chaper 4 3

33 Take he mahemaical expecaion of he las line in Equaion (4.3.1), using he properies of expecaion (see Chaper.5.1) and he assumpion ha E(e ) = 0. Eb ( ) =β c+β +β cx+ ( w+ c) Ee ( ) * 1 =β c +β +β c x 1 (4.3.) In order for he linear esimaor b = k y o be unbiased, i mus be rue ha * c = 0 and cx = 0 (4.3.3) Undergraduae Economerics, nd Ediion Chaper 4 33

34 These condiions mus hold in order for b * = ky o be in he class of linear and unbiased esimaors. So we will assume he condiions (4.3.3) hold and use hem o simplify expression (4.3.1): (4.3.4) * b = ky =β + ( w + c) e We can now find he variance of he linear unbiased esimaor b * following he seps in Equaion (4..11) and using he addiional fac ha c( x x) 1 x cw = 0 = cx c = ( x x) ( x x) ( x x) (The las sep is from condiions (4.3.3)) Undergraduae Economerics, nd Ediion Chaper 4 34

35 Use he properies of variance o obain: * ( ) var( b ) = var β + ( w + c ) e = ( w + c ) var( e ) w c w c cw =σ ( + ) =σ +σ (Noe: σ = 0) = var( b ) +σ c (4.3.5) var( b ) since c 0 (and 0) σ > The las line of Equaion (4.3.5) esablishes ha, for he family of linear and unbiased esimaors * b, each of he alernaive esimaors has variance ha is greaer han or equal o ha of he leas squares esimaor b. The only ime ha var( b* ) = var( b ) is Undergraduae Economerics, nd Ediion Chaper 4 35

36 when all he c = 0, in which case b * = b. Thus, here is no oher linear and unbiased esimaor of β ha is beer han b, which proves he Gauss-Markov heorem. Undergraduae Economerics, nd Ediion Chaper 4 36

37 4.4 The Probabiliy Disribuion of he Leas Squares Esimaors If we make he normaliy assumpion, ha he random errors e are normally disribued wih mean 0 and variance σ, hen he probabiliy disribuions of he leas squares esimaors are also normal. b b 1 ~ N β1, T ( x x) ~ N β, σ σ ( x x) x (4.4.1) This conclusion is obained in wo seps. Firs, based on assumpion SR1, if e is normal, hen so is y. Second, he leas squares esimaors are linear esimaor, of he 37 Undergraduae Economerics, nd Ediion Chaper 4

38 erm b = w y, and weighed sums of normal random variables, using Equaion (.6.4), are normally disribued hemselves. Consequenly, if we make he normaliy assumpion, assumpion SR6 abou he error erm, hen he leas squares esimaors are normally disribued. If assumpions SR1-SR5 hold, and if he sample size T is sufficienly large, hen he leas squares esimaors have a disribuion ha approximaes he normal disribuions shown in Equaion (4.4.1). Undergraduae Economerics, nd Ediion Chaper 4 38

39 4.5 Esimaing he Variance of he Error Term The variance of he random variable e is he one unknown parameer of he simple linear regression model ha remains o be esimaed. The variance of he random variable e (see Chaper.3.4) is var( e) E[( e E( e)) ] E( e ) =σ = = (4.5.1) if he assumpion E(e ) = 0 is correc. Since he expecaion is an average value we migh consider esimaing σ as he average of he squared errors, e ˆ σ = (4.5.) T Undergraduae Economerics, nd Ediion Chaper 4 39

40 The formula in Equaion (4.5.) is unforunaely of no use, since he random error e are unobservable! While he random errors hemselves are unknown, we do have an analogue o hem, namely, he leas squares residuals. Recall ha he random errors are e = y β 1 β x and he leas squares residuals are obained by replacing he unknown parameers by heir leas squares esimaors, e = y b b x ˆ 1 I seems reasonable o replace he random errors e in Equaion (4.5.) by heir analogues, he leas squares residuals, o obain Undergraduae Economerics, nd Ediion Chaper 4 40

41 ˆ e ˆ σ = (4.5.3) T Unforunaely, he esimaor in Equaion (4.5.3) is a biased esimaor of σ. Happily, here is a simple modificaion ha produces an unbiased esimaor, and ha is ˆ ˆ e σ = (4.5.4) T The ha is subraced in he denominaor is he number of regression parameers (β 1, β ) in he model. The reason ha he sum of he squared residuals is divided by T is ha while here are T daa poins or observaions, he esimaion of he inercep and slope pus wo consrains on he daa. This leaves T unconsrained Undergraduae Economerics, nd Ediion Chaper 4 41

42 observaions wih which o esimae he residual variance. This subracion makes he esimaor ˆσ unbiased, so ha ˆ E( σ ) =σ (4.5.5) Consequenly, before he daa are obained, we have an unbiased esimaion procedure for he variance of he error erm, σ, a our disposal Esimaing he Variances and Covariances of he Leas Squares Esimaors Replace he unknown error variance σ in Equaion (4..10) by is esimaor o obain: Undergraduae Economerics, nd Ediion Chaper 4 4

43 x var(? b ˆ 1) =σ, se( b1) = var( b1) T ( x x) σˆ var(? b ) =, se( b ) = var( b ) ( x x) x cov( ˆ b ˆ 1, b) =σ ( x x) (4.5.6) Therefore, having an unbiased esimaor of he error variance, we can esimae he variances of he leas squares esimaors b 1 and b, and he covariance beween hem. The square roos of he esimaed variances, se(b 1 ) and se(b ), are he sandard errors of b 1 and b. Undergraduae Economerics, nd Ediion Chaper 4 43

44 4.5. The Esimaed Variances and Covariances for he Food Expendiure Example The leas squares esimaes of he parameers in he food expendiure model are given in Chaper In order o esimae he variance and covariance of he leas squares esimaors, we mus compue he leas squares residuals and calculae he esimae of he error variance in Equaion (4.6.4). In Table 4. are he leas squares residuals for he firs five households in Table 3.1. Table 4. Leas Squares Residuals for Food Expendiure Daa ŷ= b + b x?e = y y y Undergraduae Economerics, nd Ediion Chaper 4 44

45 Using he residuals for all T = 40 observaions, we esimae he error variance o be e ˆ σ ˆ = = = T 38 The esimaed variances, covariances and corresponding sandard errors are x var( ˆ b ˆ 1) =σ = = T ( x x) 40(153463) se( b) = var( ˆ b) = = x x ˆ σ var( ˆ b ) = = = ( ) Undergraduae Economerics, nd Ediion Chaper 4 45

46 se( b ) = var( ˆ b ) = = x 698 cov( ˆ b ˆ 1, b) =σ = = ( x x) Undergraduae Economerics, nd Ediion Chaper 4 46

47 4.5.3 Sample Compuer Oupu Dependen Variable: FOODEXP Mehod: Leas Squares Sample: 1 40 Included observaions: 40 Variable Coefficien Sd. Error -Saisic Prob. C INCOME R-squared Mean dependen var Adjused R-squared S.D. dependen var S.E. of regression Akaike info crierion Sum squared resid Schwarz crierion Log likelihood F-saisic Durbin-Wason sa Prob(F-saisic) Table 4.3 EViews Regression Oupu Undergraduae Economerics, nd Ediion Chaper 4 47

48 Dependen Variable: FOODEXP Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model Error C Toal Roo MSE R-square Dep Mean Adj R-sq C.V Parameer Esimaes Parameer Sandard T for H0: Variable DF Esimae Error Parameer=0 Prob > T INTERCEP INCOME Table 4.4 SAS Regression Oupu Undergraduae Economerics, nd Ediion Chaper 4 48

49 VARIANCE OF THE ESTIMATE-SIGMA** = 149. VARIABLE ESTIMATED STANDARD NAME COEFFICIENT ERROR X E-01 CONSTANT Table 4.5 SHAZAM Regression Oupu Covariance of Esimaes COVB INTERCEP X INTERCEP X Table 4.6 SAS Esimaed Covariance Array Appendix Undergraduae Economerics, nd Ediion Chaper 4 49

50 Deriving Equaion (4.5.4) eˆ = y b b x = (β + β x + e) b b x = ( b β ) ( b β ) x + e 1 1 = ( b β ) x e ( b β ) x + e (see derivaion of var( b)) 1 = ( b β )( x x) + ( e e) eˆ = ( b β )( x x) + ( e e) ( b β )( x x)( e e) = ( b β ) ( x x) + ( e e) ( b β )( x x) e + ( b β )( x x) e eˆ = ( b β ) ( x x) + ( e e) ( b β ) ( x xe ) + ( b β ) ( x xe ) = ( b β ) ( x x) + ( e e) ( b β ) ( x x) e ( Q ( x x) e = 0 ) Undergraduae Economerics, nd Ediion Chaper 4 50

51 Noe: ( x x) e b β (from Equaions (4..1) and (4..) = we = ( x ) x ( x x) e = ( b β ) ( x x) Noe: ( e e) = ( e ee+ e ) = e e e + Te = e ( 1 e)( e) T( 1 + e) T T = ( ) 1 ( ) 1 e e e e ( e) T + = T T = e 1 ( e + ee ) = T 1 e ee T i j i j i j T T i j Undergraduae Economerics, nd Ediion Chaper 4 51

52 Now, eˆ = ( b β ) ( x x) + ( e e) = ( b β ) ( x x) + T 1 e ee T T i j i i E[ eˆ ] = E[( b β )] ( x x) + T 1 E[ e ] E( ee ) T T i j i i = E b x x T 1 + E e E ee = T Q [( β )] ( ) [ ] ( ( ) 0) i j = T 1 + T var( b ) ( x ) ( σ ) x T = σ + ( ) ( x ) ( 1)σ (from Equaion (4..11)) ( ) x T x x = σ + ( T 1)σ = ( T )σ Undergraduae Economerics, nd Ediion Chaper 4 5

53 Finally, E? = 1 E e T [σ ] [ ] = 1 ( T )σ = σ T Undergraduae Economerics, nd Ediion Chaper 4 53

54 Exercise Undergraduae Economerics, nd Ediion Chaper 4 54

### Lecture 18. Serial correlation: testing and estimation. Testing for serial correlation

Lecure 8. Serial correlaion: esing and esimaion Tesing for serial correlaion In lecure 6 we used graphical mehods o look for serial/auocorrelaion in he random error erm u. Because we canno observe he u

### Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

### Graphing the Von Bertalanffy Growth Equation

file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

### 4. The Poisson Distribution

Virual Laboraories > 13. The Poisson Process > 1 2 3 4 5 6 7 4. The Poisson Disribuion The Probabiliy Densiy Funcion We have shown ha he k h arrival ime in he Poisson process has he gamma probabiliy densiy

### INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

### Chapter 7: Estimating the Variance of an Estimate s Probability Distribution

Chaper 7: Esimaing he Variance of an Esimae s Probabiliy Disribuion Chaper 7 Ouline Review o Clin s Assignmen o General Properies of he Ordinary Leas Squares (OLS) Esimaion Procedure o Imporance of he

### Newton's second law in action

Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In

### 2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity

.6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This

### Research Question Is the average body temperature of healthy adults 98.6 F? Introduction to Hypothesis Testing. Statistical Hypothesis

Inroducion o Hypohesis Tesing Research Quesion Is he average body emperaure of healhy aduls 98.6 F? HT - 1 HT - 2 Scienific Mehod 1. Sae research hypoheses or quesions. µ = 98.6? 2. Gaher daa or evidence

### ... t t t k kt t. We assume that there are observations covering n periods of time. Autocorrelation (also called

Supplemen 13B: Durbin-Wason Tes for Auocorrelaion Modeling Auocorrelaion Because auocorrelaion is primarily a phenomenon of ime series daa, i is convenien o represen he linear regression model using as

### Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

### Lecture 12 Assumption Violation: Autocorrelation

Major Topics: Definiion Lecure 1 Assumpion Violaion: Auocorrelaion Daa Relaionship Represenaion Problem Deecion Remedy Page 1.1 Our Usual Roadmap Parial View Expansion of Esimae and Tes Model Sep Analyze

### Math 201 Lecture 12: Cauchy-Euler Equations

Mah 20 Lecure 2: Cauchy-Euler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem

### MTH6121 Introduction to Mathematical Finance Lesson 5

26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

### YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

### Renewal processes and Poisson process

CHAPTER 3 Renewal processes and Poisson process 31 Definiion of renewal processes and limi heorems Le ξ 1, ξ 2, be independen and idenically disribued random variables wih P[ξ k > 0] = 1 Define heir parial

### Issues Using OLS with Time Series Data. Time series data NOT randomly sampled in same way as cross sectional each obs not i.i.d

These noes largely concern auocorrelaion Issues Using OLS wih Time Series Daa Recall main poins from Chaper 10: Time series daa NOT randomly sampled in same way as cross secional each obs no i.i.d Why?

### A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

### Hedging with Forwards and Futures

Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures

### Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

### Chapter 7. Response of First-Order RL and RC Circuits

Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

### The Simple Linear Regression Model: Specification and Estimation

Chapter 3 The Simple Linear Regression Model: Specification and Estimation 3.1 An Economic Model Suppose that we are interested in studying the relationship between household income and expenditure on

### Vector Autoregressions (VARs): Operational Perspectives

Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101-115. Macroeconomericians

### Fakultet for informasjonsteknologi, Institutt for matematiske fag

Page 1 of 5 NTNU Noregs eknisk-naurviskaplege universie Fakule for informasjonseknologi, maemaikk og elekroeknikk Insiu for maemaiske fag - English Conac during exam: John Tyssedal 73593534/41645376 Exam

### CHARGE AND DISCHARGE OF A CAPACITOR

REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

### Markov Models and Hidden Markov Models (HMMs)

Markov Models and Hidden Markov Models (HMMs (Following slides are modified from Prof. Claire Cardie s slides and Prof. Raymond Mooney s slides. Some of he graphs are aken from he exbook. Markov Model

### Part 1: White Noise and Moving Average Models

Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical

### PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO

Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,

### Revisions to Nonfarm Payroll Employment: 1964 to 2011

Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm

### Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

### Cointegration: The Engle and Granger approach

Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

### Permutations and Combinations

Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes - ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he k-value for he middle erm, divide

### 23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

### 11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

Inroducion Chaper 14: Dynamic D-S dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuing-edge

### cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

### Chapter 8: Regression with Lagged Explanatory Variables

Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

### Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,

### Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that

Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex

### Kernfachkombinationen: Investmentanalyse. Portfoliomanagement (Volatility Prediction)

Kernfachkombinaionen: Invesmenanalyse Porfoliomanagemen (Volailiy Predicion) O. Univ.-Prof. Dr. Engelber J. Dockner Insiu für Beriebswirschafslehre Universiä Wien A-0 Brünnersrasse 7 Tel.: [43] () 477-3805

### CHAPTER 12: AUTOCORRELATION: WHAT HAPPENS IF THE ERROR TERMS ARE CORRELATED?

Basic Economerics, Gujarai and Porer CHAPTER 1: AUTOCORRELATION: WHAT HAPPENS IF THE ERROR TERMS ARE CORRELATED? 1.1 (a) False. The esimaors are unbiased bu hey are no efficien. (b) True. We are sill reaining

### Acceleration Lab Teacher s Guide

Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion

### A Mathematical Description of MOSFET Behavior

10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical

### House Price Index (HPI)

House Price Index (HPI) The price index of second hand houses in Colombia (HPI), regisers annually and quarerly he evoluion of prices of his ype of dwelling. The calculaion is based on he repeaed sales

### Chabot College Physics Lab RC Circuits Scott Hildreth

Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard

### 4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

### 23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

### Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

### Journal Of Business & Economics Research September 2005 Volume 3, Number 9

Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy Yi-Kang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo

### Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is \$613.

Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

### Measuring macroeconomic volatility Applications to export revenue data, 1970-2005

FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970-005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a

### Lecture III: Finish Discounted Value Formulation

Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal

### A Robust Exponentially Weighted Moving Average Control Chart for the Process Mean

Journal of Modern Applied Saisical Mehods Volume 5 Issue Aricle --005 A Robus Exponenially Weighed Moving Average Conrol Char for he Process Mean Michael B. C. Khoo Universii Sains, Malaysia, mkbc@usm.my

### Impact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences

S.R. No. 002 10/2015/CEFT Impac of Deb on Primary Defici and GSDP Gap in Odisha: Empirical Evidences 1. Inroducion The excessive pressure of public expendiure over is revenue receip is financed hrough

### Fourier Series Solution of the Heat Equation

Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

### DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows

### HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.

Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can

### Rotational Inertia of a Point Mass

Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha

### TEACHER NOTES HIGH SCHOOL SCIENCE NSPIRED

Radioacive Daing Science Objecives Sudens will discover ha radioacive isoopes decay exponenially. Sudens will discover ha each radioacive isoope has a specific half-life. Sudens will develop mahemaical

### Wages and Unemployment. 1. Pre-Keynesian economics. Equilibrium in all markets, FE of labor in labor market equilibrium. 2. Keynes' revolution

Wages and Unemploymen 1. Pre-Keynesian economics Equilibrium in all markes, FE of labor in labor marke equilibrium 2. Keynes' revoluion 3. Inflaion His ideas developed in he hisorical conex of pos-wwi

### The Transport Equation

The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be

### 1. The graph shows the variation with time t of the velocity v of an object.

1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially

### Understanding Sequential Circuit Timing

ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor

### Relative velocity in one dimension

Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

Chaper 8 Copyrigh 1997-2004 Henning Umland All Righs Reserved Rise, Se, Twiligh General Visibiliy For he planning of observaions, i is useful o know he imes during which a cerain body is above he horizon

### Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution

Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are

### ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS

ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,

### Morningstar Investor Return

Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

### Forecasting Malaysian Gold Using. GARCH Model

Applied Mahemaical Sciences, Vol. 7, 2013, no. 58, 2879-2884 HIKARI Ld, www.m-hikari.com Forecasing Malaysian Gold Using GARCH Model Pung Yean Ping 1, Nor Hamizah Miswan 2 and Maizah Hura Ahmad 3 Deparmen

### 3.1. The F distribution [ST&D p. 99]

Topic 3: Fundamenals of analysis of variance "The analysis of variance is more han a echnique for saisical analysis. Once i is undersood, ANOVA is a ool ha can provide an insigh ino he naure of variaion

### Chapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr

Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i

### Stability. Coefficients may change over time. Evolution of the economy Policy changes

Sabiliy Coefficiens may change over ime Evoluion of he economy Policy changes Time Varying Parameers y = α + x β + Coefficiens depend on he ime period If he coefficiens vary randomly and are unpredicable,

### Chapter 8 Student Lecture Notes 8-1

Chaper Suden Lecure Noes - Chaper Goals QM: Business Saisics Chaper Analyzing and Forecasing -Series Daa Afer compleing his chaper, you should be able o: Idenify he componens presen in a ime series Develop

### 11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.

11. Tire pressure. The graph You have a hole in your ire. You pump i up o P=400 kilopascals (kpa) and over he nex few hours i goes down ill he ire is quie fla. Draw wha you hink he graph of ire pressure

### Usefulness of the Forward Curve in Forecasting Oil Prices

Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,

### An empirical analysis about forecasting Tmall air-conditioning sales using time series model Yan Xia

An empirical analysis abou forecasing Tmall air-condiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research

### PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

### Machine Learning in Pairs Trading Strategies

Machine Learning in Pairs Trading Sraegies Yuxing Chen (Joseph) Deparmen of Saisics Sanford Universiy Email: osephc5@sanford.edu Weiluo Ren (David) Deparmen of Mahemaics Sanford Universiy Email: weiluo@sanford.edu

### 9. Capacitor and Resistor Circuits

ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren

### Technical Description of S&P 500 Buy-Write Monthly Index Composition

Technical Descripion of S&P 500 Buy-Wrie Monhly Index Composiion The S&P 500 Buy-Wrie Monhly (BWM) index is a oal reurn index based on wriing he nearby a-he-money S&P 500 call opion agains he S&P 500 index

### RESTRICTIONS IN REGRESSION MODEL

RESTRICTIONS IN REGRESSION MODEL Seema Jaggi and N. Sivaramane IASRI, Library Avenue, New Delhi-11001 seema@iasri.res.in; sivaramane@iasri.res.in Regression analysis is used o esablish a relaionship via

### RC, RL and RLC circuits

Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.

### Section 5.1 The Unit Circle

Secion 5.1 The Uni Circle The Uni Circle EXAMPLE: Show ha he poin, ) is on he uni circle. Soluion: We need o show ha his poin saisfies he equaion of he uni circle, ha is, x +y 1. Since ) ) + 9 + 9 1 P

### Time variant processes in failure probability calculations

Time varian processes in failure probabiliy calculaions A. Vrouwenvelder (TU-Delf/TNO, The Neherlands) 1. Inroducion Acions on srucures as well as srucural properies are usually no consan, bu will vary

### 5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.

5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()

### The Optimal Instrument Rule of Indonesian Monetary Policy

The Opimal Insrumen Rule of Indonesian Moneary Policy Dr. Muliadi Widjaja Dr. Eugenia Mardanugraha Absrac Since 999, according o Law No. 3/999, Bank Indonesia (BI- he Indonesian Cenral Bank) se inflaion

### Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.

### 4. International Parity Conditions

4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

### A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion

### Two Compartment Body Model and V d Terms by Jeff Stark

Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic

### The Torsion of Thin, Open Sections

EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such

### Fair games, and the Martingale (or "Random walk") model of stock prices

Economics 236 Spring 2000 Professor Craine Problem Se 2: Fair games, and he Maringale (or "Random walk") model of sock prices Sephen F LeRoy, 989. Efficien Capial Markes and Maringales, J of Economic Lieraure,27,

### ARCH 2013.1 Proceedings

Aricle from: ARCH 213.1 Proceedings Augus 1-4, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference

### Time Series Analysis Using SAS R Part I The Augmented Dickey-Fuller (ADF) Test

ABSTRACT Time Series Analysis Using SAS R Par I The Augmened Dickey-Fuller (ADF) Tes By Ismail E. Mohamed The purpose of his series of aricles is o discuss SAS programming echniques specifically designed

### ON THURSTONE'S MODEL FOR PAIRED COMPARISONS AND RANKING DATA

ON THUSTONE'S MODEL FO PAIED COMPAISONS AND ANKING DATA Alber Maydeu-Olivares Dep. of Psychology. Universiy of Barcelona. Paseo Valle de Hebrón, 171. 08035 Barcelona (Spain). Summary. We invesigae by means

### RC Circuit and Time Constant

ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisor-capacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he

### Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given

### Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.

Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,

### Advanced time-series analysis (University of Lund, Economic History Department)

Advanced ime-series analysis (Universiy of Lund, Economic Hisory Deparmen) 30 Jan-3 February and 6-30 March 01 Lecure Uni-roo esing and he consequences of non-saionariy on regression analysis..a. Why is