Lecture 8: MultCollnearty Course Outlne Econometrcs Your questons!? MULTICOLLINEARITY 8 LECTURE 1- What s the nature of IT? - What are ts consequences? 3- How to detect IT? 4- What are the remedal measures? 1 Multcollnearty The nature of Multcollnearty Consequences of Multcollnearty Detecton of Multcollnearty Remedal measures 3 4 The nature of multcollnearty In general: When there are some functonal relatonshps exstng among ndependent varables, that s λ X = 0 or λ 1 X 1 + λ X + λ 3 X 3 + + λ X = 0 Such as 1X 1 + X = 0 X 1 = -X If multcollnearty s perfect, the regresson coeffcents of the X varables, β s, are ndetermnate and ther standard errors, Se(β )s, are nfnte. 5 Example: 3-varable Case: If x 3 = λx, Smlarly If x 3 = λx β β β 3 β 3 Y = β 1 + β X + β 3 X 3 + u (Σyx )(Σx 3 ) - (Σyx 3 )(Σx x 3 ) = (Σx )(Σx 3 ) - (Σx x 3 ) = (Σyx )(λ Σx ) - (λσyx )(λσx x ) (Σx )(λ Σx ) - λ (Σx x ) (Σyx 3 )(Σx ) - (Σyx )(Σx x 3 ) = (Σx )(Σx 3 ) - (Σx x 3 ) (λσyx )(Σx ) - (Σyx )(λσx x ) = (Σx )(λ Σx ) - λ (Σx x ) = 0 0 Indetermnate = 0 0 6 Indetermnate TBD Semester - 014/015 Page 7 of 140
Lecture 8: MultCollnearty Course Outlne Econometrcs If multcollnearty s mperfect, x 3 = λ x + ν (or x 3 = λ 1 + λ x + ν ) where ν s a stochastc error Then the regresson coeffcents, although determnate, possess large standard errors, whch means the coeffcents can be estmated but wth less accuracy. β = (Σyx )(λ Σx + Σν ) - (λ Σyx + Σyν )(λ Σx x + Σx ν ) (Σx )(λ Σx + Σν ) - (λ Σx x + Σx ν ) 0 7 Large varance and covarance of OLS estmators ˆ = σ u σ u var( β) = VIF x (1 r ) x 3 Varance-nflatng factor: VIF 1 = 1 r Hgher par-wse correlaton hgher VIF larger varance where r 3 <=== OLS : X = α1 + α X 3 + v r + ' ' ' 3 <=== OLS : X 3 = α1 + α X v var( ˆ β ) j σ σ = u u = j (1 rj ) x j x VIF j 8 3 Consequences of mperfect multcollnearty 1. Although the estmated coeffcents are BLUE, OLS estmators have large varances and covarances, makng the estmaton wth less accuracy.. The estmaton confdence ntervals tend to be much wder, leadng to accept the zero null hypothess more readly. 3. The t-statstcs of coeffcents tend to be statstcally nsgnfcant. 4. The R can be very hgh. 5. The OLS estmators and ther standard errors can be senstve to small change n the data. Can be detected from regresson results 9 Detecton of Multcollnearty Hgh R but few sgnfcant t ratos Auxlary regressons Varance - nflaton factor - VIF Intercorrelaton Matrx (Evew 4) 10 Other examples: CPI <=> WPI; CD rate <=> TB rate M <=> M3 Sgnfcant Insgnfcant Snce GDP and GNP are hghly related 11 Auxlary regressons Basc dea: Multcollnearty arses because one or more of the regressors are exact or approxmately lnear combnatons of the other regressors regress each X on the remanng X varables and compute the correspondng R each one of these regressons s called an auxlary regresson, auxlary to the man regresson of Y on the X s 1 TBD Semester - 014/015 Page 73 of 140
Lecture 8: MultCollnearty Course Outlne Econometrcs Auxlary regressons Implementaton Step 1: regress the auxlary regresson Step : compute the correspondng R, desgnated as R Step 3: compute F statstcs Rx x x x ( k 1) 1,, 3... k Fstat = F ( 1 Rx, x, x... x )/( n k ) R x 1 Where, x, x3... xk s the coeffcent of determnaton 1 the regresson of varable X on the remanng X varables - k s the number of explanatory varables n the orgnal regresson ncludng the ntercept term. - n s the number of observatons n the orgnal regresson 3 k 13 Auxlary regressons Step4: compare F stat wth F crt (k-1, n-k) If F stat >F crt t means that the partcular X should be dropped from the model and vce versa n whch we may retan that varable n the model. 14 V-I-F Some authors use the VIF as an ndcator of Multcollnearty: The larger s the value of VIF j The more troublesome or collnearty s the varable X j. Accordng to ths,f the VIF of varable exceeds 10 (ths wll happen f R j exceeds 0.9), that varable s sad to be hghly collnear 15 Remedal Measures Y = β 1 + β X + β 3 X 3 + u 1. Utlse a pror nformaton = β1 + β X + 0. 1β X 3 + u gven β = β 1 + β ( X + 0.1X ) + u 3 = 0. 1β 3 = β 1 + βz + u. Combnng cross-sectonal and tme-seres data 3. Droppng a varable(s) and re-specfy the regresson 4. Transformaton of varables: ΔY = β1 + βδx + β3δx 3 + u' () Frst-dfference form Y 1 X () Rato transformaton = β 1( ) + β ( ) + β 3 + u ' X 3 X 3 X 3 5. Addtonal or new data 6. Reducng collnearty n polynomal regresson Y = β 1 + β X + β3x 3 + u' Y = β1 + β X + β3x 3 + u' 7. Do nothng (f the objectve s only for predcton) 16 () Frst-dfference form The basc dea of ths method s that f the relaton holds at tme t,t must also hold at tme t-1. Then we take the frst dfference from the regresson, then run the new regresson whch s not on the orgnal varables. The frst dfference regresson model often reduces the severty of multcollnearty, there s no a pror reason to beleve that ther dfferences wll also be hghly correlated. 17 At tme t At tme t-1 Frst dfference Where () Frst-dfference form Y + u Y t = β1 + βx t + β3x3 t t 1 = β1 + βx t 1 + β3x 3t 1 + ut 1 Y + v t Yt 1 = β1 + β( X t X t 1) + β3( X3 t X3 t 1) t = ut ut 1 v However ths method creates some mperfecton such as the new error term - v - may not satsfy one of the assumptons of the classcal lnear regresson model (CLRM). t t 18 TBD Semester - 014/015 Page 74 of 140
Lecture 9: Heteroskedastcty Course Outlne Econometrcs Heteroscedastcty HETEROSKEDASTICITY 9 LECTURE The nature of Heteroscedastcty Consequences of Heteroscedastcty Detecton of Heteroscedastcty Remedal measures 1 f(y ) Homoscedastcty Case. expendture. x 1 1 =80 x 1 =90 x 1 3 =100 ncome x 1 The probablty densty functon for Y at two levels of famly ncome, X, are dentcal. Y. Var(u ) = E(u )= σ 3 y Homoscedastc pattern of errors 0............................... The scattered ponts spread out qute equally....... x 4 f(y ) Heteroscedastcty Case. x 1 1 x 1 expendture x 1 3 The varance of Y ncreases as famly ncome, X, ncreases. Y.. Var(u ) = E(u )= σ ncome x 1 5 Heteroscedastc pattern of errors y t........................................... 0 The scattered ponts spread out qute unequally x t 6 TBD Semester - 014/015 Page 75 of 140
Lecture 9: Heteroskedastcty Course Outlne Econometrcs Defnton of Heteroscedastcty: Two-varable regresson: Y = β 1 + β X + u β = xy x = k Y = k (β 1 + β X + u ) => β = β + Σk u E(β ) = β unbased Var (β ) = E (β - β ) = E (Σk u ) = E (k 1 u 1 + k u +. + k 1 k u 1 u + ) = k 1 σ 1 + k σ +..+ 0 +... = σ Σ k = σ Σ x (Σ x ) = σ Σ x Var (β ) = Var(u ) = E(u ) = σ σ σ Σ x f σ 1 σ σ 3.e., heteroscedastcty k =0 k X =1 f σ 1 = σ = σ 3 =.e., homoscedastcty 7 Consequences of heteroscedastcty 1. OLS estmators are stll lnear and unbased. Var( β )s are not mnmum. => not the best => not effcency => not blue σ σ 3. Var ( β ) = nstead of Var( β ) = Σx Σ x Σu 4. σ = n-k s based, E(σ ) σ 5. t and F statstcs are unrelable. Y = β 0 + β 1 X + u SEE = σ RSS = Σ u Two-varable case Cannot be mn. 8 Informal Method Detecton of Heteroscedastcty Informal and formal methods 1. Graphcal method : plot the estmated resdual ( u ) or squared (u ) aganst the predcted dependent Varable (Y ) or any ndependent varable(x ). Observe the graph whether there s a systemc pattern as: u Yes, heteroscedastcty exsts 9 Y 10 GRAPHICAL METHOD u no heteroscedastcty u yes u yes Formal Methods u yes Y u yes Y u yes Y 1- Park Test - Goldfeld Quandt Test 3- Whte s Heteroscedastcty Test - No cross terms - Wth cross terms Y Y Y 11 1 TBD Semester - 014/015 Page 76 of 140
Lecture 9: Heteroskedastcty Course Outlne Econometrcs 1- PARK TEST H 0 : No heteroscedastcty exsts.e., Var( u ) = σ (homoscedastcty) H 1 : Yes, heteroscedastcty exsts.e., Var( u ) = σ Park test procedures: 1. Run OLS on regresson: Y = β 1 + β X + u, obtan u. Take square and take log : ln ( u ) 3. Run OLS on regresson: ln( u ) = β 1* + β * ln X + v Suspected varable that causes 4. Use t-test to test H 0 : β * = 0 (Homoscedastcty) heteroscedastcty Example: Studenmund (001), Equaton 10.4, pp.370 Procedure 1 If t * > t c ==> reject H 0 ==> heteroscedastcty exsts If t * < t c ==> not reject H 0 ==> homoscedastcty 13 14 Check the resduals whether they are spreadng out or not? Procedure : Obtan the resduals from prevous regresson, take squares and take logs 15 16 Procedure 3 & 4 Ln( u ) = β 1* + β * ln X + v If t > t c => reject H 0 => heteroscedastcty 17 - The Goldfeld-Quandt Test H 0 : homoscedastcty Var ( u ) = σ H 1 : heteroscedastcty Var ( u ) = σ Goldfeld-Quandt Test procedures: (1) Order or rank the observatons accordng to the values of X, begnnng wth the lowest X value. () Omt c central observatons, where c s specfed a pror, and dvde the remanng (n-c) observaton nto two groups each of (n-c)/ observatons. (3) Run the separate regresson on two sub-samples and obtan the respectve RSS 1 and RSS.. Each RSS has [(n-c)/ - k] df RSS /df (4) Compute the λ-rato: λ = RSS 1 /df (5) Compare the λ and the F c, f λ > F c (0.05, (n-c)/)-k) ==> reject the H 0 18 TBD Semester - 014/015 Page 77 of 140
Lecture 9: Heteroskedastcty Course Outlne Econometrcs Gujarat(003)Table 11.3 Re-order data 3.1- Whte s heteroscedastcty test (no cross terms) (LM test) Run regresson based on ths group of data (n 1 =13), Obtans RSS 1 H 0 : homoscedastcty Var ( u ) = σ H 1 : heteroscedastcty Var ( u ) = σ Test procedures: Omt central observatons (c = 4) (1) Run OLS on regresson: Y = β 1 + β X + β 3 X 3 +...+ β q X q + u, obtan the resduals, u () Run the auxlary regresson: u ˆ X + v = δ1 + δx +... + δqxq + δq+ 1X +... + δq 1 q Run regresson based on ths group of data (n =13), Obtans RSS RSS / df = > RSS / df c λ F19? 0 1 H 0 : δ = δ 3 = = δ q = 0 (3) Compute W (or LM) = n R ~ χ df asy (4) Compare the W and χ df(=q-1) (where the df s #(q) of regressors n ()) f W > χ df ==> reject the Ho Y = β 1 + β X + β 3 X 3 + β 4 X 4 + u 1 W = W > χ (0.05, 6) = 1.59 χ (0.10, 6) = 10.64 reject H o W < χ (0.05, 6) = 1.59 χ (0.10, 6) = 10.64 not reject H o The Whte test for a lnear model The test statstc (nr ) ndcates heteroscedastcty s exsted. 3 After transform the lnear model to a log-log model. The Whte test for a log-log model of the same data 4 The W-statstc (nr ) ndcates heteroscedastcty s not exsted TBD Semester - 014/015 Page 78 of 140
Lecture 9: Heteroskedastcty Course Outlne Econometrcs 3.- Whte s general heteroscedastcty test (wth cross terms) H 0 : homoscedastcty Var ( u ) = σ H 1 : heteroscedastcty Var ( u ) = σ Test procedures: (1) Run OLS on regresson: Y = β 1 + β X + β 3 X 3 + u, obtan the resduals, u () Run the auxlary regresson: u = α 1 + α X + α 3 X 3 + α 4 X + α 5 X 3 + α 6 X X 3 + v (3) Compute W (or LM) = n R ~ χ df asy (4) Compare the W and χ df (where the df s # of regressors n ()) f W > χ df ==> reject the Ho 5 6 Example 8.4 (Wooldrdge, pp.58) For the log-log model W = W > χ (0.05, 9) = 16.9 χ (0.10, 9) = 14.68 reject H o W < χ (0.05, 9) = 16.9 χ (0.10, 9) = 14.68 not reject H o The Whte test for a lnear model The test statstc ndcates heteroscedastcty s exsted. The Whte test for a log-log model The test statstc ndcates heteroscedastcty s not exsted 7 8 σ Remedal Measures When s known: the method of weghted least squares. When σ s unknown - Assumpton 1 - Assumpton - Assumpton 3 - Assumpton 4 General Meanng of Weghted Least Squares (WLS) Suppose : Y = β 1 + β X + β 3 X 3 + u E ( u ) = 0, E ( u u j )= 0 j Var (u ) = σ = σ Z(X ) = σ Z = σ E(Y ) If all Z = 1 (or any constant), homoscedastcty returns. But Z can be any value, and t s the proportonalty factor. In the case of σ was known:to correct the heteroscedastcty Transform the regresson: Y 1 X X 3 u =β 1 + β + β 3 + Z Z Z Z Z If Var(u )=σ Z Then each term dvded by Z 9 => Y * = β 1 X 1* + β X * + β 3 X 3 * + u * 30 TBD Semester - 014/015 Page 79 of 140
Lecture 9: Heteroskedastcty Course Outlne Econometrcs Theoretcally, n the transformed equaton where ε 1 () E ( ) = E (ε ) = 0 Z Z u 1 1 () E ( Z ) = E (u ) = Z σ = σ Z Z u u j 1 () E ( ) = E ( u u j ) = 0 Z Z j Z Z j These three results satsfy the assumptons of classcal OLS, therefore, use the Z as the weght for each regressor can correct the problem of heteroscedastcty, and can obtan the BLUE estmators. Determne to use Z or Z as the weght? 31 When σ s known: the method of weghted least squares: Let s consder the -varable model γ = β 1 + β X + u Dvde both sdes of above model by σ to obtan: Y 1 X u = β1 + β + σ σ σ σ Now, n the model, the error term s realzed by σ, hence, the varance of the error term can be rewrtten as follows: u 1 var = var 1 ( u )= σ const = 1= σ σ σ u 3 When σ s not known If the resdual plot aganst X s as followng : u u Let s consder the two-varable model: Y = β 1 + β X + u We now consder several assumptons about the pattern of heteroscedastcty. Assumpton 1: The error varance s proportonal to X : E( u ) = σ = σ X Dvde both sdes of the orgnal model by X to obtan: Y u = β 1 + β + X X X We obtan the transformed dsturbance term, whch has the varance as follows: u 1 1 var = u = X = = const X var( ) σ σ X X 33 + 0 - X 3 Ths plot suggests a varance s ncreasng proportonal to X 3. The scattered plots spreadng out as nonlnear trumpet pattern. Therefore, we mght expect σ = Z σ Z = X 3 =>Z =X 3 Hence, the transformed equaton becomes Y 1 X X 3 u = β X 1 + β X + β 3 X 3 + 3 X 3 X 3 Ths becomes => Y * = β 1 X 1* + β X * + β 3 + u * the ntercept Where u * satsfes the assumptons of classcal OLS coeffcent 34 X 3 Example: Studenmund (001), Eq. 10.7, pp.373 35 36 TBD Semester - 014/015 Page 80 of 140
Lecture 9: Heteroskedastcty Course Outlne Econometrcs When σ s not known Assumpton : The error s proportonal to X. the square root transformaton: E( u ) = σ = σ X Dvde both sdes of the orgnal model by X to obtan Y β u = + β X + X X X u Now, n the new model, the error term s realzed by X, hence, the varance of the error term can be rewrtten as follows: u 1 1 var = var( u) = σ X = σ = const X X X 37 If the resdual plot aganst X s as followng : u + 0 - u X 3 Ths plot suggests a varance s ncreasng proportonal to X 3. The scattered plots spreadng out as a lnear cone pattern Therefore, we mght expect σ = Z σ h = X 3 => h = X 3 The transformed equaton s Y 1 X X 3 u = β 1 + β + β 3 + X 3 X 3 X 3 X 3 X 3 => Y * = β 1 X 1 * + β X * + β 3 X 3* + u * X 3 38 Smple OLS result :(Gujarat(003), Example 11.10. pp.44) R&D = 19.99 + 0.0319 Sales SEE = 759 t = (0.194) (3.830) R =0.478 C.V. = 0.906 Whte Test for heteroscedastcty W W < χ (0.05, ) = 5.9914 not reject Ho W> χ (0.10, ) = 4.6051 reject Ho 39 40 A bell shape pattern of resduals: Transformaton equatons: 1. Y 1 ( ) = -46.67 X X + 0.0367 X (-0.64) (5.17) C.V. = 0.8195 =>(1) R&D = -46.67 + 0.036 Sales SEE = 7.5 (-0.64) (5.17) R = 0.3648 41. Y 1 X ( ) = β 1 + β X X () R&D = -43.49 + 0.0366 Sales SEE = 0.01 (-1.79) (5.5) R = 0.168 X Compare the C.V. To determne whch weght s approprated C.V. = 0.7467 4 TBD Semester - 014/015 Page 81 of 140
Lecture 9: Heteroskedastcty Course Outlne Econometrcs After transformaton by @sqrt(x), resduals stll spread out wder Y X = β 1 1 X +β X C.V. = 0.819543 44 After transformaton by X, resduals spread out more stable Y X 1 = β 1 + β X C.V. = 0.746745 46 When σ s not known Assumpton 3: the error varance s proportonal to the square of the mean value of Y: E( u [ ) ] ) = σ = σ E( Y Smlar to prevous two cases, dvde both sdes of the orgnal model by E( Y : ) Y β X u = + β + E( Y ) E( Y ) E( Y ) E( Y ) Smlarly, we also obtan the transformed error term s varance s constant. The problem here s E( Y ) = β 1 + βx depends on coeffcent, whch are unknown. Therefore, we may, frst, estmate Yˆ = ˆ β, 1 + ˆ βx whch s an estmator of E(Y ). Then dvde both sdes of orgnal model by estmated value of Y. When σ s not known Assumpton 4: A log transformaton such as lny = β1 + β lnx + u very often reduces here heteroscedastcty when compared wth the orgnal regresson. Why? Because log transformaton compresses the scales n whch the varables are measures, thereby reducng the dfferences among values. The log transformaton also helps coeffcent measure the elastcty of Y wth respect to X more exact. 47 48 TBD Semester - 014/015 Page 8 of 140
Lecture 9: Heteroskedastcty Course Outlne Econometrcs Alternatve remedy of heteroscedastcty: Weghted log-log model Refers to Studenmund (001), Eq.(10.31), pp.375 49 50 W < χ (0.05, 5) = 11.07 not reject H o Concludng Example 51 5 Scatter Plot of Corrupton and GDP per Capta (133 Countres) 60000 50000 PERGDP 40000 30000 0000 χ (0.05, ) = 5.9914 χ (0.10, ) = 4.60517 10000 0 1 3 4 5 6 7 8 9 10 Snce W* > χ reject Ho Heteroscedastcty s exsted CORRUPTION 53 54 TBD Semester - 014/015 Page 83 of 140
Lecture 9: Heteroskedastcty Course Outlne Econometrcs Change the functon form as the log-lnear model χ (0.05, ) = 5.9914 χ (0.10, ) = 4.60517 W* < χ (0.05,) not reject Ho It means there s no more heterscedastcty problem after the change of functonal Form. 55 56 Exercses If after the Whte test, the result stll shows the heteroscedastcty problem, then you should try the weghted least square (WLS) method to remedy t Ex 10.7 Ex 10.9 Ex 11.15 Ex 11.1 57 58 TBD Semester - 014/015 Page 84 of 140
Lecture 10: Autocorrelaton Course Outlne Econometrcs AUTOCORRELATION 10 LECTURE 1 Tme Seres Data Tme seres process of economc varables e.g., GDP, M1, nterest rate, exchange rate, mports, exports, nflaton rate, etc. 3 4 Decomposton of tme seres Statc Models X t Trend Cyclcal or seasonal random Y t = β 1 + β Z t + u t Subscrpt t ndcates tme. The regresson s a contemporaneous relatonshp,.e., how does current Y affected by current Z? Example: Statc Phllps curve model tme 5 nf t = β 1 + β unem t + u t nf: nflaton rate unem: unemployment rate 6 TBD Semester - 014/015 Page 85 of 140
Lecture 10: Autocorrelaton Course Outlne Econometrcs Fnte Dstrbuted Lag effects Effect at tme t Effect at tme t+1 Effect at tme t+. Effect at tme t+q Economc acton at tme t The Dstrbuted Lag Effect Economc acton at tme t The Dstrbuted Lag Effect (wth order q) Effect at tme t-1 Effect at tme t- Effect at tme t-3. Effect at tme t-q Y t =α 1 +δ Z t +u t Y t+1 =α 1 +δ Z t+1 +δ 3 Z t +u t or Y t =α 1 +δ Z t +δ 3 Z t-1 + u t. Y t = α 1 +δ Z t +δ 3 Z t-1 +δ 4 Z t- + +δ q Z t-q +u t Intal state: z t = z t-1 = z t- = c Y =α +δ Z +..+δ Z +u or Y =α +δ Z + +δ Z +u t+q 1 t+q q t t t 1 t q t-q t 7 8 Y = α 1 + δ Z t + δ 3 Z t-1 + δ 4 Z t- + u t Long-run propensty (LRP) = δ + δ 3 + δ 4 Permanent change n Y for 1 unt permanent change n Z. Dstrbuted Lag model n general: Y t = α 1 + δ Z t + δ 3 Z t-1 + + δ q Z t-q THE NATURE OF AUTOCORRELATION + other factors + u t LRP (or long run multpler) = δ + δ 3 +..+ δ q 10 9 10 The nature of problem Smlar to Multcollnearty and Heteroscedastcty problems, the volaton of one of the OLS assumptons gves rse to autocorrelaton. That assumpton states : there s no autocorrelaton or seral correlaton among the dsturbances (error terms, resduals) enterng nto the populaton regresson functon. It can be presented by the formula : cov ( U, U j ) = 0, j When the above assumpton s not satsfed, then comes the so-called Autocorrelaton whch can be performed by cov U, U 0, ( ) j j More detals of related assumpton The Classcal Model assumes that the dsturbance term relatng to any observaton s not nfluenced by the dsturbance term relatng to any other observaton. Ths assumpton s more related to tme-seres data. Examples : o When regressng a model of output dependng on labor and captal nputs, f there s a labor strke affectng output ths quarter, there s no reason to beleve that ths dsrupton wll be carred over to the next quarter. o If output s lower ths quarter, there s no reason to expect t to be lower next quarter 11 1 TBD Semester - 014/015 Page 86 of 140
Lecture 10: Autocorrelaton Course Outlne Econometrcs Assumpton s volated Autocorrelaton The Autocorrelaton s presented by followng to formulas : cov U, U = EU EU U EU = EUU 0, ( ) ( ( ))( ( ) ( ) j j j The above formula means that the values of the error term are not ndependent, that s, that the error n one perod n some way nfluences the error n another perod. Applyng ths phenomenon nto the prevous examples, we can say that, the dsrupton caused by a strke ths quarter may very well affect output next quarter. j j 13 Defnton: Frst-order of Autocorrelaton, AR(1) Y t = β 1 + β X t + u t t = 1,,T If Cov (u t, u s ) = E (u t u s ) 0 where t s and f u t = ρ u t-1 + v t E ( vt ) = 0 where -1 < ρ < 1 (ρ : RHO) var( vt ) = σ v and v t ~ d (0, σ v ) (whte nose) cov( vt, v s ) = 0 Ths scheme s called frst-order autocorrelaton and denotes as AR(1) Autoregressve : The regresson of u t can be explaned by tself lagged one perod. ρ(rho) : the frst-order autocorrelaton coeffcent or coeffcent of autocovarance 14 Example of seral correlaton: Year Consumpton t = β 1 + β Income t + error t 1973 30 30 u 1973..... 1995 558 714 u 1995 1996 699 8 u 1996 1997 881 907 u 1997 1998 95 1003 u 1998 1999 984 1174 u 1999 000 107 146 u 000 Error term represents other factors that affect consumpton TaxRate 1999 TaxRate 000 The year Tax Rate may be determned by year rate TaxRate 000 = ρ TaxRate 1999 + v 000 u t = ρ u t-1 + v t ν t ~ d(0, σ v ) 15 If u t = ρ 1 u t-1 + v t t s AR(1), frst-order autoregressve If u t = ρ 1 u t-1 + ρ u t- + v t t s AR(), second-order autoregressve. If u t = ρ 1 u t-1 + ρ u t- + + ρ n u t-n + v t t s AR(n), n th -order autoregressve Cov (u t u t-1 ) > 0 => 0 < ρ < 1 postve AR(1) Cov (u t u t-1 ) < 0 => -1 < ρ < 0 negatve AR(1) -1 < ρ < 1 Hgh order autocorrelaton 16 0 u 0 u x x x x x x x x x x x x x x x x x x x tme 0 u x x x x x x x x x x tme x x x x tme The current error term tends to have the same sgn as the prevous one. 17 u x x x x x x x x x x x x x x tme The current error term tends to have the opposte sgn from the prevous. u x x x x x 0 x x x x x x x x x x x x x x x x x xx x x tme The current error term tends to be randomly appeared from the prevous. 18 TBD Semester - 014/015 Page 87 of 140
Lecture 10: Autocorrelaton Course Outlne Econometrcs ρ The error term at tme t s a lnear combnaton of the current and past dsturbance. 0 < ρ < 1-1 < ρ < 0 ρ = 1 The further the perod s n the past, the smaller s the weght of that error term (u t-1 ) n determnng u t The past s equal mportance to the current. The Consequences of Seral Correlaton ρ > 1 The past s more mportance than the current. 19 0 The consequences of seral correlaton: 1. The estmated coeffcents are stll unbased. E(β k ) = β k. The varances of the β k s no longer the smallest 3. The standard error of the estmated coeffcent, Se(β k ) becomes large Therefore, when AR(1) s exstng n the regresson, The estmaton wll not be BLUE 1 Two varable regresson model: Y t = β 1 + β X t + u t The OLS estmator of β, Σ x y ===> β = Σ x If E (u t u t-1 ) = 0 then Var (β σ ) = Σ x t If E(u t u t-1 ) 0, and u t = ρu t-1 + v t,then Var (β σ ) AR1 = σ + Σ x ρ t x t+1 + ρ Σx t x t+ Σ x t Σ x t Σ x t Σx t -1 < ρ < 1 If ρ = 0, zero autocorrelaton, than Var(β ) AR1 = Var(β ) If ρ 0, autocorrelaton, than Var(β ) AR1 > Var(β ) +. The AR(1) varance s not the smallest Autoregressve scheme: u t = ρu t-1 + v t ==> u t = ρ[ρu t- + v t-1 ] + v t ==> u t-1 = ρu t- + v t-1 u t = ρ u t- + ρv t-1 + v t ==> u t- = ρu t-3 + v t- => u t = ρ [ρu t-3 + v t- ] + ρv t-1 + v t E(u t u t-1 ) = σ 1 - ρ E(u t u t- ) = ρσ E(u t u t-3 ) = ρ σ. E(u t u t-k ) = ρ k-1 σ u t = ρ 3 u t-3 + ρ v t- + ρv t-1 + v t It means the more perods past, the less effect on current perod 3 How to detect autocorrelaton? 4 TBD Semester - 014/015 Page 88 of 140
Lecture 10: Autocorrelaton Course Outlne Econometrcs Gujarat(003) Table1.4, pp.460 How to detect autocorrelaton? Durbn-Watson d Test The Breusch-Godfrey (BG) test of hgher order autocorrelaton Durbn h-test 5 6 Run OLS: u ˆ ρuˆ + v t = t 1 t and check the t-value of the coeffcent Durbn-Watson Autocorrelaton test From OLS regresson result: where d or DW * = 0.19 (At 5% level of sgnfcance, k = 1, n=40) d L = 1.44 d u = 1.544 Reject H 0 regon H 0 : no autocorrelaton ρ = 0 H 1 : yes, autocorrelaton exsts. or ρ > 0 postve autocorrelaton d L d u DW 0.1904 0.91445 = ˆ ρ 1 = 1 = 0.9385 7 0 1.44 1.544 DW * 0.19 8 Durbn-Watson test OLS : Y = β 1 + β X + + β k X k + u t obtan u t, DW-statstc(d) Assumng AR(1) process: u t = ρu t-1 + v t I. H 0 : ρ = 0 no autocorrelaton -1 < ρ < 1 H 1 : ρ > 0 yes, postve autocorrelaton DW * Compare d * and d L, d u (crtcal values) f d * < d L ==> reject H 0 f d * > d u ==> not reject H 0 f d L d * d u ==> ths test s nconclusve 9 Durbn-Watson d Test (cont ) Example 1 : The DW = 1.8756; N=3; K = excludng the ntercept The crtcal values are D L =1.168 D U =1.534 at 5% In ths case we would not reject the null hypothess of no autocorrelaton snce 1,534 < 1, 8756 < 4-1,534 Example : The DW=0.1380; N=3; K =1 (explanatory varable) The crtcal values are D L =1.37 D U =1.50 at 5% In ths case, we can not reject the hypothess that there s postve seral correlaton n the resduals snce 0.1380 < 1.37 30 TBD Semester - 014/015 Page 89 of 140
Lecture 10: Autocorrelaton Course Outlne Econometrcs Durbn-Watson test(cont.) T (u t -u t-1 ) t= DW = (1 -ρ) T u (d) t t=1 Snce -1 ρ 1 d = (1- ρ) d ==> = 1 - ρ ==> ρ = 1- d Durbn-Watson test(cont.) II. H 0 : ρ =0 no negatve autocorrelaton H 1 : ρ < 0 yes, negatve autocorrelaton we use (4-d) (when d s greater than ) f (4 - d) < d L or 4 - d L < d < 4 ==> reject H 0 f d L (4 - d) d u or 4 - d u > d > d u ==> not reject H 0 mples 0 d 4 f d L (4 - d) d u or 4 - d u d 4 - d L ==> nconclusve 31 3 Durbn-Watson test(cont.) II. H 0 : ρ =0 No autocorrelaton H 1 : ρ 0 two-taled test for auto correlaton ether postve or negatve AR(1) If d < d L ==> reject H 0 or d > 4 - d L If d u < d < 4 - d u ==> not reject H 0 H 0 : ρ = 0 postve autocorrelaton H 1 : ρ > 0 reject H 0 nconclusve not reject not reject H 0 : ρ = 0 negatve autocorrelaton H 1 : ρ < 0 nconclusve reject H 0 If d L d d u or 4 - d u d 4 - d L ==> nconclusve 33 0 d L d u 1.37 1.546 1.55 1.703 0.3 4-d DW u 4-d L 4 (d).45.63 1% & 5%.97.475 Crtcal values 34 For example : UM t = 3.1-0.078 CAP t - 0.146 CAP t-1 + 0.043T t (15.6) (.0) (3.7) (10.3) _ R = 0.78 F = 78.9 σ u = 0.677 RSS = 9.3 DW = 0.3 n = 68 () observed K = 3 (number of ndependent varable) () n = 68, α= 0.01 sgnfcance level 0.05 () d L = 1.55, d u = 1.703 0.05 d L = 1.37, d u = 1.546 0.01 Reject H 0, postve autocorrelaton exsts 35 The assumptons underlyng the d(dw) statstcs : 1. Intercept term must be ncluded n OLS regresson.. X s are nonstochastc 3. Only test AR(1) : u t = ρu t-1 + v t where v t ~ d (0, σ v ) 4. Not nclude the lagged dependent varable, Y t = β 1 + β X t + β 3 X t3 + + β k X tk + γ Y t-1 + u t (autoregressve model) 5. No mssng observaton 1970 Y 100 X 15 1980 35 0 mssng 81 N.A. N.A. 8 N.A. N.A. 93 53 37 94 81 41 95............... 36 TBD Semester - 014/015 Page 90 of 140
Lecture 10: Autocorrelaton Course Outlne Econometrcs Lagged Dependent Varable and Autocorrelaton Y t = β 1 + β X t + β 3 X 3t + + β k X k.t + α 1 Y t-1 +u t Durbn h Test DW statstc wll often be closed to or DW does not converge to (1 -ρ) Durbn-h Test: Compute h * = ρ DW s not relable n 1 - n*var (α 1 ) Compare h * to Z where Z c ~ N (0,1) normal dstrbuton If h * > Z c => reject H 0 : ρ = 0 (no autocorrelaton) 37 38 Breusch-Godfrey (BG) test of hgher-order autocorrelaton or called Durbn s m test (Lagrange Multpler, LM, Test) The Breusch-Godfrey (BG) test of hgher order autocorrelaton 39 Test Procedures: (1) Run OLS and obtan the resduals u t. () Run u t aganst all the regressors n the model plus the addtonal regressors, u t-1, u t-, u t-3,, u t-p. u t = β 1 + β X t + u t-1 + u t- + u t-3 + + u t-p + v Obtan the R value from ths regresson. (3) compute the BG-statstc: (n-p)r (4) compare the BG-statstc to the χ p (p s # of degree-order) (5) If BG > χ p, reject Ho (No autocorrelaton), t means there s a hgher-order autocorrelaton If BG < χ p, not reject Ho, t means there s a no hgher-order autocorrelaton 40 Clck VIEW Compare wth the crtcal values Check on the t-statstcs To see The order of autocorrelaton 41 4 TBD Semester - 014/015 Page 91 of 140
Lecture 10: Autocorrelaton Course Outlne Econometrcs REMEDIAL MEASURES 1. Frst-dfference transformaton. Add T = trend 3. Cochrane-Orcutt Two-step procedure (CORC) 4. Cochrane-Orcutt Iteratve Procedure 5. Generalzed least Squares 6. Durbn s Two-step method 43 Remedy: 1. Frst-dfference transformaton Y t = β 1 + β X t + u t Y t-1 = β 1 + β X t-1 + u t-1 assume ρ = 1 ==> Y t -Y t-1 = β 1 - β 1 + β (X t -X t-1 ) + (u t -u t-1 ) ==> ΔY t = β ΔX t + u t no ntercept Y. Add T = trend t = β 1 + β X t + β 3 T + u t Y t-1 = β 1 + β X t-1 + β 3 (T -1) + ε t-1 ==> (Y t -Y t-1 ) = (β 1 - β 1 ) + β (X t -X t-1 ) + β 3 [T - (T -1)] + (u t -u t-1 ) ==> ΔY t = β ΔX t + β 3 *1 + u t ==> ΔY t = β 1* + β ΔX t +u t If β 1* > 0 => an upward trend n Y (β > 0) 44 3. Cochrane-Orcutt Two-step procedure (CORC) (1). Run OLS on Y t = β 1 + β X t + u t and obtans ε t (). Run OLS on u t = ρ u t-1 + v t Generalzed Least Squares (GLS) and obtans ρ method (3). Use the ρ to transform the varables : Y t* = Y t - ρ Y t-1 X t* = X t - ρ X t-1 -) Y t = β 1 + β X t + u t ρ Y t-1 = β 1 ρ + β ρ X t-1 + ρu t-1 (Y t - ρy t-1 )= β 1 (1-ρ) +β (X t - ρx t-1 ) + (u t -ρu t-1 ) (4). Run OLS on Y t* = β 1* + β * X t* + u t * 45 4. Cochrane-Orcutt Iteratve Procedure (5). If DW test shows that the autocorrelaton stll exstng, than t needs to terate the procedures from (4). Obtans the u * t (6). Run OLS u t* = ρ u t-1* + v t DW ρ (1 - ) and obtans ρ whch s the second-round estmated ρ (7). Use the ρ to transform the varable Y ** t = Y t - ρ Y t-1 Y t = β 1 + β X t + u t X ** t = X t - ρ X t-1 ρ Y t-1 = β 1 ρ + β ρx t-1 + ρu t-1 (8). Run OLS on Y ** t = β ** 1 + β ** X ** t + u ** t Where s (Y t - ρ Y t-1 ) = β 1 (1 - ρ) + β (X t - ρ X t-1 ) + (u t - ρ u t-1 ) 46 Cochrane-Orcutt Iteratve procedure(cont.) (9). Check on the DW 3 -statstc, f the autocorrelaton s stll exstng, than go nto thrd-round procedures and so on. Untl (ρ - ρ < 0.01) the estmated ρ s dffers a lttle. Generalzed least Squares (GLS) 5. Pras-Wnsten transformaton Y t = β 1 + β X t + u t t = 1,,T (1) Assume AR(1) : u t = ρu t-1 + v t -1 < ρ < 1 ρy t-1 = ρβ 1 + ρβ X t-1 + ρu t-1 () (1) - () => (Y t - ρy t-1 ) = β 1 (1 - ρ) + β (X t - ρx t-1 ) + (u t - ρu t-1 ) To avod the loss of the frst observaton of each varable, the frst observaton of Y * and X * should be transformed as : Y t=1* = 1 - ρ (Y t=1 ) X t=1* = 1 - ρ (X t=1 ) but Y t=* = Y t= - ρ Y t=1 ; X t=* = X t= - ρ X t=1 Y t=3* = Y t=3 - ρ Y t= ; X t=3* = X t=3 - ρ X t=.................. Y * t = Y t - ρ Y t-1 ; X * t = X t - ρ X t-1 GLS => Y t* = β 1* + β * X t* + u* t 47 48 TBD Semester - 014/015 Page 9 of 140
Lecture 10: Autocorrelaton Course Outlne Econometrcs 6. Durbn s Two-step method : Y t = β 1 + β X t + u t Snce (Y t - ρy t-1 ) = β 1 (1 - ρ) + β (X t - ρx t-1 ) + V t => Y t = β 1* + β X t - ρβ X t-1 + ρy t-1 + V t I. Run OLS => ths specfcaton Y t = β 1 * + β * X t - β 3* X t-1 + β 4* Y t-1 + v t Obtan II. Transformng the varables : III. Run OLS on model : β 4* as an estmated ρ (RHO) Y t* = Y t - β 4* Y t-1 as Y t* = Y t - ρ Y t-1 and X t* = X t - β 4* X t-1 as X t* = X t - ρ X t-1 Y t* = α 1 + α X t* + u t where α 1 = β 1 (1 - ρ) and α = β Compare 49 Example 50 Example: Gujarat(003) Table 1-4, p.460 Wage(Y t ) = β 1 + β Output(X t )+ u t u t = ρ u t-1 + v t DW (1 - ρ) 0.19 (1-0.9385) 51 ρ 1 - DW 0.914 (1-0.0614) 5 Cochrane-Orcutt Two-step procedure () Runnng the Cochrane-Orcutt teratve procedure n EVIEWS Crtcal values: D u =1.337 D L =1.37 Snce DW > D u No Autocorrelaton After CO-correcton 53 54 TBD Semester - 014/015 Page 93 of 140
Lecture 10: Autocorrelaton Course Outlne Econometrcs Exercses 1.15 1.34 1.35 ρ THE END 55 56 TBD Semester - 014/015 Page 94 of 140
Course Outlne Econometrcs ECONOMETRICS APPENDIX TUTORIAL EXERCICES PROJECT GUIDELINE TBD Semester - 014/015 Page 95 of 140
Appendx 1: Varance Analyss Course Outlne Econometrcs Hypothess Testng One Populaton ANALYSIS OF VARIANCE 1 APPENDIX Z Test (1 & tal) Mean Proporton Varance t Test (1 & tal) Z Test (1 & tal) Test (1 & tal) 1 Hypothess Testng The F test and Analyss of Varance Indep. Z Test (Large sample) Mean t Test (Small sample) Two Populatons Pared t Test (Pared sample) Proporton Z Test Varance F Test -For the comparson of two varances or standard devatons, an F test s used. -F test can only ndcate whether or not a dfference exsts among the means. It can not ndcate where the dfference les. -If the F test ndcates that there s a dfference among the means, other statstcal tests are used to fnd where the dfference exsts. The most commonly used tests are the Scheffe test and the Tukey test. 3 4 The F test If two ndependent samples are selected from two normally dstrbuted populatons n whch the varances are equal ( 1 = ) and f the varances s 1 and s are compared as s 1 /s, the samplng dstrbuton of the varances s called the F dstrbuton. Characterstcs of the F Dstrbuton The values of F cannot be negatve, because varances are always postve or zero. The dstrbuton s postvely skewed The mean value of F s approxmately equal to 1 The F dstrbuton s a famly of curves based upon the degrees of freedom of the varance of the numerator and the degrees of freedom of the varance of the denomnator. 5 6 TBD Semester - 014/015 Page 96 of 140
Appendx 1: Varance Analyss Course Outlne Econometrcs Formula for the F Test s1 F = s Where _ s 1 _ s the larger of the two varances. The F test has two terms for the degrees of freedoms: that the numerator, n1 1, and that of the denomnator, n 1, where n1 s the sample sze from whch the larger varance was obtaned. The larger of the varances s placed n the numerator of the F formula. Testng the dfference between two varances One-Taled Rght One-Taled Left Two-Taled H 0 : 1 H a : 1 > H 0 : 1 H a : 1 H 0 : 1 = H a : 1 7 8 Notes for the Use of the F Test The larger varance should always be desgnated as s 1 and be placed n the numerator of the formula. F = s 1 /s For two-taled test, the value must be dvded by and the crtcal value be placed on the rght sde of the F curve. If the standard devatons nstead of the varances are gven n the problem, they must be squared for the formula for the F test. When the degrees of freedom cannot be found n F Tables, the closest value on the smaller sde should be used. Assumptons for Testng the Dfference Between Two Varances The populatons from whch the samples were obtaned must be normally dstrbuted. The samples must be ndependent of each other. 9 10 The followng steps should be used: Step 1: State the hypotheses, and dentfy the null and alternatve hypotheses Step : Fnd the crtcal value. Step 3: Compute the test value. Step 4: Make the decson. Step 5: Summarze the results. ANOVA When the F test s used to test a hypothess concernng the means of three or more populatons, the technque s called Analyss of Varances ANOVA 11 1 TBD Semester - 014/015 Page 97 of 140
Appendx 1: Varance Analyss Course Outlne Econometrcs Several reasons why the t test should not be done When one s comparng two means at a tme, the rest of the means under study are gnored. The probablty of rejectng the null hypothess when t s true s ncreased. The greater the number of means there are to compare, the greater s the number of t tests that are needed. 13 Assumptons for the F Test for Comparng Three or More Means The populatons from whch the samples were obtaned must be normally or approxmately normally dstrbuted. The samples must be ndependent of each other. The varances of the populatons must be equal. 14 Steps should be taken Fnd the between-group varance Fnd the wthn-group varance Step 1:? Step :? Step 3:? Step 4: Make decson Steps 5: Summarze the results. 15 16 ANOVA Table Example 13-5 Do t? Source Analyss of Varance Summary Table Sum of d.f Mean Squares Square F Source Analyss of Varance Summary Table Sum of d.f Mean Squares Square F Between SSB k 1 MSB Wthn SSW N-k MSW Total Between 160.13 80.07 9.17 Wthn 104.8 1 8.73 Total 64.93 14 17 18 TBD Semester - 014/015 Page 98 of 140
Appendx 1: Varance Analyss Course Outlne Econometrcs TWO-WAY ANALYSIS OF VARIANCE There s only one ndependent varable n one-way analyss of varance. The two-way analyss of varance nvolves two ndependent varables. Assumptons for the Two-Way ANOVA The populatons form whch the samples were obtaned must be normally or approxmately normally dstrbuted. The samples must be ndependent. The varances of the populatons from whch the samples were selected must be equal. The groups must be equal n sample sze. 19 0 HYPOTHESES The two-way ANOVA desgn has several null hypotheses. There s one for each ndependent varable and one for the nteracton. In the plant food-sol type problem, the hypotheses are as follows: Ho: There s no nteracton between the type of plant food used and the type of sol used on the plant growth. H1: There s nteracton between the type of plant food used and the type of sol used on the plant growth. 1 HYPOTHESES Ho: There s no dfference n the means of the heghts of the plants grown usng dfferent plant foods H1: There s dfference n the means of the heghts of the plants grown usng dfferent plant foods Source ANOVA two-way TABLE Analyss of Varance Summary Table Sum of d.f Mean Squares Square A SSA a 1 MSA FA B SSB b - 1 MSB FB A x B SSAxB (a-1).(b-1) MSAxB FAxB Wthn SSW ab(n-1) MSW (error) Total F 3 4 TBD Semester - 014/015 Page 99 of 140
Course Outlne Econometrcs Tutoral #1 1 TBD Semester - 014/015 Page 100 of 140
Course Outlne Econometrcs TBD Semester - 014/015 Page 101 of 140
Course Outlne Econometrcs 1. How to Use the EVews? Tutoral # Some basc steps to entry the EVews: Open EVews Import data Specfy sample range Generate varables Run regresson estmaton. Understand dfferent types of data Tme seres data: Cross-secton data Pooled (Panel) data 3. Computer exercse: Gujarat (003): Queston #1.7 The data presented n Table 1.5 s related to the advertsng budget of 1 frms for 1983 and mllons of mpressons retaned per week by the vewers of the products of these frms. The data are based on a survey of 4000 adults n whch users of the products were asked to cte a commercal they had seen for the product category n the past week. a. Plot mpressons on the vertcal axs and advertsng expendture on the horzontal axs. b. What can you say about the nature of the relatonshp between the two varables? c. Lookng at your graph, do you thnk t pays to advertse? Thnk about all those commercals shown on Super Bowl Sunday or durng the World Seres? d. What s the numercal relatonshp between the two varables? That means you run the OLS regresson as: IMPRESSION S 1 EXPEDITURE u 3 TBD Semester - 014/015 Page 10 of 140
Course Outlne Econometrcs Tutoral # 3 1. Basc Tools of EVIEWS: Workng wth data: Generate a new seres Edtng a seres Seres expressons: Addton (+), subtracton (-), multplcaton (*), dvson (/) and rase to a power (), etc. Seres functons: log, trend, seasonal dummy, square root, exponental, etc Lead, Lags, and Dfferences.. Computer Exercses: Ths s a regresson applcaton of Phllps Curve theory. Usng the past several year data of Hong Kong s CPI and unemployment from the fle named "HK_cp_unemploy.xls". We would try to measure how does the average unemployment rate s related to the deflaton rate. The estmate smple regresson equaton s: Unemployment = 1 + Deflaton (or nflaton) + u 1 (1) (a) Interpret the ntercept n your equaton (1) (b) Interpret the coeffcents n equaton (1). (c) How much of the varaton n dependent varable s explaned by the ndependent varables? Is ths a lot n your opnon? (d) If we add one more varable, say Exports s added nto the equaton as (), whch varable has a greater effect on the dependent varable? Unemployment = 1 + Deflaton + 3 Exports + u () 4 TBD Semester - 014/015 Page 103 of 140
Course Outlne Econometrcs 1. Basc Tools of EVIEWS: Tutoral # 4 Understand the estmated regresson output: Coeffcents (sgn and sze) R-squared Standard Error of the regresson (SEE) Sum of Squared resdual (SSR). Computng Exercse: Gujarat(003): Problem #3.0, pp. 96-98 Table 3.6 gves data on ndexes of output per hour (X) and real compensaton per hour (Y) for the busness and non-farm busness sectors of the US economy for 1959-1997. The base year of the ndexes s 199=100 and the ndexes are seasonally adjusted: () () () (v) Plot Y aganst X for the two sectors separately. What s the economc theory behnd the relatonshp between the two varables? (What s the expected sgn of the β.) Does the scatter graph support the theory? Estmate the OLS regresson of Y on X for the two sectors separately and compare the two results? Interpret the estmated results for the two sectors. 5 TBD Semester - 014/015 Page 104 of 140
Course Outlne Econometrcs Tutoral # 5 1. Computng Exercse: Gujarat (003) #5.9, pp.153-4. Table 5.5 gves data on average publc teacher pay (annual salary n dollars) and spendng on publc schools per pupl (dollars) n 1985 for 50 states and the Dstrct of Columba, USA. To fnd out f there s any relatonshp between teacher s pay and per pupl expendture n publc schools, the followng model was suggested: Pay = β 1 + β Spend + u Where Pay stands for teacher s salary and Spend stands for per pupl expendture. () () () (v) (v) Plot the data and eyeball a regresson lne. Suppose on the bass of a you decde to estmate the above regresson model. Obtan the estmates of the parameters, ther standard errors, R, RSS and ESS. How good s ths regresson result? Interpret the regresson. Does t make economc sense? Does the ntercept make any economc sense? Establsh a 95% confdence nterval for β. Would you reject the hypothess that the true slope coeffcent s 3.0? (.e., H 0 : β = 3.0). 6 TBD Semester - 014/015 Page 105 of 140
Course Outlne Econometrcs 1. Basc Tools n EVIEWS: Generate the Trend varable Generate the log varable Tutoral # 6. Computng Exercse: Gujarat (003) # 7.17 Wldcats actvty: Wldcats are wells drlled to fnd and produce ol and/or gas n an mproved area or to fnd a new reservor n a fled prevously found to be productve of ol or gas or to extend the lmt of a known ol or gas reservor. Table 7.7 gves data on the followng varables: Y = the number of wldcats drlled (Thousands) X = prce at the wellhead n the prevous perod (n constant dollars, 197=100) X3 = domestc output ($mllons of barrels per day) X4 = GNP constant dollars ($bllons) X5 = trend varable See whether the followng model fts that data: Y t 1 X t 3 ln X 3t 4X 4t 5X 5t a. Can you offer an a pror ratonale to ths model? b. Assumng the model s acceptable, estmate the parameters of the model and ther estmated standard errors, and obtan R and R. c. Comment on your results n vew of your pror expectatons. u d. How would you nterpret the coeffcent of lnx 3t n ths model? e. What other specfcaton would you suggest to explan wldcat actvty? Why? f. Whch of the coeffcents are ndvdually statstcally sgnfcant at 5% level? g. What s the overall sgnfcance of the regresson? t Extra exercse: 7.1, 7., 7.7, 7.0. 7 TBD Semester - 014/015 Page 106 of 140
Course Outlne Econometrcs Tutoral # 7 Computng Exercses: Gujarat (003) #7.1, pp. 40 Consder the followng demand functon for money n the Unted States for the perod 1980:1 003:4: Where Y = real GDP 3 M Y r e t 1 t t u t M = real money demand, usng the M defnton of money r = nterest rate To estmate the above demand for money functon, you are gven the data n the fle named tutor06.xls. Note: To convert nomnal quanttes nto real quanttes, dvde M and GDP by CPI or deflator. There s no need to dvde the nterest rate varable by CPI or deflator. Also note that you have gven dfferent nterest rates, a short-term rate s measured by the one-month federal fund rate or the 3-month treasury bll rate, and the long-term rate s measured by the government bond yeld on 10-year treasury bond, both types of nterest rates have been used as n pror emprcal studes. a. Gven the data, estmate the above demand functon. What are the ncome elastcty and nterest rate elastcty of demand for money? (Comparng the regresson results of Hong Kong s demand for money, how dfferent s the real money supply relatonshp n HK and n the US?) b. What are the results f you use the dfferent defnton of money and nterest rate? Justfy your results whether they have a better performance than n (a)? c. Instead of estmatng the above demand functon, suppose you were to ft the functon ( ) Y M t r 1 t e u t. How would you nterpret the results? d. How wll you decde whch s a better specfcaton? (a) or (c) and whch nterest rate (short-term or long-term) s better to ft the model? 8 TBD Semester - 014/015 Page 107 of 140
Course Outlne Econometrcs Tutoral # 8 Computng Exercse: Gujarat (003) #8.6, pp.88 The demand for cable. Table 8.10 gves data used by a telephone cable manufacturer to predct sales to a major customer for the perod 1968-1983. The varables n the table are defned as follows: Y = annual sales n MPF, mllon pared feet X = gross natonal product (GNP), $, bllons X 3 = housng starts, thousands of unts X 4 = unemployment rate, % X 5 = prme rate lagged 6 months X 6 = customer lne gans, % You are to consder the followng model: Y t 1 X t 3X 3t 4 X 4t 5X 5t 6X 6t a. Estmate the precedng regresson b. What are the expected sgns of the coeffcents of ths model? c. Are the emprcal results n accordance wth pror expectatons? u d. Are the estmated partal regresson coeffcents ndvdually statstcally sgnfcant at the 5% level of sgnfcance? e. Suppose you frst regress varable Y on X, X 3, and X 4 only and then decde to add the varables X 5 and X 6. How would you fnd out f t s worth addng the varables X 5 and X 6? Whch test do you use? Show the necessary calculatons. t Extra exercse:8.5, 8.7, 8.10, 8.11, 8.13, 8.17, 8.18, [8.8]. 9 TBD Semester - 014/015 Page 108 of 140
Course Outlne Econometrcs Computng Exercse: Tutoral # 9 (1) Gujarat (003) #9.3, pp. 35, Table 9.7 gves the quarterly data on unemployment rate and job vacancy rate n the US. Besdes the questons from the textbook, we may also want to understand whether there s any seasonal effect and dfferences on the unemployment rate n the US. () Frst to defne the seasonal (or quarterly) dummy varables () Rune the regresson wth dummy varables () Measure the estmated result for each quarter. () Gujarat (003) #9.4, pp.331, Table 9.8 gves data on quadrennal presdental electons n the Unted States from 1916 to 1996. The varables are defned as: Year = Electon year V = Democratc share of the two-party presdental vote I = Indcator varable (1 f there s a Democratc ncumbent at the tme of the electon,-1 f there s a Republcan ncumbent) D = Indcator varable (1 f a Democratc ncumbent us runnng for electon, -1 f a Republcan ncumbent s runnng for electon and 0 otherwse) W = Indcator varable (1 for the electons of 190, 1944, 1948, and 0 otherwse) G = Growth rate of real per capta GDP n the frst 3 quarters of the electon year. P = Absolute value of the growth rate of the GDP deflator n the frst 15 quarters of the admnstraton N = Number of quarters n the frst 15 quarters of the admnstraton n whch the growth rate of real per capta GDP s greater than 3.% a. Usng the data gven n Table 9.8 to develop a sutable model and predct the Democratc share of the two-party presdental vote. b. How would you use ths model to predct the outcome of a presdental electon? c. Chatteejee et al. (000) suggested consderng the followng model as a tral model to predct presdental electons: V 1 3 4 5 ) 6 7 I D W ( G* I P N u Estmate ths model and comment on the results n relatons to the results of the model you have chosen n (a). Extra exercse: 9., 9.3, 9.8, 9.11. 10 TBD Semester - 014/015 Page 109 of 140
Course Outlne Econometrcs Tutoral # 10 Computng Exercse: Gujarat (003) #10.7, pp.38 Table 10.1 gves data on mports, GDP and the CPI for the Unted States over the perod 1970 1998. (Smlarly, there s also a data fle hk_gdp_cp_mports.xls for Hong Kong over the perod 1973:1 003:6) You are asked to consder the followng model: ln Im ports t 1 ln GDPt 3 ln CPI a. Estmate the parameters of ths model usng the data gven n the table. (Compare the result from the Hong Kong s data) t u b. Do you suspect that there s multcollnearty n the US s data? (Also, do you suspect that n the HK s data?) c. Regress: (1) ln Imports t = A 1 + A ln GDP t () ln Imports t = B 1 + B ln CPI t (3) ln GDP t =C 1 + C ln CPI t On the bass of these regressons, (1), (), and (3), whch can you say about the nature of multcollnearty n the data? (Run the same regresson for the HK s data and compare the results) d. Suppose there s multcollnearty n the data but ˆ and ˆ 3 are ndvdually sgnfcant at the 5 percent level and the overall F test s also sgnfcant. In ths case should we worry about the collnearty problem? (Is there the same problem n the HK s data?) Extra exercse: 10.3, 10.5, 10.1, 10.4. t 11 TBD Semester - 014/015 Page 110 of 140
Course Outlne Econometrcs Tutoral # 11 Computng Exercse: Gujarat (003), #11.15, pp.43 Table 11.17 gves data on 81 cars about MPG (average mles per gallons), HP (engne horsepower), VOL (cubc feet of cab space), SP (top speed, mles per hour) and WT (vehcle weght n 100lb.) a. Consder the followng model: MPG 1 SP 3HP 4WT u t Estmate the parameters of ths model and nterpret the results. Do they make economc sense? b. Would you expect the error varance n the precedng model to be heteroscedastc? Why? c. Use the Whte test to fnd out f the error varance s heteroscedastc. d. Obtan Whte s heteroscdastcty-consstent standard errors and t-values and compare your results wth those obtaned from OLS. e. If heteroscedastcty s establshed, how would you transform the data so that n the transformed data the error varance s homoscedastcty? Show the necessary calculatons. Extra exercse: 11., 11.4, 11.6, 11.0. 1 TBD Semester - 014/015 Page 111 of 140
Course Outlne Econometrcs Tutoral # 1 Computng Exercse: Gujarat (003) #1.6, pp.498 Refer to the data on the copper ndustry gven n Table 1.7. The varables are defned as follows: C = 1-month average US domestc prce of copper (cents per pound) G = annual gross natonal product ($, bllons) I = 1-month average ndex of ndustral producton L = 1-month average London Metal Exchange prce of copper (pounds sterlng) H = number of housng starts per year (thousands of unts) A = 1-month average prce of alumnum (cents per pound) a. Estmate the followng regresson model and nterpret the results: lnct 1 ln It 3 ln Lt 4 ln Ht 5 ln A u b. Obtan the resduals and standardzed resduals from the precedng regresson and plot them. What can you surmse about the presence of autocorrelaton n these resduals? c. Estmate the Durbn-Watson d statstc and comment on the nature of autocorrelaton n these resduals? d. How would you fnd out f an AR(p) process better descrbes autocorrelaton than AR(1) process? t t Extra exercse: 1.1, 1.3, 1.5, 1.7. 13 TBD Semester - 014/015 Page 11 of 140