Dynamic Models, Autocorrelation and Forecasting

Transcription

1 ECON 4551 Economerics II Memorial Universiy of Newfoundland Dynamic Models, Auocorrelaion and Forecasing Adaped from Vera Tabakova s noes

2 9.1 Inroducion 9.2 Lags in he Error Term: Auocorrelaion 9.3 Esimaing an AR(1) Error Model 9.4 Tesing for Auocorrelaion 9.5 An Inroducion o Forecasing: Auoregressive Models 9.6 Finie Disribued Lags 9.7 Auoregressive Disribued Lag Models Slide 9-2

3 Figure 9.1 Slide 9-3

4 The model so far assumes ha he observaions are no correlaed wih one anoher. This is believable if one has drawn a random sample, bu less likely if one has drawn observaions sequenially in ime Time series observaions, which are drawn a regular inervals, usually embody a srucure where ime is an imporan componen.

5 If we canno compleely model his srucure in he regression funcion iself, hen he remainder spills over ino he unobserved componen of he saisical model (is error) and his causes he errors be correlaed wih one anoher.

6 In general, here are many siuaions where ineria affecs a ime series Or lagged effecs of explanaory variables Or auocorrelaion is inroduced by massaging of he original daa (inerpolaion, smoohing when going from monhly o quarerly, ec.) Or nonsaionariy

7 Bu, as wih heeroskedasiciy, we should suspec firs ha here is a specificaion issue Wrong funcional form or missing variables

8 Three ways o view he dynamics: y f ( x, x, x,...) 1 2 (9.1) y f ( y, x ) 1 (9.2) y f ( x ) e e f ( e 1) (9.3) Slide 9-8

9 Assume Saionariy For now Figure 9.2(a) Time Series of a Saionary Variable Slide 9-9

10 Figure 9.2(b) Time Series of a Nonsaionary Variable ha is Slow Turning or Wandering Slide 9-10

11 Figure 9.2(c) Time Series of a Nonsaionary Variable ha Trends Slide 9-11

12 9.2.1 Area Response Model for Sugar Cane ln A ln P 1 2 ln ln A P e 1 2 (9.4) y x e 1 2 (9.5) e e v 1 (9.6) Slide 9-12

13 y x e 1 2 (9.7) e e v 1 Bu assume v well behaved: (9.8) E v v v v s 2 ( ) 0 var( ) v cov(, s) 0 for (9.9) Assume also 1 1 (he series is saionary (9.10) Slide 9-13

14 Ee ( ) 0 OK (9.11) var( ) 2 e e 2 v 1 2 OK, consan (9.12) 2 e e k cov, k e k 0 No zero!!! (9.13) bu since we sick o saionary series, sill consan Slide 9-14

15 corr( e, e ) k cov( e, e ) cov( e, e ) var( e)var 2 k k k e 2 e var( e) k e Focus on k=1 k (9.14) corr( e, e 1) aka auocorrelaion coefficien (9.15) Running simple OLS On sugarcane example yˆ x (se) (.061) (.277) (9.16) Slide 9-15

16 Slide 9-16

17 We see runs Wha if we saw bouncing? Figure 9.3 Leas Squares Residuals Ploed Agains Time Slide 9-17

18 General covariance formula: r xy ( x x)( y y) cov( x, y ) 1 var( x)var( y) ( x x) ( y y) Consan variance T T T Null expecaion of x and y if hey are errors (9.17) r 1 cov( e, e ) 1 2 T var( e ) T 2 ee ˆˆ eˆ (9.18) Slide 9-18

19 The exisence of AR(1) errors implies: The leas squares esimaor is sill a linear and unbiased esimaor, bu i is no longer bes. There is anoher esimaor wih a smaller variance. The sandard errors usually compued for he leas squares esimaor are incorrec. Confidence inervals and hypohesis ess ha use hese sandard errors may be misleading. Slide 9-19

20 As wih heeroskedasic errors, you can salvage OLS when your daa are auocorrelaed. Now you can use an esimaor of sandard errors ha is robus o boh heeroskedasiciy and auocorrelaion proposed by Newey and Wes. This esimaor is someimes called HAC, which sands for heeroskedasiciy auocorrelaed consisen. Slide 9-20

21 HAC is no as auomaic as he heeroskedasiciy consisen (HC) esimaor. Wih auocorrelaion you have o specify how far away in ime he auocorrelaion is likely o be significan The auocorrelaed errors over he chosen ime window are averaged in he compuaion of he HAC sandard errors; you have o specify how many periods over which o average and how much weigh o assign each residual in ha average. Tha weighed average is called a kernel and he number of errors o average is called bandwidh. Principles of Economerics, 3rd Ediio Slide 9-21

22 Usually, you choose a mehod of averaging (Barle kernel or Parzen kernel) and a bandwidh (nw1, nw2 or some ineger). Ofen your sofware (example GRETL) defauls o he Barle kernel and a bandwidh compued based on he sample size, N. Trade-off: Larger bandwidhs reduce bias (good) as well as precision (bad). Smaller bandwidhs exclude more relevan auocorrelaions (and hence have more bias), bu use more observaions o increase precision (smaller variance). Choose a bandwidh large enough o conain he larges auocorrelaions. The choice will ulimaely depend on he frequency of observaion and he lengh of ime i akes for your sysem o adjus o shocks. Slide 9-22

23 Once grel recognizes ha your daa are ime series, hen he robus command will auomaically apply he HAC esimaor of sandard errors wih he defaul values of he kernel and Bandwidh (or wih your cusomized choices) Again, remember ha here are several choices abou how o run hese correcions (so differen sofware packages can yield slighly differen resuls) In a way, his correcion is jus an exension of Whie s correcion for heeroskedasiiciy I is based on a formula ha has he OLS formula as a special case Slide 9-23

24 Sugar cane example The wo ses of sandard errors, along wih he esimaed equaion are: yˆ x (.061) (.277) 'incorrec' se's (.062) (.378) 'correc' se's The 95% confidence inervals for β 2 are: (.211,1.340) (incorrec) (.006,1.546) (correc) Slide 9-24

25 However, as in he case of robus esimaion under heeroskedasiciy, he HAC correcion correcs or accouns for auocorrelaion bu does no exploi i We can ge a more precise esimaor if we exploi he Idea ha some observaions are differen no because of he effec of he regressor values, bu because of he effec of he adjacen errors Slide 9-25

26 y x e 1 2 (9.19) e e v 1 (9.20) y x e v (9.21) e y x (9.22) Slide 9-26

27 e y x (9.23) y (1 ) x y x v (9.24) ln( A ) ln( P) e.422e v 1 (se) (.092) (.259) (.166) (9.25) Now his ransformed nonlinear model has errors ha are uncorrelaed over ime Slide 9-27

28 Solving his nonlinear regression is sill based on minimizing he sum leas squares, bu i canno be done wih close formulae now (because of he nonlineariy in he parameers) so we call i Generalised Leas Squares again I can be shown ha nonlinear leas squares esimaion of (9.24) is equivalen o using an ieraive generalized leas squares esimaor called he Cochrane-Orcu procedure. Deails are provided in Appendix 9A. Slide 9-28

29 The nonlinear leas squares esimaor only requires ha he errors be sable (no necessarily saionary). Oher mehods commonly used make sronger demands on he daa, namely ha he errors be covariance saionary. Furhermore, he nonlinear leas squares esimaor gives you an uncondiional esimae of he auocorrelaion parameer and yields a simple -es of he hypohesis of no serial correlaion. Mone Carlo sudies show ha i performs well in small samples as well. Slide 9-29

30 Bu nonlinear leas squares requires more compuaional power han linear esimaion, hough his is no much of a consrain hese days. Nonlinear leas squares (and oher nonlinear esimaors) use numerical mehods raher han analyical ones o find he minimum of your sum of squared errors objecive funcion. The rouines ha do his are ieraive. Slide 9-30

31 y (1 ) x x y v (9.26) y x x y v (9.27) (1 ) We should hen es he above resricions (esing nonlinear resricions is a possibiliy) yˆ x.611 x.404y 1 1 (se) (.656) (.280) (.297) (.167) (9.28) Slide 9-31

32 y (1 ) x x y v (9.26) y x x y v (9.27) This resriced model could be esimaed wih OLS as long as v is well behaved, because here are no nonlineariies Check ha our inuiion of having dynamics given by lagged effecs of errors boils down o having lagged effecs of dependen and independen variables!!! Slide 9-32

33 9.4.1 Residual Correlogram H : 0 H : z Tr1 N(0,1) (9.29) z (9.30) Slide 9-33

34 9.4.1 Residual Correlogram (more pracically ) r or r T T 1 1 r k or rk T T (9.31) k cov( e, e ) E( e e ) k k 2 var( e) E( e ) (9.32) Slide 9-34

35 Figure 9.4 Correlogram for Leas Squares Residuals from Sugar Cane Example Slide 9-35

36 Residual ACF /T^ In GRETL lag Residual PACF /T^ lag Figure 9.4 Correlogram for Leas Squares Residuals from Sugar Cane Example Slide 9-36

37 y x e 1 2 y (1 ) x y x v For his nonlinear model, hen, he residuals should be uncorrelaed Slide 9-37

38 Figure 9.5 Correlogram for Nonlinear Leas Squares Residuals from Sugar Cane Example Slide 9-38

39 Anoher way o deermine wheher or no your residuals are auocorrelaed is o use an LM (Lagrange muliplier) es. For auocorrelaion, his es is based on an auxiliary regression where lagged OLS residuals are added o he original regression equaion. If he coefficien on he lagged residual is significan hen you conclude ha he model is auocorrelaed. Slide 9-39

40 y x e v (9.33) = F = p-value =.021 We can derive he auxiliary regression for he LM es: y x eˆ vˆ (9.34) b b x eˆ x eˆ vˆ Firs resul from GRETL Slide 9-40

41 eˆ ( b ) ( b ) x eˆ vˆ x eˆ vˆ (9.35) LM T R Cenered around zero, so he power of his es now would come from The lagged error, which is wha we care abou Second resul from GRETL Slide 9-41

42 eˆ ( b ) ( b ) x eˆ vˆ x eˆ vˆ (9.35) LM T R This was he Breusch-Godfrey LM es for auocorrelaion and i is disribued chi-sq In STATA: regress la lp esa bgodfrey Slide 9-42

43 There are oher varians of his es Noe ha i is only valid in large samples (heoreically infiniely big ones) GRETL also repors Ljung-Box Q es The number of lags gives you he df for he chi-sq and he smaller df for he F disribuions you need Slide 9-43

44 In general, p should be large enough so ha v is whie noise y y y y v p p (9.36) CPI CPI 1 y ln( CPI ) ln( CPI 1) CPI 1 Adding lagged values of y can serve o eliminae he error auocorrelaion INFLN INFLN.2179 INFLN.1013 INFLN (se) (.0253) (.0615) (.0645) (.0613) (9.37) Slide 9-44

45 Helps decide How many lags To include in he model Figure 9.6 Correlogram for Leas Squares Residuals from AR(3) Model for Inflaion Slide 9-45

46 y y y y v (9.38) y y y y v T 1 1 T 2 T 1 3 T 2 T 1 yˆ ˆ ˆ y ˆ y ˆ y T 1 1 T 2 T 1 3 T Slide 9-46

47 yˆ ˆ ˆ yˆ ˆ y ˆ y T 2 1 T 1 2 T 3 T (9.39) u y yˆ ( ˆ ) ( ˆ ) y ( ˆ ) y ( ˆ ) y v 1 T 1 T T 2 2 T T 2 T 1 Slide 9-47

48 Slide 9-48

49 u 1 vt 1 (9.40) u ( y yˆ ) v u v v v 2 1 T 1 T 1 T T 2 1 T 1 T 2 (9.41) u u u v ( ) v v v T T 1 1 T 2 T 3 (9.42) Slide 9-49

50 var( u ) v var( u ) (1 ) v 1 var( u ) [( ) 1] v yˆ 1.96 ˆ, yˆ 1.96 ˆ T j j T j j (9.43) Slide 9-50

51 This model is jus a generalizaion of he ones previously discussed. In his model you include lags of he dependen variable (auoregressive) and he conemporaneous and lagged values of independen variables as regressors (disribued lags). The acronym is ARDL(p,q) where p is he maximum disribued lag and q is he maximum auoregressive lag Slide 9-51

52 y 0x 1x 1 2x 2 qx q v, q 1,, T (9.44) Ey ( ) x s s WAGE WAGE 1 x ln( WAGE ) ln( WAGE 1) WAGE 1 Slide 9-52

53 Slide 9-53

54 Slide 9-54

55 y x x x y y v q q 1 1 p p (9.45) y x x x x e x e s 0 s s (9.46) Slide 9-55

56 Figure 9.7 Correlogram for Leas Squares Residuals from Finie Disribued Lag Model Slide 9-56

57 INFLN PCWAGE.0377 PCWAGE.0593PCWAGE 1 2 (se) (.0288) (.0761) (.0812) (.0812).2361 PCWAGE.3536 INFLN.1976 INFLN (.0829) (.0604) (.0604) (9.47) Slide 9-57

58 Figure 9.8 Correlogram for Leas Squares Residuals from Auoregressive Disribued Lag Model Slide 9-58

59 y x x x x y y v ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Slide 9-59

60 Figure 9.9 Disribued Lag Weighs for Auoregressive Disribued Lag Model Slide 9-60

61 auocorrelaion auoregressive disribued lag models auoregressive error auoregressive model correlogram delay muliplier disribued lag weigh dynamic models finie disribued lag forecas error forecasing HAC sandard errors impac muliplier infinie disribued lag inerim muliplier lag lengh lagged dependen variable LM es nonlinear leas squares sample auocorrelaion funcion sandard error of forecas error oal muliplier form of LM es Slide 9-61

62 Slide 9-62

63 y x e e e v y x y x v (9A.1) 1 y y x x v (9A.2) y y y x x x x Slide 9-63

64 y x x v (9A.3) y x ( y x ) v (9A.4) Slide 9-64

65 y x e y 1 1 x 1 e y x x e (9A.5) y 1 y x x 1 x e 1 e (9A.6) Slide 9-65

66 v var( e ) (1 )var( ) (1 ) e1 2 v Slide 9-66

67 H : 0 H : d T eˆ ˆ 2 e 1 2 T 1 eˆ 2 (9B.1) Slide 9-67

68 d T T T 2 2 eˆ ˆ ˆ ˆ e 1 2 ee T 2 eˆ 1 T T T 2 2 eˆ ˆ ˆ ˆ e 1 ee T T T eˆ ˆ ˆ e e (9B.2) 1 1 2r 1 Slide 9-68

69 d 21 r 1 (9B.3) d d c Slide 9-69

70 Figure 9A.1: Slide 9-70

71 The Durbin-Wason es. used o be he sandard and is exac (i works in small samples) comes ou of every sofware package s defaul oupu i is easy o calculae bu he criical values depend on he values of he variables, unless you have a compuer o look up he exac disribuion you are suck DW did provide he upper and lower disribuions bu hen es may end up being inconclusive Slide 9-71

72 Figure 9A.2: Slide 9-72

73 The Durbin-Wason bounds es. if d d, rejec H : 0 and accep H : 0; Lc 0 1 if d d, do no rejec H : 0; Uc if d d d he es is inconclusive. Lc Uc, 0 The size of he inconclusive area shrinks wih sample size Slide 9-73

74 The Durbin-Wason es. used o be he sandard, bu : i canno handle lagged values of he dependen variable (i is biased for auoregressive moving average models, so ha auocorrelaion is underesimaed) Bu for large samples one can easily compue he unbiased normally disribued h-saisic) Bu hen Breusch-Godfrey es is much more powerful insead Slide 9-74

75 The Durbin-Wason es. used o be he sandard, bu : i assumes normaliy of he errors i canno handle more han one lag (i deecs only AR1 auocorrelaion) Slide 9-75

76 Also remember ha he Durbin-Wason es assumes ha ha here is an inercep in he model ha he X regressors are nonsochasic ha here are no missing values in he series Slide 9-76

77 y x x x x e x e s s s 0 y x x x y y v q q 1 1 p p Slide 9-77

78 y x y v (9C.1) y x y (9C.2) y x y x ( x y ) x x y Slide 9-78

79 y x x ( x y ) x x x y y 2 j x x x x y 2 j j j 1 ( j 1) (9C.3) j 2 j s j x s 1 y ( j 1) s 0 (1 ) Slide 9-79

80 y s 0 s 0 1 x s (9C.4) 2 (1 1 1 ) 1 1 y x e s s s 0 Slide 9-80

81 s s s 0(1 1 1 ) s Slide 9-81

82 y x x x x y y v (9C.5) (9C.6) s 1 s 1 2 s 2 for s 4 Slide 9-82

83 ˆ y y y 3 T T 1 T 2 yt 1 yˆ y (1 ) y (1 ) y 1 2 T 1 T T 1 T 2 (9D.1) (1 ) yˆ (1 ) y (1 ) y (1 ) y T T 1 T 2 T 3 (9D.2) yˆ y (1 ) yˆ T 1 T T Slide 9-83

84 Figure 9A.3: Exponenial Smoohing Forecass for wo alernaive values of α Slide 9-84