1. Lecture 7 Stephen G Hall IDENTIFICATION

2. Standard Identification (without cointegration) The Structural Model The reduced form Model The identification problem: If we estimate a reduced form model can we uniquely determine the structural parameters? In the absence of any restrictions on the model the answer is no! There are more parameters in the structural form than the reduced form

3. Suppose there are two X variables and 2 Y variables, for one value of each X variable we define a point in the space of the two Y variables. This point obviously does not allow us to know the actual lines in the Y1 Y2 space. We can put any pair of lines we like through this point, adding more points does not help, each point can have an infinite number of lines through it. The system is not identified

4. But suppose we know something about the structural system then we can use this information to identify it from the reduced form. X 1 is excluded from the Y 2 equation and X 2 is excluded from the Y 1 equation. This now allows us to identify the structural interrelationships. The reduced form has 4 parameters and given the restrictions above the structural form also has 4 parameters The next slide shows what happens as X 1 varies. Clearly as X 1 does not affect Y 2 the Y 2 line will not move.

5. Now as we know the Y 2 line will not move we know that the different points must be tracing out the Y 1 relationship So excluding a variable allows us to uniquely determine one of the relationships Y1Y1 Y2Y2

6. 2 conditions to asses identification. The Order condition, necessary but not sufficient Let G be the total number of endogenous variables, G 1 be the number of endogenous variables in an equation, K be the number of exogenous variables in the system and K 1 be the number of exogenous variables in an equation. Then the equation is identified if K-K 1 =G 1 -1 That is the number of excluded exogenous variables is equal to the number of endogenous variables in the equation minus 1. If K-K 1 <G 1 -1 then the equation is not identified If K-K 1 >G 1 -1 then the equation is over-identified

7. The Rank Condition Necessary AND sufficient (but harder to check) We want a unique solution to one of the equations in The rank condition requires us to partition the matrices and impose restrictions thus. The equation is exactly identified if the rank of is equal to G 1 -1

8. If a model is: Under identified; the structural form is not uniquely determined. Exactly identified; there is a unique mapping between the reduced form and structural form. BUT we can not test the identifying restrictions as they are simply a rewriting of the model. They do not represent a restriction on the reduced form. Over identified; there are more restrictions than we need to identify the structural model. We can then test the over identifying restrictions, this is the way we test economic theories.

9. WE HAVE INCREASINGLY COME TO UNDERSTAND THAT THE IDENTIFICATION PROBLEM IS EVEN MORE IMPORTANT WHEN WE MODEL USING COINTEGRATION TECHNIQUES UNDERSTANDING LONG RUN STRUCTURES IS PARTLY A MATTER OF FINDING COINTEGRATING VECTORS. BUT UNDERSTANDING THESE REQUIRES A PROCESS OF IDENTIFICATION. AN IMPORTANT RECENT CONTRIBUTION IS `LONG RUN STRUCTURAL MODELLING' BY H M PESARAN AND Y SHIN, CAMBRIDGE MIMEO, 1994.

10. Identification and Cointegration The structural form The reduced form Two parts to identification now: the short run and the long run

11. Even knowing the A 0 matrix does not help us to know the long run structural cointegrating vector as; To identify the long run we must first know r, the cointegrating rank and then impose some structure on the model. We know how to test r. There are a number of ways of then identifying the structure

12. IDENTIFICATION IN THE JOHANSEN PROCEDURE JOHANSEN IS ESTIMATING THE COINTEGRATING VECTORS IN AN UNIDENTIFIED VAR, HOW DOES IT PRODUCE A UNIQUE ANSWER? RECALL, AT THE END OF THE DERIVATION IN JOHANSEN WE FOUND TWO CONDITIONS THESE ARE THE ARBITRARY IDENTIFICATION CONDITIONS. THEY PICK OUT ORTHOGONAL SETS OF COINTEGRATING VECTORS WITH AN ARBITRARY NORMALIZATION THAT IS BASED ON THE SCALING OF THE DATA. THESE ARE DATA BASED RESTRICTIONS NOT THEORY BASED.

13. PHILLIPS IDENTIFICATION PHILLIPS (1991) OPTIMAL INFERENCE IN COINTEGRATED SYSTEMS ECONOMETRICA 59 283-306. PROPOSED A SYSTEM WHICH IDENTIFIED THE COINTEGRATING STRUCTURE IN THE FOLLOWING WAY. WHERE X 1 IS A VECTOR OF r VARIABLES AND X 2 IS AN m-r VECTOR. THIS IDENTIFIES THE SYSTEM BY ASSUMING. 1) X 1 ONLY ENTERS ITS OWN EQUATION, SO THERE ARE EXCLUSION RESTRICTIONS 2) THE LEVELS TERMS ARE ALL EXCLUDED FROM THE DETERMINATION OF X 2

14. JOHANSEN (1992) IDENTIFYING RESTRICTIONS OF LINEAR EQUATIONS, MIMEO COPENHAGEN ATTEMPTED TO INCORPORATE A PRIORI, THEORETICAL RESTRICTIONS INTO COINTEGRATING SYSTEMS. HE GENERALISED HIS OWN METHOD FOR IMPOSING AND TESTING RESTRICTIONS HE THEN GENERALISES HIS ESTIMATION TECHNIQUE TO ITERATIVELY IMPOSE THESE RESTRICTIONS. HOWEVER THE RESTRICTIONS ARE STILL RATHER ARBITRARY AND LIMITED, NON-HOMOGENEOUS RESTRICTIONS ARE NOT ALLOWED AND RESTRICTIONS CAN NOT BE IMPOSED ACROSS COINTEGRATING VECTORS.

15. PESARAN & SHIN ESTABLISH A SET OF FORMAL CONDITIONS FOR IDENTIFICATION WHICH PARALLEL THE STANDARD FORM OF IDENTIFICATION. THEY POINT OUT THAT WE CAN CONSIDER THE IDENTIFICATION OF LONG RUN TERMS AND THE DYNAMICS SEPARATELY THEY ESTABLISH THE NEED FOR K=r 2 A PRIORI RESTRICTIONS. IF K<r 2 THE MODEL IS UNDERIDENTIFIED IF K=r 2 THE MODEL IS EXACTLY IDENTIFIED IF K>r 2 THE MODEL IS OVER IDENTIFIED THIS DIFFERS FROM STANDARD IDENTIFICATION BECAUSE WE DO NOT HAVE TO CONSIDER THE EXOGENEITY OF THE VARIABLES AND THE DYNAMIC TERMS DO NOT HELP IN IDENTIFICATION.

16. EXAMPLES r=1, r 2 =1 r=2, r 2 =4

17. r=3, r 2 =9

18. Estimating the exactly identified cointegrating vectors (k=r 2 ) let H be the r 2 xmr jacobian matrix of the restrictions h (first derivatives) then and we can derive the restricted coefficients from the unrestricted ones as This takes the unrestricted estimates (J) and maps them into the restricted ones (R)

19. The over identified case k>r 2 In this case we need to maximise the likelihood function subject to the over identifying restrictions we can form the following lagrangian Pesaran and Shin then give a numerical algorithm for solving this problem. Alternatively it may be solved in any non-linear system estimation package (eg TSP) by setting up the system and estimating subject to the full set of restrictions and the cointegrating rank r of the unrestricted system.

20. Finally they show that the standard likelihood ratio test, has a standard distribution with k-r 2 degrees of freedom.

21. An Example Data was generated from the following model Where

22. the following vector error correction model was estimated using Johansen's ML technique using MICROFIT for a sample size of 100 for two values of (0 and 0.8, implying no dynamics and the presence of dynamics) How do the results relate to the underlying model? Is the estimation procedure appropriate?

23. statistic95%CV Max eigen r=085.322.0 r=164.815.6 r=21.49.2 trace r=0151.634.9 r=166.319.9 1=21.459.2 Results for =0 So correct answer

24. CVVec 1Vec 2 X1-.105 (-1).024(-1) X2.113(1.07).076(-3.1) X3 -0.007(-0.06) -.098(4.0) Alpha X17.2(.76)-10(.245) X2-3.4(-.36)-8.6(.21) x3.087(0.009).308(-.007) Estimated vectors and adjustment coefficients, (normalized in brackets) Not much like the originals!

25. statistic95%CV Max eigen r=0114.322.0 r=144.615.6 r=21.269.2 trace r=0160.234.9 r=145.819.9 1=21.269.2 Results for =0.8 So correct answer again

26. CVVec 1Vec 2 X1-.014(-1).013(-1) X2.085(5.9).056(-4.3) X30.013(0.95)-.19(14.7) Alpha X14.1(0.06)-6.6(.08) X2-8.5(-.122)-4.6(.06) x3-1.4(-.02)1.7(-.022) Estimated vectors and adjustment coefficients, (normalized in brackets) Again not much like the originals!

27. But we need to identify the target relationships from the space spanned by these cointegrating vectors. We now illustrate this process

28. X1X2X3 True 110 True 201 CV11.07-0.6 CV2-3.14.0 Comparing the true and estimated relationships for

29. CV2*0.15=-.150.4650.6 -CV1+CV2*0.15=1.150.720 normalized10.620 Imposing by first multiplying CV2 by 0.15 then subtracting CV1 and normalizing Identifying target relationship 1

30. CV1-CV2=04.174.6 normalized01-1.1 Imposing by first subtracting CV2 from CV1 then normalizing Identifying target relationship 2

31. X1X2X3 True 11-50 True 201 CV15.9-0.9 CV2-4.314.8 Comparing the true and estimated relationships for

32. CV2*0.0606=-.0608-0.260.9 +CV1=-1.06085.640 normalized1-5.30 Imposing by first multiplying CV2 by 0.0606 then subtracting CV1 and normalizing Identifying target relationship 1

33. CV1-CV2=0-10.215.7 normalized01-1.5 Imposing by first subtracting CV2 from CV1 then normalizing Identifying target relationship 2

34. Irreducibility As an underlying concept for identification

35. Irreducible cointegrating vectors Davidson proposed the notion of irreducibility as a way of detecting and identifying structural relations. An irreducible vector is one where you can not remove any variables without loosing cointegration

36. the main points of this approach are formalised in five theorems. Definition I. A set of I(1) variables will be called irreducibly cointegrated (IC) if they are cointegrated, but dropping any of the variables leaves a set that is not cointegrated.

37. n Theorem 1 (Davidson, 1994). If a column of b (say b 1 ) is identified by the rank condition, the OLS regression which includes just the variables having unrestricted non-zero coefficients in b 1 is consistent for b 1. Theorem 2. An IC vector is unique, up to the choice of normalisation.

38. n Theorem 3 If and only if a structural cointegrating relation is identified by the rank condition, it is irreducible. n So a structural relationship is always irreducible

39. – Definition 2 – A solved vector is a linear combination of structural vectors from which one or more common variables are eliminated by choice of offsetting weights such that the included variables are not a superset of any of the component relations. n Lemma 1 n Provided b is restricted only by zero and normalisation restrictions, a solved IC relation contains at least as many variables as each of the identified structural relations from which it derives. n Solved vectors have more variables in them

40. In general, therefore, the fewer variables an IC relation contains, and the fewer it shares with other IC relations, the better the chance that it is structural and not a solved form. In the extreme cases, we can actually draw definite conclusions, as the following pair of results show. n Theorem 4 n If an IC relation contains strictly fewer variables than all those others having variables in common with it then, subject to the condition of Lemma l, it is an overidentified structural relation. Theorem5 If an IC relation contains a variable, which appears in no other IC relation, it is structural.

41. n So in some circumstances we can directly detect an identified structural relationship just by the logic of irreducibility n An example n Assume 4 I(1) variables n x, y, w, z n if r=2 n we test for irreducible sets n Suppose we found 2 ICV n x co integrated with y z co integrated with w n these are then by the previous theorems the structural theoretical relationships

42. Worse Case Example n 3 I(1) variables, x, y, z, n in the structural model r=2 and the structural relationships are – x co integrates with y y co integrates with z n Then by construction x co integrates with z n this is the solved vector, we find three ICVs two structural ones and the solved one n So irreducibility does not help here

43. Conclusion If we want to understand structure identification is crucial We now have the technology to identify both the long and the short run structure of a model

44. An Example Bargaining models and identifying the wage equation. G Chamberlin, S G Hall, and S G B Henry.

45. Introduction Standard models of wage determination have followed the bargaining framework for many years Manning 1993 argued that the wage equation was not identified in this model A major problem not resolved

46. Introduction this paper uses modern cointegration identification theory to argue that this view is wrong It –Outlines identification –Outlines wage theory –Show how this identifies the wage equation –Apply these ides to the UK

47. The Manning problem Suppose the demand for employment is E d =f(x) And supply is E s =g(z) Then a bargain over wages will result such that, the real wage w will be W=h(x,z)

48. But problem The wage equation now contains all the variables in the system – nothing is excluded Hence it can not be identified by the standard order condition

49. The paper then outlines identification theory as earlier in this lecture

50. The key thing is that these restrictions are on the cointegrating vectors. Not on the equations. Nothing needs to be excluded from any equations to identify the long run

51. Bargaining theory Does the theory identify the long run relationships? go back to the key paper in this area McDonald –Solow(1981) outline various cases

52. Case 1 Firm; Maximise Profits R (L)-wL Union Maximise Utility L(U(w) - Un ) Which implies R’(L)-w=0 (U(w)-Un )/wU´(w)+LR´´(L)/R´(L)=0 2 cointegrating vectors

53. Case 2 This is a true bargaining model. A contract curve is derived (U(w)-U(wr))/U(w) = w- R (L) Which may imply (U(w)-U(wr))/U(w) = B w- R (L) = B 2 cointegrating relationships

54. The message The more structure or theory we can bring to bare on a problem the more likely we are to be able to properly identify the system we are interested in.

55. estimation We now estimate a model of the wage bargaining process in the UK. We assume cobb-douglas technology and estimate both the production function, the marginal revenue condition and the labour supply condition

56. The core model Note: this system is heavily over identified as it contains both many exclusion restrictions in the CVs and also has many cross equation restrictions. Despite the fact that the wage equation has all the variables in it.

57. Stationarity tests Variable DF DF(4) Q -1.65 -2.37 L -0.86 -3.2 W -2.76 Pp -7.3 -3.2 Pc -5.56 -3.2 U -0.13 -1.68 K -2.7 -2.8 Wd -3.0 -2.8 Rr -1.03 -1.51 Ud -6.4 -2.79

58. Weak exogeneity VariablesWeakly exogenousWald Test W, pc, pp, u, l, y, k U 7.2 (7.8) W, pc, pp, l, y, k K 1.39(7.8)

59. Testing r Conclusion: 3 cointegrating vectors Variables LR (V) LR(T) q,l,k 26.4(25.4) 63.3(42.3) w, pp,k,l 24.1(25.4) 52.9(42.3) w, pc, u, wd 35.9(31.8) 70.2(63.0)

60. Model structure This gives us a 6-equation system to estimate with 3 long run relationships, which we hope to identify as the production function and the marginal product condition.

61. The estimated long run structure ECM1Y= 0.0036 T + 0.78 E + 0.22 K (62.3) (-) (21.3) ECM2W= PPI + 0.0036T – 0.22 E + 0.22 K - - - ECM3W=CPI –0.4Wd –0.02 U + 0.012T (0.7) (3.87) (2.1)

62. The wage equation Both the firm side of the Bargain and the union side have significant and correctly signed adjustment coefficients, although they both suggest rather slow adjustment, which is perhaps not surprising for the case of UK. The other dynamic equations will not be reported here.

63. Some tests of restrictions Restriction Number of restrictions (k) LR(k ) 1Restricting the adjustment matrix 5 9.5 11.07 2Within equation restrictions 2 0.24 5.99

64. conclusion The bargaining wage equation may be identified We have outlined the theory of identification and bargaining We have applied this with success to UK data.