3681:. The simultaneous equation model requires a theory of reciprocal causality that includes special features if the causal effects are to be estimated as simultaneous feedback as opposed to one-sided 'blocks' of an equation where a researcher is interested in the causal effect of X on Y while holding the causal effect of Y on X constant, or when the researcher knows the exact amount of time it takes for each causal effect to take place, i.e., the length of the causal lags. Instead of lagged effects, simultaneous feedback means estimating the simultaneous and perpetual impact of X and Y on each other. This requires a theory that causal effects are simultaneous in time, or so complex that they appear to behave simultaneously; a common example are the moods of roommates. To estimate simultaneous feedback models a theory of equilibrium is also necessary – that X and Y are in relatively steady states or are part of a system (society, market, classroom) that is in a relatively stable state.
3294:
3005:
1350:
2198:
2773:
3069:
1923:
1111:
2735:
250:
1530:
3423:
3000:{\displaystyle {\Big (}{\begin{bmatrix}y_{i}\\Y_{-i}\end{bmatrix}}M_{i}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big )}{\Big (}{\begin{bmatrix}y_{i}\\Y_{-i}\end{bmatrix}}M{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big )}^{\!-1}}
3289:{\displaystyle {\Big |}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}'M_{i}{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}-\lambda {\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}'M{\begin{bmatrix}y_{i}&Y_{-i}\end{bmatrix}}{\Big |}=0}
57:
model: whilst typically one would determine the quantity supplied and demanded to be a function of the price set by the market, it is also possible for the reverse to be true, where producers observe the quantity that consumers demand
3664:
Across fields and disciplines simultaneous equation models are applied to various observational phenomena. These equations are applied when phenomena are assumed to be reciprocally causal. The classic example is supply and demand in
2400:
1765:
495:
1985:
3490:
822:
3555:
1345:{\displaystyle {\begin{aligned}y_{1}&=\gamma _{12}y_{2}+\delta _{11}z_{1}+\delta _{12}z_{2}+\delta _{13}z_{3}+u_{1}\\y_{2}&=\gamma _{21}y_{1}+\delta _{21}z_{1}+\delta _{22}z_{2}+u_{2}\end{aligned}}}
932:
1116:
1742:
1669:
599:
864:
3627:
estimator. Here K represents the number of instruments, n the sample size, and α a positive constant to specify. A value of α=1 will yield an estimator that is approximately unbiased.
2591:
2465:
2254:. For this, the structural system of equations is transformed into the reduced form first. Once the coefficients are estimated the model is put back into the structural form.
3615:
1359:
variables. Without further restrictions, the first equation is not identified because there is no excluded exogenous variable. The second equation is just identified if
1403:
641:
matrix Γ, which describes the relation between the dependent variables, has a complicated structure. It has ones on the diagonal, and all other elements of each column
4777:
2537:
2505:
115:
2579:
2559:
2280:
2274:
in 1949. It is used when one is interested in estimating a single structural equation at a time (hence its name of limited information), say for observation i:
17:
1607:. It is an equation-by-equation technique, where the endogenous regressors on the right-hand side of each equation are being instrumented with the regressors
4767:
3310:
1918:{\displaystyle y_{i}={\begin{pmatrix}Y_{-i}&X_{i}\end{pmatrix}}{\begin{pmatrix}\gamma _{i}\\\beta _{i}\end{pmatrix}}+u_{i}\equiv Z_{i}\delta _{i}+u_{i},}
2193:{\displaystyle {\hat {\delta }}_{i}={\big (}{\hat {Z}}'_{i}{\hat {Z}}_{i}{\big )}^{-1}{\hat {Z}}'_{i}y_{i}={\big (}Z'_{i}PZ_{i}{\big )}^{-1}Z'_{i}Py_{i},}
1090:
by crossing out those columns which correspond to the excluded endogenous variables, and those rows which correspond to the included exogenous variables.
383:
4437:
Xie, F.; Levinson, D. (2010-05-01). "How streetcars shaped suburbanization: a
Granger causality analysis of land use and transit in the Twin Cities".
4033:
Girshick, M. A.; Haavelmo, Trygve (1947). "Statistical
Analysis of the Demand for Food: Examples of Simultaneous Estimation of Structural Equations".
3434:
4517:
Wong, Chi-Sum; Law, Kenneth S. (1999-01-01). "Testing
Reciprocal Relations by Nonrecursive Structuralequation Models Using Cross-Sectional Data".
2271:
3056:
1027:, which can be phrased as “the number of excluded exogenous variables is greater or equal to the number of included endogenous variables”.
745:
3669:. In other disciplines there are examples such as candidate evaluations and party identification or public opinion and social policy in
1582:) on the right hand side. Therefore, cross equation restrictions in place of within-equation restrictions can achieve identification.
3496:
41:
are functions of other dependent variables, rather than just independent variables. This means some of the explanatory variables are
77:
of the regressors is violated. And while it would be natural to estimate all simultaneous equations at once, this often leads to a
4005:
Vajda, S.; Valko, P.; Godfrey, K.R. (1987). "Direct and indirect least squares methods in continuous-time parameter estimation".
942:
1102:
is by imposing within-equation parameter restrictions. Yet, identification is also possible using cross equation restrictions.
90:
4726:
4663:
4592:
3899:
Theil, H. (1953). Estimation and
Simultaneous Correlation in Complete Equation Systems (Memorandum). Central Planning Bureau.
3759:
3734:
1105:
To illustrate how cross equation restrictions can be used for identification, consider the following example from
Wooldridge
4312:
Page, Benjamin I.; Jones, Calvin C. (1979-12-01). "Reciprocal
Effects of Policy Preferences, Party Loyalties and the Vote".
38:
4742:
896:
4704:
4682:
4641:
4622:
4267:
4158:
4130:
4076:
3883:
3841:
3784:
74:
1705:
1632:
3705:
1099:
975:} should be iid, with zero mean and some covariance matrix Σ (which is unknown). In particular, this implies that
3695:
3649:
560:
4482:
Marini, Margaret Mooney (1984-01-01). "Women's
Educational Attainment and the Timing of Entry into Parenthood".
838:
4296:
3829:
3802:
3775:
Quandt, Richard E. (1983). "Computational
Problems and Methods". In Griliches, Z.; Intriligator, M. D. (eds.).
89:
in the 1940s and 1950s, of various techniques that estimate each equation in the model seriatim, most notably
3641:
2730:{\displaystyle {\hat {\delta }}_{i}={\Big (}Z'_{i}(I-\lambda M)Z_{i}{\Big )}^{\!-1}Z'_{i}(I-\lambda M)y_{i},}
2415:
1595:
The simplest and the most common estimation method for the simultaneous equations model is the so-called
3926:(1957). "A generalized classical method of linear estimation of coefficients in a structural equation".
1611:
from all other equations. The method is called “two-stage” because it conducts estimation in two steps:
870:
is quite complicated, and therefore the reduced form is more suitable for prediction but not inference.
4772:
4748:
3800:
Christ, Carl F. (1994). "The Cowles
Commission's Contributions to Econometrics at Chicago, 1939–1955".
1003:
82:
42:
3573:
4363:
Wlezien, Christopher (1995-01-01). "The Public as
Thermostat: Dynamics of Preferences for Spending".
3673:; road investment and travel demand in geography; and educational attainment and parenthood entry in
3859:
Wooldridge, J.M., Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass.
2267:
1525:{\displaystyle y_{1}-\delta _{12}z_{2}=\gamma _{12}y_{2}+\delta _{11}z_{1}+\delta _{13}z_{3}+u_{1}}
999:
78:
70:
4400:"Positive Returns and Equilibrium: Simultaneous Feedback Between Public Opinion and Social Policy"
1596:
94:
4632:
Fomby, Thomas B.; Hill, R. Carter; Johnson, Stanley R. (1984). "Simultaneous Equations Models".
4072:"Estimator of the parameters of a single equation in a complete system of stochastic equations"
3565:
2263:
2251:
1692:
832:
671:
contains all exogenous regressors from all equations, but without repetitions (that is, matrix
4582:
4284:
4255:
4122:
3648:
is common to all equations. If all regressors are in fact predetermined, then 3SLS reduces to
3645:
2471:
1561:
1035:
245:{\displaystyle y_{it}=y_{-i,t}'\gamma _{i}+x_{it}'\;\!\beta _{i}+u_{it},\quad i=1,\ldots ,m,}
4614:
4607:
3984:
Park, S-B. (1974) "On Indirect Least Squares Estimation of a Simultaneous Equation System",
2512:
2480:
1564:
to estimate the coefficients in the above equation since there are one endogenous variable (
550:
equation. Finally, we can move all endogenous variables to the left-hand side and write the
3690:
828:
50:
2395:{\displaystyle y_{i}=Y_{-i}\gamma _{i}+X_{i}\beta _{i}+u_{i}\equiv Z_{i}\delta _{i}+u_{i}}
8:
1356:
4219:(1962). "Three-stage least squares: simultaneous estimation of simultaneous equations".
4714:
4564:
4542:
4499:
4380:
4345:
4337:
4236:
4194:
4095:
4052:
3989:
3968:
3945:
3811:
3418:{\displaystyle {\hat {\delta }}={\Big (}Z'(I-\kappa M)Z{\Big )}^{\!-1}Z'(I-\kappa M)y,}
2564:
2544:
4722:
4700:
4678:
4659:
4637:
4618:
4588:
4546:
4534:
4464:
4419:
4349:
4329:
4292:
4263:
4154:
4126:
4115:
4018:
3879:
3837:
3780:
3755:
3730:
3670:
86:
54:
34:
4526:
4491:
4454:
4446:
4411:
4372:
4321:
4228:
4186:
4085:
4044:
4014:
3937:
3906:
2226:
is the projection matrix onto the linear space spanned by the exogenous regressors
66:
4177:(1977). "Some Properties of a Modification of the Limited Information Estimator".
490:{\displaystyle y_{i}=Y_{-i}\gamma _{i}+X_{i}\beta _{i}+u_{i},\quad i=1,\ldots ,m,}
339:(since it is already present on the left-hand side). The regression coefficients
3910:
3485:{\displaystyle \delta ={\begin{bmatrix}\beta _{i}&\gamma _{i}\end{bmatrix}}}
4567:
and Ralph O. Mueller, 2nd ed., 41–79. Charlotte, NC: Information Age Publishing
4212:
3923:
4090:
4071:
1397:
as known for the purpose of identification. Then, the first equation becomes:
1093:
4761:
4651:
4559:
2013. “Reverse Arrow Dynamics: Feedback Loops and Formative Measurement.” In
4538:
4468:
4423:
4333:
4530:
1972:)-dimensional vector of regression coefficients, then the 2SLS estimator of
4692:
4602:
4174:
4035:
3928:
3700:
2247:
2239:
735:
4450:
4216:
3963:
2243:
4459:
4752:
4503:
4384:
4341:
4240:
4198:
4099:
4056:
3993:
3986:
The Canadian Journal of Statistics / La Revue Canadienne de Statistique
3949:
3815:
3678:
4415:
1034:, a stronger condition which is necessary and sufficient, is that the
1013:, a necessary condition for identification, is that for each equation
3674:
3666:
2262:
The “limited information” maximum likelihood method was suggested by
46:
4495:
4376:
4325:
4232:
4190:
4048:
3941:
1098:
In simultaneous equations models, the most common method to achieve
817:{\displaystyle Y=X\mathrm {B} \Gamma ^{-1}+U\Gamma ^{-1}=X\Pi +V.\,}
4587:(Second ed.). Basingstoke: Palgrave Macmillan. p. 395.
2405:
The structural equations for the remaining endogenous variables Y
3550:{\displaystyle Z={\begin{bmatrix}X_{i}&Y_{-i}\end{bmatrix}}}
2257:
697:, and each of its columns consists of the components of vectors
893:(this later requirement means that in the limit the expression
4121:. Cambridge, Massachusetts: Harvard University Press. p.
835:. Unfortunately, the task of decomposing the estimated matrix
546:
matrix of endogenous regressors on the right-hand side of the
85:. This situation prompted the development, spearheaded by the
4744:
Lecture on the Identification Problem in 2SLS, and Estimation
2409:
are not specified, and they are given in their reduced form:
3836:(Second ed.). New York: McGraw-Hill. pp. 376–423.
1950:) matrix of both endogenous and exogenous regressors in the
3653:
1094:
Using cross-equation restrictions to achieve identification
306:
1 vector of all other endogenous variables which enter the
4262:(Second ed.). New York: Macmillan. pp. 695–701.
3636:
The three-stage least squares estimator was introduced by
2470:
Notation in this context is different than for the simple
1369:, which is assumed to be true for the rest of discussion.
4654:; Lahiri, Kajal (2009). "Simultaneous Equations Models".
3903:
Henri Theil’s Contributions to Economics and Econometrics
2246:
in a simultaneous equations model are estimated from the
3869:
3867:
3865:
1388:. Since the second equation is identified, we can treat
938:
matrix). Matrix Γ is also assumed to be non-degenerate.
4399:
4144:
4142:
377:
equation, we can write each equation in vector form as
69:
of the statistical parameters of interest, because the
4658:(Fourth ed.). New York: Wiley. pp. 355–400.
3640:. It can be seen as a special case of multi-equation
3511:
3449:
3304:
The LIML is a special case of the K-class estimators:
3237:
3189:
3141:
3086:
2943:
2897:
2842:
2789:
1828:
1787:
1709:
1636:
900:
842:
81:
non-linear optimization problem even for the simplest
3862:
3832:(1971). "Simultaneous-equation Methods: Estimation".
3576:
3499:
3437:
3313:
3072:
2776:
2594:
2567:
2547:
2515:
2483:
2418:
2283:
1988:
1768:
1708:
1635:
1406:
1114:
990:
Lastly, assumptions are required for identification.
899:
841:
748:
704:
and zeros, depending on which of the regressors from
563:
386:
118:
4139:
3652:(SUR). Thus it may also be seen as a combination of
619:
matrix of dependent variables. Each of the matrices
4721:(Fifth ed.). South-Western. pp. 554–582.
3631:
2767:is the smallest characteristic root of the matrix:
4606:
4114:
3967:
3609:
3549:
3484:
3417:
3288:
2999:
2729:
2573:
2553:
2531:
2499:
2459:
2394:
2192:
1917:
1736:
1663:
1524:
1344:
926:
858:
816:
593:
489:
244:
4778:Mathematical and quantitative methods (economics)
4148:
4004:
3725:Martin, Vance; Hurn, Stan; Harris, David (2013).
3659:
3374:
3368:
3331:
3275:
3075:
2988:
2982:
2887:
2880:
2779:
2674:
2668:
2619:
1724:
1651:
1590:
941:Secondly, error terms are assumed to be serially
919:
187:
4759:
4673:Ruud, Paul A. (2000). "Simultaneous Equations".
4631:
4291:. Princeton University Press. pp. 276–279.
4032:
3878:(5th ed.). Prentice Hall. pp. 398–99.
3779:. Vol. I. North-Holland. pp. 699–764.
3724:
1372:Now we impose the cross equation restriction of
1355:where z's are uncorrelated with u's and y's are
927:{\displaystyle \scriptstyle {\frac {1}{T}}X'\!X}
4675:An Introduction to Classical Econometric Theory
4149:Davidson, Russell; MacKinnon, James G. (1993).
4581:Asteriou, Dimitrios; Hall, Stephen G. (2011).
1737:{\displaystyle \scriptstyle {\hat {Y}}_{\!-i}}
1664:{\displaystyle \scriptstyle {\hat {Y}}_{\!-i}}
100:
49:usually is the consequence of some underlying
4677:. Oxford University Press. pp. 697–746.
4650:
4561:Structural Equation Modeling: A Second Course
4069:
3749:
2258:Limited information maximum likelihood (LIML)
2147:
2113:
2062:
2013:
886:, both in finite samples and in the limit as
269:is the observation index. In these equations
4768:Simultaneous equation methods (econometrics)
4580:
4211:
3754:(Fourth ed.). Wiley. pp. 355–357.
3637:
4699:. Oxford: Basil Blackwell. pp. 68–89.
4436:
3855:
3853:
3729:. Cambridge University Press. p. 159.
729:Postmultiplying the structural equation by
594:{\displaystyle Y\Gamma =X\mathrm {B} +U.\,}
369:1 correspondingly. Vertically stacking the
4713:
3570:κ=1: 2SLS. Note indeed that in this case,
859:{\displaystyle \scriptstyle {\hat {\Pi }}}
186:
4717:(2013). "Simultaneous Equations Models".
4458:
4311:
4089:
3560:Several estimators belong to this class:
2238:Indirect least squares is an approach in
2233:
882:of exogenous regressors must be equal to
831:, and it can be estimated for example by
813:
590:
4636:. New York: Springer. pp. 437–552.
4153:. Oxford University Press. p. 649.
4151:Estimation and inference in econometrics
3850:
3828:
1006:be solvable for the unknown parameters.
652:or zeros, depending on which columns of
645:are either the components of the vector
4697:Lectures on Advanced Econometric Theory
4516:
4397:
4362:
4282:
4112:
3922:
3617:the usual projection matrix of the 2SLS
1604:
1573:) and one excluded exogenous variable (
943:independent and identically distributed
14:
4760:
4691:
4481:
4253:
4173:
3873:
3799:
3774:
3750:Maddala, G. S.; Lahiri, Kajal (2009).
3727:Econometric Modelling with Time Series
3624:
2585:The explicit formula for the LIML is:
91:limited information maximum likelihood
65:Simultaneity poses challenges for the
45:with the dependent variable, which in
18:Limited information maximum likelihood
4365:American Journal of Political Science
3962:
3898:
3299:
3060:
1600:
321:” notation indicates that the vector
310:equation on the right-hand side, and
4672:
4601:
1759:equation in the model is written as
675:should be of full rank). Thus, each
604:This representation is known as the
554:equations jointly in vector form as
532:matrix of exogenous regressors, and
2561:: The instrument(s) (often denoted
2460:{\displaystyle Y_{-i}=X\Pi +U_{-i}}
1599:method, developed independently by
934:should converge to a nondegenerate
733:, the system can be written in the
24:
4613:. New York: McGraw-Hill. pp.
4574:
4070:Anderson, T.W.; Rubin, H. (1949).
2438:
866:into the individual factors Β and
846:
801:
783:
764:
759:
577:
567:
373:observations corresponding to the
25:
4789:
4736:
4314:American Political Science Review
4077:Annals of Mathematical Statistics
993:
284:1 vector of exogenous variables,
109:regression equations of the form
3706:Parameter identification problem
3632:Three-stage least squares (3SLS)
3610:{\displaystyle I-\kappa M=I-M=P}
3055:is the smallest solution of the
1629:and obtain the predicted values
964:, then the sequence of vectors {
878:Firstly, the rank of the matrix
4553:
4519:Organizational Research Methods
4510:
4475:
4430:
4391:
4356:
4305:
4276:
4247:
4205:
4167:
4106:
4063:
4025:
3998:
3978:
3956:
3696:Seemingly unrelated regressions
3650:seemingly unrelated regressions
708:were included or excluded from
462:
217:
4256:"System Methods of Estimation"
3916:
3892:
3822:
3803:Journal of Economic Literature
3793:
3768:
3743:
3718:
3660:Applications in social science
3406:
3391:
3359:
3344:
3320:
3057:generalized eigenvalue problem
2711:
2696:
2652:
2637:
2602:
2083:
2047:
2025:
1996:
1717:
1644:
1591:Two-stage least squares (2SLS)
1084:matrix which is obtained from
873:
849:
13:
1:
4439:Journal of Economic Geography
3711:
2507:: The endogenous variable(s).
1585:
31:Simultaneous equations models
4656:Introduction to Econometrics
4634:Advanced Econometric Methods
4484:American Sociological Review
4398:Breznau, Nate (2016-07-01).
4019:10.1016/0005-1098(87)90027-6
3752:Introduction to Econometrics
656:were included in the matrix
633:-columned submatrix of this
259:is the equation number, and
7:
3911:10.1007/978-94-011-2546-8_6
3874:Greene, William H. (2002).
3684:
2539:: The exogenous variable(s)
2266:in 1947, and formalized by
726:matrix of the error terms.
317:are the error terms. The “−
291:is the dependent variable,
101:Structural and reduced form
10:
4794:
3970:Principles of Econometrics
3638:Zellner & Theil (1962)
3010:where, in a similar way,
1004:system of linear equations
83:system of linear equations
4719:Introductory Econometrics
4113:Amemiya, Takeshi (1985).
1000:identification conditions
827:This is already a simple
27:Type of statistical model
4260:Elements of Econometrics
3777:Handbook of Econometrics
4531:10.1177/109442819921005
4285:"Multiple-Equation GMM"
4283:Hayashi, Fumio (2000).
4091:10.1214/aoms/1177730090
3974:. New York: John Wiley.
3654:two-stage least squares
1597:two-stage least squares
1009:More specifically, the
689:-columned submatrix of
328:may contain any of the
95:two-stage least squares
71:Gauss–Markov assumption
4715:Wooldridge, Jeffrey M.
4404:Policy Studies Journal
3646:instrumental variables
3611:
3551:
3486:
3419:
3290:
3001:
2731:
2575:
2555:
2533:
2532:{\displaystyle X_{-i}}
2501:
2500:{\displaystyle Y_{-i}}
2461:
2396:
2252:ordinary least squares
2234:Indirect least squares
2194:
1919:
1738:
1693:ordinary least squares
1665:
1526:
1346:
928:
860:
833:ordinary least squares
818:
595:
491:
246:
79:computationally costly
4117:Advanced Econometrics
4031:First application by
3612:
3552:
3487:
3420:
3291:
3002:
2732:
2576:
2556:
2534:
2502:
2462:
2397:
2195:
1920:
1739:
1666:
1527:
1347:
929:
861:
819:
596:
492:
247:
51:equilibrium mechanism
4584:Applied Econometrics
4254:Kmenta, Jan (1986).
3876:Econometric analysis
3691:General linear model
3574:
3497:
3435:
3311:
3070:
2774:
2592:
2565:
2545:
2513:
2481:
2416:
2281:
1986:
1766:
1706:
1633:
1404:
1112:
897:
839:
829:general linear model
746:
693:. Matrix Β has size
561:
384:
116:
3834:Econometric Methods
2695:
2636:
2173:
2130:
2097:
2039:
608:. In this equation
185:
156:
53:. Take the typical
39:dependent variables
4565:Gregory R. Hancock
4451:10.1093/jeg/lbp031
3905:(Springer, 1992),
3607:
3547:
3541:
3482:
3476:
3415:
3300:K class estimators
3286:
3267:
3219:
3171:
3116:
2997:
2973:
2929:
2872:
2821:
2727:
2683:
2624:
2571:
2551:
2529:
2497:
2457:
2392:
2190:
2161:
2118:
2076:
2018:
1915:
1857:
1817:
1734:
1733:
1661:
1660:
1522:
1342:
1340:
945:. That is, if the
924:
923:
856:
855:
814:
591:
487:
353:are of dimensions
242:
170:
135:
105:Suppose there are
43:jointly determined
4773:Regression models
4728:978-1-111-53104-1
4665:978-0-470-01512-4
4594:978-0-230-27182-1
4416:10.1111/psj.12171
3761:978-0-470-01512-4
3736:978-0-521-19660-4
3671:political science
3656:(2SLS) with SUR.
3644:where the set of
3623:κ=λ - α / (n-K):
3323:
2605:
2574:{\displaystyle Z}
2554:{\displaystyle X}
2086:
2050:
2028:
1999:
1979:will be given by
1720:
1647:
1535:Then, we can use
1002:require that the
909:
852:
87:Cowles Commission
75:strict exogeneity
55:supply and demand
35:statistical model
16:(Redirected from
4785:
4745:
4732:
4710:
4688:
4669:
4647:
4628:
4612:
4603:Chow, Gregory C.
4598:
4568:
4557:
4551:
4550:
4514:
4508:
4507:
4479:
4473:
4472:
4462:
4434:
4428:
4427:
4395:
4389:
4388:
4360:
4354:
4353:
4320:(4): 1071–1089.
4309:
4303:
4302:
4280:
4274:
4273:
4251:
4245:
4244:
4209:
4203:
4202:
4171:
4165:
4164:
4146:
4137:
4136:
4120:
4110:
4104:
4103:
4093:
4067:
4061:
4060:
4029:
4023:
4022:
4002:
3996:
3982:
3976:
3975:
3973:
3960:
3954:
3953:
3920:
3914:
3900:
3896:
3890:
3889:
3871:
3860:
3857:
3848:
3847:
3826:
3820:
3819:
3797:
3791:
3790:
3772:
3766:
3765:
3747:
3741:
3740:
3722:
3616:
3614:
3613:
3608:
3556:
3554:
3553:
3548:
3546:
3545:
3538:
3537:
3523:
3522:
3491:
3489:
3488:
3483:
3481:
3480:
3473:
3472:
3461:
3460:
3424:
3422:
3421:
3416:
3390:
3382:
3381:
3372:
3371:
3343:
3335:
3334:
3325:
3324:
3316:
3295:
3293:
3292:
3287:
3279:
3278:
3272:
3271:
3264:
3263:
3249:
3248:
3228:
3224:
3223:
3216:
3215:
3201:
3200:
3176:
3175:
3168:
3167:
3153:
3152:
3135:
3134:
3125:
3121:
3120:
3113:
3112:
3098:
3097:
3079:
3078:
3051:In other words,
3047:
3006:
3004:
3003:
2998:
2996:
2995:
2986:
2985:
2978:
2977:
2970:
2969:
2955:
2954:
2934:
2933:
2926:
2925:
2909:
2908:
2891:
2890:
2884:
2883:
2877:
2876:
2869:
2868:
2854:
2853:
2836:
2835:
2826:
2825:
2818:
2817:
2801:
2800:
2783:
2782:
2762:
2736:
2734:
2733:
2728:
2723:
2722:
2691:
2682:
2681:
2672:
2671:
2664:
2663:
2632:
2623:
2622:
2613:
2612:
2607:
2606:
2598:
2580:
2578:
2577:
2572:
2560:
2558:
2557:
2552:
2538:
2536:
2535:
2530:
2528:
2527:
2506:
2504:
2503:
2498:
2496:
2495:
2466:
2464:
2463:
2458:
2456:
2455:
2431:
2430:
2401:
2399:
2398:
2393:
2391:
2390:
2378:
2377:
2368:
2367:
2355:
2354:
2342:
2341:
2332:
2331:
2319:
2318:
2309:
2308:
2293:
2292:
2225:
2199:
2197:
2196:
2191:
2186:
2185:
2169:
2160:
2159:
2151:
2150:
2143:
2142:
2126:
2117:
2116:
2107:
2106:
2093:
2088:
2087:
2079:
2075:
2074:
2066:
2065:
2058:
2057:
2052:
2051:
2043:
2035:
2030:
2029:
2021:
2017:
2016:
2007:
2006:
2001:
2000:
1992:
1924:
1922:
1921:
1916:
1911:
1910:
1898:
1897:
1888:
1887:
1875:
1874:
1862:
1861:
1854:
1853:
1840:
1839:
1822:
1821:
1814:
1813:
1802:
1801:
1778:
1777:
1743:
1741:
1740:
1735:
1732:
1731:
1722:
1721:
1713:
1670:
1668:
1667:
1662:
1659:
1658:
1649:
1648:
1640:
1581:
1572:
1559:
1531:
1529:
1528:
1523:
1521:
1520:
1508:
1507:
1498:
1497:
1485:
1484:
1475:
1474:
1462:
1461:
1452:
1451:
1439:
1438:
1429:
1428:
1416:
1415:
1396:
1387:
1368:
1351:
1349:
1348:
1343:
1341:
1337:
1336:
1324:
1323:
1314:
1313:
1301:
1300:
1291:
1290:
1278:
1277:
1268:
1267:
1251:
1250:
1237:
1236:
1224:
1223:
1214:
1213:
1201:
1200:
1191:
1190:
1178:
1177:
1168:
1167:
1155:
1154:
1145:
1144:
1128:
1127:
1089:
1083:
1066:
1056:
1047:
1026:
986:
978:
933:
931:
930:
925:
918:
910:
902:
892:
869:
865:
863:
862:
857:
854:
853:
845:
823:
821:
820:
815:
794:
793:
775:
774:
762:
732:
721:
614:
600:
598:
597:
592:
580:
496:
494:
493:
488:
458:
457:
445:
444:
435:
434:
422:
421:
412:
411:
396:
395:
268:
251:
249:
248:
243:
213:
212:
197:
196:
181:
166:
165:
152:
131:
130:
21:
4793:
4792:
4788:
4787:
4786:
4784:
4783:
4782:
4758:
4757:
4743:
4739:
4729:
4707:
4685:
4666:
4644:
4625:
4595:
4577:
4575:Further reading
4572:
4571:
4558:
4554:
4515:
4511:
4496:10.2307/2095464
4480:
4476:
4435:
4431:
4396:
4392:
4377:10.2307/2111666
4371:(4): 981–1000.
4361:
4357:
4326:10.2307/1953990
4310:
4306:
4299:
4281:
4277:
4270:
4252:
4248:
4233:10.2307/1911287
4213:Zellner, Arnold
4210:
4206:
4191:10.2307/1912683
4172:
4168:
4161:
4147:
4140:
4133:
4111:
4107:
4068:
4064:
4049:10.2307/1907066
4030:
4026:
4003:
3999:
3988:, 2 (1), 75–82
3983:
3979:
3961:
3957:
3942:10.2307/1907743
3921:
3917:
3897:
3893:
3886:
3872:
3863:
3858:
3851:
3844:
3827:
3823:
3798:
3794:
3787:
3773:
3769:
3762:
3748:
3744:
3737:
3723:
3719:
3714:
3687:
3662:
3634:
3575:
3572:
3571:
3540:
3539:
3530:
3526:
3524:
3518:
3514:
3507:
3506:
3498:
3495:
3494:
3475:
3474:
3468:
3464:
3462:
3456:
3452:
3445:
3444:
3436:
3433:
3432:
3383:
3373:
3367:
3366:
3365:
3336:
3330:
3329:
3315:
3314:
3312:
3309:
3308:
3302:
3274:
3273:
3266:
3265:
3256:
3252:
3250:
3244:
3240:
3233:
3232:
3218:
3217:
3208:
3204:
3202:
3196:
3192:
3185:
3184:
3183:
3170:
3169:
3160:
3156:
3154:
3148:
3144:
3137:
3136:
3130:
3126:
3115:
3114:
3105:
3101:
3099:
3093:
3089:
3082:
3081:
3080:
3074:
3073:
3071:
3068:
3067:
3044:
3037:
3030:
3023:
3016:
3011:
2987:
2981:
2980:
2979:
2972:
2971:
2962:
2958:
2956:
2950:
2946:
2939:
2938:
2928:
2927:
2918:
2914:
2911:
2910:
2904:
2900:
2893:
2892:
2886:
2885:
2879:
2878:
2871:
2870:
2861:
2857:
2855:
2849:
2845:
2838:
2837:
2831:
2827:
2820:
2819:
2810:
2806:
2803:
2802:
2796:
2792:
2785:
2784:
2778:
2777:
2775:
2772:
2771:
2741:
2718:
2714:
2687:
2673:
2667:
2666:
2665:
2659:
2655:
2628:
2618:
2617:
2608:
2597:
2596:
2595:
2593:
2590:
2589:
2566:
2563:
2562:
2546:
2543:
2542:
2520:
2516:
2514:
2511:
2510:
2488:
2484:
2482:
2479:
2478:
2474:case. One has:
2448:
2444:
2423:
2419:
2417:
2414:
2413:
2408:
2386:
2382:
2373:
2369:
2363:
2359:
2350:
2346:
2337:
2333:
2327:
2323:
2314:
2310:
2301:
2297:
2288:
2284:
2282:
2279:
2278:
2260:
2236:
2204:
2181:
2177:
2165:
2152:
2146:
2145:
2144:
2138:
2134:
2122:
2112:
2111:
2102:
2098:
2089:
2078:
2077:
2067:
2061:
2060:
2059:
2053:
2042:
2041:
2040:
2031:
2020:
2019:
2012:
2011:
2002:
1991:
1990:
1989:
1987:
1984:
1983:
1977:
1970:
1966:
1959:
1948:
1944:
1933:
1906:
1902:
1893:
1889:
1883:
1879:
1870:
1866:
1856:
1855:
1849:
1845:
1842:
1841:
1835:
1831:
1824:
1823:
1816:
1815:
1809:
1805:
1803:
1794:
1790:
1783:
1782:
1773:
1769:
1767:
1764:
1763:
1749:
1723:
1712:
1711:
1710:
1707:
1704:
1703:
1700:
1689:
1682:
1650:
1639:
1638:
1637:
1634:
1631:
1630:
1623:
1593:
1588:
1580:
1574:
1571:
1565:
1557:
1550:
1543:
1536:
1516:
1512:
1503:
1499:
1493:
1489:
1480:
1476:
1470:
1466:
1457:
1453:
1447:
1443:
1434:
1430:
1424:
1420:
1411:
1407:
1405:
1402:
1401:
1395:
1389:
1386:
1379:
1373:
1366:
1360:
1339:
1338:
1332:
1328:
1319:
1315:
1309:
1305:
1296:
1292:
1286:
1282:
1273:
1269:
1263:
1259:
1252:
1246:
1242:
1239:
1238:
1232:
1228:
1219:
1215:
1209:
1205:
1196:
1192:
1186:
1182:
1173:
1169:
1163:
1159:
1150:
1146:
1140:
1136:
1129:
1123:
1119:
1115:
1113:
1110:
1109:
1096:
1085:
1081:
1074:
1068:
1065:
1058:
1054:
1049:
1046:
1039:
1023:
1019:
1014:
1011:order condition
996:
980:
976:
974:
963:
911:
901:
898:
895:
894:
887:
876:
867:
844:
843:
840:
837:
836:
786:
782:
767:
763:
758:
747:
744:
743:
730:
716:
713:
702:
687:
680:
661:
650:
631:
624:
609:
606:structural form
576:
562:
559:
558:
544:
537:
530:
523:
512:
505:
453:
449:
440:
436:
430:
426:
417:
413:
404:
400:
391:
387:
385:
382:
381:
366:
358:
351:
344:
337:
326:
315:
303:
296:
289:
281:
274:
260:
205:
201:
192:
188:
174:
161:
157:
139:
123:
119:
117:
114:
113:
103:
62:set the price.
28:
23:
22:
15:
12:
11:
5:
4791:
4781:
4780:
4775:
4770:
4756:
4755:
4738:
4737:External links
4735:
4734:
4733:
4727:
4711:
4705:
4689:
4683:
4670:
4664:
4652:Maddala, G. S.
4648:
4642:
4629:
4623:
4599:
4593:
4576:
4573:
4570:
4569:
4552:
4509:
4490:(4): 491–511.
4474:
4445:(3): 453–470.
4429:
4410:(4): 583–612.
4390:
4355:
4304:
4297:
4275:
4268:
4246:
4204:
4185:(4): 939–953.
4166:
4159:
4138:
4131:
4105:
4062:
4024:
4013:(6): 707–718.
3997:
3977:
3955:
3924:Basmann, R. L.
3915:
3891:
3884:
3861:
3849:
3842:
3821:
3792:
3785:
3767:
3760:
3742:
3735:
3716:
3715:
3713:
3710:
3709:
3708:
3703:
3698:
3693:
3686:
3683:
3661:
3658:
3633:
3630:
3629:
3628:
3621:
3618:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3568:
3558:
3557:
3544:
3536:
3533:
3529:
3525:
3521:
3517:
3513:
3512:
3510:
3505:
3502:
3492:
3479:
3471:
3467:
3463:
3459:
3455:
3451:
3450:
3448:
3443:
3440:
3426:
3425:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3389:
3386:
3380:
3377:
3370:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3342:
3339:
3333:
3328:
3322:
3319:
3301:
3298:
3297:
3296:
3285:
3282:
3277:
3270:
3262:
3259:
3255:
3251:
3247:
3243:
3239:
3238:
3236:
3231:
3227:
3222:
3214:
3211:
3207:
3203:
3199:
3195:
3191:
3190:
3188:
3182:
3179:
3174:
3166:
3163:
3159:
3155:
3151:
3147:
3143:
3142:
3140:
3133:
3129:
3124:
3119:
3111:
3108:
3104:
3100:
3096:
3092:
3088:
3087:
3085:
3077:
3042:
3035:
3028:
3021:
3014:
3008:
3007:
2994:
2991:
2984:
2976:
2968:
2965:
2961:
2957:
2953:
2949:
2945:
2944:
2942:
2937:
2932:
2924:
2921:
2917:
2913:
2912:
2907:
2903:
2899:
2898:
2896:
2889:
2882:
2875:
2867:
2864:
2860:
2856:
2852:
2848:
2844:
2843:
2841:
2834:
2830:
2824:
2816:
2813:
2809:
2805:
2804:
2799:
2795:
2791:
2790:
2788:
2781:
2738:
2737:
2726:
2721:
2717:
2713:
2710:
2707:
2704:
2701:
2698:
2694:
2690:
2686:
2680:
2677:
2670:
2662:
2658:
2654:
2651:
2648:
2645:
2642:
2639:
2635:
2631:
2627:
2621:
2616:
2611:
2604:
2601:
2583:
2582:
2570:
2550:
2540:
2526:
2523:
2519:
2508:
2494:
2491:
2487:
2468:
2467:
2454:
2451:
2447:
2443:
2440:
2437:
2434:
2429:
2426:
2422:
2406:
2403:
2402:
2389:
2385:
2381:
2376:
2372:
2366:
2362:
2358:
2353:
2349:
2345:
2340:
2336:
2330:
2326:
2322:
2317:
2313:
2307:
2304:
2300:
2296:
2291:
2287:
2268:T. W. Anderson
2264:M. A. Girshick
2259:
2256:
2235:
2232:
2201:
2200:
2189:
2184:
2180:
2176:
2172:
2168:
2164:
2158:
2155:
2149:
2141:
2137:
2133:
2129:
2125:
2121:
2115:
2110:
2105:
2101:
2096:
2092:
2085:
2082:
2073:
2070:
2064:
2056:
2049:
2046:
2038:
2034:
2027:
2024:
2015:
2010:
2005:
1998:
1995:
1975:
1968:
1964:
1957:
1954:equation, and
1946:
1942:
1931:
1926:
1925:
1914:
1909:
1905:
1901:
1896:
1892:
1886:
1882:
1878:
1873:
1869:
1865:
1860:
1852:
1848:
1844:
1843:
1838:
1834:
1830:
1829:
1827:
1820:
1812:
1808:
1804:
1800:
1797:
1793:
1789:
1788:
1786:
1781:
1776:
1772:
1753:
1752:
1747:
1730:
1727:
1719:
1716:
1698:
1695:regression of
1687:
1680:
1672:
1657:
1654:
1646:
1643:
1621:
1605:Basmann (1957)
1592:
1589:
1587:
1584:
1578:
1569:
1555:
1548:
1541:
1533:
1532:
1519:
1515:
1511:
1506:
1502:
1496:
1492:
1488:
1483:
1479:
1473:
1469:
1465:
1460:
1456:
1450:
1446:
1442:
1437:
1433:
1427:
1423:
1419:
1414:
1410:
1393:
1384:
1377:
1364:
1353:
1352:
1335:
1331:
1327:
1322:
1318:
1312:
1308:
1304:
1299:
1295:
1289:
1285:
1281:
1276:
1272:
1266:
1262:
1258:
1255:
1253:
1249:
1245:
1241:
1240:
1235:
1231:
1227:
1222:
1218:
1212:
1208:
1204:
1199:
1195:
1189:
1185:
1181:
1176:
1172:
1166:
1162:
1158:
1153:
1149:
1143:
1139:
1135:
1132:
1130:
1126:
1122:
1118:
1117:
1100:identification
1095:
1092:
1079:
1072:
1060:
1052:
1041:
1032:rank condition
1021:
1017:
995:
994:Identification
992:
968:
957:
953:is denoted by
949:row of matrix
922:
917:
914:
908:
905:
875:
872:
851:
848:
825:
824:
812:
809:
806:
803:
800:
797:
792:
789:
785:
781:
778:
773:
770:
766:
761:
757:
754:
751:
711:
700:
685:
678:
659:
648:
629:
626:is in fact an
622:
602:
601:
589:
586:
583:
579:
575:
572:
569:
566:
542:
535:
528:
521:
510:
503:
498:
497:
486:
483:
480:
477:
474:
471:
468:
465:
461:
456:
452:
448:
443:
439:
433:
429:
425:
420:
416:
410:
407:
403:
399:
394:
390:
364:
356:
349:
342:
335:
332:’s except for
324:
313:
301:
294:
287:
279:
272:
253:
252:
241:
238:
235:
232:
229:
226:
223:
220:
216:
211:
208:
204:
200:
195:
191:
184:
180:
177:
173:
169:
164:
160:
155:
151:
148:
145:
142:
138:
134:
129:
126:
122:
102:
99:
33:are a type of
26:
9:
6:
4:
3:
2:
4790:
4779:
4776:
4774:
4771:
4769:
4766:
4765:
4763:
4754:
4750:
4746:
4741:
4740:
4730:
4724:
4720:
4716:
4712:
4708:
4706:0-631-14956-2
4702:
4698:
4694:
4693:Sargan, Denis
4690:
4686:
4684:0-19-511164-8
4680:
4676:
4671:
4667:
4661:
4657:
4653:
4649:
4645:
4643:0-387-90908-7
4639:
4635:
4630:
4626:
4624:0-07-010847-1
4620:
4616:
4611:
4610:
4604:
4600:
4596:
4590:
4586:
4585:
4579:
4578:
4566:
4562:
4556:
4548:
4544:
4540:
4536:
4532:
4528:
4524:
4520:
4513:
4505:
4501:
4497:
4493:
4489:
4485:
4478:
4470:
4466:
4461:
4456:
4452:
4448:
4444:
4440:
4433:
4425:
4421:
4417:
4413:
4409:
4405:
4401:
4394:
4386:
4382:
4378:
4374:
4370:
4366:
4359:
4351:
4347:
4343:
4339:
4335:
4331:
4327:
4323:
4319:
4315:
4308:
4300:
4294:
4290:
4286:
4279:
4271:
4269:9780023650703
4265:
4261:
4257:
4250:
4242:
4238:
4234:
4230:
4226:
4222:
4218:
4214:
4208:
4200:
4196:
4192:
4188:
4184:
4180:
4176:
4175:Fuller, Wayne
4170:
4162:
4160:0-19-506011-3
4156:
4152:
4145:
4143:
4134:
4132:0-674-00560-0
4128:
4124:
4119:
4118:
4109:
4101:
4097:
4092:
4087:
4083:
4079:
4078:
4073:
4066:
4058:
4054:
4050:
4046:
4043:(2): 79–110.
4042:
4038:
4037:
4028:
4020:
4016:
4012:
4008:
4001:
3995:
3991:
3987:
3981:
3972:
3971:
3965:
3959:
3951:
3947:
3943:
3939:
3935:
3931:
3930:
3925:
3919:
3912:
3908:
3904:
3901:Reprinted in
3895:
3887:
3885:0-13-066189-9
3881:
3877:
3870:
3868:
3866:
3856:
3854:
3845:
3843:0-07-032679-7
3839:
3835:
3831:
3825:
3817:
3813:
3809:
3805:
3804:
3796:
3788:
3786:0-444-86185-8
3782:
3778:
3771:
3763:
3757:
3753:
3746:
3738:
3732:
3728:
3721:
3717:
3707:
3704:
3702:
3699:
3697:
3694:
3692:
3689:
3688:
3682:
3680:
3676:
3672:
3668:
3657:
3655:
3651:
3647:
3643:
3639:
3626:
3625:Fuller (1977)
3622:
3619:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3569:
3567:
3563:
3562:
3561:
3542:
3534:
3531:
3527:
3519:
3515:
3508:
3503:
3500:
3493:
3477:
3469:
3465:
3457:
3453:
3446:
3441:
3438:
3431:
3430:
3429:
3412:
3409:
3403:
3400:
3397:
3394:
3387:
3384:
3378:
3375:
3362:
3356:
3353:
3350:
3347:
3340:
3337:
3326:
3317:
3307:
3306:
3305:
3283:
3280:
3268:
3260:
3257:
3253:
3245:
3241:
3234:
3229:
3225:
3220:
3212:
3209:
3205:
3197:
3193:
3186:
3180:
3177:
3172:
3164:
3161:
3157:
3149:
3145:
3138:
3131:
3127:
3122:
3117:
3109:
3106:
3102:
3094:
3090:
3083:
3066:
3065:
3064:
3062:
3058:
3054:
3049:
3045:
3038:
3031:
3024:
3017:
2992:
2989:
2974:
2966:
2963:
2959:
2951:
2947:
2940:
2935:
2930:
2922:
2919:
2915:
2905:
2901:
2894:
2873:
2865:
2862:
2858:
2850:
2846:
2839:
2832:
2828:
2822:
2814:
2811:
2807:
2797:
2793:
2786:
2770:
2769:
2768:
2766:
2760:
2756:
2752:
2748:
2744:
2724:
2719:
2715:
2708:
2705:
2702:
2699:
2692:
2688:
2684:
2678:
2675:
2660:
2656:
2649:
2646:
2643:
2640:
2633:
2629:
2625:
2614:
2609:
2599:
2588:
2587:
2586:
2568:
2548:
2541:
2524:
2521:
2517:
2509:
2492:
2489:
2485:
2477:
2476:
2475:
2473:
2452:
2449:
2445:
2441:
2435:
2432:
2427:
2424:
2420:
2412:
2411:
2410:
2387:
2383:
2379:
2374:
2370:
2364:
2360:
2356:
2351:
2347:
2343:
2338:
2334:
2328:
2324:
2320:
2315:
2311:
2305:
2302:
2298:
2294:
2289:
2285:
2277:
2276:
2275:
2273:
2269:
2265:
2255:
2253:
2249:
2245:
2241:
2231:
2229:
2223:
2219:
2215:
2211:
2207:
2187:
2182:
2178:
2174:
2170:
2166:
2162:
2156:
2153:
2139:
2135:
2131:
2127:
2123:
2119:
2108:
2103:
2099:
2094:
2090:
2080:
2071:
2068:
2054:
2044:
2036:
2032:
2022:
2008:
2003:
1993:
1982:
1981:
1980:
1978:
1971:
1960:
1953:
1949:
1938:
1934:
1912:
1907:
1903:
1899:
1894:
1890:
1884:
1880:
1876:
1871:
1867:
1863:
1858:
1850:
1846:
1836:
1832:
1825:
1818:
1810:
1806:
1798:
1795:
1791:
1784:
1779:
1774:
1770:
1762:
1761:
1760:
1758:
1750:
1728:
1725:
1714:
1701:
1694:
1690:
1683:
1676:
1673:
1655:
1652:
1641:
1628:
1624:
1617:
1614:
1613:
1612:
1610:
1606:
1602:
1598:
1583:
1577:
1568:
1563:
1554:
1547:
1540:
1517:
1513:
1509:
1504:
1500:
1494:
1490:
1486:
1481:
1477:
1471:
1467:
1463:
1458:
1454:
1448:
1444:
1440:
1435:
1431:
1425:
1421:
1417:
1412:
1408:
1400:
1399:
1398:
1392:
1383:
1376:
1370:
1363:
1358:
1333:
1329:
1325:
1320:
1316:
1310:
1306:
1302:
1297:
1293:
1287:
1283:
1279:
1274:
1270:
1264:
1260:
1256:
1254:
1247:
1243:
1233:
1229:
1225:
1220:
1216:
1210:
1206:
1202:
1197:
1193:
1187:
1183:
1179:
1174:
1170:
1164:
1160:
1156:
1151:
1147:
1141:
1137:
1133:
1131:
1124:
1120:
1108:
1107:
1106:
1103:
1101:
1091:
1088:
1082:
1075:
1063:
1055:
1044:
1037:
1033:
1028:
1025:
1012:
1007:
1005:
1001:
991:
988:
984:
972:
967:
961:
956:
952:
948:
944:
939:
937:
920:
915:
912:
906:
903:
890:
885:
881:
871:
834:
830:
810:
807:
804:
798:
795:
790:
787:
779:
776:
771:
768:
755:
752:
749:
742:
741:
740:
738:
737:
727:
725:
719:
714:
707:
703:
696:
692:
688:
681:
674:
670:
666:
662:
655:
651:
644:
640:
636:
632:
625:
618:
612:
607:
587:
584:
581:
573:
570:
564:
557:
556:
555:
553:
549:
545:
538:
531:
524:
517:
513:
506:
484:
481:
478:
475:
472:
469:
466:
463:
459:
454:
450:
446:
441:
437:
431:
427:
423:
418:
414:
408:
405:
401:
397:
392:
388:
380:
379:
378:
376:
372:
368:
360:
352:
345:
338:
331:
327:
320:
316:
309:
305:
297:
290:
283:
275:
267:
263:
258:
239:
236:
233:
230:
227:
224:
221:
218:
214:
209:
206:
202:
198:
193:
189:
182:
178:
175:
171:
167:
162:
158:
153:
149:
146:
143:
140:
136:
132:
127:
124:
120:
112:
111:
110:
108:
98:
96:
92:
88:
84:
80:
76:
72:
68:
63:
61:
56:
52:
48:
44:
40:
37:in which the
36:
32:
19:
4718:
4696:
4674:
4655:
4633:
4609:Econometrics
4608:
4583:
4563:, edited by
4560:
4555:
4525:(1): 69–87.
4522:
4518:
4512:
4487:
4483:
4477:
4460:11299/179996
4442:
4438:
4432:
4407:
4403:
4393:
4368:
4364:
4358:
4317:
4313:
4307:
4289:Econometrics
4288:
4278:
4259:
4249:
4227:(1): 54–78.
4224:
4221:Econometrica
4220:
4217:Theil, Henri
4207:
4182:
4179:Econometrica
4178:
4169:
4150:
4116:
4108:
4084:(1): 46–63.
4081:
4075:
4065:
4040:
4036:Econometrica
4034:
4027:
4010:
4006:
4000:
3985:
3980:
3969:
3964:Theil, Henri
3958:
3936:(1): 77–83.
3933:
3929:Econometrica
3927:
3918:
3902:
3894:
3875:
3833:
3830:Johnston, J.
3824:
3810:(1): 30–59.
3807:
3801:
3795:
3776:
3770:
3751:
3745:
3726:
3720:
3701:Reduced form
3663:
3635:
3559:
3427:
3303:
3052:
3050:
3040:
3033:
3026:
3019:
3012:
3009:
2764:
2758:
2754:
2750:
2746:
2742:
2739:
2584:
2469:
2404:
2261:
2250:model using
2248:reduced form
2244:coefficients
2240:econometrics
2237:
2227:
2221:
2217:
2213:
2209:
2205:
2202:
1973:
1962:
1955:
1951:
1940:
1936:
1929:
1927:
1756:
1754:
1745:
1696:
1685:
1678:
1674:
1626:
1619:
1615:
1608:
1601:Theil (1953)
1594:
1575:
1566:
1552:
1545:
1538:
1534:
1390:
1381:
1374:
1371:
1361:
1354:
1104:
1097:
1086:
1077:
1070:
1061:
1050:
1042:
1031:
1029:
1015:
1010:
1008:
997:
989:
982:
970:
965:
959:
954:
950:
946:
940:
935:
888:
883:
879:
877:
826:
736:reduced form
734:
728:
723:
717:
709:
705:
698:
694:
690:
683:
676:
672:
668:
664:
657:
653:
646:
642:
638:
634:
627:
620:
616:
610:
605:
603:
551:
547:
540:
533:
526:
519:
515:
508:
501:
499:
374:
370:
362:
354:
347:
340:
333:
329:
322:
318:
311:
307:
299:
292:
285:
277:
270:
265:
261:
256:
254:
106:
104:
64:
59:
30:
29:
3063:, p. 503):
3061:Theil (1971
1677:: Estimate
1562:instruments
874:Assumptions
715:. Finally,
518:1 vectors,
4762:Categories
4753:Mark Thoma
4298:1400823838
4007:Automatica
3712:References
3679:demography
2242:where the
1618:: Regress
1586:Estimation
1357:endogenous
264:= 1, ...,
67:estimation
4547:122284566
4539:1094-4281
4469:1468-2702
4424:1541-0072
4350:144984371
4334:0003-0554
3675:sociology
3667:economics
3620:κ=λ: LIML
3596:−
3584:κ
3581:−
3532:−
3466:γ
3454:β
3439:δ
3401:κ
3398:−
3376:−
3354:κ
3351:−
3321:^
3318:δ
3258:−
3210:−
3181:λ
3178:−
3162:−
3107:−
2990:−
2964:−
2920:−
2863:−
2812:−
2706:λ
2703:−
2676:−
2647:λ
2644:−
2603:^
2600:δ
2522:−
2490:−
2450:−
2439:Π
2425:−
2371:δ
2357:≡
2335:β
2312:γ
2303:−
2154:−
2084:^
2069:−
2048:^
2026:^
1997:^
1994:δ
1891:δ
1877:≡
1847:β
1833:γ
1796:−
1726:−
1718:^
1653:−
1645:^
1491:δ
1468:δ
1445:γ
1422:δ
1418:−
1307:δ
1284:δ
1261:γ
1207:δ
1184:δ
1161:δ
1138:γ
850:^
847:Π
802:Π
788:−
784:Γ
769:−
765:Γ
568:Γ
476:…
438:β
415:γ
406:−
231:…
190:β
159:γ
141:−
47:economics
4695:(1988).
4605:(1983).
3966:(1971).
3685:See also
3388:′
3341:′
3226:′
3123:′
2693:′
2634:′
2272:H. Rubin
2171:′
2128:′
2095:′
2037:′
1057:, where
916:′
183:′
154:′
60:and then
4749:YouTube
4615:117–121
4504:2095464
4385:2111666
4342:1953990
4241:1911287
4199:1912683
4100:2236803
4057:1907066
3994:3314964
3950:1907743
3816:2728422
1961:is an (
1755:If the
1691:by the
1048:equals
667:matrix
615:is the
298:is the
276:is the
4725:
4703:
4681:
4662:
4640:
4621:
4591:
4545:
4537:
4502:
4467:
4422:
4383:
4348:
4340:
4332:
4295:
4266:
4239:
4197:
4157:
4129:
4098:
4055:
3992:
3948:
3882:
3840:
3814:
3783:
3758:
3733:
3428:with:
3059:, see
2763:, and
2740:where
2203:where
1928:where
1675:Step 2
1616:Step 1
979:, and
663:. The
637:. The
500:where
361:1 and
255:where
4543:S2CID
4500:JSTOR
4381:JSTOR
4346:S2CID
4338:JSTOR
4237:JSTOR
4195:JSTOR
4096:JSTOR
4053:JSTOR
3990:JSTOR
3946:JSTOR
3812:JSTOR
3564:κ=0:
3020:I − X
2747:I − X
1935:is a
1071:k − k
1067:is a
977:E = 0
722:is a
682:is a
539:is a
525:is a
4723:ISBN
4701:ISBN
4679:ISBN
4660:ISBN
4638:ISBN
4619:ISBN
4589:ISBN
4535:ISSN
4465:ISSN
4420:ISSN
4330:ISSN
4293:ISBN
4264:ISBN
4155:ISBN
4127:ISBN
3880:ISBN
3838:ISBN
3781:ISBN
3756:ISBN
3731:ISBN
2270:and
1744:and
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