BEBR FACULTY WORKING PAPER NO. 1176 Tests for Serial Dependence and Other Specification Analysis in Models of Markets in Disequilibrium Anil K. Bera Peter M. Robinson College of Commerce and Business Administration Bureau of Economic and Business Research University of Illinois. Urbana-Champaign BEBR FACULTY WORKING PAPER NO. 1176 College of Commerce and Business Administration University of Illinois at Urbana-Champaign September, 1985 Tests for Serial Dependence and Other Specification Analysis in Models of Markets in Disequilibrium Anil K. Bera, Assistant Professor Department of Economics Peter M. Robinson London School of Economics and Political Science An earlier version of this paper was presented at the Econometric Society Winter Meetings, San Francisco, 1983. We would like to thank Professor Lung-Fei Lee for providing us with the Fair and Jaffee (1972) data set and Mr. Shigetaka Miyazaki for very competent research assistance. The financial support of the Bureau of Economic and Business Research of the University of Illinois through an IBE Summer Grant is gratefully acknowledged Abstract The assumption of serial independence of the disturbance terms is the starting point of almost all the work, that has been done on analyzing market disequilibrium models. Under serial dependence the usual maximum likelihood estimator (MLE) will be inefficient although they may remain consistent. Two other assumptions about the distur- bance terras, namely normality and homoscedastici ty , are also usually made. Violation of these is likely to make the MLE to be inconsistent. In this paper, we first derive tests for serial dependence given nor- mality and homoscedastici ty using the Lagrange multiplier (LM) test principle. The likelihood function under serial dependence is very complicated and involves multiple integrals of dimensions equal to the sample size, ruling out the possibility of using the Wald and likeli- hood ratio tests. However, the test statistic we obtain through the LM principle is very simple. Next, we relax the normality assumption using the Box-Cox transformation family and generalize the above test to such non-normal cases. We also suggest a general specification test for the disequilibrium market model, namely white's information matrix test, which can detect non-normality, heteroscedastici ty and serial dependence jointly. This procedure will be useful when there is not much information about the alternative hypothesis. In the last part of the paper we present an analysis of disequilibrium models assuming that the disturbances are logistic rather than normal. This modification does simplify the computations quite a lot since for the logistic distribution a simple closed form expression for its distribution func- tion is available. The relative performance of this distribution com- pared to the normal distribution is also investigated. Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/testsforserialde1176bera 1. Introduction Models of markets in disequilibrium (MMD) have been studied, both theoretically and empirically, by many authors in recent years [see Quandt (1982) or Maddala (1983), for a review]. Unlike many other econometric models, however, MMD have yet to be subjected to rigorous specification analysis [though see Quandt (1981) and Lee (1982, 1984)]. In most studies, for example, disturbances are assumed to be serially independent (I), whereas MMD are designed to describe time series data. Since autocorrelation in economic time series is not always accounted for by the exogenous variables , serial dependence might be expected to be present. Fair and Jaffee (1972) incorporated serial correlation in their model, but for estimation they had to make an assumption that there was an equilibrium (that is, both the demand and supply were equal to the actual quantity transacted) before every switching point. Fair and Kelejian (1974) noted that when the disturbances are serially correlated, the coefficients in the demand and supply equations are not identified. Ameraiya (1974) and Maddala and Nelson (1974) commented that serial correlation is a serious problem and, when incorporated into the model, it renders maximum likelihood estimation com- putationally intractable. The usual maximum likelihood estimators (MLEs) maximize a likelihood based on serially independent, identically distributed, normal disturbances. We continue to call these MLEs even when one or more of the assumptions are in fact breached by the data to hand. Presence of serial dependence (I) will render the MLEs asymptotically inefficient, though they may still be consistent. Failure of the other assumptions, normality (N) and homoscedasticity (H), may lead to inconsistency of the MLEs. -2- In the present paper, we develop some specification tests, with an emphasis on I, for a particular type of disequilibrium model, namely Model 1 of Maddala and Nelson (1974). Other MMD, such as Model 2 of Maddala and Nelson, which assume knowledge of sample separation will be easier to handle. Throughout, the Lagrange multiplier (LM) or score test principle is adopted, partly because of its good asymptotic power properties but mostly because the likelihood under I is vastly more tractable than that under I. As noted by Maddala and Nelson (1974), the latter involves multiple integrals of dimension equal to the sample size. [Quandt (1981) gives an expression under first-order autoregressive (AR(1)) disturbances, that involves only univariate integrals but, as noted by Lee (1984), and as we indicate in the next section, this is not the likelihood.] Thus Wald and likelihood ratio tests are not feasible. In the next section we derive the LM test for I assuming NH, our analysis being along the lines of Robinson et al. (1985) where a similar test for the Tobit model was proposed. The class of possible alter- natives to I is, in certain respects, broader than those in Lee (1984) where he considered only AR(1) processes, but Lee's maintained assump- tion of contemporaneously correlated residuals is more general than ours. In another paper Lee (1982) also derived a test for non-normality (N) with alternatives belonging to the bivariate Edgeworth series of distributions. His test strongly rejected the normality assumption when applied to the housing data of Fair and Jaffee (1972). The tests suggested in Section 2 will not be valid under N. Therefore, in Sec- tion 3, we relax the normality assumption, using the Box-Cox transfor- -3- mation, and test for I given NH. Subsequently, the possibility of heteroscedasticity (H) is also introduced. Then usiag White (1982) information matrix test principle, we suggest a general specification test. This procedure can be viewed as a simultaneous test of INH. Such a test is not likely to be very powerful when only one or two of I, N or H are violated, but it does seem useful as a first step when there is little information concerning possible departures from the classical assumptions. A strategy to be adopted should any of the tests reject would be desirable, as would be a development of tests of N and H given I. Unfortunately both objectives are hampered by the difficulty, as previously mentioned, of ML estimation under I. It may be worth investigating whether at least consistent estimation of serial correlation is possible, for example by moment methods or methods simi- lar to those of Robinson (1982b), which apply to the Tobit model. It is now well known that normality of the disturbances is not an assumption that can be taken for granted in the context of MMD; nor, unlike in the usual regression model, does it lead to neat statistical properties or to computational convenience. It is, therefore, worth considering alternative distributions which may be just as relevant as the normal but make computation easier. Goldfeld and Quandt (1981) considered a Laplace type distribution which they called the Sargan density. However, for the simple disequilibrium model this did not lead to much computational gain, perhaps due to the "piecewise" nature of their likelihood function. In our view, in this situation, a better alternative is the logistic distribution. The shape of the logistic distribution is very similar to the normal, with a slightly -4- thicker tail, and, most important, it has a closed from expression for its distribution function. In Section 4, we study the likelihood function assuming a logistic distribution, and generate data under the normal, logistic and Student's t distributions and make a comparative study of the performances of the normal and logistic distributions. 2. Tests for Serial Dependence under Normality 2.1 Likelihood Function with Serial Dependence We consider the following simple disequilibrium model D t = X lt 3 l + U lt' (2,1) S t = X 2t 3 2 + U 2t' (2 ' 2) Q t = min(D t ,S t ) t-1,2,..., (2.3) where D , S and Q denote respectively the quantity demanded, supplied and transcated at period t, x and x are column vectors of k, and k„ exogenous variables and u, and u„ are the disturbance 1 2 5 It 2t terms such that E(u. |x, , s=l,2,...) = 0, i = 1,2. We also assume, it is like Quandt (1981), but unlike Lee (1984), that u and u are inde- pendent for al 1 t and s. Quandt (1981) assumed that the u are also AR(1) and set up the function T n h (Q |D ,S ), (2.4) t=l l t ° ° where h (Q Id ,S ) is the conditional density of Q given D ,S , and t x t ' o' o J t & o' o involves only univariate integrals. Quandt (1981) described (2.4) as the likelihood function, but in fact the latter is of the form -5- T h(Q 1 ,Q 2 ,...Q T ) = h (Q ) H h (Q |Q ,...,Q ). (2.5) t=2 Although estimates maximizing (2.4) may be consistent, they are not efficient, and indeed the autocorrelation coefficients seem likely to be poorly identified by (2.4) because h (Q Id ,S ) * as t * °°. Perhaps this explains the seeming contradiction between the significance of the likelihood ratio test for I and the insignificance of the indi- vidual autocorrelation coefficients, mentioned in Quandt (1981), though this might as well be due to the presence of other misspecif ica- tion(s) such as N or a wrong functional form in (2.1) and/or (2.2). Let f(D,S;9) be the joint probability density function (p.d.f) of the lx2T vector (D,S) = ^ ,D„ , . . . ,D ,S ,S 2 , . . . ,S ) , which is a given function of D, S and an unknown parameter vector 8. The log- likelihood function is T w 1-w L(6) - log E J f(D,S;6) II (dD ) mt (dS ) mt m R t=l (2.6) Q where Q = (Q 1 ,Q 2 »---»Q T ). / = / ••• / . R Q x Q T w = 1 if Q = D , mt x t t' = if Q = S , x t t ' T and the sura is over all 2 choices of (w , ,w „,..., w „,) , that is, over ml mz mi all possible demand/supply sequences. Under our setup D and S are independent, i.e., -6- f(D,S;6) = f 1 (D;e i )f 2 (S;8 2 ), (2.7) i r where 9' = (99). We now introduce I explicitly into the model. As a class of alternatives to independence we take, as in Robinson (1982a), u to be generated by a stationary Gaussian process. The j-th autocorrelation of u. , P..(^.), is a uniquely defined, dif f erentiable function of a it ij i p. -dimensional column vector ^ . , i=1.2. Let R.(^.) be the TxT *i l 11 Toeplitz matrix with (j , j+k)-th element P (^.). If 4> (y;M,^) denotes IK. 1 X the p.d.f of a T-variate normal distribution with mean u and variance- covariance matrix ft, the log-likelihood function can be written as * L(9) = log Z / i(D;Xi o 2 R(^)) ■ * (S;X J> 2 ,oh ( # ) ) m R T w 1-w n (dDj mt (dS ) mt | , (2.8) t-1 C C : Q 2 where o. = V(u. ) and X. is the Txk . matrix with t-th row x. , i=l,2. l it i l it 2.2 The Test Statistics We assume, as in Robinson et al. (1985), for a unique value of i> . , which we take with no Loss of generality to be the vector of zeros, that P..(0) = 0, i=l,2; j=l,2,... . So a test of I can be expressed as H : ■-P. = 0, i = 1,2 against H, : * J * 0, for some i. (2.9) o l ° 1 1 The alternatives we shall emphasize are the p.-th order autoregression [AR(p.)] and the p.-th order moving average [MA(p.)] processes. As in -7- the classical regression case, the same LM statistic will result from both AR(p.) and MA(p.) alternatives. The statistic falls out quickly from the general form which we shall derive. Statistics for various other types of alternative can be readily obtained from our general form. It is clear from the expression (2.8) that the likelihood function involves normal integrals of dimension equal to the sample size, and in general it cannot be expressed in terras of normal integrals of unit dimension. Therefore, maximization of (2.8) will be hampered by the need to compute integrals of very large dimension. Thus the Wald and likelihood ratio tests are not at all practically feasible. It can be shown that under H , when R (V.) = R_('>»0 = I™, where I is an identity matrix of dimension T, (2.8) reduces to the log-likelihood function given in Maddala and Nelson (1974, p. 1015). To see that and for our future use, we introduce the following notations: t t o t o t t t r i 9 A- (B r°r 8 2'V' V'W' 9 ' =2 - oo ■ F l(Qt ) - ) vv*itV a i )d V and F 2 (Q t ) - J W X i t 3 2'°2 )dS t ^t For convenience, we will drop the symbol Q and write ^(Q*.) ds f etc. Now from (2.8) -8- we ) W . 1-W . mt . . » mt log 2 / II {f a (D )f 2 (S )(dD ) (dS ) m R t=l ■ log I il (w mt F u f 2t + U-W mt )f u F 2t > m t=l Hog (If + f F 2t ) . t = l (2.10) Since the log-likelihood function simplifies considerably under H , we turn to the LM approach and develop tests for I. Let us consider the partition 9' = (9 .9 ) where we want to test A B H : 9=0. There are many asymptotically equivalent forms of the LM o B test statistic and we will use the following form: LM = — 77! M ay B B where " indicates the quantities have been evaluated at the restricted MLE of 9 say 9 and M satisfies plim [I E C --l^i£i}| - M] = 0. B B ,y The above form is valid since under our setup E (£&1) - « ay,; (2.11) as shown in the subsequent derivation, -9- To derive LM we require Che derivatives (under H ) -~- log (z ;0,a )dz t 1 t' ' 1 t V*u B i + (l-w mc )(Q t -x u3l ) • f (Q ) J Q r x 2t 3 2 V Z t ;0, °2 )dz t W n,tVlt f 2t + (I -V> ( V X l t B l )f l t F 2 t -10- Thus T [exp L(« )] >-^r £ £ P. 3V 1J ° 2a2 C)U =i l.t"»J * ' {w mt a i f lt f 2t + °-V )( V X it B l )f n F 2t } m • (w a*f f + (i- w )(q - x ' 3 )f F_ } mu 1 lu 2u mu u lu 1 lu 2u n { w F., f_. + (1-w , )f F , 1. i ^ t mk Ik 2k mk Ik 2k Following our derivation of (2.10), we can write i T "rrr exp L( 8 ) - -~ L I 'p Y Y. d ^T • o /J .2 l,t-u,j It lu lj 2o t,u=l ,J " tF Hc f 2 k + f lk F 2k } > k=l k*t,u where r jt = ^ lt [ \i 2t + ( V X ltV F 2t ]# Therefore, £ £ P, Y Y II (F,,f +f F ) . l,t-u,j T lt r lu . . Ik 1 2k lk*2k ; t,u=l J k=i i_ [ L (8 )] --3 _^ u Jc*t^u 3ip o 2 T T-l T — r- 2 P L V V ~2 . . lki . . It l,t-k, o k=l J t=k+l ' ' -11- where it °l f lt £ 2t + ( V x it 8 l )f lt F 2t 1C F U £ 2t + f lt F 2t F lt f 2t + f !t F 2t Under H , the v are independent, and noting that the denominator of v is the p.d.f of Q , say h(Q ), we have «v lt ) " / It h(Q t ) h ] -- I P lk * + c-sg- W-**- o. k=l t=k+l 1 l ' 33 3^ i i 2 T-l T [L(8 )1 = -^— L(6 rt ) + -i Z p Z {v (— v ) ^2 ' l v o yj 2 3v. v (T 2 . , ik , * t\2 i.t-k' 3a. 3^. a. i a. k=l t=k+l 3a. * li l l l i and these have zero expectations, by the independence of v over t and E(v. ) = 0. Of course it E{ ^r [L(8 )]} = 0, and El * ; [L(6 )]} = 0, 33.3^. 3a z 3\|;. ° J i J i for i * j , verifying equation (2.11) and E(— — ,[L(8 )]} = 0, 1 J for i * j. Finally, -13- Ei-n-^- L(d ) -tA- L(t> )} a^. o 0^ o , T-l T-l T T E[ ^ Epp I E V v t v v ] a. 4 , . , ikj ImA . . . . ... it i,t-k iu i,u-m' l k=l m=l J t=k+l u=k+l T-l T 1 9 9 E p., .p., . E[ ^ vf vf ] a. 4 Ikj iki \ . _, It i,t-k' l k=l J t=k+l T-l T Z P...P... 2 A A where A = -t E(v 2 ). (2.13) k=1 ikj ik* t=k+1 it i,t-k it Q 2 it/ i 2 A closed form expression for E(v ) does not appear to be available, however, so we replace (2.13) by T-l T V = /, p ikj p iw c ik> c ik =a - I , v it u i,t-k> J k=l J a. 4 t=k+l 2 "2 where v. is v. with tf. and 0. replaced by their MLEs 3. and a . it it l l K J 11 The LM statistic is thus given by 2 , LM = l l.H.l. , i-1 2 J * , T-l T - - where L. =- — £ P.,d., with d., = Z *> v and M. = ((M. .,,)). 1 a. 2 k=l lk lk lk t-k+1 1C 1,t_k * 1J " l When the null hypothesis H is true, under certain regularity condi tions V LM + x 2 • (2.14) P l + P 2 -14- These regularity conditions may be established using the results of Hartley and Mallela (1977) and combining the conditions of Robinson et al. (1985) on the P..(>K). We now deduce the form of LM for testing against AR(p.) and MA(p.) alternatives. For both the AR(p ) and MA(p.) P., . =1 for k = j lkj J = for k * j, so that in either case, the LM test statistic reduces to LM(p p ) = T I I rf (2.15) 1 l i=l k=l llc 1 d ik where r., = — -s- . Note that (2.15) is analogous to the familiar lk ,- 2 l/n /T c cf/2 i ik Box-Pierce portmanteau statistic for testing for serial correlation. 2.3 Empirical Illustration We apply the test to the housing start model of Fair and Jaffee (1972). Their versions of (2.1 )— (2.3) are [see Maddala and Nelson (1974, p. 1025)] D t =a + Vlt +U 2 Z 2t + V3t + U lt' (2 ' 16) S t = 3 + Vlt + d 2 Z 4t + y 3 Z 5t + Vet + U 2f (2 ' 17) Q t = min(D ,S ), (2.18) -15- where Q is the observed number of housing starts in period t, z is a time trend, z is a measure of the stock of houses, z~ is the mortgage rate lagged two periods, z, is the six month moving average of the flow of private deposits in savings and loan associations (SLAs) and mutual savings banks, lagged one period, z is the flow of borrowings by the SLAs from the Federal Home Loan Bank lagged two periods, and z = z . Using the Fair and Jaffee (1972) data with o t J , t+1 a sample of size 126, Maddala and Nelson (1974, Table II) obtained the MLEs which are reproduced below: n - I a n a i a o a T S n t*i 3 o S-j *, a ? a o Parameter 1 2 3 12 3 4 1 2 Estimate j 223.74 2.52 -.022 -.090 15.55 -.153 .053 .053 .093 350.0 80.2 Lee (1982) found evidence of no contemporaneous correlation between the u. for these data, so the setting for an LM test for I seems appropriate here. We use the above estimates and obtain the following components of the LM statistic given in (2.15): Tr ll = 4 * 4733 Tr 21 = 23.9101 It 1 = 6.0109 Tr^ = .4636 TrJ_ = 10.3031 TrJ, = .9269 i~l 11 26 Tr^ = 1.7940 Tr^ = 9.0344 Tr^ - .3906 Tr^ 4 = .0003 Tr^ = 6.7310 Tr^g = .2724 It seems, the length of the disturbance AR process is short for the demand equation whereas for the supply equation it is quite the oppo- site. Using the above values we can form a number of test statistics, e.g. , -16- LM(1,0) = 4.4733 LM(0,1) = 23.9101 LM(1,1) = 28.3834 LM(1,2) = 38.6865 LM(2,1) = 28.8470 LM(2,2) = 39.1501 2 The X critical values for 1, 2, 3, and 4 degrees of freedom at one per- cent significant level are respectively 6.63, 9.21, 11.34, and 13.28. Therefore, we strongly reject the null hypothesis of no serial dependence. The inference remains the same even if we consider higher values of p and p ? . Here we should note that our sample size is 2 126, and the asymptotic X critical values are unlikely to be appro- priate. However, for this particular example, the statistic values for the supply equation are so large that the conclusion of u being serially correlated seems justified insofar as the maintained assumption of correct specification, normality and horaoscedasticity are valid. 3. Tests for Serial Dependence in the Presence of Non-normality and A General Specification Test 3.1 The Test Statistics for Serial Dependence under Non-normality In the last section, a test was derived under the assumption that the disturbances are normally distributed. When this assumption is violated, the usual MLEs will be inconsistent and the test will be invalid. We nowallow for N using the Box-Cox transformation family, generalizing our previous test. We consider the model as given in (2.1)— (2.3) except now assume that initially the u. 's are not normally distributed but a Box-Cox it ' transformation of the dependent variables restores normality. Therefore, now we have where -17- D t " Vl + U lt' (3 ' 1} (V S t =X 2t 6 2 +U 2t' (3 - 2) Q t = mln(D t ,S t ), (3.3) (* ) D / - 1 D, = T(D t ,A ) --S^- if \ * 0, (3.4) = log D t if X ± = 0, ( V ( V 2 and the p.d.f. of D is (D ; X^., a (R (♦- )). Similarly we (A ) i i i 1 i 1 define S and its p.d.f. Strictly speaking, the u ' s cannot be normal because of the nature of the transformation in (3.1) and (3.2). For (3.4) to be well defined, D and S should always be positive. This implies the u. 's ' t t it cannot be normal. An appropriate assumption in this situation would be that the u 's are truncated normal [see Poirier (1978)] T(0,A.) - x! 8. < u. < T(°°,A.) - x! 3. . l it l it ' l it l However, here we are dealing with macro demand and supply functions; t therefore, the x. tf.'s are positive and can be expected to be much ' it l larger than ju. |. Hence, for all practical purposes, we can neglect the effects of truncation. = ( l5 . Q A d . - W v a r A 1> t$ 2 , a 2 , X 2 ), a g = ( V;L , * 2 ), d ' = (^a» ^ n ) an d the log-likelihood function for the model (3.1)— (3-3) A B ° can be written as -18- L(8) = log 2 / C T (D; X8, A p a*R (ij; )) m R C T (S; X 2 3 2 , X 2 , o^R 2 (^ 2 )) w 1-w 4 mt y ._ v mt n (dD t ) mu (ds t ) where € T (D; Vl' V a iW } = (27r)' T/2 |a2 Ri (^)| i Then proceeding as in the previous section, we have ^ «- «v'i -i t i -i.t,.j ; »t v ■ 1 x ,rf,)(D J - x 3 ) 1 1 1 u lu 1 t*u w mn, 1-w II (f.(D )f 9 (S )(dD ) (dS ) 1 n L n n n n=l mn ( V ■ 2 V 1 where fjCD^ = ^^ ; x } ^ y a 1 )D t . Now -19- V^ 1 ' - x it 8 i )f 1 ( v f 2 (s t )(dD t )Wmt(ds t )1 v | Q t l Q c where f - ♦ (Q X ; «J » 1) and I - / ♦ (. j «j t » 2 .«*>d. . since / (D 2 - * lt V f l (D t )dD t = Vl (Q t ; x lt S l ,1)# Therefore, the derivations and the resulting test statistic are essentially the same as in the previous section with new definitions of f,(Q«.)t F -(CL) a nd v, . i=l,2. For this case v will be i x t i x t it It 2- ( V " lA, I , r > 2, (r-1) where 9 v is the value of 9 at the r-th iteration, S is a scalar (r) chosen at each iteration step and A = ((A )) = (( oL ( 9)/ 39 . ) ) , L (9) being the log-density function for the t-th observation. We experi- mented with many starting values 9 for the iterations, such as estimates obtained from the Directional Method I considered by Fair and Jaffee (1972) and final iterated values from an arbitrary replication. (In many cases, the iterations converged to different points which were close to the starting values indicating the presence of locally optimal points). After much trial and error, we found that the latter option gave better results. For the logistic log-likelihood we used numerical derivatives, whereas for the normal log-likelihood we had to provide the analytical derivatives in order for the iteration process to converge. This indicates one advantage of using the logistic distribution. After obtaining the estimates from 100 replications we calculated the root mean square errors (RMSE's) for different cases, and these are reported in Table 4.1. The results are quite mixed and Che RMSE's do not differ much for the six cases. Use of the correct distribution does not always lead to lower RMSEs . This may be, apart from con- vergence to suboptimal points, due to finite-sample errors and/or the small number of replications. Under the correct setup, the logistic distribution does slightly better than the normal, which is -26- Table 4.1 Comparison of Root Mean Square Errors True Distribution Normal Logistic Student's t Assumed Distribution Normal Logistic Normal Logistic Normal Logistic Parameters a 10.6150 9.5568 10.6073 10.2584 10.5496 11.3700 a l .1532 .1382 .1531 .1487 .1523 .1654 a 2 .0014 .0013 .0014 .0014 .0014 .0015 a 3 .1750 .1768 .1750 .1770 .1751 .1774 6 o .9774 .8912 .9768 .9613 .9715 1.0711 h .7816 .6293 .7804 .6767 .7756 .6634 h .2860 .2224 .2856 .2217 .2839 .2186 3 3 .2558 .1950 .2553 .1951 .2536 .1941 \ .3521 .3419 .3507 .3503 .3481 .4123 -27- not the case when we use the normal distribution correctly. When the distribution is misspecif ied, the performance of the normal distribution is better, though the differences are not substantial. Therefore, bearing in mind its computational advantages, the logistic distribution might be preferred to the normal in the analysis of market disequilibrium models with independent disturbances. -28- Ref erences Chesher, A. (1983), "The Information Matrix Test, Simplified Calcula- tion Via A Score Test Interpretation," Economics Letters , 13, 45-48. Fair, R. C. and D. M. Jaffee (1972), "Methods of Estimation for Markets in Disequilibrium," Econometrica , 40, 497-514. Fair, R. C. and H. H. Kelejian (1974), "Methods of Estimation for Markets in Disequilibrium: A Further Study," Econometrica , 42, 177-190. Goldfeld, S. M. and R. E. Quandt (1981), "Econometric Modelling with Non-normal Disturbances," Journal of Econometrics , 17, 141-155. Hartley, M. J. and P. Mallela (1977), "The Asymptotic Properties of the Maximum Likelihood Estimator for a Model in Disequilibrium," Econometrica , 45, 1205-1220. Lee, L-F. 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