The numerical values of some key parameters in econometric models

Journal

of Econometrics

21 (1983) 229243.

THE NUMERICAL

North-Holland

Publishing

Company

VALUES OF SOME KEY PARAMETERS ECONOMETRIC MODELS*

IN

T.W. ANDERSON Stanford University, Stanford, CA 94305, USA

Kimio

MORIMUNE

and Takamitsu

SAWA

Kyoto University, Sakyo-ku, Kyoto, Japan Received

November

1979, final version

received

May 1982

In the case of two endogenous variables, exogenous predetermined variables, and normally distributed disturbances, the distributions of the Two-Stage Least Squares (TSLS) and Limited Information Maximum Likelihood (LIML) estimators can be compared on the basis of three key parameters: the non-centrality parameter, a standardization of the structural coefficient, and the number of excluded exogenous variables. In this paper the values of these parameters are estimated in eleven structural equations from various actual econometric models. The distribution functions of the normalized TSLS and LIML estimators are given for the first two key parameters set at approximately their trimmed means, and the third at its median.

1. Introduction In recent years various theoretical studies have been carried out to shed light on the finite-sample properties of alternative single-equation estimation methods in a complete system of simultaneous structural equations. Although much of this work has been limited to estimators of the coehicient of one endogenous variable in an equation with two endogenous variables in a modzl with normally distributed errors and all predetermined variables fixed and well-behaved asymptotically, the results have thrown considerable further light on comparing alternative estimators. Subject to the limitations stated above, the recent theoretical analysis has shown that the exact distributions of the Two-Stage Least Squares (TSLS) and Limited Information Maximum Likelihood (LIML) estimators are essentially characterized by three quantities; that is, the values taken by structural and reduced-form parameters as well as the exogenous variables influence the distributions of the estimators only through three key *This work was supported by National Science Foundation Grants SOC77-14944-A01 at the Institute for Mathematical Studies in the Social Sciences, Stanford University, and SOC76-22232 at the University of Illinois. The authors thank the anonymous referees, Arnold Zellner, and Naoto Kunitomo for their valuable comments.

03044076/83/O

/$03.00 0

1983 North-Holland

230

TW Anderson et al., Key parameters in econometric models

parameters: the so-called non-centrality parameter, a standardization of the structural coefficient, and the number of the excluded exogenous variables whose definitions will be given in the next section. In addition the distribution of the LIML estimator is affected to a lesser extent by the number of degrees of freedom in the estimated covariance matrix of the reduced form. We have analyzed the properties of the TSLS and LIML estimators with systematic computation for a number of suitably chosen cases with different values of these parameters [see Anderson and Sawa (1977, 1979) and Anderson et al. (1982)]. These computations show that the distributional properties of both estimators depend crucially on the values taken by the key parameters. This fact, of course, prevents us from deducing general conclusions about the comparison of alternative estimators. Therefore, we are motivated to find typical values of these parameters in empirical econometric models. Moreover, in view of the facts that many papers have been written on asymptotic expansions and the entire inference in econometric models has been based on asymptotic normality, it is important to know the values of non-centrality parameters in actual studies in order to appraise the relevance of the mathematical results. From past literature we selected eleven models for this evaluation. We also treated two studies with hypothetical models to determine why different Monte Carlo experiments have given different comparative statements.

2. The model ,and estimators In this section we define the three quantities the TSLS and LIML estimators depend. coefficient p in the equation Yl =yzB+Z,y,

+u,

on which the distributions of We estimate the structural

(2.1)

where y1 and yZ are T-component (column) vectors of T observations on two endogenous variables, Z, is a T x K, matrix of observations on K 1 exogenous variables, p is a scalar parameter, y1 is a K,-component vector of parameters, and u is a T-component vector of (unobservable) disturbances. The reduced form of the system of structural equations includes

(2.2) where Z, is a T x K, matrix of observations on K, exogenous variables that are excluded from the structural equation (2.1), rrll and nr2 are K,component vectors, 7rZl and nZ2 are K,-component vectors of reduced form

T.W Anderson et al., Key parameters in econometric models

231

coefficients, and (vi v2) is a T x 2 matrix of (unobservable) disturbances. make the following conventional assumptions about the reduced form: Assumption 1. The rows of (vi v2) are independently each row having mean 0 and (non-singular) covariance a =

i,j=

Cwij),

Assumption 2. K, and T>K.

normally matrix

distributed,

1,2.

The T x K matrix

(2.3) Z consists

of known

Assumption 3. The matrix non-zero component.

(nZ1 n2J

Then variance

value of /I so (2.1) holds. of u is

there is a unique of each component

We

is of rank

numbers,

is of rank

one and zZ2 has at least one

Since U= vi -fir,,

the

a2=o,,-2po,,+/?%0,,. Define two matrices i,j= 1,2,

G= (gij)=(p’,J,2,iP2j),

(2.4)

where A,,.,

=z;~2-z;~,~z;~,~-‘z;~2~

and P2i is the least squares

estimator

of rr2i, i = 1,2, and

n=(CGij)=f~;[z-z(zz,-‘a]~j),

For estimating /I the TSLS estimator T1,cG,,]/Cg,,estimator &iMi_ is [g,,root of the determinantal equation

I

;G-Afi

i, j= 1,2.

&sLs is g,,/g,,, T&C&~], where

I

=O.

The three key parameters d2 =42A22.,n22b22,

(2.5)

f,

and the LIML is the smaller

(2.6) are the non-centrality

parameter, (2.7)

232

TW. Anderson et al., Key parameters in econometric models

the standardized structural coefficient,

(2.8) and the number of the excluded exogenous variables K,. The non-centrality

parameter

divided

by 7;

II/=S2/7;

(2.9)

is called the reduced non-centrality (1974, 1977)-j,

parameter.

In some

work

[Anderson

(2.10)

is used instead of ~3~. The determinantal equation

definition

of 1,

is the

larger

root

of the

(2.11)

where 0 is the right-hand side of (2.4) with z2i and z22 in place of p2i and p22. Eq. (2.11) is the population analogue of the determinantal equation (2.6). It is also interesting to see that the correlation coefficient between v2 and u is w2-w22B p=

--c(

OJOZZ

We immediately

(2.12)

=p’

see that 1011is large if v2 is highly correlated

with u.

The LIML estimator does not have any finite moment, including the mean, but the TSLS estimator has finite moments to order equal to the degree of over-identifiability [Kinal (1980)]. The reason that the LIML estimator has no finite moment is that its distribution has ‘fat’ tails; nevertheless the main body of an LIML distribution may be more concentrated than the main body of the corresponding TSLS distribution. (The idea can be illustrated by comparison of a Cauchy distribution with a small scale parameter, say 1, and a normal distribution with a large standard deviation, say 1000.) The infinite moments are due to the behavior of the distributions outside of the range of practical interest to the econometrician. The asymptomic expansions of the distributions approximate the

Tl4! Anderson et al., Key parameters

in econometric

models

233

distributions over the range of interest and the moments of these approximations describe their properties and, as well, the properties of the original distributions. (However, the moments of the approximate distributions are not necessarily approximations to the moments of the exact distributions - particularly when they are infinite!) The cdf’s for the TSLS and LIML estimators were expanded in terms of the reciprocal of the square root of ,u’ [Anderson and Sawa (1973) and Anderson (1974)]. Based on the asymptotic expansions of the cdf’s, Anderson (1974) made some comparative and &,,,. The probability in every symmetric statements about &,,s interval around /3 is greater for the LIML estimator if

or equivalently p2 > 2/(K, + 1). Another measure may be the Mean Squared Error (MSE) of the asymptotic expansion [the expected value of (p-p)“, where the integral is taken with respect to the derivative of the asymptotic expansion of the cdf, which is usually identical with the asymptotic expansion of the density function]. The MSE of the asymptotic expansion to order l/p’ of the LIML estimator is smaller than that of the TSLS estimator if K, > 7 and

Ial>JWX

(2.14)

or equivalently p2 >2/(Kz - 5); the MSE of the TSLS estimator is smaller if K,>7 and (t$
D

234

2W Anderson et al., Key parameters in econometric models

normal approximation to the econometric investigati0ns.r

3. Estimation

TSLS

distribution

is inadequate

for most

of a and p*

On the basis of a thorough review of the literature, we selected eleven structural equations from various actual econometric models. A selected equation satisfied the following criteria: First, it includes exactly two endogenous variables. Second, it must be over-identified; otherwise, the TSLS and LIML estimators are identical. Third, specification of the equation must be plausible in the sense that the hypothesis of the specified over-identifying restrictions is statistically accepted. Fourth, the whole system including the equation must be linear. Fifth, the number of degrees of freedom in the estimator of $2 must not be too small. (Specifically 2 and 4 degrees of freedom were too small.) We have examined as many equations as possible to determine whether or not they are acceptable on the basis of the above criteria. The number of acceptable equations has turned out to be surprisingly small relative to the number of equations examined (11 out of 29). In particular, many equations failed to satisfy the third criterion at the one percent significance level; over-identifying restrictions turned out to be doubtful in those equations.’ The structural parameters were estimated by the LIML and TSLS methods. As far as the selected equations are concerned, the two estimation methods gave almost identical estimated values except in cases 8 and 11 (table 1).3 Of course, the two methods yield corresponding results for the standardized structural coeffkient CI according to the formula (2.8), though differences in B tend to be amplified (particularly when d is near singular). Several alternative ways may be suggested to estimate $. A naive estimator is ~*=P;~A~~,~P~~/(T~~~). However, (T-K)$*/K, is distributed as a non‘For given G( and 6’ the distribution of the Ordinary Least Squares (OLS) estimator is the distribution of the TSLS estimator with K, replaced by T-K,. The effect of increasing K, applies to replacing K, by a larger value T-K,. The larger value T-K, will make for a larger bias in the estimator; it tends to estimate oi2/oz2 instead of p. ‘Anderson and Rubin (1949) found that the smaller root fir of (2.6) is equivalent to the likelihood ratio statistic for testing the hypothesis that the over-identifying restrictions implied by the structural equation (2.1) are true. In a later article Anderson and Rubin (1950) found that TX, has a large-sample asymptotic X2-distribution with K, - 1 degrees of freedom. (In general, the number of degrees of freedom is the number of excluded exogenous variables minus one less than the number of explanatory endogenous variables.) Kadane (1970) found that (T-K)I,/(K, - 1) has a limiting F-distribution with K, - 1 and T-K degrees of freedom as IR approaches 0 (proportionate to a fixed no); this notion of limit (‘small-disturbance’) is identical to large 6’ for T fixed [Anderson (1977)]. 3The study 8 is the estimation of an investment function. The coefficient b is for the profit variable, and the negative value of the LIML estimate may be unreasonable from an economic viewpoint. However, the total effect of profit (current plus lagged) is positive. The study 11 is about a meat demand function. The estimated coefficient is that of price. Both the TSLS and LIML estimates may be reasonable.

3

20

“For oi,,, bTrimmed

4

20

Median Trimmed meanb

0.48 0.22 0.41 0.13 4.01 1.68 0.25 0.003 5.77 10.57 0.02 0.41 1.03

0.96 0.43 2.03 0.67 8.02 10.09 0.98 0.01 23.07 31.70 0.03

0.98

3.31

0.29

116

0.09 0.08 1.47 0.09 0.13 0.27 0.09 0.62 0.38 0.34 0.13 0.13

2.883 0.556 0.434 0.075 0.515 0.517 0.393 -0.538 0.715 0.001 -4.844

1,

4.50

2.11

values.

0.49

0.90

0.40

0.96 1.99 - 0.05 ~ 2.07 0.29 0.40 -0.30 -9.16 0.32 - 0.02 -7.01

0.87 1.65 -0.01 ~ 1.83 0.29 0.39 -0.28 -0.04 0.32 -0.01 - 1.25 0.32

6mLa

&xc.=

equations.

1.85 2.11 2.03 3.57 8.71 11.67 1.07 0.87 25.46 31.71 1.54

12

structural

43

2.788 0.435 0.439 0.150 0.512 0.514 0.397 0.062 0.711 0.001 - 1.579

37 42 43 75 78 595 21 17 509 634 34

LLMI.

in ‘acceptable’

means are calculated with respect to their absolute .,xcgj,where xu, is the jth order statistic.

92

20

19 9 43 14 72 515 20 0.2 461 634 0.68

B mu

P2

values of key parameters 6’

and the trimmed mean of x(s), xt+

$I&

4

and c&,, the medians mean is the arithmetic

2 2 5 5 2 6 4 4 4 3 2

3 3 3 3 1 2 3 3 2 3 1

20 20 21 21 9 51 20 20 20 20 22

1 2 3 4 5 6 7 8 9 10 11

K,

K,

T

Eq. no.

Numerical

Table 1

G-H (3.5) G-H (3.4) Klein 1 W Klein 1 I Lesser Clark C-I Klein 3 (3.3.38) Klein 3 (3.3.39) Klein 3 (3.3.41) Klein 3 (3.3.46) Tintner demand

236

TW! Anderson et al., Key parameters in econometric models

central F with non-centrality parameter ?i2 and K, and T-K degrees of freedom. Then a($*) = (S2 + K,)/( T - K - 2). Hence, the unbiased estimator of a2 (=T$) is {(T-K-2)$*)-K,. W e may also note that $*/(l +$*) is the coefficient of determination regressing [Z-Z,(Z’lZ,)-‘Z’l]y, onto [Z-Z,(Z’lZ,)-‘Z’l]Z, (that is, it is the proportion of variance explained by Z, after taking account of Z,). If the equation is over-identified, the reduced-form coefficients can be estimated taking the over-identifying restrictions into account; the least squares estimator p22 of 7t22 is no longer the maximum likelihood estimator. The formulas for the (limited-information) maximum likelihood estimators for the reduced-form coefficients are given in Anderson and Rubin (1949). Comparing (2.6) and (2.11), we realize that the larger root, say i2, of (2.6) is a reasonable estimator of its population analogue 3,,. In fact, it can be seen that i2 is equivalent to an estimator of (1 +a2)d2/T with the (limitedinformation) maximum likelihood estimators substituted for rr22r 0 and /I. We can estimate $ by dividing I2 by (1 +oi2) where oi is the right-hand side of (2.8) with fi replaced by its maximum likelihood estimate and a replaced by d given by (2.5). The tabulated values in table 1 are estimated values based on i2. It might be noted that [T/(T-K)]fi may be a better estimator of a than fi and hence (T- K)z2 may be a better estimator of Ti, than T12. Correspondingly the estimates of J2 and ,ii2 in table 1 might be multiplied by (T-K)/T. However, since this factor of (T-K)/T is near 1 among models in table 1 we did not adjust the estimates. In order to make the cross-equations comparison reasonable, we computed and tabulated g/K, (=$‘/(TK,)). This quantity may be regarded as the average explanatory power of each of the excluded exogenous variables relative to the disturbance variance. As contemporary macro-econometric models are more or less nonlinear, our selection of the models was confined to the historical models published in the 1950’s or early 1960’s. The selected models are relatively small-scale. In contemporary models, K, and T take much larger values than in these historical models. In using the information in our study G/K, seems more appropriate for generalizing the conclusions, because the values of $/K2 are in some vague sense independent of T and K,, but d2 and p2 depend on the size of the models. Numerical estimates of relevant parameter values are tabulated in table 1 for eleven selected actual equations. Some details about those selected The similar table is given in the quantities are given in the appendix. appendix for eighteen equations which are not acceptable according to the criteria described at the beginning of this section. Nine of the eleven acceptable equations were estimated from samples of size ranging from 20 to 22. The number of excluded exogenous variables K, ranges from 2 to 6. The discussion of IX is in terms of the LIML estimate

TW Anderson et al., Key parameters in econometric models

237

because it is relatively unbiased (while the TSLS estimate is far from medianunbiased) and is used in the estimate of $. However, as noted above the LIML and TSLS estimates differ substantially in only two cases. (See footnote 3.) The estimated values of a are very scattered, ranging from 0.02 to 9.16 in their absolute values. The median of Ioil is 0.40, while the trimmed mean (deleting the two smallest and the two largest) is 0.90. We can consider the latter figure as a ‘typical value’. The estimated normalized non-centrality parameter q/K, ranges from 0.00 to 10.57. The median is 0.41, while the trimmed mean is 1.03. We take the latter as a ‘typical value’. This suggests that the average explanatory power of exogenous variables is not as great as might be intuitively expected. To put it differently, the systematic variation of an endogenous variable may depend heavily upon the ‘included’ exogenous variables, though not so much on the ‘excluded’ variables. Another estimated parameter fi2 also varies over a wide range, from 17 to 634. The median is 43 and the trimmed mean is 116. In fig. 1 we give the distribution functions of the normalized LIML and TSLS estimators where d2 and Ial are set at approximately their trimmed means, and K, to its median (T-K=30). The distribution of the LIML estimator is better approximated by the normal distribution at least in the range of one sigma than the TSLS estimator is. Further, the first and third quartiles of LIML are (-0.649,0.715), while those of TSLS are (-0.831,0.474) which is not centered at 0. This characterization gets more conspicuous as the value of K, increases. Fig. 2 gives distributions of the TSLS estimator for four different values of K,. These TSLS distributions with the trimmed mean of /a( IS extremely skewed when K, is as large as and greater than 10 ([%I= 1 implies p2 =0.5). First and third quartiles of some distributions are tabulated in table 3. These quartiles are badly biased toward negative values (for positive a). The LIML distributions for these cases are robust for different values of K,. It agrees with the TSLS distribution for K,= 1. (The structural equation is just identified when K, = 1.) Interquartile ranges vary negligibly with K,. It is obvious from these discussions that the TSLS estimate is inaccurate when K, is as large as 10 when lal=l. It seems desirable to exclude the TSLS estimator since the TSLS distribution can be skewed more with smaller values of K, than 10 if the value of ICC[is larger. If we want to avoid strongly biased estimates, we should prefer the LIML estimator. (Because of the long tail of the exact distribution of the LIML estimator there is a small probability of an LIML estimator yielding an extreme value.) Now we look at the asymptotic criteria. Four of the eleven equations (numbers 2, 4, 8, and 11) satisfy the condition (2.13); for each pair of values of K, and tl there is an interval of greater concentration of the LIML estimator if d2 is large enough to guarantee the accuracy of the approximate cdf’s. Note that the condition (2.14) is not relevant when K, is as small as in

TM Anderson et al., Key parameters

238

-2

-1

in econometric

models

0

i

Fig. 1. CDF’s; G(= 1, K, = 4. a2 = 100.

K2=l

2

1

Fig. 2. CDF’s; TSLS, a: = 1, 6’ = $ x Kz = 2OK,. The distributions

of LIML

for these cases almost agree K2 = 1 and are not given.

with that

of TSLS

with

T.W Anderson et al., Key parameters

in econometric

models

239

the selected equations. At first sight this may seem to favor the TSLS procedure. It should be noted, however, that in every selected equation K, is extremely small as compared with contemporary large-scale models. In fact, for the trimmed mean value 0.90 of jtll (the corresponding value of p2 is 0.44), the condition (2.13) is fulfilled if K, is at least 4; (2.14) is fulfilled if K, is at least 10. Unless our estimation of the values of c( and d2 is biased, we may safely conclude that the LIML method should be preferred to the TSLS procedure when we estimate a structural equation in a large-scale system ordinarily dealt with in contemporary econometric analysis.4 However, it is fair to note that the conditions (2.13) and (2.14) are derived from asymptotic expansions of the cdf’s of the estimators, the accuracy of which depends on the magnitude of d2. [For instance, at CY = 1.0 the maximum absolute error of the approximation to the TSLS distribution is less than 0.01 at K, = 3 for d2 2 10, at K, =7 for S2z 30, and at K,= 10 for d2 270. See Anderson and SaGa (1977).]

4. Monte Carlo experiments In past Monte Carlo experiments it was taken for granted that a whole system, composed of several equations, needs to be completely specified. However, the specification of the system influences the result of experiment only through the key parameters K,,c(and d2 (and T-K for LIML). If we compute these values for each equation with two endogenous variables and an arbitrary number of exogenous variables, we can explain why a particular Monte Carlo experiment has yielded its particular conclusion. Comparing Wagner’s and Neiswanger-Yancey’s Monte Carlo experiments (table 2), we realize that they have the same value of K, and almost the same value of Ial; the main difference exists in their values of the non-centrality parameter. These two experiments aim primarily to compare the LIML with the ordinary least squares estimator. Wagner concludes that ‘least squares’ generally gives more biased but less variable estimators than the LIML method, and that the t-distribution may be used in constructing confidence intervals with LIML estimators of the parameters and the sampling variance of the estimators [Wagner (1958, p. 117)]. On the contrary, Neiswanger and Yancey (1959), among other things, conclude that ‘(for the LIML) confidence intervals established under the assumption of a ‘t’ distribution and tests of hypotheses made under the same assumption may lead to serious error of inferences’ (p. 402). It is simply due to the difference in the non-centrality parameter that they have led them to opposite conclusions. The normal approximation of the LIML distribution becomes better as pLz increases; 11’ 4Many econometric models may be nearly block diagonal. of several equations and the degree of over-identifiability inequalities (2.13) and (2.14) may be satisfied in many cases.

However, each block may consist can be greater than ten. The

T.W Anderson et al., Key parameters in econometric models

240

Table 2 Numerical

values of key parameters

in Monte

Carlo experiments. G(

Eq. no.a

T

K,

$I

*/K,

P

$

1,

R,

M-l M-2 M-3

20 25 25

2 2 2

10.65 3.48 2.18

5.33 1.74 1.09

213 87 65

375 155 115

0 0 0

18.77 6.20 4.60

“The equations M-l, M-2, M-3 are taken from Monte values are true values computed from the specified parameters

-0.86 -0.87 0.64

Wagner Neiswanger-Yancey Neiswanger-Yancey

Carlo experiments. The tabulated and exogenous variables.

Table 3 First

K2 1 10 30 50

and

third

quartiles; c(= 1, 6’ = $ x K, =20K,, T-K = lOK,. TSLS”

LIML” -0.62 -0.66 -0.67 -0.67

0.77 0.71 0.70 0.69

-0.620 - 1.053 - 1.426 - 1.673

0.767 0.228 -0.151 -0.400

“The two numbers for each case are the lower the upper quartiles, respectively.

and

in Wagner’s model is 375 while those in Neiswanger-Yancey’s are 155 and 115. Also, it is pointed out by both authors that the bias of the ordinary least squares is quite serious. This is fully explained by a fairly large value assigned to Itl(.

20 10 83 83 51 51 32 18 20 18 20 18 18 20 18 20 18 18

1 2 3 4 5 6 1 8 9 10 11 12 13 14 15 16 17 18

“When T-K

T

Eq. no.

Appendix

2 2 6 6 6 6 6 13 13 15 15 15 13 13 13 13 14 13

K,

0.35 5.3 6.53 5.80 10.14 8.59 11.81 35.9 38.45 260.33 322.2 7184.56 27.83 21 21.94 21.5 647.78 15.17

I,&

is small, better estimates

2 2 2 1 1 3 3 1 1 1 3 3 3 3 2 3

1

4

K, 7 53 542 481 517 438 378 646 769 4686 6444 129322 501 420 395 430 11660 273

“2

8 1224 542 828 620 591 688 703 777 15464 11534 221141 706 751 5111 516 87447 35863

P

0.23 3.97 0.31 2.02 0.13 2.74 1.42 7.81 1.46 76.96 34.09 77.9 6.03 4.22 6.41 3.44 181.3 14.31 the tabulated

-0.246 0.453 0.116 0.232 0.607 0.102 0.023 0.862 0.850 0.678 0.702 0.152 0.343 0.333 0.099 -0.782 0.009 0.317

by multiplying

-0.307 0.596 0.116 0.195 0.604 0.091 0.029 0.843 0.848 0.717 0.718 0.139 0.375 0.370 ~ 0.984 -1.121 0.022 0.088

%LS

K)/7:

0.151 ~ 2.654 -0.010 0.591 0.444 0.417 -0.801 0.231 0.087 ~ 1.146 -0.80 -0.832 -0.512 - 0.723 2.659 0.374 - 1.960 5.862 by (T-

0.40 122.44 6.53 9.98 12.15 11.58 21.49 39.08 38.84 859.1 576.68 12285.6 39.20 37.54 283.92 25.81 4858.2 1992.4

models.”

estimates

econometric

Ij7.SI.SiL,Pm_

in ‘unacceptable’

Table A.1 values of key parameters

of a2 and p2 may be obtained

0.18 2.65 1.09 0.97 1.69 1.43 1.97 2.76 2.96 17.36 21.48 478.97 2.14 1.62 1.69 1.65 46.27 1.17

&I&

Numerical

0.379 -4.695 -0.011 0.848 0.449 0.592 - 0.904 0.294 0.090 ~ 1.517 -0.89 -0.841 ~ 0.643 -0.894 3.454 0.443 -2.55 11.41

%.IML

G-H (3.1) Johnston cons. Clark G Clark H Clark C Clark I Clark I K-G(3) p. 51 K-G(3) p. 91 K-G(4) p. 52 K-G(4) p. 91 K-G(5) p. 52 K-G(6) p. 52 K-G(6) p. 91 K-G(8) p. 52 K-G(8) p. 91 K-G(9) p. 52 K-G(lO) p. 52

TW Anderson et al., Key parameters in econometric models

242

Appendix-reference

to models.

‘Acceptable’ structural equations (table 1) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

The equation The equation The equation The equation Leser, p. 59. The equation The equation The equation The equation The equation The equation Goldberger.

(3.5) of (3.4) is (3.1.33) (3.1.32)

Girshick and Haavelmo is used. used. of p. 71 of Klein is used. of p. 71 is used.

(8’) of p. 108 of Clark is re-estimated. (3.3.38) on p. 108 of Klein is used. (3.3.39) on p. 109 is used. (3.3.41) on p. 109 is used. (3.3.46) on p. 109 is used. (21) of p. 176 of Tintner is used. The same model

is seen on p. 303 of A.

Monte Carlo experiments (table 2) (M-l) (M-2) (M-3)

The equation (1) on p. 118 of Wagner is used. The equation (2) is just identified. The first equation of Model-l on p. 391 of Neiswanger-Yancey is used. The second equation of Model-l on p. 391 is used.

‘Unacceptable’ structural equations (table A.l) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)

The The The The The The The The The The The The The The The The The The

equation (3.1) of Girshick and Haavelmo. consumption function is on p. 271 and the data are on p. 269 of Johnston. equation (3’) on p. 108 of Clark. equation (4’) on p. 108 of Clark. consumption function in the footnote on p. 104. of Clark. import function (1921-1933) in Figure 5 of Clark, p. 103. import function (1934-41) in Figure 5 of Clark, p. 103. equation (3) on p. 51 of Klein and Goldberger. equation (3) on p. 91. equation (4) on p. 52. equation (4) on p. 91. equation (5) on p. 52. equation (6) on p. 52. equation (6) on p. 91. equation (8) on p. 52. equation (8) on p. 91. equation (9) on p. 52. equation (10) on p. 52.

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