On Testing the Significance of a Subset of Coefficients in a Set of Seemingly Unrelated Regressions Using Mixed Estimation

April 1977

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ON TESTING THE SIGNIFICANCE OF A SUBSET OF COEFFICIENTS IN A SET OF SEEMINGLY UNRELATED REGRESSIONS USING MIXED ESTIMATION

by

P.A.V.B. Swamy and R. Berner

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

On Testing the Significance of a Subset of Coefficients in a Set of Seemingly Unrelated Regressions Using Mixed Estimation

By P.A.V.B. Swamy aiid R. Berner

Theil's (1971), p. 314 F-test for linear restrictions in the context of joint GLS is generalized in this note to the case in which the mixed

estimator is used.

Consider the stacked linear model, in which observations for each equation are adjacent, (1) y= Xi+u where y is an nT vector of T observations for n dependent variables, X is an nTxK matrix of observations on K independent variables, 8 is a Kxl coefficient vector, and u is an nTxl vector of errors such that E(u) = 0, E(uu') = 0°, an nTxnT covariance matrix defined as (2) 022 = ¢Z@l, | so that o%£ is nm, the contemporaneous covariance matrix of the errors. Suppose that the analyst has prior information concerning some of the 6's formulated as

= = 1’ = (3) XX) R,8 + Vy» E(v,) 0, E(vjvj) Vo

where r, is a qxl vector of unbiased estimators of linear combinations

1

of the B, R, is a known qxK matrix (of rank q), XY is a qxl vector of random

1

errors with known (and uncorrelated with u) covariance matrix Voe As is well

known, the mixed estimator of 8 under 1-3 is (see Theil (1971), p. 346-352):

1 ’ ~1 taal -1,1 -1 -1 = (— + mn Xt ' (4) G2 X'2Q “X RY RD (Sox 2 YIRVo rj)»

b —m where o* is defined by

1 nT-K

and 8 is the GLS or Aitken estimator of 8 in (1).

(5) o2 = (y-xa) '0} (y-xe),

The mixed estimator is obtained by applying Aitken estimation to

(6) w= 2B +e,

where w = > 2 | ls £ =! > xy Ly | { “a 2 0 so that E(e) = 9, E(ee') = 07) yg = OnE

(0 a |

By analogy to Theil (1971), p. 314, it is evident that, under the

null hypothesis ro * ROB,

=-1 -1 -1 ' 71 27 b_) [R, @ ae Z) Ro]

9 (r,~-R

—2 ob,

(nT+q-K) (x,-R — ' ~ — P (w-Zb_)"E * a-Zb

(7)

is distributed as F(p, nT+q-K) where p is the full row rank of Ros assuming that « ~ N(0, o*2_).

Generalized least squares estimation subject to the constraint ro*R8 yields

(8)

y " at Qt x * ! Q ¢ R } ' 2

6 —m

b+(2'2 tz) RI ER, (ZteT 2) RS

(x,-R,b_)-

From (8) it follows that

I

, tater (th _T ' atyttoyn~lay yal (9) (b, DD 2 an Z(b b. (rp Rob [R,@ an Z) Ry] . -1,.-1 -1,.-1,,,-1 ' . trp 7 ' = Ry (Z'E 2) RSLR, (Z"EZ) RZD (ty-Ryb,)

-1_.-1.,.-1 (x, Rob) [R,@ x 2) Ro] (r5 Rob).

Ut

inserting (¥) into (7) gives

(nT+q-K) (b (10) —_—_————— — _7 ty ~ p (w-Zb_) "2" (w-Zb_)

-p )tztyvte dh -b) —m m m1 —m

~3-

which is F distributed with p and nI+q-K degrees of freedom.

We may use either (7) or (10) to test the hypothesis r, = R In

2 = Rob. applying the above test it should be remembered that the null hypothesis,

tr, = RB should not contradict the a priori assumptions r.

1

If the statements r, = R68 and r, = R, BY. refer to the same linear

1 1

combination of the elements of 8, then they are contradictory.

An example of the application of this test is in estimation of demand systems as in Paulus (1975) and Berner (1975), in which off-diagonal price coefficients are constrained to be zero in some variants.

Berner and Paulus consider the relative price version of the Rotter-~

dam demand model,

* (11) w.,Dq

it i=l,...,n,

at MePde FOAL) + Figg PSE PPGL) F fae where the HW, are marginal budget shares and the V,4 are price coefficients. In Berner (1975), the fifteen goods are divided into five groups with three geographic origins for goods in each group. The system is thus a complete system of consumer import and domestic demand equations. Imposing block-additivity across the groups means that for two groups r aud s, 1 Paulus' results indicate that, at the grouping level used here, the

v.. = 0 for ier and jes, but “45 # 0 for i,jer or s.

block~additivity assumption may be inappropriate. A simple extension of the model involves adding off-diagonal blocks of price coefficients to

allow for specific substitution or complementarity among all the goods

thor details, see Theil (1975), Volume I.

-4-

in the pairwise groupings clothing/other and durables/shelter. Nine coefficients are added for each block, for a total of eighteen. The block-additive model thus has 30 free parameters, while the extended model has 48.

Block-additive sample and mixed results for the Netherlands for the period 1954-70 are presented in Table I. Table II presents the prior marginal shares and their standard errors used in estimation, The fact that some of the marginal shares for the sample estimates are negative is disturbing, and indicates a possible misspecification.

Table III presents the estimated parameters of the extended model. Fourteen of the eighteen additional price coefficients are more than twice their standard errors, and many are five to ten times the standard errors. However, the F-test developed here has a value of 1.18 with 18 and 204 degrees of freedom, which is less than the critical value of 1.93 at the 5% level of significance. Hence, block-additivity cannot be rejected in favor of the extended model. It is clear, of course, that

other extensions of the model may dominate the block-additive version.

ltheil (1975) in Volume II, Chapter 8, suggests weak separability rather than the strong separability of the block-additive model. This would be stronger than the specification used here for the extended portion of the model, and yet would allow for off-diagonal price action among more groups than the present extended model does.

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TABLE II

Netherlands, 1954-1970

Average Value Shares and Prior Estimates of Marginal Shares for 15 Products

Average Prior Estimate and Standard Deviation of: Product by Value Income Marginal Orgin Shares Elasticity Share (1) (2) (3) (4) a FOOD 1. Domestic 334 .300(.192) .100(.064) 2. EEC .008 1.88(.750) .015(.006) 3. Rest of World .013 -462(.154) .006(.002) CLOTHING 4. Domestic 132 .758(.380) .100(.050) 5. EEC .016 .313(3.13) .005(.050) 6. Rest of World .005 1.20(.400) .006(.002) SHELTER 7. Domestic . 127 .394(. 787) .050(.10) 8. EEC -003 3.33(3.33) .010(.010) 9. Rest of world .002 5.00(5.00) -010(.010) DURABLES 10. Domestic .115 .522(.235) .060(.027) 11. EEC .008 -625(.625) .005(.005) 12. Rest of World .005 1.60(1.00) .008(.005) OTHER 13. Domestic 215 2.33(.700) -500(.150) 14. EEC .010 -500(.500) .050(.050)

15 Rest of World .006 -- --

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-5- References

Berner, R. (1975), “Estimating Consumer Import Demand Equations," presented at the Third World Congress of the Econometric Society, August,

Paulus, J.D. (1975), "Mixed Estimation of a Complete System of Consumer

Demand Equations," Annals of Economic and Social Measure= ment, 4, No. 1 (Winter), 117-132.

Theil, H. (1971), Principles of Econometrics, New York: Wiley.

(1975), Theory and Measurement of Consumer Demand, Two volumes, Amsterdam: North-Holland.