# Seemingly Unrelated Regressions with Unequal Observations

Srivastava and Dwivedi (1979) surveyed the developments in the SUR model and described the extensions of this model to the serially correlated case, the nonlinear case, the misspecified case, and that with unequal number of observations. Srivastava and Giles (1988) dedicated a monograph to SUR models, and surveyed the finite sample as well as asymptotic results. More recently, Fiebig (2001) gives a concise and up to date account of research in this area. In this section, we consider one extension to focus upon. This is the case of SUR with unequal number of observations considered by Schmidt (1977), Baltagi, Garvin and Kerman (1989) and Hwang (1990).     Let the first firm have T observations common with the second firm, but allow the latter to have N extra observations. In this case, (10.2) will have y1 of dimension T x 1 whereas y2 will be of dimension (T + N) x 1. In fact, y2 = (уЕуОО and X’2 = (X^X") with * denoting the T common observations for the second firm, and o denoting the extra N observations for the second firm. The disturbances will now have a variance-covariance matrix

GLS on (10.2) will give anX[Xi a12XX2* 1 _1

a12X2’Xi a22X2’X2 + (X2o/X2o)/a22) _

(10.26)

a11 Xi yi + a12 Xi y2 a12Xl’yi + a22X2!y*2 + (X^y®^) _

where £_1 = [aij] for i, j = 1, 2. If we run OLS on each equation (T for the first equation, and T + N for the second equation) and denote the residuals for the two equations by e1 and e2, respectively, then we can partition the latter residuals into e’2 = (e2,e2). In order to estimate Q, Schmidt (1977) considers the following procedures:

(1) Ignore the extra N observations in estimating Q. In this case

?ii = sn = e’iei/T; ai2 = S12 = eie2/T and?22 = s22 = e2’e2/T (10.27)

(2) Use T + N observations to estimate a22. In other words, use s11, s12 and <r22 = s22 = e’2e2/(T + N). This procedure is attributed to Wilks (1932) and has the disadvantage of giving estimates of Q that are not positive definite.

(3) Use s11 and s22, but modify the estimate of a12 such that Q is positive definite. Srivastava and Zaatar (1973) suggest <r12 = s12(s22/s22)1/2.

(4) Use all (T + N) observations in estimating Q. Hocking and Smith (1968) suggest using aii = sii – (N/N + T)(si2/s22)2(s22 – s°22) where s°22 = e2e2/N; <712 = si2(s22/s22) and a 22 = s22.

(5) Use a maximum likelihood procedure.

All estimators of Q are consistent, and вfgls based on any of these estimators will be asymp­totically efficient. Schmidt considers their small sample properties by means of Monte Carlo experiments. Using the set up of Kmenta and Gilbert (1968) he finds for T = 10,20, 50 and N = 5,10,20 and various correlation of the X’s and the disturbances across equations the fol­lowing disconcerting result: “..it is certainly remarkable that procedures that essentially ignore the extra observations in estimating £ (e. g., Procedure 1) do not generally do badly relative to procedures that use the extra observations fully (e. g., Procedure 4 or MLE). Except when the disturbances are highly correlated across equations, we may as well just forget about the extra observations in estimating £. This is not an intuitively reasonable procedure.”

Hwang (1990) re-parametrizes these estimators in terms of the elements of £_1 rather than £. After all, it is £_1 rather than £ that appears in the GLS estimator of в. This re-parametrization shows that the estimators of £_1 no longer have the ordering in terms of their use of the extra observations as that reported by Schmidt (1977). However, regardless of the parametrization chosen, it is important to point out that all the observations are used in the estimation of в whether at the first stage for obtaining the least squares residuals, or in the final stage in computing GLS. Baltagi et al. (1989) show using Monte Carlo experiments that better estimates of £ or its inverse £_1 in Mean Square Error sense, do not necessarily lead to better GLS estimates of в.

10.4 Empirical Examples

Example 1: Baltagi and Griffin (1983) considered the following gasoline demand equation:

і Gas, я і Y, д і pmg R, Car

logCar = ° вi‘ogN + e2logPcdp + ftt‘og N +

where Gas/Car is motor gasoline consumption per auto, Y/N is real per capita income, PMG/PGDP is real motor gasoline price and Car/N denotes the stock of cars per capita. This data consists of annual observations across 18 OECD counties, covering the period 1960­1978. It is provided as GASOLINE. DAT on the Springer web site. We consider the first two countries: Austria and Belgium. OLS on this data yields . Y Pmg, Car

3.727 + 0.761 logN – 0.793 logp – 0.520 log N

(0.373) (0.211) (0.150) GDP (0.113)

. Y Pmg, Car

3.042 + 0.845 logN – 0.042 logp – 0.673 log N

(0.453) (0.170) (0.158) GDP (0.093) where the standard errors are shown in parentheses. Based on these OLS residuals, the estimate of E is given by

0.0012128 0.00023625 0.00092367

The Seemingly Unrelated Regression estimates based on this E, i. e., after one iteration, are given by , , Y Pmg, Car

3.713 + 0.721 log 0.754 log———- – 0.496 log

N PgDP N

(0.372) (0.209) (0.146) (0.111)

, , Y Pmg, Car

2.843 + 0.835 logN – 0.131 log – 0.686 log N

(0.445) (0.170) (0.154) GDP (0.093)

The Breusch-Pagan (1980) Lagrange multiplier test for diagonality of E is Tr21 = 0.947 which is distributed as xl under the null hypothesis. The Likelihood Ratio test for the diagonality of E, given in (10.23), yields a value of 1.778 which is also distributed as x1 under the null hypothesis. Both test statistics do not reject H0. These SUR results were run using SHAZAM and could be iterated further. Note the reduction in the standard errors of the estimated regression coefficients is minor as we compare the OLS and SUR estimates.

Suppose that we only have the first 15 observations (1960-1974) on Austria and all 19 ob­servations (1960-1978) on Belgium. We now apply the four feasible GLS procedures described by Schmidt (1977). The first procedure which ignores the extra 4 observations in estimating E yields sn = 0.00086791, si2 = 0.00026357 and s2 = 0.00109947 as described in (10.27). The

resulting SUR estimates are given by

. . , Gas, Y, Pmg, Car

Austria log^— = 4.484 + 0.817 log— – 0.580 log—————— 0.487 log——

Car N PGDP N

(0.438) (0.168) (0.176) (0.098)

Gas Y Pmg, Car

Belgium log = 2.936 + 0.848 log—– 0.095 log— – 0.686 log—

Car N PGDP N

(0.436) (0.164) (0.151) (0.090)

The second procedure, due to Wilks (1932) uses the same sn and s12 in procedure 1, but <r22 = s22 = e2e2/19 = 0.00092367. The resulting SUR estimates are given by

. . , Gas, Y, Pmg, Car

Austria log = 4.521 + 0.806 log—– 0.554 log— – 0.476 log—

Car N PGDP N

(0.437) (0.167) (0.174) (0.098)

Gas Y Pmg, Car

Belgium log = 2.937 + 0.848 log—– 0.094 log— – 0.685 log—

Car N PGDP N

(0.399) (0.150) (0.138) (0.082)

The third procedure based on Srivastava and Zaatar (1973) use the same s11 and s22 as proce­

dure 2, but modify <r12 = s12(s22/s22)^2 = 0.00024158. The resulting SUR estimates are given by

. . , Gas, Y _ Pmg, Car

Austria log^— = 4.503 + 0.812 log— – 0.567 log———- 0.481 log——

Car N PGDP N

(0.438) (0.168) (0.176) (0.098)

Gas Y Pmg, Car

Belgium log^— = 2.946 + 0.847 log— – 0.090 log———- 0.684 log——

Car N PGDP N

(0.400) (0.151) (0.139) (0.082)

The fourth procedure due to Hocking and Smith (1968) yields <r11 = 0.00085780, <r12 = 0.0022143 and <r22 = s22 = 0.00092367. The resulting SUR estimates are given by

. . , Gas, Y _ Pmg, Car

Austria log– = 4.485 + 0.817 log— – 0.579 log———- 0.487 log——

Car N PGDP N

(0.437) (0.168) (0.176) (0.098)

Gas Y Pmg, Car

Belgium log^— = 2.952 + 0.847 log— – 0.086 log———- 0.684 log——

Car N PGDP N

(0.400) (0.151) (0.139) (0.082)

In this case, there is not much difference among these four alternative estimates.

Example 2: Growth and Inequality. Lundberg and Squire (2003) estimate a two equation model of growth and inequality using SUR. The first equation relates Growth (dly) to education (adult years schooling: yrt), the share of government consumption in GDP (gov), M2/GDP (m2y), Inflation (inf), Sachs-Warner measure of openness (swo), changes in the terms of trade (dtot), initial income (f pcy), dummy for 1980s (d80) and dummy for 1990s (d90). The second equation relates the Gini coefficient (gih) to education, M2/GDP, civil liberties index (civ), mean land Gini (mlg), mean land Gini interacted with a dummy for developing countries (mlgldc). The data contains 119 observations for 38 countries over the period 1965-1990, and can be obtained from http://www. res. org. uk/economic/datasets/datasetlist. asp.

Table 10.1 Growth and Inequality: SUR Estimates

 . sureg (Growth: dly = yrt gov m2y inf swo (Inequality: gih = yrt m2y civ mlg mlgldc), Seemingly unrelated regression dtot f pcy d80 d90) corr Equation Obs Parms RMSE “R-sq” chi2 P Growth 119 9 2.313764 0.4047 80.36 0.0000 Inequality 119 5 6.878804 0.4612 102.58 0.0000 Coef. Std. Err. z P> z [95% Conf. Interval] Growth yrt – .0497042 .1546178 -0.32 0.748 -.3527496 .2533412 gov – .0345058 .0354801 -0.97 0.331 -.1040455 .0350338 m2y .0084999 .0163819 0.52 0.604 -.023608 .0406078 inf – .0020648 .0013269 -1.56 0.120 -.0046655 .000536 swo 3.263209 .60405 5.40 0.000 2.079292 4.447125 dtot 17.74543 21.9798 0.81 0.419 -25.33419 60.82505 f pcy – 1.038173 .4884378 -2.13 0.034 -1.995494 -.0808529 d80 – 1.615472 .5090782 -3.17 0.002 -2.613247 -.6176976 d90 – 3.339514 .6063639 -5.51 0.000 -4.527965 -2.151063 cons 10.60415 3.471089 3.05 0.002 3.800944 17.40736 Inequality yrt – 1.000843 .3696902 -2.71 0.007 -1.725422 -.2762635 m2y – .0570365 .0471514 -1.21 0.226 -.1494516 .0353785 civ .0348434 .5533733 0.06 0.950 -1.049748 1.119435 mlg .1684692 .0625023 2.70 0.007 .0459669 .2909715 mlgldc .0344093 .0421904 0.82 0.415 -.0482823 .117101 cons 33.96115 4.471626 7.59 0.000 25.19693 42.72538 Correlation matrix of residuals Growth Inequality Growth 1.0000 Inequality 0.0872 1.0000 Breusch-Pagan test of independence: chi2(1) = 0.905, Pr = 0.3415.

Table 10.1 gives the SUR estimates reported in Table 1 of Lundberg and Squire (2003, p. 334) using the sureg command in Stata. Among other things, these results show that openness enhances growth and education reduces inequality. The correlation among the residuals of the two equations is weak (0.0872) and the Breusch-Pagan test for diagonality of the variance – covariance matrix of the disturbances of the two equations is statistically insignificant, not rejecting zero correlation among the two equations.

Problems

1. When Is OLS as Efficient as Zellner’s SUR?

(a) Show that OLS on a system of two Zellner’s SUR equations given in (10.2) is the same as OLS on each equation taken separately. What about the estimated variance-covariance matrix of the coefficients? Will they be the same?

(b) In the General Linear Model, we found a necessary and sufficient condition for OLS to be equivalent to GLS is that X’U-1 PX = 0 for every y where PX = I — PX. Show that a neces­sary and sufficient condition for Zellner’s GLS to be equivalent to OLS is that aijX[PX – = 0 for i = j as described in (10.10). This is based on Baltagi (1988).

(c) Show that the two sufficient conditions given by Zellner for SUR to be equivalent to OLS both satisfy the necessary and sufficient condition given in part (b).

(d) Show that if Xi = XjC’ where C is an arbitrary nonsingular matrix, then the necessary and sufficient condition given in part (b) is satisfied.

2. What Happens to Zellner’s SUR Estimator when the Set of Regressors in One Equation Are a Subset of Those in the Second Equation? Consider the two SUR equations given in (10.2). Let X1 = (X2,Xe), i. e., X2 is a subset of X1. Prove that

(a) P2,sur = e2,OLS.

(b) f31SUR = eiOLS — Ae2,oLS, where A = si2(X1 X1)-1 X1 /s22. &2,ols are the OLS residuals from the second equation, and the S’ij’s are defined in (10.14).

3. What Happens to Zellner’s SUR Estimator when the Set of Regressors in One Equation Are Or­thogonal to Those in the Second Equation? Consider the two SUR equations given in (10.2). Let X1 and X2 be orthogonal, i. e., X1X2 = 0. Show that knowing the true £ we get (a) e1,GLS = e1,OLS + ((j12/a11)(X1 X1) 1X1 y2 and e2,GLS = @2,OLS

x2 У1.

(b) What are the variances of these estimates?

(c) If X1 and X2 are single regressors, what are the relative efficiencies of вi OLS with respect

to ei, GLS for i = 1 2?

4. An Unbiased Estimate of jij. Verify that sij, given in (10.13), is unbiased for aij. Note that for computational purposes tr(B) = tr(PXiPXj).

5. Relative Efficiency of OLS in the Case of Simple Regressions. This is based on Kmenta (1986, pp. 641-643). For the system of two equations given in (10.15), show that Et=1(Xit —

Deduce that var(/?12,GLsJ = (J11J22 — J?2)J11to®2®2/[j11J22mX2X2mXlXl — a212ml1x2] and

(c) Using p = j12/(j11j22)1^2 and r = mxix2/(mxiximx2x2)1/2 and the results in parts (a) and

(b) , show that (10.16) holds, i. e., var(/?12,GLS)/var(^12,ols) = (1 — P2)/[1 — P2r2].

(d) Differentiate (10.16) with respect to в = p2 and show that (10.16) is a non-increasing function of в. Similarly, differentiate (10.16) with respect to A = r2 and show that (10.16) is a non­decreasing function of A. Finally, compute this efficiency measure (10.16) for various values of p2 and r2 between 0 and 1 at 0.1 intervals, see Kmenta’s (1986) Table 12-1, p. 642.

6. Relative Efficiency of OLS in the Case of Multiple Regressions. This is based on Binkley and Nelson (1988). Using partitioned inverse formulas, verify that var(et GLS) = Ац given below (10.17). Deduce (10.18) and (10.19).

7. Consider the multiple regression case with orthogonal regressors across the two equations, i. e., X[X2 = 0. Verify that R2 = R*2, where R^ and R*2 are defined below (10.20) and (10.21), respectively.

8. (a) SUR With Unequal Number of Observations. This is based on Schmidt (1977). Derive the

GLS estimator for SUR with unequal number of observations given in (10.26).

(b) Show that if a2 = 0, SUR with unequal number of observations reduces to OLS on each equation separately.

9. Grunfeld (1958) considered the following investment equation:

Iit = a + PFit + e2Cit + uit

where Iit denotes real gross investment for firm i in year t, Fit is the real value of the firm (shares outstanding) and Cit is the real value of the capital stock. This data set consists of 10 large U. S. manufacturing firms over 20 years, 1935-1954, and are given in Boot and de Witt (1960). It is provided as GRUNFELD. DAT on the Springer web site. Consider the first three firms: G. M., U. S. Steel and General Electric.

(a) Run OLS of I on a constant, F and C for each of the 3 firms separately. Plot the residuals against time. Print the variance-covariance of the estimates.

(b) Test for serial correlation in each regression.

(c) Run Seemingly Unrelated Regressions (SUR) for the first two firms. Compare with OLS.

(d) Run SUR for the three assigned firms. Compare these results with those in part (c).

(e) Test for the diagonality of V across these three equations.

(f) Test for the equality of all coefficients across the 3 firms.

10. (Continue Problem 9). Consider the first two firms again and focus on the coefficient of F. Refer to the Binkley and Nelson (1988) article in The American Statistician, and compute R2, R*q , Veq and Vx‘2q.

(a) What would be equations (10.20) and (10.21) for your data set?

(b) Substitute estimates of <тц and в2 and verify that the results are the same as those obtained in problems 9(a) and 9(c).

(c) Compare the results from equations (10.20) and (10.21) in part (a). What do you conclude?

11. (Continue Problem 9). Consider the first two firms once more. Now you only have the first 15 observations on the first firm and all 20 observations on the second firm. Apply Schmidt’s (1977) feasible GLS estimators and compare the resulting estimates.

12. For the Baltagi and Griffin (1983) Gasoline Data considered in section 10.5, the model is

і Gas, n і Y, я і Pmg, о і Car,

logCar = “ + e>logN + ,3’2logPgwp +g N + u

where Gas/Car is motor gasoline consumption per auto, Y/N is real per capita income, PMG/PGDp is real motor gasoline price and Car/N denotes the stock of cars per capita.

(a) Run Seemingly Unrelated Regressions (SUR) for the first two countries. Compare with OLS.

(b) Run SUR for the first three countries. Comment on the results and compare with those of part (a). (Are there gains in efficiency?)

(c) Test for Diagonality of £ across the three equations using the Breusch and Pagan (1980) LM test and the Likelihood Ratio test.

(d) Test for the equality of all coefficients across the 3 countries.

(e) Consider the first 2 countries once more. Now you only have the first 15 observations on the first country and all 19 observations on the second country. Apply Schmidt’s (1977) feasible GLS estimators, and compare the results.

13. Trace Minimization of Singular Systems with Cross-Equation Restrictions. This is based on Baltagi (1993). Berndt and Savin (1975) demonstrated that when certain cross-equation restrictions are imposed, restricted least squares estimation of a singular set of SUR equations will not be invariant to which equation is deleted. Consider the following set of three equations with the same regressors:

Vi = aiiT + + ti i =1, 2, 3.

where Vi = (Vii, Vi2,…, ViT)’, X = (xi, X2,…,xt)’, and ti for (i = 1, 2, 3) are T x 1 vectors and iT is a vector of ones of dimension T. ai and ei are scalars, and these equations satisfy the adding up restriction ^3=1 y-it = 1 for every t = 1, 2,…,T. Additionally, we have a cross-equation restriction: в1 = в2.

(a) Denote the unrestricted OLS estimates of fii by bi where bi = ^T=1(xt — X)yit/^2T=i(xt — X)2 for i = 1, 2, 3, and X = ^T=1 xt/T. Show that these unrestricted bi’s satisfy the adding up restriction в1 + в2 + в3 = 0 on the true parameters automatically.

(b) Show that if one drops the first equation for i = 1 and estimate the remaining system by trace minimization subject to @1 = в2, one gets @1 = 0.4b1 + 0.6b2.

(c) Now drop the second equation for i = 2, and show that estimating the remaining system by trace minimization subject to @1 = в2, gives @1 = 0.6b1 + 0.4b2.

(d) Finally, drop the third equation for i = 3, and show that estimating the remaining system by trace minimization subject to @1 = в2 gives в 1 = 0.5b1 + 0.5b2.

Note that this also means the variance of в 1 is not invariant to the deleted equation. Also, this non-invariancy affects Zellner’s SUR estimation if the restricted least squares residuals are used rather than the unrestricted least squares residuals in estimating the variance covariance matrix of the disturbances. Hint: See the solution by Im (1994).

14. For the Natural Gas data considered in Chapter 4, problem 16. The model is

logConSit = во + e1logPgit + в2 logPoit + e^fogPeit + e^ogHDDit

+e5logPIit + uit

where i = 1, 2,…, 6 states and t = 1, 2,…, 23 years.

(a) Run Seemingly Unrelated Regressions (SUR) for the first two states. Compare with OLS.

(b) Run SUR for all six states. Comment on the results and compare with those of part (a). (Are there gains in efficiency?)

(c) Test for Diagonality of £ across the six states using the Breusch and Pagan (1980) LM test and the Likelihood Ratio test.

(d) Test for the equality of all coefficients across the six states.

15. Equivalence of LR Test and Hausman Test. This is based on Qian (1998). Suppose that we have the following two equations:

ygt ag + ugt g 1, 2? I 1, 2,***T

where (u1t, u2t) is normally distributed with mean zero and variance U = £ <g> IT where £ = [<rgs] for g, s = 1, 2. This is a simple example of the same regressors across two equations.

(a) Show that the OLS estimator of ag is the same as the GLS estimator of ag and both are equal to yg = £)f=1 Vgt/T for g =1, 2.

(b) Derive the maximum likelihood estimators of ag and ags for g, s, = 1, 2. Compute the log – likelihood function evaluated at these unrestricted estimates.

(c) Compute the maximum likelihood estimators of ag and ags for g, s = 1, 2 under the null hypothesis H0; 011 = <^22.

(d) Using parts (b) and (c) compute the LR test for H0; a11 = a22.

(e) Show that the LR test for H0 derived in part (c) is asymptotically equivalent to the Hausman test based on the difference in estimators obtained in parts (b) and (c). Hausman’s test is studied in Chapter 12.

16. Estimation of a Triangular, Seemingly Unrelated Regression System by OLS. This is based on Sentana (1997). Consider a system of three SUR equations in which the explanatory variables for the first equation are a subset of the explanatory variables for the second equation, which are in turn a subset of the explanatory variables for the third equation.

(a) Show that SUR applied to the first two equations is the same (for those equations) as SUR applied to all three equations. Hint: See Schmidt (1978).

(b) Using part (a) show that SUR for the first equation is equivalent to OLS.

(c) Using parts (a) and (b) show that SUR for the second equation is equivalent to OLS on the second equation with one additional regressor. The extra regressor is the OLS residuals from the first equation. Hint: Use Telser’s (1964) results.

(d) Using parts (a), (b) and (c) show that SUR for the third equation is equivalent to OLS on the third equation with the residuals from the regressions in parts (b) and (c) as extra regressors.

17. Growth and Inequality. Lundberg and Squire (2003). See example 2, section 10.5. The data con­tains 119 observations for 38 countries over the period 1965-1990, and can be obtained from http://www. res. org. uk/economic/datasets/datasetlist. asp.

(a) Estimate these equations using SUR, see Table 10.1, and verify the results reported in Table 1 of Lundberg and Squire (2003, p. 334). These results show that openness enhances growth and education reduces inequality.

(b) Report the Breusch-Pagan test for diagonality of the variance-covariance matrix of the dis­turbances of the two equations. Compare the SUR estimates in part (a) to OLS on each equation separately.

References

This chapter is based on Zellner(1962), Kmenta(1986), Baltagi (1988), Binkley and Nelson (1988), Schmidt (1977) and Judge et al. (1982). References cited are:

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CHAPTER 11