ESTIMATION OF RISK EFFECTS WITH SEEMINGLY

UNRELATED REGRESSIONS AND PANEL DATA

Guang H. Wan, William E. Griffiths

and Jock R. Anderson

No. 40 - September, 1989

ISSN

ISBN

0157-0188

0 85834 843 8

Department of EconometricsUniversity of New EnglandARMIDALE NSW 2351 AUSTRALIA

ESTIMATION OF RISK EFFECTS WITH SEEMINGLY

UNRELATED REGRESSIONS AND PANEL DATA

Guang H. Wan and William E. Griffiths

University of New England, Armidale, NSW 2351, Australia

:lock R. Anderson

The World Bank, Washington, D.C., USA

Abstract

In this paper, production functions in the form of seemingly unrelated

regressions (SUR) with errors that are heteroscedastic and that contain

cross-section and time components are proposed. These functions are dis-

tinguished from others in that they allow the risks (indicated by variances)

of outputs to change in any direction in response to input changes. The

SUR are then applied in the analysis of cross-section time-series data for

rice, wheat, and maize production in China.

1 Introduction

The importance of risk (variance of output) has long been recognised in theanalysis of production functions, particularly agricultural production functions.

See, for example, Anderson, Dillon and Hardaker (1977). It is recognised thatsome inputs, e.g., investment in improving environmental conditions, are in-

versely or negatively reiated So the variance of crop outputs; whereas a positiverelationship may exist between other inputs, e.g., areas sown with modern eul-tivars (el., Anderson et al. (1989)), and the output variabilities of agricultural

crops. As long as a decision maker is not risk neutral, the relationship between

the variance of output and each of the inputs is an important ingredient into anydecisions concerning optimal input allocation. When increasing the level of an

input leads to an increase in the variance of output, we say the the marginal risk

of that input is positive. Conversely, we say an input has a negative marginalrisk when increasing its level leads to a decline in the variance of output. :lust

and Pope (1978) show that these different relationships cannot be correctly han-

dled by the commonly-used functions, no matter whether the function has an

additive error or a multiplicative error and no matter whether the function is

linear or nonlinear. For example, the widely-used Cobb-Douglas, transcendental

and CES functions, restrict the marginal product and marginal risk to be of the

same sign, normally positive. Other restrictions of these functions are detailed

by Just and Pope (1978).

To relax these restrictions, Just and Pope propose models with heteroscedas-

tic disturbances such as

Y = f(X) + h(X)~, (1)

Y = f(X) + h(X,~), (2)

Y = f(X,~), (3)

where Y and X are output and input variables respectively, e is usually a vector

of random disturbances, and h, f represent functional forms. Since equations

(2) and (3) are rather too general for insightful discussion of their estimation,

Just and Pope (1978) focus on equation (!) and suggest a four-step procedure for

estimating/1), where both f and h are assumed to be log-linear in parameters.

Using a Cobb-Douglas function for f and h, Griffiths and Anderson (1982)

consider an error component version of equation (1) and develop and apply

corresponding estimation techniques.

This paper presents an extension of the model considered by Griffiths and

Anderson (1982) into seemingly unrelated regressions (SUR). The SUR specifi-

cation is relevant when a number of different outputs exist, such as the output

of different crops, and where the disturbance terms from the functions for each

crop are correlated. This specification, which includes components for tiIne

series and cross-sectional units, is outlined in section 2. Discussion of an econo-

metric estimation procedure is presented in section 3. Some empirical results

based on Chinese data are provided in section 4 and the paper is concluded with

a summary in section 5.

2 The model

If there are N cross-sectional firms over T time periods producing M crops, a

set of M nonlinear stochastic equations of the form

K K

t’~ = 7,n H X~m’~ +em o H X~ (4)k=l k=l

can be established, where m = 1,2,..., M, 1’~ is the NT x 1 vector of ob-

servations on the output variable, ~.~ is an NT x 1 disturbance vector, X.~

is the NT x 1 vector of observations on the k-¢h input variable for the m-th

equation and as,3s are parameters to be estimated. The symbols, o and 11,

denote component multiplication of matrices. Raising the vector X.~ to the

power of 3,,~ (or a,~) means that each element in the vector is raised to the

power 3,~ (or ~,~).Assumingi= 1,2,..., N andt= 1,2,..., T, let

.K

Xmkit,

Hm = diag(h,~l, h,~,..., hmyT), (6)

H = diao(H1, H~,..., HM), (7)

~m = Z~#r~. + Z~A.~ + u,,~, (8)

where

Z. = IN ® eT, (9)

Z~ = eN ® IT, (10)

and ® denotes the Kronecker operation; IN, I~, denote N x N and T x T unit

matrices, ear,e~ are N x 1 and T x 1 vectors of ones. The model can then bewritten as

K

= 1I x +

where the i-¢h element of the vector ~,~ = [/~,~l,/~m~,-",/~m~r]’ and the i-th

element of the vector A,~ = [)wni, ;k,~.,, .. ¯, X,~T]~ represent the error components

specific to the i-th firm and t-th period in the m-*h equation, respectively; the

NT × 1 vector v,~ = [V-~l, ~-~2, "" ’, v,~NT]’ contains the error component which

is random over time and space for the m-th equation. Further, define

and

Ul

UM

k=l k=l k=l

the SUR models can be written as

Y = Xc7 + u. (13)

Following Avery (1977) and Baltagi (1980), the three components of u (i.e.,

#, ~ and v) are, as seems reasonable, assumed to be stochastically independent

from each other and

However, it is assuIned that the 3 components can be correlated with their

counterparts in the equations for different crops. That is,

or in matrix notation,

O’tzml IN 0

0 cr~mtIT0 0

0

0 (17)o’~,ml I NT

for m and 1 =

subscripts and 4, s time subscripts.

By defining

�1

�2

CM

the covariance matrix for (13) can be expressed as

1, 2, ..- , M, where i,j are section subscripts, m,l equation

~2 = E(uu’) = HE(��’)H

(18)

(19)

where

~-~M1 ~M2 " "" ~-~MM

The typical element of fl denoted by ~,~z has the form

where A = IN ~ eTe~ and B = eye~ @ IT. Let

A B JNT

T N NT’]NT : eNTe~T,

then equation (21) can be alternatively expressed as (cf., B~tagi (1980))

+ ~ NT

where

(20)

(21)

(22)(23)

(24)

O’lrnl : O’vmI ÷ (25)

(26)(27)

and o’umt are the distinct characteristic roots of flint of multiplicity N- 1, T- 1,

1 and (N - 1)(T - 1), respectively. These eigenvalues of a,,~l can be computed

according to Nerlove (1971), if necessary.After obtaining the values of ~rl,~z, ~r;,~l, ~a~t and ¢~mt by equations (25)

to (27) for m, l = 1, 2,-.., M, Baltagi (1980) shows that

+ (2S)

where

all of dimension M x M. As shown later, this expression will be useful for

computing f~ - 1

Under the above model specification,/3s represent production elasticities andc~s "risk elasticities" or risk effects of inputs, where risk is defined as the variance

of Y. Since as can be of any sign, the proposed SUR are distinguished from

more conventional ones in that they allow risks of outputs to change in any direc-tion in response to input changes. Also, the three error components in the model

are al! heteroscedastic in the sense that the variances ofand H,~u,~ depend on the input levels. This implies that the magnitude of both

firm and time effects will be influenced by the measured input levels, a situation

which may be more realistic than otherwise.

3 The multi-stage estimation procedure

Given the covariance matrix of (13) in (19), it can be seen that to use nonlinear

least squares to estimate 7 = (7~, 7~,’" ’, 7M)’ and/3 = (!3t,/3~., -..,/3M)’, where

/3.,. = (/3,~i,/3.~,’" ", fl,~K)’, the objective function to be nfinimised is

where it’ = u’H-1.

However, H cannot be computed without estimates ofa = (al, as, " ¯, aM)’,

and ~ depends on the unknown variances of the error components. To estimate

these various quantities, we begin by minimising

---- ~ ~ y ~’ (34)~U mit -- ~m Xmkit

m=l i=l t=l k

and obtain ~ and ~. Since cross-equation erroris not considered here, the

estimation can be undertaken for each m separately. Due to the existence of

heteroscedasticity and cross-equation error, the estimates will be asymptoticMly

inefficient, but they are generally consistent. Therefore, the estimated residual

~it = Ymit - ~ ~ X~t will converge in distribution to ~t under appro-

priate assumptions.

The second step is to estimate a. To do so, rewrite equation (12) in a slightly

different form as

u~ = h~, (~ + ~, + ,~,). (35)

Squaring the above equation and taking logarithms yields

K

In u~t = ln(,m~ + ~mt + "~it)2 + 2 ~ a~ In X~t. (36)

Let

then

K

E(ln u~t) = a,~0 ÷ 2 ~ a,~ In X,~u~t, (39)

~,~, = ln(.,~ + ~.~, + ~,.~)~ - ~o. (40)

Thus

ln(g.~i ÷ Ant ÷ r’..{t)2 : ~,~o ÷ ~.~t (41)

and equation (36) reduces to

Combining the set of M equations,

(42)

(43)

is obtained, where a = (al,a2,’’’,aM)’, am = (a,~o,2a,n,’’’,2a,~K)’, { :

({1,(2,’’’,~MNT)’,~=diag(~,~2,’’’,~M), ~m is a NTx (K + I) matr~

with 1.0s in the first column and lnXm} in the other columns, ~ is defined

similarly to ~ with ~ (In ~ ~ ~ )’.

When umit is replaced by its consistent estimator ~i~, equation (42) can be

used for estimation of a~. However, properties of (~it have to be investigated

in order to discover the properties of the estimates and to employ an appropriate

estimation technique.

If #mi, A~t and ~mi* are assumed to be normally distributed, the random

variables defined as

(44)

(45)

where

become standard normal variables with zero means and unit variance. Moreover,2q,~it, m = 1, 2,.-., M are each X2 random variables with one degree of freedom.

Taking the logarithm of the square of equation (44) produces

In 2 = ln(/z,~ + Amt + v,~)2 - lno’~qmit

am0 + (mi* In ~ (46)~ -- crm ,

where the second equality is obtained by use of equation (.1l). This variable

is thus distributed as the logarithm of a distribution with one degree of

freedom. Furthermore, this relationship between In q~it and ~mit gives us the

opportunity to derive the properties of the ~,,u, from those of the In q2mit. Since

both a,~0 and In o’~ are constant and ~t is defined by equation (38), it can be

shown (Harvey (1976)) that

Var(lnq2m,,) = Vav((,~it) = 4.9348, (47)

E(lnq2m,t) -- -1.2704

a.~o In : (48)

According to equations (38) and (47), ~,~i, has zero mean and a constant

variance. Therefore, a,~ can be estimated by applying OLS to equation (42) for

m = 1, 2,..., M separately and this produces no bias or inconsistency. But, it

does result in inefficiency since the M sets of equations are related and each of

them has a composite error structure similar to that of (17) as shown below.

When i = j and/or t = s, q,,~t and qljs will be correlated. This implies that

Inqmit" and In q~s will be also correlated when i = j and/or t = s. It can be

shown that

~(~.~, ~,~,)= E [~ q~,, ~ ~-,] - ~.2~0~~.

Since

E(qmitqlj,) - ~,,,m, i=j and t¢ s, (50)

E(qmitqlj~) = ~ t = s and i ~ j, (51)

E(qmitq,j,) = ~ i=s and i=j, (52)

E(q~,tq~j,) = O, 1�s and iCj, (53)

where ~r.~t = a~,nt + o’;~.~i + ~rv,~t, the following can be derived (Griffiths and

Anderson (1982), Johnson and Kotz (1972)):

(54)

(55)

Thus, ~mit can be viewed as having an error components structure similar to

that of e,~t and (42) can be estimated via a generalised least squares procedure

that utilises this structure. Specifically, ~ can be more efficiently estimated

by modifying the procedure and formulae in Baltagi (1980). This modification

produces generalised least squares(GLS) estimates of ~, namely 5, where

(59)

The As in the above expression are similar to fls defined by (25) to (32) with

o’s replaced by 3s. Estimated As can be specified after calculating estimated 6s

according to (54) to (57). However, such a calculation requires the estimation

of the ~rs, as will be discussed in (68) to (72). Alternatively, one can obtain the

best unbiased estimates of As directly (el., Baltagi (1980)) from

where ~ = (~, ~:,..., (M) is an NT × M matrix of residuals, which can be

obtained in two ways: (a) applying OLS to equation (42) for m = 1, 2,..-, M

separately and calculating the corresponding residuals; or (b) performing least

squares with dummy variables (LSDV) on (42) for m = 1, 2,..., M separately

10

and computing the corresponding residuals (Amemiya (1971)). It is noted that

both sets of residuals can be used to replace ~ for estimating the As and theresulting a has the same asymptotic efficiency in each case. However, As es-

timated tiom the LSDV residuals are asymptotically more efficient than those

from OLS residuals (Prucha (1984)). Thus, LSDV is used in this study to obtain

¢.Referring to both Baltagi (1980)and Prucha (1984), it can be shownthat

^Once d is obtained, generalisedleast squares residuals ~mit can befound

fronlK

^ ^’2 2~^ in X,,~,. (64)~*’nit ~ In umit - ~m0 -k=l

It is now possible to find efficient estimates of/3, which correct for het-

eroscedastieity, error components and correlation across equations. This is thetask of the third step.

According to equations (46) and (48),

= ~.~it - 1.2704,

O,~it = ~/exp(~,~it - 1.2704). (65)

It can be shown that

E(qmit qlit)

(66)

11

where O’mt = O’umt + O’:.~Z + Cq.mt and P,~z is the correlation coefficient between

em and et, which can be estimated byN T

i=1 t=l

From equation (48),

^2O’m = exp(&mo + 1.2704). (68)

These results give

&mZ = tSmt&~&t. (69)

Now, ~r,mt, ~Xmt and o’~mz can be estimated by

T N-1 N2

&~mt - NT(N 1) E E E umit ~tjt (71)- htctt=l i=1 j=i+l

~T~,ml ~ ~ml -- ~ml -- ~’~ml, (72)

whereK

~’,~, : l~ x~,. (73)

If these computations are being made with a view towards using (54) to

(57), then the estimate for &m~ would be from OLS or LSDV, rather than

GLS, because (54) to (57) are required before GLS estimation is employed.

Substituting &~mt, ~,mt, and &~mZ into equations (25) to (27) enables the

computation of 5 via equations (28) to (32). According to equation (19),

(74)

where ~ can be obtained through equations (5) to (7) with h,~t replaced by

~m~t. Thus, to obtain the efficient estimates of/3, represented by /~, it is a

matter of minimising

~__ Ut4--1U

== i~,’h-li~, (75)

12

where it = 9-1u.

For the purpose of programming, it is necessary to find a transformation

of the error term, say pi~, such that iz’p’pi, = i~’~-lit. When ~ is of small

dimension, one of the methods is to find c and ^ such that p = A-~c’, where

c is an orthogonal matrix consisting of the characteristic vectors of ~2 and /x

is a diagonal matrix consisting of eigenvalues of ~. However, f~ is of order

(MNT × MNT), which could well exceed a dinaension of 200. In this case,

finding c and A from f~ requires solving a polynomial equation of degree of over

200. This is a difficult task and unreliable results may occur. To tackle this

problem, a two step procedure is developed: (a) Decomposing ~-1 according to

the suggestion of Baltagi (1980) gives

+ (76)

where ill, h~, ha and ~u can be calculated according to equations (25) to (27)

and (29) to (32) with ~rs replaced by their estimated counterparts. It is noted

that these matrices only have dimensions of M x M. (b) Let h:~ = P4P~ and

define

for i = 1, 2, 3, 4. Further, define

A (78)D1 -T NT’B ~NT (79)D= -N NT ’&+r (80)Da -NT ’

D~ = Q. (81)

Then, since the Dis are all idempotent and DiDj = 0 for i ~ j, equation (76)

can be written as4

= ~ (P~P~ ® DiDo)i----1

® (82)

13

Therefore, an equivalent operation for minimising ~ is to minimise ii’ii, where

To summarise, the estimation of seemingly unrelated regression models,

which carry risk implications and incorporate composite errors, can be carried

out using the following steps:

(1) Find 3 and 5" by using nonlinear least squares either to minimise

for ’m :-- l, 2,..., M, or to minimise u’u; denote the corresponding residuals by

{l.

(2) Obtain & by applying the GLS techniqne on the SUR models with error

components, where in fi~,i, is regressed linearly on the ln X,,~i,s; denote the

corresponding residues by ~.

(3) Use ~ to estimate q via (65) and then p,.t,~r .......via (67), (68). This

enables the estimation of ~r,~t via (69).

(4) Use &,~t and & to find ~z,,~ from (73) and subsequently &umt, &.x,,,~ and

b~,~ from (70) to (72). Meanwhile, H can be estimated via (6) and (7).

(5) Construct ~1, ~,-, ~a and ~ by replacing o-s in (25) to (27) and (32) by

their estimated counterparts computed in step (4).

(6) Find Pi from 5, for i-- 1,2,3,4 and then obtain ~-~ from (76)

(7) Use ~ from step (4) and ~-~ from step (6) to find 5’ and ~ by employing

nonlinear least squares to minimise u’ft-~ fi-~ fiI-~u.

4 Empirical application

Chinese survey data for 28 regions (i.e., firms) for a 4-year period from 1980 to

1983 are utilised to estimate the disturbance-related production functions, as

proposed in the preceding sections. The data, covering three crops (rice, wheat

and maize), comprise output (jin*), sown-area (mu~ ), organic fertitiser (yuan~),

chemica! fertiliser (yuan), machinery cost yuan), irrigation cost (~luan), labour

"1 jin = 0.5 kg.

?1 rnu = ~ ha./As at the end of June, 1989, 3.7 yuan approximately equalled US$1.

14

input (mandays) and other costs (yuan). Those variables in value terms are

deflated by a weighted index of agricultural prices in state and free markets.

The Marquardt-Levenberg-Nash approach was used to find the nonlinear

least squares estimates offls (Marquardt (1963), Nash and Walker-Smith (1987)).

Estimates for the mean output functions and the output variance functions are,

respectively, presented in Tables 1 and 2. Estimates are given for different error

structures and different estimators. In each case the results were obtained from

several different sets of starting values.

In Table 1, estimated coefficients for the SUR heteroscedastic models are

reported in the third column. For comparison only, results from assuming uit =

~it are also presented.

From Table l(a), it is seen that, among the eight variables included in the

model, four of them have coefficients with negative signs. That is, rice pro-

duction elasticities with respect to labour, chemical fertiliser, animal cost and

machinery cost are less than zero. Since rice is mainly planted in Southern

China, where substantial underemployment or over-supply of labour exists in

the rural areas, it may be possible that negative returns with respect to labour

starts occuring, particularly after the resumption of double cropping (after triple

cropping) since the late 1970s. The negative elasticity with respect to chemi-

cal fertiliser is consistent with the findings of Wiens (1982). Large increases in

the application of nitrogen without corresponding increases in potassium and

phosphorus might be one of the most important reasons for the negative elas-

ticity (Stone (1986)). The negative elasticity associated with machinery cost

is plausible as replacement of labour by machines "destroys" the traditional

tabour-intensive farming technique. This is particularly true with rice produc-

tion since rice requires fine soil preparation and flat land, but machine operation

cannot meet these requirments as well as labour does. As for the animal cost,

the negative sign is implausible. However, except for labour, all the negative co-

efficients have ninety-five per cent confidence intervals which include a positive

range.

[Table 1 (a) near here]

Among the remaining variables, all but irrigation are significant contributors

to rice output. An examination of the magnitudes of the estimates indicates

15

that sown area asserted the greatest positive impact on rice output, followed by

organic fertiliser. The insignificance of irrigation may result from the fact that

almost all the rice area sown is irrigated and thus the effect of irrigation cannot

be estimated precisely.

The estimates of the mean maize output function are tabulated in Table l(b).

From the asymptotic t-ratios, all positive estimates are statistically significant

at the five per cent level. On the other hand, the three negative coefficients for

labour, chemical fertiliser and animal cost are implausible, but they do have

ninety-five per cent confidence intervals which include a positive range. Unlike

the case of rice, machinery cost is positively related to maize production. A

possible explanation is that, for maize production, machine operation is mainly

involved with cultivation and planting. Thus, there is less post-harvest loss when

harvesting by machines. More importantly, timing of planting is more crucial

for maize production than for rice and the requirement for seedbed preparation

is not as great as for rice. Maize is mainly grown in the central and north of

China, where farming techniques are relatively poorer than in the south. In

other words, replacement of labour by machines is likely to create a positive

impact on maize output. Moreover, in the far north the excess labour problem

is less severe if it exists at all. This may also help explain the positive sign of

[Table 1 (b) near here]

Area sown is the dominant factor influencing maize output. The production

elasticity with respect to sown area is 0.68, followed by 0.16 with respect to

organic fertiliser and 0.15 with respect to other costs. The elasticity is only 0.02

for machinery cost and 0.016 for irrigation.

The wheat production function is the function that seems to be estimated

most successfully (Table l(c)). The only negative estimate is the elasticity

with respect to irrigation. Wheat is largely planted in the far north of China,

where water supply relys heavily on rainfall. It is noted that the negative value

has a small t-statistic. Thus, the true elasticity of wheat output with respect

to irrigation nfight be very small and its estimate could well turn out to be

nonpositive.

16

Contrary to both rice and maize, the coefficients of labour, chemical fertiliser

and animal cost are all positive, although the estimate associated with animalcost is not significant at the 5 per cent level. Chemical fertiliser has the smallest

positive elasticity and organic fertiliser has the largest elasticity. The elasticitywith respect to labour is not only positive, but substantia! relative to that for the

other inputs. This comes as no surprise since wheat is predominantly planted

in the far north of China where labour is relative scarce. The above-mentioned

reason could also explain the relatively large elasticity of machinery cost inwheat production.

[Table 1 (¢) near here]

Overall, Table 1 indicates that, where labour is relatively scarce, machinery

generates a positive and significant impact on crop yield. For example, when

labour input has negative returns in rice production, machinery creates a nega-

tive effect on production and the effect is significant at a 10 per cent level. In the

case of maize production, labour had no significant impact and machinery gen-

erated a limited, though a significant effect on yield (the coefficient is only 0.02),

whereas when the labour effect is significantly positive in wheat production, the

machinery effect becomes positive, significant and substantial (the elasticity is

0.08).

The parameters whose signs determine the signs of marginal risks are pre-

sented in Table 2. Although attention will be focused on the estimates given

by GLS, parameters estimated by other techniques are also shown in Table 2.

Goodness-of-fit for each equation is measured by the R: from separate OLS ap-

plied to each equation. The goodness of fit for the SUR system was calculated

according to

2 ~(2-1 @ INT)~RsuR = 1 -}:,(E_1 ® DhrT)g (84)

where E-1 is the variance covariance matrix of the SUR models, ~ is a MNTx 1

vector containing the GLS residuals, and DNT = INT -- JNT/NT. The FsUR

statistic is obtained based on R~un (Judge et al. (1985, p. 478)). The results

(not reported in the tables) were R~SUR = 0.55 and Fs[zR = 16.01. Noting

that the data used for estimation are basically cross-sectional (the time span

is relatively short), 0.55 indicates a reasonable goodness of fit. The Fsr~R is

17

statistically significant at any conventional level, which suggests the existence

of heteroscedasticity. This may imply the inadequacy of conventional functions

or the superiority of the heteroscedastic SUR models.

Machinery, organic fertiliser and other costs seem to have stabilising effects

on rice output (Table 2(a)). The coefficient for machinery is insignificant. It is

reasonable to have a negative &s since the major component of other costs is

expenditure on management. The significance of both positive ~7 and negative

&7 imply the importance of organic fertiliser in achieving a high and stable yield

in China’s rice production. The variance of production is positively related to

chemical fertiliser application, though there is a lack of statistical significance.

This is in line with the expectation of Hazell (1984), who suspected that, as

seed-fertiliser technology advances with the adoption of high yielding varieties,

increased use of chemical fertiliser may bring about higher production vari-

ability. As far as animal cost is concerned, the significantly positive sign is

implausible and thus needs further investigation. While irrigation is expected

to help stabilise production, the empirical result here does not seem support-

able. Noting that &5 is significant and of considerable magnitude, it seems that

better management of the irrigation system in China is urgently needed. The

positive d5 and negative ~ could well be the result of a malfanctioning of the

irrigation system due to a collapsed management of water resource and irriga-

tion facilities after the introduction of the agricultural production responsibility

system in late 1978. The impact of area sown on production risks basically

depends on the correlation coefficients among rice outputs of different seasons

and on management skills. In general, a positive relationship is expected. Fi-

nally, labour does not produce a significant impact on production risk. This is

primarily because the labour input in China was near "saturation" long before

1980. Thus its changes may not generate any effect on either mean output or

output risk.

[Table 2 (a) near here]

Contrary to the case of rice production, animal cost and irrigation were

estimated to be stabilising factors in maize production in China (Table 2(b)).

This may be due to the relative insensitivity of maize to water supply in timing,

quantity and frequency. In other words, irrigation can help to stabilise maize

18

production and, while there are irrigation problems in China, these may only

generate a very limited impact on maize yield variability. The variables other

than animal cost and irrigation are all positively related to maize production

variance. This is plausible for labour, area sown and chemical fertiliser for the

reasons discussed earlier. The positive signs of machinery, organic fertiliser and

other costs are implausible. However, all the positive estimates have ninety-five

per cent confidence intervals which include negative values.

[Table 2 (b) near here]

The relationship estimated between wheat output variance and inputs can

be found in Table 2(c). All the slope parameters are insignificant at the 5 per

cent level. The negative value for area sown is unexpected as is that for organic

fertiliser. The estimates associated with labour and other costs are not only

positive, but also quite large in magnitude. It should be stressed that all the

slope coefficients could well be zeros in accordance with the asymptotic ~-ratios.

[Table 2 (c) near here]

It is difficult to generalise findings from the estimates of the three equations

because (a) most of the estimates are not encouraging in terms of statisticzd

significance; and (b) the magnitudes of and especially the signs of parameters

are often inconsistent across equations. However, as far as the relationship

between the ’green revolution’ and production risks is concerned, the empirical

results indicate that there is a positive link between seed-fertiliser techno!ogy

and output variability. This may be due to the introduction of modern cultivars

which have a narrower genetic base than their predecessors (Hazell (1984)). The

nature of irrigation in the context of output variability crucially depends on the

reliability of the water supply. Taking into account the fact that the irrigation

systems in many parts of China are severely damaged, a nonnegative effect of

irrigation on output risk may be understandable. The machinery input possibly

brought about higher risks, which could arise from the poor quality of both

tools and operations.

The estimated matrices [~u-~t], [~vmt], [~,~t] and [~,~1] are given in Table 3.

A comparison of the corresponding values of [~x-~t] and [~u,~t] indicates that

the time effect may be negligible. Cross-equation covarianees are given by the

19

off-diagonal values of [~,~z]. If the assumption of a normal distribution for each

of the three (time, firm and random) errors holds, the diagonal elements of [~mZ]

should be close to 4.9348. See equation (47). Statistical tests (F statistics) are

undertaken and it is found that all three values are not significantly different

from 4.9348 at a five per cent level. In passing, it is noted that the negative

values on the diagonals of these matrices are possible and they can be set to

zero in practice if necessary (Fuller and Battese (1974, p. 72)).

[Table 3 near here]

The variance-covariance matrices of the mean output functions are given inTable 4. The lack of time effects is again seen by the small ratios of

The contemporary covariances across equations are all positive and substantial.

[Table 4 near here]

5 Summary

In this paper, SUR models which incorporate time-specific and firm-specific er-

ror components and permit the marginal variances of outputs to have either

positive or nonpositive signs are proposed. An estimation procedure is sug-

gested. This attempt is of empirical significance, particularly in agro-economic

research, since outputs of various agricultural activities tend to be influenced by

some common factors, notably weather and policy changes. Also, increases of

different inputs can either enhance or reduce output risks. Conventional SUR

models restrict the marginal risks to be positive.

Using combined time-series (4 years) and cross-section (28 regions) data

on Chinese rice, maize and wheat production, heteroscedastic SUR production

functions were estimated. The results indicate that, as chemical fertiliser, sown

area and irrigation costs increase, output variances generally rise. On the other

hand, organic fertiliser, machinery cost and the other costs may help stabilise

Chinese cereal production. The labour input does not create significant impacts

on either mean outputs or output variances. These results suggest the possible

superiority of the heteroscedastic SUR models over more conventional ones.

2O

It must be noted that most of the inputs considered in the models are not

necessarily siginificantly related to production risks. This is not to suggest that

these and other inputs are, in fact, unimportant or unnecessary in production

and its riskiness. It may, however, imply the importance of weather and govern-

inent intervention in agriculture in determining the variability of Chinese cereal

production.

21

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23

Table 1: Parameter Estimates for the Mean Output Function(a) Rice

Error Structure

nit = ~it nit = (gi+At÷~it)hit

6.836 269.663(3.29) (3.37)

~1 -0.181 0.728(*tea) (-2.47) (S.TS)

~2 0.347 --0.125(Labour) (10.52) (-2.38)

~3 -0.007(Chemical Fertiliser) (-0.18)

~4 -0.031 -0.035(Animal Cost) (-1.61) (-1.43)

~5 -0.010 0.017(Irrigation) (-0.75) (0.73)

~6 -0.031(Machinery Cost) (-2.59)

~7 0.905(Organic Fertihser) (11.98)

0.379(5.46)

~s 0.120 0.098(Other Costs) (7.30) (3.00)

Note: Figures in brackets are asymptotic t-ratios.

Table 1: Parameter Estimates for the Mean Output Function(b) Maize

Error Structure

~t =~t u~t=(~+At+~)h~t325.277 415.365(4.02) (6.42)

~1 0.584 0.676(Area) (15.10) (13.24)

~2 0.012 -0.027(Labour) (0.32) (-0.70)

~a 0.0002(Chemical Fertihser) (0.01)

~4 -0.062 -0.017(Animal Cost) (-5.72) (-0.81)

~6 0.013 0.016(Irrigation) (1.79) (1.94)

~6 0.009(Machinery Cost) (0.88)

0.022(2.13)

~7 0.106(Organic Fertihser) (2.71)

0.161(4.55)

~s 0.330 0.147(Other Costs) (8.61) (5.36)

Note: Figures in brackets are asymptotic t-ratios.

Table 1: Parameter Estimates for the Mean Output Function(c) Wheat

Error StructureUit = bit Uit = (l~i + At + ~it)hit103.304 136.950(3.79) (3.75)

81 0.382 0.198(Area) (4.41) (1.96)

82 0.212 0.140(Labour) (4.95) (2.14)

83 -0.014(Chemical Fertiliser) (-0.82)

0.049(1.98)

84 -0.044 0.064(Animal Cost) (-2.37) (1.79)

85 0.011 -0.024(Irrigation) (0.80) (- 1.21)

86 0.074(Machinery Cost) (2.90)

0.084(3.29)

87 0.230(Organic Fertihser) (7.60)

0.261(4.09)

8s 0.148 0.187(Other Costs) (2.78) (3.13)

Note: Figures in brackets are asymptotic t-ratios.

Table 2: Parameter Estimates for the Output-Variance Function(a) Rice

Estimation TechniqueOLS LSDV GLS

22.432 - 18.813(14.90) - (7.66)

251 3.443 1.339 2.072(Area) (8.02) (0.67) (2.56)

2&2 -0.910 0.651 0.174(Labour) (-3.03) (0.44) (0.32)

253 0.303 1.017 0.515(Chemical Fertiliser) (2.82) (1.16) (1.92)

254 0.470 1.228 1.005(Animal Cost) (2.63) (1.36) (2.84)

2&~ 0.273 !.049 0.796(Irrigation) (1.69) (1.27) (2.40)

256 0.101 -0.179 -0.005(Machinery Cost) (1.08) (-0.33) (-0.03)

257 -2.003 -1.987 -1.850(Organic Fertiliser) (-5.59) (-1.22) (-2.84)

253 -0.450 -1.894 -1.433(Other Costs) (-1.89) (-1.66) (-3.17)

R2 0.637F-ratio 22.609

Note: Figures in brackets are asymptotic t-ratios.

Table 2: Parameter Estimates for the Output-Variance Function(b) Maize

Estimation TechniqueOLS LSDV GLS9.138 - 9.107(6.77) - (3.91)

2&I 0.377 1.675 1.056(Area) (0.83) (0.45) (1.34)

2&: 0.368 0.013 0.058(Labour) (1.03) (0.004) (0.11)

2&3 0.040 0.054 0.099(Chemical Fertiliser) (0.35) (0.08) (0.56)

2~4 --0.055 -0.785 -0.440(Animal Cost) (-0.39) (-0.70) (-1.77)

2&s -0.160 -0.436 -0.182(Irrigation) (-2.22) (-0.79) (-1.28)

2&6 0.282 0.390 0.349(Machinery Cost) (2.87) (0.51) (1.86)

2&7 0.340 0.338 0.437(Organic Fertiliser) (1..08) (0.16) (0.76)

2&s 0.250 0.484 0.261(Other Costs) (0.82) (0.29) (0.49)

R2 0.533F-ratio 14.684

Note: Figures in brackets are asymptotic t-ratios.

Table 2: Parameter Estimates for the Output-Variance Function

(c) Wheat

Estimation TechniqueOLS LSDV GLS8.310 7.226(6.99) (2.54)

(Area)

(Labour)

0.016 -0.709 -0.407(0.04) (--0.37) (-0.41)

0.886 -0.207 0.700(3.40) (--0.17) (1.26)

2&3 -0.331 0.290 0.018(ChemicM Fertiliser) (-3.01) (0.96) (0.09)

254(Animal Cost)

-0.005 0.003 -0.073(-0.03) (0.01) (-0.24)

(Irrigation)-0.002 0.184 0.056(-0.02) (0.42) (0.24)

25’6 0.444 -0.011 0.247(Machinery Cost) (4.14) (-0.03) (1.13)

2&7 -0.141 0.682 0.113(Organic Fertiliser) (-0.47) (0.67) (0.19)

2&g 0.482 1.270 0.878(Other Costs) (1.64) (1.58) (1.65)

R~ 0.596F-ratio 19.010

Note: Figures in brackets are asymptotic t-ratios.

Table 3: Covariance Matrices of Output-Variance Functions

-0.010 -0.043 -0.168:0.043 0.058 -0.046-0.168 -0.046 0.148

1.660 1.336 -0.6971.336 3.373 -0.402

-0.697 -0.402 3.321

5.293 -0.227 -0.088

[~~~z] -0.227 2.803 0.122-0.088 0.122 2.009

6.093 0.833 -0.576

0.833 5.035 -0.118-0.576 -0.118 3.981

Table 4: Covariance Matrices of Mean Output Functions

1570.784 619.503 -131845.170619.503 246.119 -277397.578

-131845.170-277397.578 76840813.036

15928.522 5809.082 840678.539[&~mZ] 5809.082 4593.849 271162.822

840678.539 271162.822 186857666.081

14630.159 3448.611 1982192.8153448.611 56.530 985374.936

1982192.815 985374.936 263829703.787

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