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Estimation of a model containing unobservable variables using grouped observations
Journal of Econometrics
A MODEL
6 (1977) 51-63. 0 North-Holland
Publishing Company
ESTIMATION OF CONTAINING UNOBSERVABLE VARIABLES USING GROUPED OBSERVATIONS
An Application
to the Permanent Income Hypothesis
Clifford
L.F. ATTFIELD*
University of East Anglia, Norwich, England Received February 1975, final version received July 1976 The method of estimating relationships in a model containing unobservable variables is combined with the technique of estimation from a sample of grouped observations to analyse the permanent income hypothesis. From a sample of group means the paper measures the relative importance of such factors as wealth, occupation, age and family-size as determinants of permanent income in the United Kingdom, and also considers the mechanism by which transitory receipts influence total consumption expenditure.
1. Introduction Mayer (1972) devotes a considerable part of his book to tests of the permanent income hypothesis (PIH). Although the results enable him to reject the full permanent income theory at one extreme and the measured income theory at the other, he concludes that for ‘the much more demanding question of quantifying the error of the two extreme positions. . . tests still give vague results’. [Mayer (1972, p. 356)]. There appear to be two major reasons for this difficulty in assessing the PIH. Firstly the scarcity of data-particularly cross-section budget studies - and secondly, even when data exist, the problem of obtaining direct estimates of the theoretical constructs. Recent developments in the econometric literature indicate how these problems may be overcome. With regard to the latter, the application of the Zellner method (1970) for obtaining estimates in systems where some variables are ‘unobservable’ permits a direct estimation of the marginal propensity to consume (permanent) out of permanent income [Friedman’s ‘k’ (1957)], whilst simultaneously allowing some inference to be made concerning the existence of a correlation between permanent and transitory components. Secondly, the data problem may be alleviated somewhat *I am greatly indebted to a referee and an Associate Editor, both anonymous, for correcting a number of mistakes in an earlier draft. All remaining errors are of course my own responsibility.
52
C.L.F. AttjTeld,Model containing unobservablevariables
by the technique suggested by Haitovsky (1973) for obtaining Aitken estimates from grouped data. In this paper it is shown that the grouping technique can be easily extended to a system of recursive equations and hence data tabulated in the form of sample means can be utilised to obtain estimates of the structural parameters of the permanent income model. 2. Model specification The structural equations for an individual consumer unit in a cross-section sample of Nindividuals are
cp= k,yP,
c = cp+ul,
y =
yp+u2,
ye =
X~S+E,
(1)
where c is measured consumption and y measured income; cp, yp, u1 and u2 are permanent and transitory components of consumption and income respectively; x is a (K x 1) vector of those variables of which ‘the permanent component of income is itself a resultant . . . such as location, age, occupation, education and the like’ [Friedman (1957, p. 216)]; /? is a (K x 1) vector of structural coefficients; k1 is the marginal propensity to consume out of permanent income, and E is a stochastic error. Throughout, the transitory components are treated as random errors with the stochastic specifications :
Ul
E
[I u2
MY 41 =
z;or,,
E(E) = E(u,) = E(u,) = 0, E(Eu;) = 0 (j = 1,2), E(d) = cJN, where E, uI,u2 are (Nx 1) vectors. The force of these assumptions is that for any individual in the sample, transitory components may be correlated, oi2 # 0, but correlation between transitory components between individuals is excluded. In addition it is assumed that E(d)
= 0
and
E(xu$) = 0,
j=
1,2.
After substitution the reduced form for the model becomes
(2) y = xgi-s+u;!
= xn2+v2,
53
C.L.F. Attfield, Model containing unobservable variables
where
where
and so
E
Ul
b;,
v;] = sz@I,.
[I v2 At this stage the specification is identical with the one given by Goldberger (1972, pp. l-8). The (K+l) structural coefficients contained in j? and k, are overidentified whilst the remaining structural parameters in Sz are underidentified unless there are restrictions on either aEEor in X. (As it stands there are only three unique reduced form parameters in Sz from which we wish to extract four structural parameters viz. oEE, c12, 022, c1 1.) One restriction could be that transitory components are uncorrelated so CJ~~= 0 and hence z is diagonal, or, alternatively, that permanent income is an exact function of the variables in xi so that gEE= 0 in which case 52 = C, estimates of the structural coeficients will of course be identical in both cases [cf. Goldberger (1972, pp. 6-8)]. Estimates of the full set of structural parameters, on one of the above assumptions, can then be obtained by the Zellner procedure ZEF or, equivalently, by full information maximum likelihood (FlML) - assuming that ul, u2 and E are normally distributed. Suppose that instead of a full cross-section sample only group means are available. Specifically, imagine the sample has been grouped into M classifications (by occupation for example) and the sample mean of each variable within the classification is the only information obtainable, e.g., for consumption the data given would be %7
cg=
[ i=l
ng being the number
Rearranging
CC&
1
-l l$J
3
of consumer
g
=
1,. . ., M,
units in the gth occupational
each block of equations
group, and so
in (2) into the M groups,
summing
and
54
C.L.F. Attjield, Model containing unobservable oariables
taking sample means for each group results in Gc = GX/Ik,+k,Gs+Gu,
= GXn,+Gu,, (3)
Gy = GXjI+Gs+Gu2
= GXn,+Gv,,
where
.O
-0 . 1 .nhi_ Since the original transitory components were assumed homoskedastic this transformation introduces heteroskedasticity, the covariance matrix of the reduced form errors becoming
;;l
E [
2
[u;G',
u;G']
= QQGG'.
1
To overcome this problem decompose the positive definite matrix (GG’)-1 into T’T = (GG’)-I, where 0 .......... 0 0
dn2 0
O...O z/nj...
0
Premultiplying the equations in (3) by T, the covariance matrix of the reduced fotm errors becomes E
;;:I1
[v;G’T’,
[
v;G’T’]
= sZ@l,.
(4)
2
Hence, with the exception of the order of the identity matrix on the righthand side of the Kronecker product [in (4)], the covariance matrix of the reduced form (and structural) disturbances in the ungrouped case is the same as the covariance matrix of the disturbances in the transformed, grouped case. Since
C.L.F. Aftfield, ModeI containing unobservable variables
55
the reduced form errors, after applying the G and T transformations, are linear functions of the ungrouped structural disturbances, if these latter are normally distributed then FIML may be applied to the transformed system to obtain estimates which have the same asymptotic properties as Aitken estimates.
3. Application of the method to U.K. data Some of the most comprehensive budget studies undertaken in the U.K. were the Oxford Savings Surveys which spanned the years 1952-1954. In his article, Fisher (1956) classifies the 1953 sample survey into age within occupational groups, and presents a table giving sample means for savings, income, liquid assets and family size for these categories (p. 264). Another table (p. 275) gives the unweighted frequencies of the cell groupings. In obtaining the published sample means, a weighting scheme was used whereby larger weights were allocated to those consumer units residing in housing with a rateable value of less than &30 [Fisher (1956, pp. 204 and 275)]. Fortunately it is possible to use the cell means to solve a set of homogeneous linear equations to obtain the bveighted frequencies used to calculate the means given by Fisher.l The weighted frequencies were used to compute the transformation matrix T. To obtain a consumption series the data on savings (including durables) were converted into consumption (excluding durables) using the income identity.’ The variables included in the x vector- the determinants of permanent income -were age (x,), square of age (x,), occupation (x3), liquid assets (x,), and family size (x,), which are assumed to be the determinants of permanent income. The coding for the age variable was the same as the one used in the survey, i.e., 1 to 6 for age groups 18-24, 25-34, 35-44, 45-54, 55-65, and 65 plus, respectively. Income, consumption and liquid assets were measured in pounds sterling and family size by the number of individuals in a consumer unit. The occupation variable was coded as follows: retired and unoccupied: 1, manual: 2, clerical and sales : 3, managers: 4, self-employed: 5. The estimates obtained using this coding should be regarded as more illustrative than substantive, a more detailed study would incorporate dummy variables for the occupations. Theoretically, permanent income is defined as the return a consumer expects to receive from human and non-human sources [Friedman (1957, p. lo)]. ‘Since the weighting scheme is given by Fisher (1956, p. 275) and one of the frequencies is unity it is possible to solve the homogeneous equations uniquely. For details of the method, see Attfield (1973). *If income is measured with some error, when saving is converted to consumption this latter will incorporate the same error. This merely implies, however, that the elements of the reduced form error matrix will include the variance of the measurement error. The model specification can easily be extended to such a formulation without affecting the properties of the estimates of kl and 8. On the other hand the definition of income used by Fisher in this study has been criticised by Friedman (1957a).
56
C.L.F. Attfield, Model containing unobservable variables
Human wealth might be considered a function of age, occupation and family size - the second-degree polynomial in age reflecting the time path of permanent income over an individual’s lifetime. 3 Liquid assets on the other hand are assumed to be representative of non-human assets. Whilst it is realised that in general these are not really an adequate proxy for non-human wealth [cf. Mayer (1972, p. 168)], Fisher (1956, p. 241) shows that in the Oxford Savings Survey for certain occupational categories ‘liquid assets are an important ingredient of the assets total’. 4. Results Estimates were obtained by FIML using an algorithm similar to the one developed by Eisenpress and Greenstadt (1966). From initial starting values for k, and fi the procedure converges, by iteration, to a maximum of the likelihood function. To be reasonably certain that the maximum was global, the starting values were procured from the ZEF estimation procedure [Zellner (1970)]. Table 1 (A) gives estimates of the fully unrestricted reduced form coefficients, i.e., estimates of the vectors n1 and n;2 in (2). Table 1 (B) gives estimates of the structural coefficients, i.e., k, and the vector fi, for the case of Q diagonal and the case of Q unconstrained. The likelihood ratio test was used to test the overall model specification. First, we wish to test whether the reduced form error covariance matrix is diagonal. If it is, then transitory components of consumption and income are uncorrelated, permanent income is uncorrelated with transitory income and permanent income is a non-stochastic function of the variables in xq4 The likelihood for the reduced form model in table 1 (A) with error covariance matrix constrained to be diagonal and the likelihood for the same model but with the error covariance matrix unrestricted, were computed. The null hypothesis that the reduced form errors are contemporaneously uncorrelated may then be tested against the alternative of correlated errors by means of the likelihood ratio 1. In large samples the statistic - 2 log,l is distributed as chi-square with one degree of freedom. Since the value of the test statistic is 41.59 the null hypothesis of a diagonal reduced form error covariance matrix can be rejected at the 1 ‘A significance level. This implies that either transitory components are correlated or that permanent income is correlated with transitory income (either of which leads to rejection of the strict PIH), or that permanent income is a 3Friedman (1963, p. 6) comments: ‘It is worth recording that we do have some relevant data on human capacity - age, occupation, education, etc. One interesting direction of research would be to try to estimate human wealth from such indices thereof and to use such estimates in analyzing consumption’. Family size - more strictly ‘number in income unit’ - captures the human wealth aspect of children; see end of next section. 4Assuming no measurement errors -see footnote 2.
64.153 (45.530)
29.802 (51.46)
Consumption
Income
45.847 (14.532)
- 7.706 (7.680)
0.408 (0.098)
0.330 (0.083)
Liquid assets
- 10.168 (5.443) -11.528 (7.674)
50.516 (36.540)
61.989 (52.660)
0.924 (0.035)
0.925 (0.014)
(i) Diagonal s2
(ii) Unconstrained
33.291 (15.799)
37.797 (10.410)
Occupation (83)
0.367 (0.097)
0.382 (0.069)
92.576 (28.800)
95.399 (21.161)
Family size (85)
100.340 (29.812)
83.965 (26.376)
0.936
0.833
0.656
0.678 0.667
0.922
0.870
$;oe)
_
~.-
_
^ _,, _ _ _
. ,. -
-
-
__ -
___^,
0.939
0.779
r,
..I
and income
0.942
0.652
Estimates of
R2
MSE (x 106)
a11 (x 106) $;oe,
Family size
“The figuresin brackets are estimates of asymptotic standard errors obtained from the (inverse of) the information matrix. br, is one minus the ratio of the generalised variance of the reduced form residuals to the generalised variance of consumption [cf. Dhrymes (1970, p. 254)], being the simultaneous equation analogue of the coefficient of determination of a single equation.
(Age)’ (82)
Age (81)
Liquid assets (84)
Estimates of structural parameters.aSb
28.122 (12.858)
- 11.472 (6.7952)
(B)
Occupation
(Age)’
Explanatory variables
Estimates of reduced form parameters.
Permanent income (kr)
Q
Age
Dependent variable
(A)
Table 1
.
Y
58
C.L.F. Attfield, Model containing unobservable variables
stochastic function of the variables in x, or a combination of these factors. A further analysis of the error covariance matrix is undertaken in the next section. Secondly, we wish to test the specification of the structural model. For this, the likelihood of the reduced form model of table 1 (A) with an unrestricted error covariance matrix and the likelihood of the structural model of table 1 (B. ii), also with an unrestricted error covariance matrix, were computed. The null hypothesis that the structural model co&cients are correctly specified may then be tested by forming the ratio of these likelihoods. In this case the value of the chi-square test statistic is 12.17 and since this is less than 13.28, which is the value of a chi-square variate with four degrees of freedom at the 1% significance level, the null hypothesis that the structural model is correctly specified cannot be rejected. The results imply that the M.P.C. (permanent) out of permanent income is 0.925 compared with the M.P.C. (measured) out of measured income of 0.798, obtained by Fisher (1956, p, 270) from a simple linear regression of consumption on income. The value of 0.925 is essentially the same as the consistent estimate of k, of 0.941 computed as the ratio of mean consumption to mean income from Fisher’s paper (1956, p. 264). The point estimates of the coefficients on age imply that permanent income increases with age in the cross-section for individuals (i.e., consumer-unit heads) aged up to the range 34-45 and thereafter declines. Thus it might be inferred that this is the time profile of permanent income over an average individual’s lifetime. Similarly the coefficient estimate on liquid assets-the proxy for the value of physical assets - can be taken as the average individual’s subjective rate of discount (Friedman’s u) on non-human wealth. The reciprocal of this coefficient (2.725), is an estimate of Friedman’s (1963, pp. 5-11) ‘horizon’ and is strikingly close to his estimate of ‘about three years’. The family size coefficient may be interpreted as the rate of return on a single addition to the consumer unit, e.g., the birth of a child increases permanent income by &92 (recall that the estimates refer to 1953). ‘Children are, after all, a way of achieving security for old age; indeed in many cultures, the primary way. The raising of children can be viewed as a form of capital accumulation, only of human rather than non-human capital.’ [Friedman (1957, p. 122)].
5. Analysis of the reduced form covariance matrix The hypothesis of a zero off-diagonal element in s2 was rejected in the previous section. Assuming no measurement errors it follows that either transitory elements are correlated, or that permanent income is stochastic, or that permanent income is correlated with transitory income, or a combination of these factors. Since liquid assets have been used as a proxy for non-human wealth and variables such as education and location have been omitted from the equation determining permanent income, we can be quite certain that oEe # 0. To ascer-
C.L.F. Attfield, Model containing unobservable variables
59
tain whether correlations between transitory components or between permanent and transitory components also contribute to the non-zero value of w1 2 requires further analysis. Before proceeding with this analysis it is necessary to distinguish between three separate model specifications. Firstly there is Friedman’s (1957) original permanent income hypothesis (PIHl) in which transitory components are assumed to be uncorrelated (or 2 = 0) and transitory components are assumed to be uncorrelated with permanent components. In this model the MPC (measured) out of transitory income is zero. Secondly there is the modified model developed by Friedman (1963) in which the whole of a windfall is considered to be transitory income which is invested in an asset (PIH2). The return on this asset in a particular accounting period contributes toward total permanent income for the period. Formally, the permanent income equation now becomes yp = x’~+E+ru2, where u2 is transitory income and r the return on each unit of invested transitory income per accounting period, an estimate of which is the return on non-human assets of 0.367 in table 1. This formulation implies that while transitory components are uncorrelated, permanent income is correlated with transitory income and hence the MPC (measured) out of transitory income is given by k,r. 5 Thirdly, there is a variant which includes a relationship between the transitory components (PIH3). Many writers have argued that it is invalid to assume that transitory components are uncorrelated. On the contrary, part of a windfall which is all transitory income will be spent on transitory consumption. The difficulty lies in testing this proposition since as Mayer (1972, p. 40) argues
sit has to be assumed that there is a lag of, say, one accounting period between the receipt of a windfall and the inclusion of the asset nurchased with the windfall in the individual’s total wealth stock. Otherwise, that element of the x’ vector which refers to total wealth will include YU~.Also, since the Survey under consideration took one year as the accounting period, it is assumed that all windfalls (both positive and negative) occurred at the beginning of the year.
C.L.F. Attjield, Model containing unobservable variables
60
u1
=
kzu,+I,
where
E(1;)= O, E(iU2) = OS E(CiCj)
=
= -wt;)
O,
i #j,
cJ&
i=j,
= 0,
and where k2 is the MPC (transitory) out of transitory income such that 0 < k, < 1. With this specification, however, if kzuz is consumed, only (1 - k,)u, remains for saving. Thus the permanent income equation becomes yp = x’/I+e+(l
-kz)ruz.
In this model the MPC (measured) out of transitory income is given by k,f (1 -k,)rk,. The three specifications may be summarised by the structural equations
cp= klyp,
c = cp+ul,
y = yp+uz,
yp = x’/?+&+6yuz, ~1 =
where E, u2 and 1: are assumed to be uncorrelated and 6 = 0 or 1. The reduced form becomes
k,u,+L and where y = (1 -k,)r
c = x’~k,+kle+(6kly+k2)u2+1;, JJ =
(5)
X’j+E+(6y+l)U2.
If k, = 0 and 6 = 0, we have the first model PIHl (with crll = rr
(k,a,,+@k,y+kz)
klo,,+(6k,y+k2)(6y+l)a22
G+(~Y+
(h+
W22)
1)20,,
I
61
C.L.F. Attfield, Model containing unobservable variables
that k, # k, these equations
Assuming
d
ee=
have the solution
w1,(6~+1)-o,,(6k,y+kz) (4-U ’ k,o,,-u,,
c22 = (++l)(k,-k,)’
(~k,y+k,)(~y+l)(k:~,,-~,,)+(~k,y+k,)2(~,,--k,~,,) +k,(h-+-U2(u,,-k,o,J
o&c=
(6~f W, -k,)
From table 1 we have the estimates of k, and r as 0.925 and 0.367, respectively. In addition, the reduced form residuals from the unconstrained Q estimation reported in table 1 can be used to form estimates of wrl, wlz and w22. Then the remaining structural parameters in models PIHI and PIH2 can be uniquely obtained; for model PIH3 the equations in (6) may be solved for arbitrary values of kZ. Estimates of the remaining structural parameters are reported in table 2.
Table 2 Estimates of structural parameters for models PIHI, PIH2 and PIH3.” Estimates
PIHl PIH2 PIH3 PIH3 PIH3 PIH3
(i) (ii) (iii) (iv)
Value of8
Value ofkz
Ull (x105)
gzz (x105)
o,e (x105)
c”:(lO5,
0 1 1 1 1 1
0 0 0.20 0.45 0.55 0.70
0.393 0.856 1.152 2.043 2.809 5.399
2.016 1.474 1.988 3.266 4.268 7.466
7.206 6.466 5.894 4.503 3.427 0.021
1.073 1.381 1.518 1.740
‘cr22, oeeEand qii were computed from (6) using the values wrr = 655930, wr z = 666570, 022 = 922230, kl = 0.925, r = 0.367, and the values of 6 and k, given in this table. c~rr was computed from the relation cl1 = kzZoz2 + Q.
To attempt to choose between the models we have to turn to evidence from other empirical work. Mayer (1972, p. 349) cites seven studies which have estimated the MPC (measured) out of transitory income for different economies and different time periods. He gives the results of these studies as the ratio of the MPC (measured) out of transitory income to the MPC (permanent) out of permanent income. Using the notation in this paper, this ratio corresponds E
62
C.L.F. Attfield, Model containing unobservable variables
exactly to
k,+dk,(l-k,)r k,
= ”
In the case of the first model, PIHl, the a-ratio is zero since k, = 6 = 0. For PIH2 where k, = 0, S = 1, the cc-ratio is given by r and for PIH3, a = k2/k, + (1 - k,)r. Of the twelve a-ratios computed in the seven studies listed by Mayer, all are greater than 0.4 but less than 0.87 with a mean value of 0.69 [Mayer (1972, p. 353 and p. 353, fn. 8].6 Clearly neither PIHl with a = 0 nor PIH2 with LX= 0.367 are compatible with this body of evidence. On the other hand PIH3 is quite consistent with the twelve a-ratios since the lower limit of a = 0.4 implies k2 = 0.05, the upper limit of a = 0.87 implies k, = 0.7 and the mean ratio of a = 0.69 gives a value of k2 of 0.45.
6. Concluding
remarks
This paper has attempted to show that provided the classical statistical properties are valid for the original set of observations it is possible to utilise the ‘unobservable’ and grouped variables procedures to analyse the impact on consumption of both qualitative and quantitative variables within the permanent income framework.7 The resulting estimates may then be used to evaluate the crucial assumptions of the permanent income model. What emerges is evidence which is more consistent with the hypothesis that both permanent and transitory income are correlated and that transitory consumption is correlated with transitory income than it is with the ‘strict’ permanent income model. The estimates obtained allow some inference to be made regarding the magnitude of transmission effects between a windfall, permanent income and measured consumption on the one hand and between the windfall, transitory consumption and measured consumption on the other. Although this evidence is not by any means definitive it is suggestive of the manner in which the ‘unobservable’ variables procedure might be used to completely evaluate the assumptions of the PIH. If, for example, data were available on a variable (or variables) considered to be a determinant of transitory income, then a further equation in the form of a stochastic relationship between (unobservable) transitory income and this 61t should be noted that some of the studies referred to by Mayer were based on the concept of ‘normal’ rather than permanent income. [See Mayer (1972, pp. 349,353-354).] 7The assumption that the covariance matrix of the structural disturbances is homoskedastic is crucial for the grouping procedure. This is of course difficult to justify in any budget study, and for the data used in this paper Fisher (1956, p. 216) does find some evidence of heteroskedasticity. For the model specified here this would probably be related to the occupation variable, but whereas the transitory variance might be considered larger for the self-employed, the exact functional form for all occupations would be extremely difficult to specify.
C.L.F. Attfield, Model containing unobservable variables
63
variable could be added to the model specification.’ This should suffice to identify all structural parameters since the MPC (transitory) out of transitory income would be estimated directly from the reduced form equations, and estimates of the remaining parameters could be extracted uniquely from the estimated reduced form covariance matrix. Moreover, in this specification the structural errors of the model are formed from the disturbance in the equation relating transitory income to transitory consumption; the disturbance in the equation determining permanent income and the disturbance in the equation linking transitory income to its determinants. This allows relaxation of the assumption - always difficult to justify theoretically - that the transitory components themselves satisfy the stochastic specifications imposed throughout this paper. %lein and Liviatan (1957) suggest such variables as gambling gains, cash legacies, cash gifts, postwar credits, while Kreinin (1961) has used the German reparations payments to Israelis in 1957-1958.
Attfield, C.L.F., 1973, A test of the permanent income hypothesis in the United Kingdom, Discussion Paper 16 (University of East Anglia, Norwich). Dhrymes, P., 1970, Econometrics: Statistical foundations and applications (Harper & Row, New York). Eisenpress, H. and J. Greenstadt, 1966, The estimation of non-linear econometric systems, Econometrica 34. Fisher, M., 1956, Explorations in savings behaviour, Institute of Economics and Statistics Bulletin 18 (Oxford University, Oxford). Friedman, M., 1957, A theory of the consumption function (Princeton University Press, Princeton, NJ). Friedman, M., 1957a, Savings and the balance sheet, Institute of Economics and Statistics Bulletin 19 (Oxford University, Oxford). Friedman, M., 1963, Windfalls, the ‘horizon and’ related concepts in the permanent income hypothesis, in: C. Christ, ed., Measurement in economics (Stanford University Press, Stanford, CA). Goldberger, A.S., 1972, Maximum likelihood estimation of regressions containing unobservable independent variables, International Economic Review 13. Haitovsky, Y., 1973, Regression estimation from grouped observations, Griffin’s Statistical Monographs and Courses (Griffin, Los Angeles, CA). Klein, L. and M. Liviatan, 1957, The significance of income variability on savings behaviour, Institute of Economics and Statistics Bulletin 19 (Oxford University, Oxford). Kreinin, M., 1961, Windfall income and consumption - Additional evidence, American Economic Review LI. Mayer, T., 1972, Permanent income, wealth and consumption (University of California Press, Berkeley, CA). Zellner, A., 1970, Estimation of regression relationships containing unobservable independent variables, International Economic Review 11.
A MODEL
6 (1977) 51-63. 0 North-Holland
Publishing Company
ESTIMATION OF CONTAINING UNOBSERVABLE VARIABLES USING GROUPED OBSERVATIONS
An Application
to the Permanent Income Hypothesis
Clifford
L.F. ATTFIELD*
University of East Anglia, Norwich, England Received February 1975, final version received July 1976 The method of estimating relationships in a model containing unobservable variables is combined with the technique of estimation from a sample of grouped observations to analyse the permanent income hypothesis. From a sample of group means the paper measures the relative importance of such factors as wealth, occupation, age and family-size as determinants of permanent income in the United Kingdom, and also considers the mechanism by which transitory receipts influence total consumption expenditure.
1. Introduction Mayer (1972) devotes a considerable part of his book to tests of the permanent income hypothesis (PIH). Although the results enable him to reject the full permanent income theory at one extreme and the measured income theory at the other, he concludes that for ‘the much more demanding question of quantifying the error of the two extreme positions. . . tests still give vague results’. [Mayer (1972, p. 356)]. There appear to be two major reasons for this difficulty in assessing the PIH. Firstly the scarcity of data-particularly cross-section budget studies - and secondly, even when data exist, the problem of obtaining direct estimates of the theoretical constructs. Recent developments in the econometric literature indicate how these problems may be overcome. With regard to the latter, the application of the Zellner method (1970) for obtaining estimates in systems where some variables are ‘unobservable’ permits a direct estimation of the marginal propensity to consume (permanent) out of permanent income [Friedman’s ‘k’ (1957)], whilst simultaneously allowing some inference to be made concerning the existence of a correlation between permanent and transitory components. Secondly, the data problem may be alleviated somewhat *I am greatly indebted to a referee and an Associate Editor, both anonymous, for correcting a number of mistakes in an earlier draft. All remaining errors are of course my own responsibility.
52
C.L.F. AttjTeld,Model containing unobservablevariables
by the technique suggested by Haitovsky (1973) for obtaining Aitken estimates from grouped data. In this paper it is shown that the grouping technique can be easily extended to a system of recursive equations and hence data tabulated in the form of sample means can be utilised to obtain estimates of the structural parameters of the permanent income model. 2. Model specification The structural equations for an individual consumer unit in a cross-section sample of Nindividuals are
cp= k,yP,
c = cp+ul,
y =
yp+u2,
ye =
X~S+E,
(1)
where c is measured consumption and y measured income; cp, yp, u1 and u2 are permanent and transitory components of consumption and income respectively; x is a (K x 1) vector of those variables of which ‘the permanent component of income is itself a resultant . . . such as location, age, occupation, education and the like’ [Friedman (1957, p. 216)]; /? is a (K x 1) vector of structural coefficients; k1 is the marginal propensity to consume out of permanent income, and E is a stochastic error. Throughout, the transitory components are treated as random errors with the stochastic specifications :
Ul
E
[I u2
MY 41 =
z;or,,
E(E) = E(u,) = E(u,) = 0, E(Eu;) = 0 (j = 1,2), E(d) = cJN, where E, uI,u2 are (Nx 1) vectors. The force of these assumptions is that for any individual in the sample, transitory components may be correlated, oi2 # 0, but correlation between transitory components between individuals is excluded. In addition it is assumed that E(d)
= 0
and
E(xu$) = 0,
j=
1,2.
After substitution the reduced form for the model becomes
(2) y = xgi-s+u;!
= xn2+v2,
53
C.L.F. Attfield, Model containing unobservable variables
where
where
and so
E
Ul
b;,
v;] = sz@I,.
[I v2 At this stage the specification is identical with the one given by Goldberger (1972, pp. l-8). The (K+l) structural coefficients contained in j? and k, are overidentified whilst the remaining structural parameters in Sz are underidentified unless there are restrictions on either aEEor in X. (As it stands there are only three unique reduced form parameters in Sz from which we wish to extract four structural parameters viz. oEE, c12, 022, c1 1.) One restriction could be that transitory components are uncorrelated so CJ~~= 0 and hence z is diagonal, or, alternatively, that permanent income is an exact function of the variables in xi so that gEE= 0 in which case 52 = C, estimates of the structural coeficients will of course be identical in both cases [cf. Goldberger (1972, pp. 6-8)]. Estimates of the full set of structural parameters, on one of the above assumptions, can then be obtained by the Zellner procedure ZEF or, equivalently, by full information maximum likelihood (FlML) - assuming that ul, u2 and E are normally distributed. Suppose that instead of a full cross-section sample only group means are available. Specifically, imagine the sample has been grouped into M classifications (by occupation for example) and the sample mean of each variable within the classification is the only information obtainable, e.g., for consumption the data given would be %7
cg=
[ i=l
ng being the number
Rearranging
CC&
1
-l l$J
3
of consumer
g
=
1,. . ., M,
units in the gth occupational
each block of equations
group, and so
in (2) into the M groups,
summing
and
54
C.L.F. Attjield, Model containing unobservable oariables
taking sample means for each group results in Gc = GX/Ik,+k,Gs+Gu,
= GXn,+Gu,, (3)
Gy = GXjI+Gs+Gu2
= GXn,+Gv,,
where
.O
-0 . 1 .nhi_ Since the original transitory components were assumed homoskedastic this transformation introduces heteroskedasticity, the covariance matrix of the reduced form errors becoming
;;l
E [
2
[u;G',
u;G']
= QQGG'.
1
To overcome this problem decompose the positive definite matrix (GG’)-1 into T’T = (GG’)-I, where 0 .......... 0 0
dn2 0
O...O z/nj...
0
Premultiplying the equations in (3) by T, the covariance matrix of the reduced fotm errors becomes E
;;:I1
[v;G’T’,
[
v;G’T’]
= sZ@l,.
(4)
2
Hence, with the exception of the order of the identity matrix on the righthand side of the Kronecker product [in (4)], the covariance matrix of the reduced form (and structural) disturbances in the ungrouped case is the same as the covariance matrix of the disturbances in the transformed, grouped case. Since
C.L.F. Aftfield, ModeI containing unobservable variables
55
the reduced form errors, after applying the G and T transformations, are linear functions of the ungrouped structural disturbances, if these latter are normally distributed then FIML may be applied to the transformed system to obtain estimates which have the same asymptotic properties as Aitken estimates.
3. Application of the method to U.K. data Some of the most comprehensive budget studies undertaken in the U.K. were the Oxford Savings Surveys which spanned the years 1952-1954. In his article, Fisher (1956) classifies the 1953 sample survey into age within occupational groups, and presents a table giving sample means for savings, income, liquid assets and family size for these categories (p. 264). Another table (p. 275) gives the unweighted frequencies of the cell groupings. In obtaining the published sample means, a weighting scheme was used whereby larger weights were allocated to those consumer units residing in housing with a rateable value of less than &30 [Fisher (1956, pp. 204 and 275)]. Fortunately it is possible to use the cell means to solve a set of homogeneous linear equations to obtain the bveighted frequencies used to calculate the means given by Fisher.l The weighted frequencies were used to compute the transformation matrix T. To obtain a consumption series the data on savings (including durables) were converted into consumption (excluding durables) using the income identity.’ The variables included in the x vector- the determinants of permanent income -were age (x,), square of age (x,), occupation (x3), liquid assets (x,), and family size (x,), which are assumed to be the determinants of permanent income. The coding for the age variable was the same as the one used in the survey, i.e., 1 to 6 for age groups 18-24, 25-34, 35-44, 45-54, 55-65, and 65 plus, respectively. Income, consumption and liquid assets were measured in pounds sterling and family size by the number of individuals in a consumer unit. The occupation variable was coded as follows: retired and unoccupied: 1, manual: 2, clerical and sales : 3, managers: 4, self-employed: 5. The estimates obtained using this coding should be regarded as more illustrative than substantive, a more detailed study would incorporate dummy variables for the occupations. Theoretically, permanent income is defined as the return a consumer expects to receive from human and non-human sources [Friedman (1957, p. lo)]. ‘Since the weighting scheme is given by Fisher (1956, p. 275) and one of the frequencies is unity it is possible to solve the homogeneous equations uniquely. For details of the method, see Attfield (1973). *If income is measured with some error, when saving is converted to consumption this latter will incorporate the same error. This merely implies, however, that the elements of the reduced form error matrix will include the variance of the measurement error. The model specification can easily be extended to such a formulation without affecting the properties of the estimates of kl and 8. On the other hand the definition of income used by Fisher in this study has been criticised by Friedman (1957a).
56
C.L.F. Attfield, Model containing unobservable variables
Human wealth might be considered a function of age, occupation and family size - the second-degree polynomial in age reflecting the time path of permanent income over an individual’s lifetime. 3 Liquid assets on the other hand are assumed to be representative of non-human assets. Whilst it is realised that in general these are not really an adequate proxy for non-human wealth [cf. Mayer (1972, p. 168)], Fisher (1956, p. 241) shows that in the Oxford Savings Survey for certain occupational categories ‘liquid assets are an important ingredient of the assets total’. 4. Results Estimates were obtained by FIML using an algorithm similar to the one developed by Eisenpress and Greenstadt (1966). From initial starting values for k, and fi the procedure converges, by iteration, to a maximum of the likelihood function. To be reasonably certain that the maximum was global, the starting values were procured from the ZEF estimation procedure [Zellner (1970)]. Table 1 (A) gives estimates of the fully unrestricted reduced form coefficients, i.e., estimates of the vectors n1 and n;2 in (2). Table 1 (B) gives estimates of the structural coefficients, i.e., k, and the vector fi, for the case of Q diagonal and the case of Q unconstrained. The likelihood ratio test was used to test the overall model specification. First, we wish to test whether the reduced form error covariance matrix is diagonal. If it is, then transitory components of consumption and income are uncorrelated, permanent income is uncorrelated with transitory income and permanent income is a non-stochastic function of the variables in xq4 The likelihood for the reduced form model in table 1 (A) with error covariance matrix constrained to be diagonal and the likelihood for the same model but with the error covariance matrix unrestricted, were computed. The null hypothesis that the reduced form errors are contemporaneously uncorrelated may then be tested against the alternative of correlated errors by means of the likelihood ratio 1. In large samples the statistic - 2 log,l is distributed as chi-square with one degree of freedom. Since the value of the test statistic is 41.59 the null hypothesis of a diagonal reduced form error covariance matrix can be rejected at the 1 ‘A significance level. This implies that either transitory components are correlated or that permanent income is correlated with transitory income (either of which leads to rejection of the strict PIH), or that permanent income is a 3Friedman (1963, p. 6) comments: ‘It is worth recording that we do have some relevant data on human capacity - age, occupation, education, etc. One interesting direction of research would be to try to estimate human wealth from such indices thereof and to use such estimates in analyzing consumption’. Family size - more strictly ‘number in income unit’ - captures the human wealth aspect of children; see end of next section. 4Assuming no measurement errors -see footnote 2.
64.153 (45.530)
29.802 (51.46)
Consumption
Income
45.847 (14.532)
- 7.706 (7.680)
0.408 (0.098)
0.330 (0.083)
Liquid assets
- 10.168 (5.443) -11.528 (7.674)
50.516 (36.540)
61.989 (52.660)
0.924 (0.035)
0.925 (0.014)
(i) Diagonal s2
(ii) Unconstrained
33.291 (15.799)
37.797 (10.410)
Occupation (83)
0.367 (0.097)
0.382 (0.069)
92.576 (28.800)
95.399 (21.161)
Family size (85)
100.340 (29.812)
83.965 (26.376)
0.936
0.833
0.656
0.678 0.667
0.922
0.870
$;oe)
_
~.-
_
^ _,, _ _ _
. ,. -
-
-
__ -
___^,
0.939
0.779
r,
..I
and income
0.942
0.652
Estimates of
R2
MSE (x 106)
a11 (x 106) $;oe,
Family size
“The figuresin brackets are estimates of asymptotic standard errors obtained from the (inverse of) the information matrix. br, is one minus the ratio of the generalised variance of the reduced form residuals to the generalised variance of consumption [cf. Dhrymes (1970, p. 254)], being the simultaneous equation analogue of the coefficient of determination of a single equation.
(Age)’ (82)
Age (81)
Liquid assets (84)
Estimates of structural parameters.aSb
28.122 (12.858)
- 11.472 (6.7952)
(B)
Occupation
(Age)’
Explanatory variables
Estimates of reduced form parameters.
Permanent income (kr)
Q
Age
Dependent variable
(A)
Table 1
.
Y
58
C.L.F. Attfield, Model containing unobservable variables
stochastic function of the variables in x, or a combination of these factors. A further analysis of the error covariance matrix is undertaken in the next section. Secondly, we wish to test the specification of the structural model. For this, the likelihood of the reduced form model of table 1 (A) with an unrestricted error covariance matrix and the likelihood of the structural model of table 1 (B. ii), also with an unrestricted error covariance matrix, were computed. The null hypothesis that the structural model co&cients are correctly specified may then be tested by forming the ratio of these likelihoods. In this case the value of the chi-square test statistic is 12.17 and since this is less than 13.28, which is the value of a chi-square variate with four degrees of freedom at the 1% significance level, the null hypothesis that the structural model is correctly specified cannot be rejected. The results imply that the M.P.C. (permanent) out of permanent income is 0.925 compared with the M.P.C. (measured) out of measured income of 0.798, obtained by Fisher (1956, p, 270) from a simple linear regression of consumption on income. The value of 0.925 is essentially the same as the consistent estimate of k, of 0.941 computed as the ratio of mean consumption to mean income from Fisher’s paper (1956, p. 264). The point estimates of the coefficients on age imply that permanent income increases with age in the cross-section for individuals (i.e., consumer-unit heads) aged up to the range 34-45 and thereafter declines. Thus it might be inferred that this is the time profile of permanent income over an average individual’s lifetime. Similarly the coefficient estimate on liquid assets-the proxy for the value of physical assets - can be taken as the average individual’s subjective rate of discount (Friedman’s u) on non-human wealth. The reciprocal of this coefficient (2.725), is an estimate of Friedman’s (1963, pp. 5-11) ‘horizon’ and is strikingly close to his estimate of ‘about three years’. The family size coefficient may be interpreted as the rate of return on a single addition to the consumer unit, e.g., the birth of a child increases permanent income by &92 (recall that the estimates refer to 1953). ‘Children are, after all, a way of achieving security for old age; indeed in many cultures, the primary way. The raising of children can be viewed as a form of capital accumulation, only of human rather than non-human capital.’ [Friedman (1957, p. 122)].
5. Analysis of the reduced form covariance matrix The hypothesis of a zero off-diagonal element in s2 was rejected in the previous section. Assuming no measurement errors it follows that either transitory elements are correlated, or that permanent income is stochastic, or that permanent income is correlated with transitory income, or a combination of these factors. Since liquid assets have been used as a proxy for non-human wealth and variables such as education and location have been omitted from the equation determining permanent income, we can be quite certain that oEe # 0. To ascer-
C.L.F. Attfield, Model containing unobservable variables
59
tain whether correlations between transitory components or between permanent and transitory components also contribute to the non-zero value of w1 2 requires further analysis. Before proceeding with this analysis it is necessary to distinguish between three separate model specifications. Firstly there is Friedman’s (1957) original permanent income hypothesis (PIHl) in which transitory components are assumed to be uncorrelated (or 2 = 0) and transitory components are assumed to be uncorrelated with permanent components. In this model the MPC (measured) out of transitory income is zero. Secondly there is the modified model developed by Friedman (1963) in which the whole of a windfall is considered to be transitory income which is invested in an asset (PIH2). The return on this asset in a particular accounting period contributes toward total permanent income for the period. Formally, the permanent income equation now becomes yp = x’~+E+ru2, where u2 is transitory income and r the return on each unit of invested transitory income per accounting period, an estimate of which is the return on non-human assets of 0.367 in table 1. This formulation implies that while transitory components are uncorrelated, permanent income is correlated with transitory income and hence the MPC (measured) out of transitory income is given by k,r. 5 Thirdly, there is a variant which includes a relationship between the transitory components (PIH3). Many writers have argued that it is invalid to assume that transitory components are uncorrelated. On the contrary, part of a windfall which is all transitory income will be spent on transitory consumption. The difficulty lies in testing this proposition since as Mayer (1972, p. 40) argues
sit has to be assumed that there is a lag of, say, one accounting period between the receipt of a windfall and the inclusion of the asset nurchased with the windfall in the individual’s total wealth stock. Otherwise, that element of the x’ vector which refers to total wealth will include YU~.Also, since the Survey under consideration took one year as the accounting period, it is assumed that all windfalls (both positive and negative) occurred at the beginning of the year.
C.L.F. Attjield, Model containing unobservable variables
60
u1
=
kzu,+I,
where
E(1;)= O, E(iU2) = OS E(CiCj)
=
= -wt;)
O,
i #j,
cJ&
i=j,
= 0,
and where k2 is the MPC (transitory) out of transitory income such that 0 < k, < 1. With this specification, however, if kzuz is consumed, only (1 - k,)u, remains for saving. Thus the permanent income equation becomes yp = x’/I+e+(l
-kz)ruz.
In this model the MPC (measured) out of transitory income is given by k,f (1 -k,)rk,. The three specifications may be summarised by the structural equations
cp= klyp,
c = cp+ul,
y = yp+uz,
yp = x’/?+&+6yuz, ~1 =
where E, u2 and 1: are assumed to be uncorrelated and 6 = 0 or 1. The reduced form becomes
k,u,+L and where y = (1 -k,)r
c = x’~k,+kle+(6kly+k2)u2+1;, JJ =
(5)
X’j+E+(6y+l)U2.
If k, = 0 and 6 = 0, we have the first model PIHl (with crll = rr
(k,a,,+@k,y+kz)
klo,,+(6k,y+k2)(6y+l)a22
G+(~Y+
(h+
W22)
1)20,,
I
61
C.L.F. Attfield, Model containing unobservable variables
that k, # k, these equations
Assuming
d
ee=
have the solution
w1,(6~+1)-o,,(6k,y+kz) (4-U ’ k,o,,-u,,
c22 = (++l)(k,-k,)’
(~k,y+k,)(~y+l)(k:~,,-~,,)+(~k,y+k,)2(~,,--k,~,,) +k,(h-+-U2(u,,-k,o,J
o&c=
(6~f W, -k,)
From table 1 we have the estimates of k, and r as 0.925 and 0.367, respectively. In addition, the reduced form residuals from the unconstrained Q estimation reported in table 1 can be used to form estimates of wrl, wlz and w22. Then the remaining structural parameters in models PIHI and PIH2 can be uniquely obtained; for model PIH3 the equations in (6) may be solved for arbitrary values of kZ. Estimates of the remaining structural parameters are reported in table 2.
Table 2 Estimates of structural parameters for models PIHI, PIH2 and PIH3.” Estimates
PIHl PIH2 PIH3 PIH3 PIH3 PIH3
(i) (ii) (iii) (iv)
Value of8
Value ofkz
Ull (x105)
gzz (x105)
o,e (x105)
c”:(lO5,
0 1 1 1 1 1
0 0 0.20 0.45 0.55 0.70
0.393 0.856 1.152 2.043 2.809 5.399
2.016 1.474 1.988 3.266 4.268 7.466
7.206 6.466 5.894 4.503 3.427 0.021
1.073 1.381 1.518 1.740
‘cr22, oeeEand qii were computed from (6) using the values wrr = 655930, wr z = 666570, 022 = 922230, kl = 0.925, r = 0.367, and the values of 6 and k, given in this table. c~rr was computed from the relation cl1 = kzZoz2 + Q.
To attempt to choose between the models we have to turn to evidence from other empirical work. Mayer (1972, p. 349) cites seven studies which have estimated the MPC (measured) out of transitory income for different economies and different time periods. He gives the results of these studies as the ratio of the MPC (measured) out of transitory income to the MPC (permanent) out of permanent income. Using the notation in this paper, this ratio corresponds E
62
C.L.F. Attfield, Model containing unobservable variables
exactly to
k,+dk,(l-k,)r k,
= ”
In the case of the first model, PIHl, the a-ratio is zero since k, = 6 = 0. For PIH2 where k, = 0, S = 1, the cc-ratio is given by r and for PIH3, a = k2/k, + (1 - k,)r. Of the twelve a-ratios computed in the seven studies listed by Mayer, all are greater than 0.4 but less than 0.87 with a mean value of 0.69 [Mayer (1972, p. 353 and p. 353, fn. 8].6 Clearly neither PIHl with a = 0 nor PIH2 with LX= 0.367 are compatible with this body of evidence. On the other hand PIH3 is quite consistent with the twelve a-ratios since the lower limit of a = 0.4 implies k2 = 0.05, the upper limit of a = 0.87 implies k, = 0.7 and the mean ratio of a = 0.69 gives a value of k2 of 0.45.
6. Concluding
remarks
This paper has attempted to show that provided the classical statistical properties are valid for the original set of observations it is possible to utilise the ‘unobservable’ and grouped variables procedures to analyse the impact on consumption of both qualitative and quantitative variables within the permanent income framework.7 The resulting estimates may then be used to evaluate the crucial assumptions of the permanent income model. What emerges is evidence which is more consistent with the hypothesis that both permanent and transitory income are correlated and that transitory consumption is correlated with transitory income than it is with the ‘strict’ permanent income model. The estimates obtained allow some inference to be made regarding the magnitude of transmission effects between a windfall, permanent income and measured consumption on the one hand and between the windfall, transitory consumption and measured consumption on the other. Although this evidence is not by any means definitive it is suggestive of the manner in which the ‘unobservable’ variables procedure might be used to completely evaluate the assumptions of the PIH. If, for example, data were available on a variable (or variables) considered to be a determinant of transitory income, then a further equation in the form of a stochastic relationship between (unobservable) transitory income and this 61t should be noted that some of the studies referred to by Mayer were based on the concept of ‘normal’ rather than permanent income. [See Mayer (1972, pp. 349,353-354).] 7The assumption that the covariance matrix of the structural disturbances is homoskedastic is crucial for the grouping procedure. This is of course difficult to justify in any budget study, and for the data used in this paper Fisher (1956, p. 216) does find some evidence of heteroskedasticity. For the model specified here this would probably be related to the occupation variable, but whereas the transitory variance might be considered larger for the self-employed, the exact functional form for all occupations would be extremely difficult to specify.
C.L.F. Attfield, Model containing unobservable variables
63
variable could be added to the model specification.’ This should suffice to identify all structural parameters since the MPC (transitory) out of transitory income would be estimated directly from the reduced form equations, and estimates of the remaining parameters could be extracted uniquely from the estimated reduced form covariance matrix. Moreover, in this specification the structural errors of the model are formed from the disturbance in the equation relating transitory income to transitory consumption; the disturbance in the equation determining permanent income and the disturbance in the equation linking transitory income to its determinants. This allows relaxation of the assumption - always difficult to justify theoretically - that the transitory components themselves satisfy the stochastic specifications imposed throughout this paper. %lein and Liviatan (1957) suggest such variables as gambling gains, cash legacies, cash gifts, postwar credits, while Kreinin (1961) has used the German reparations payments to Israelis in 1957-1958.
Attfield, C.L.F., 1973, A test of the permanent income hypothesis in the United Kingdom, Discussion Paper 16 (University of East Anglia, Norwich). Dhrymes, P., 1970, Econometrics: Statistical foundations and applications (Harper & Row, New York). Eisenpress, H. and J. Greenstadt, 1966, The estimation of non-linear econometric systems, Econometrica 34. Fisher, M., 1956, Explorations in savings behaviour, Institute of Economics and Statistics Bulletin 18 (Oxford University, Oxford). Friedman, M., 1957, A theory of the consumption function (Princeton University Press, Princeton, NJ). Friedman, M., 1957a, Savings and the balance sheet, Institute of Economics and Statistics Bulletin 19 (Oxford University, Oxford). Friedman, M., 1963, Windfalls, the ‘horizon and’ related concepts in the permanent income hypothesis, in: C. Christ, ed., Measurement in economics (Stanford University Press, Stanford, CA). Goldberger, A.S., 1972, Maximum likelihood estimation of regressions containing unobservable independent variables, International Economic Review 13. Haitovsky, Y., 1973, Regression estimation from grouped observations, Griffin’s Statistical Monographs and Courses (Griffin, Los Angeles, CA). Klein, L. and M. Liviatan, 1957, The significance of income variability on savings behaviour, Institute of Economics and Statistics Bulletin 19 (Oxford University, Oxford). Kreinin, M., 1961, Windfall income and consumption - Additional evidence, American Economic Review LI. Mayer, T., 1972, Permanent income, wealth and consumption (University of California Press, Berkeley, CA). Zellner, A., 1970, Estimation of regression relationships containing unobservable independent variables, International Economic Review 11.