The Consumption Function: The Permanent Income Versus the Habit Persistence Hypothesis
Author(s): Balvir Singh and Aman Ullah
Source: The Review of Economics and Statistics, Vol. 58, No. 1 (Feb., 1976), pp. 96-103
Published by: The MIT Press
Stable URL: http://www.jstor.org/stable/1936014
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THE CONSUMPTION FUNCTION: THE
PERMANENT INCOME VERSUS
THE HABIT PERSISTENCE
HYPOTHESIS
Balvir Singh and Aman Ullah*
in problems in the estimation of a distributed
lag model which although quite well known are
SINCE the publication of Friedman'sA often ignored in the empirical research related
Theory of Consumption Function (1957), to PIH.
a number of papers have appeared testing the
This paper intends to discuss the basic difpermanent income hypothesis (PIH). With ferences in the two hypotheses and thereby
only a few exceptions (see Singh & Drost, 1971; show that it is wrong to identify HPH and PIH
Singh, 1975), most of the researchershave es- with each other. In section II, we set up the
timated permanent income through the appli- PIH and HPH models, outline various assumpcation of Cagan's (1956) or Koyck's (1954) tions and examine in what respect the two moddistributed lag model.1 As is well known, this els are different. In section III we analyze and
leads to a regressionof current consumptionon compare the properties of the estimators in the
personal disposable income (PDI) and lagged two models.
consumption- a form very similar to the consumption function implied by Brown's habit
II. The PIH and HPH Models and
persistence hypothesis (HPH). Indeed, after
Their Assumptions
the Klein-Friedman debate it has become almost a practice to consider HPH to be "truly
Let c represent consumption and y personal
a complete anticipation of" PIH notwithstand- disposable income, the superscripts * and **
ing the fact that the focus of the two hypoth- their permanent and transitory components,
eses is quite different. In fact, the application respectively, and let the subscript t denote time,
of Koyck's model to PIH by its nature violates t= 1 ... ., T. We can write the PIH model as
some of the basic assumptionsof the latter. Not
(1)
Yt = Y*t+ Y**t,Ct= C*t + C**t
only does it bring about dependence between
(2)
C*t ky*t
the transitory and permanentcomponentsof inand
come, but also yields transitory components of
y**t)
cov(y*t,
C**t)
cov(c*t,
income and consumptionthat may be mutually
(3)
cov(y**t, C**t) 0.
correlated (see Holmes, 1971; Singh, 1969;
Walters, 1968). Thus, the tests of PIH that use
Further, if we assume
00
Koyck's distributedlag model in the time series
0< X< 1
o
( 1X
rytTN
)Z
Y*t
context (see Holmes, 1970, 1972; Singh and
Drost, 1971; Singh, 1975) clearly become of
(4)
dubious value. In addition, there are some built- and add c**t on both sides of (2), we can write
the consumption function using Koyck's transReceived for publication April 16, 1973. Revision acformation as
cepted for publication December 30, 1974.
I.
Introduction
* The authors thank Nanda K. Choudhry for reading an
earlier draft of this paper and making useful suggestions.
They are also grateful to two anonymous referees for
making useful comments and suggestions which considerably improved the exposition of the paper. They are, however, fully responsible for any error that may still remain.
1 Laumas (1969), Holmes (1970, 1972), Wright (1969),
Friedman (1957), Klein (1958), Friedman and Becker
(1957, 1958).
[ 96 ]
ct = alyt+
a2Ct-1
where
-a, = k(l-X);
-A*t-l
+
a2-
(5)
Wt
X;
and wt
= C**t -at2C**t-l-
c**t
(6)
Earlier than Friedman (1957), a consumption funGtionsimilar to (5) was also suggested
PERMANENT INCOME AND HABIT PERSISTENCE HYPOTHESES
by Brown (1-952) as an implication of his
"habit persistence" hypothesis, i.e.,
Ct=P,o + PIYt + P2Ct-1 + Ut.
(7)
Brown used lag consumption as one of the
regressors to account for the slowness in the
reaction of consumer demand to the changes in
income.2This slowness is caused by the inertia
"the habits, customs, standards; and the levels
associated with real consumption previously
enjoyed" are likely to produce on "the human
physiological and psychological system" (for
details, see Brown, 1952).
While (7) represents a postulated theory,
(5) is a derived form. Clearly, one can obtain
(5) from PIH only if one assumes that permanent income is a weighted average of the
incomes of all past years and that the weights
decline geometrically over time. Even though
this assumption is not compatible with some
crucial assumptionsof PIH (see Holmes, 1971,
1972; Singh, 1969; Wright, 1969), the remarkable similarity between (5) and (7) has led
people to interpret them as "truly a complete
anticipation" of one another. However, a careful analysis would no doubt reveal that the two
consumptionfunctions differ with respect to the
underlying theories, scope, and the properties
of their estimators.
Let us now try to highlight some of the basic
differences between the PIH and HPH models
and their implied consumption functions.
First of all, habit persistence is not the main
focus of PIH. According to Friedman's theory,
if somebody's permanent income falls by $100,
his permanent consumption will fall by k X
2 Recently, Houthakker-Taylor
(1970) extended HPH
to the "state" and "flow" adjustment dynamic models.
Thus, in generalizing the inventory adjustment model
long adopted in the analysis of demand for durables, they
gave a far more attractive theoretical foundation to the
habit formation model than originally offered by Duesenberry (1949), Modigliani (1949), and Brown (1952). This,
in fact, made the concept of habit formation "amenable to,
more conclusive analysis" (see Houthakker and Taylor,
1970, p. 3) than before. However, the Houthakker-Taylor
general function is given by
Ct = 0 + /1Yt + I2Ctt-1 + I83Yt-1.
Clearly, both the Brown model (7) and a static model
with autoregressive errors (see Griliches, 1967, p. 34) are
special cases of the above model. Nevertheless, in this
paper we shall restrict ourselves to only Brown's model,
which Klein (1958) considered as "truly a complete anticipation of" PIH and Friedman and Becker (1958) in their
reply, accepted as "an error of omission."
97
$100. According to HPH, however, the consumption will fall less than proportionately to
the fall in income owing to hysteresis in consumer habits. Since hysteresis is only a short
run phenomenon and the "rational" consumer
does adjust (however slowly) to the changes
in income in the long run,3 one may.interpret
HPH as essentially a short run hypothesis and
PIH as a long run hypothesis. This may explain
why (7) has an intercept term and (5) does
not.4
Secondly, while the record of consumers' incomes in the past and their expectations with
regard to their future earnings both play important roles in PIH, only the past levels of
consumption presumably subsuming the effect
of past incomes are important in HPH.5 In
other words, one might say that. Friedman's
consumer can rationally plan his consumption
in that he has both prudence and farsightedness, whereas Brown's consumer, governed by
his habits, has neither.
In addition to these basic theoretical differences which get blurred by the use of Koyck's
3 Persistence of habits for a long time may cause permanent changes in consumer habits and this may be taken
into account by shifting the consumption function. For
details, see Wold and Jureen (1953).
4 See Ackley (1961) and Singh and Drost (1971). It may
be important to mention that it is nevertheless possible to
work out long run implications of HPH as well as short
run implications of PIH. The former is done by assuming
C = ct-, and the latter is done by carrying out the analysis either for smaller periods or for different groups in a
sample (see Klein (1962)). Indeed, one may find that the
various assumptions embedded in PIH may change in the
short run. For example, the victims of theft would (not
be able to) cut their coat according to their cloth in the
short run, although, they may do so in the long run.
5 Permanent income, defined as the present value of
future earnings discounted at the present subjective rate
of interest, is based on the concept which defines income
as "the amount a consumer unit could consume (or believes that it could) while maintaining its wealth intact,"
cf. Friedman (1957, p. 10). While the permanent income
theory emphasizes that change in income can affect consumption only through its effects on wealth, Friedman's
consumption function does not introduce wealth so explicitly as the macro-analogue of the Modigliani-Brumberg
(1954) life cycle hypothesis (LCH) attempted by AndoModigliani (1963) which also leads to similar conclusions.
The latter can even be written as (5) by using the
Ball-Drake (1964) framework. Nevertheless, LCH is not
identical with PIH (see Modigliani, 1966, p. 170 and
Modigliani and Ando, 1960). Indeed, it has better theoretical foundation than PIH. But since the point at issue
is not the comparison of LCH with PIH, or HPH, we shall
not refer to LCH in the later analysis.
98
THE REVIEW OF ECONOMICS AND STATISTICS
distributed lag scheme, the two models (5) and
(7) have the following three distinguishing features.
(3) Nonlinearity in Parameters
Unlike (7), (5) is nonlinear in parameters.
For example, the coefficient of yt in (5) is
k( 1- X). This, however, creates no additional
problem in (5) as long as one estimates k (1X) as a joint parameter and obtains k from
(1) Nature of the Regressors
In (5) the regressors Yt and ct-1 may be
consideredstochastic as they contain transitory k ( - X)/1 - X which is clearly uniquely decomponents. Moreover, ct-i also contains the termined. But it is well known that separating
error in equation - a stochastic component. In
X) in this way is not proper un(7), however, only one of the regressors, cti1, k from k(ltechnique is such
is stochastic and that also because of the sto- less perhaps the estimation
minimum sum of
chastic component in the equation.6 Not only that it finds the absolute
with respect to k and X sepwould this have important repercussionson the squared residuals
and Martin, 1961; Rao
samplingpropertiesof the estimators of the two arately. (See Fuller
1969.) Further, even if one obmodels, but it also would imply that they repre- and Griliches,
AA
X), it may only be
sent two different structures. This will become tains k as k(1 - X)/(1consistent but not unbiased. In fact, the bias
clear as we proceed.
of k in small samples may sometimes be large
enough to influence the conclusions. The variTerm
Error
the
of
(2) Interpretation
a
(5)
is
in
term
error
As noted in (6) the
ance of k, under certain conditions (see Kendall
w,
consumption
transitory
it is not
linear function of the
and Stuart, 1963), can be derived. But
A
be
would
it
of the past two years. As a result,
easy to deal with the distribution of k which is
correlatedwith cti, irrespective of the presence the ratio of two random variables. And even if
or absence of autocorrelation anmongw's. In we are able to obtain the distribution, in some
(5), however, w's are necessarily negatively special cases the tables of its probabilities may
autocorrelated since transitory components of not be known, with the result that we cannot
A
consumptionwould be by definition temporally draw inferences about estimated parameter k
independent.
by employing z or t-statistics, whichever may
The situation is slightly different in (7). be applicable.7 Consequently, one requires a
While this can as well be written in Koyck's nonlinear estimation technique to obtain esformat, the error term would not be correlated timates with least squares properties. Clearly
with ct-1 unless u's were autocorrelated,clearly then, with respect to estimation procedures,
because Brown's model does not distinguish be- (5) is more complicated than (7).
tween the permanentand transitory components
in the observed data.
III
The kinds of problems one generally enLet us now analyze the bias in Liviacounters in the estimation of such models are
tan's
(1963b) instrumental variable estimators
too well known to be mentioned here. However,
of a's and ,/'s in (5) and (7), respec(LIVE)
the effect of these characteristics of the error
tively.8
term on the sampling properties of the estimators will be given in the next section.
7
been general practice in empirical research.
8 In the later part of his paper, Brown (1952) considers
Yt as a jointly dependent variable thus recognizing that
the fixed regressor assumption on Yt is grossly unreasonable. On this ground, one might say that Yt is stochastic
in the Brown model as well. But, considering Yt as a
jointly dependent variable is not the same thing as assuming an element of transitoriness in Yt, hence the difference
between the two hypotheses. Indeed, this introduces another dimension in the analysis, which although quite important, is considered outside the scope of this study.
This has
In a recent study (Lianos and Rausser, 1972), however, it
has been shown that standardized distribution of the estimator of type k is positively skewed and does not closely
approximate z or t-distribution.
8 While there exist some other estimators which also
have the same asymptotic properties, e.g., Koyck's etc., we
only examine LIVE, mainly because of its simplicity. Further we apply the same estimators to both in order to
demonstrate the effect of the above differences in the two
models, particularly on the sampling bias in their estimates
when the estimation procedure is subjected to the same set
of assumptions.
PERMANENT INCOME AND HABIT PERSISTENCE HYPOTHESES
The LIV estimator of a is given by
The Application of LIVE
(A) Let us first consider the model (5) which
in matrix notation is given by
&= (Z' X) -1 Z' coo.
where cooand wooare T - 1 X 1 vectors with
E&-a1) = -
{Z2 Y t2
kcr02
-
~I2
-
Z2y*ty*t-l
Y*ty*t-1
[Z2
2y*t
y*t2
(Z3
(15)
And, using the result (A.6) in the appendix
and (12-14) the bias of LIV estimates of a,
and a2, to order 1/T, is given by
(8)
=co- Xa + woo
Y*tY*t-2
Y*t-lY*t-2
2
-
3 Y*ty*t -I
y*ty*t21
-
2 Y*t-2)
(16)
a2 2Y*tY*t-l1}
3 Y*t-12
99
and
a2)
E(a-
{
A
-
I2
Y*t
[2
(
2 Y*ty*t-
-
(
y*t2
2 y*t2
[
Y*tY*t-
3y*t-1
3 Y*t-1Y*t-2
2
-
ct and wt (t = 2, - . . , T), respectively, X = [yoo co] is a T - 1 X 2 matrix
whose first column (yoo) is defined by yt and
second column (co) by ct-i (t - 2, . . . , T)
components
and a is a 2 X 1 vector with elements a1 and
a2
Further, let
Z= [Yoo Yo]
(9)
be a T - 1 X 2 matrix of instrumental vari-
ables with yoo and yo column vectors of components yt and yt-.
(t
2,
. . .,
T), respec-
tively. In view of (1), however, we can write
X
X* + X** = [y*oo
2Z2
y*ty*t-
A
+ Z**[Y*oo Y*o]
+ [y**oo Y**oI
Coo- c*00 + c**0.
(10)
In addition to (3) we assume
(11)
Ec**t = 0 = Ey**t
and
0 if t7
= cr**2 if t-
0
=-
(17)
2Y*t2
Z2C*t-Iy*t-I
CooB
Coo, XB
Coo-
[YooY-oo
-
co-
X
-
X
Co]
(19)
cooand X are as defined in (8). Further, /8 is a
2 X 1 vector of elements /81 and /82, and uo0
is a T - 1 X 1 vector of elements ut (t =
2,
1TJ.
. . .,
Next, if the matrix of instrumentsin this case
is defined as
Z
Z=
[yoo _ Yoo Yo -Yo]
(20)
with Z being defined in (9), the LIV estimator
of /8 may be given by
cC**2 if t
( 12)
Then, using (1), (3), (11), and (12),
Ect - c*t; Eyt = y*t; Eytwt = 0 Eytct_i;
(13)
Ect-lw
22o**2
and
Ewtwt-,,
},
l)
Y*ty*t-lI
-Z2 Y*t*t-l2
(B) Let us now consider the model (7) which
after taking deviations from the means can be
written in compact form as9
(18)
cooB=XB3 + uoo
where
ZB
Ec**tc**t- _ 0 if t # f;
Ey**ty**t-,o
-
-
Z=Z*
2y*t-l2)
where
c*O]
+ [y**oo c**o]
-
a2
-
3 Y*ty*t- 1]
2 Y*ty*t-1
3Y*tY*t-2
>
I
a2u0**2 2=
-( 1 + a22)uO**2 ,i.=0.
(14)
/3=
(ZB'X)-1
(21)
ZB'CooB*
Using Nagar and Gupta (1968), the bias of
,81and /32, to order 1/T can then be written as
bias(/31) =
-
2
[Z
cru2S2{
(Yt
9 We want to compare the bias in estimates of coefficients
of Yt and ct-1, viz., /3, and 82 in (7) with the bias in estimates of a, and a2 given in (16) and (17).
THE REVIEW OF ECONOMICS AND STATISTICS
100
A2
come is defined according to (4), steady growth
in either measure of income would yield'0
O)
-Yoo)
(yt-i
T
Z2(Yt
T
XZ<2(Yt-i-yyO)2}
-_
(.2
OU,(S2[py'2
-
1)(12
(22
and
bias(82)-
E
A 2
yO
(yt
(Yt
){2
_2
Yoo) (Yt-1
-
YO)
T
--[Z
(yt - (Yt-) 2
2(t
T
+ /32
(Y
/
-1 5o)-
(y0)2-
2 y1
Cg92y1(y_
X
A1
X*-
Y*tAX*Yt
-oo)
t)
(23)
where pt, is the serial correlation among y's, i.e.,
1
/1
.+
(25)
This clearly implies that both y* and y will
grow at the same rate (g in the present case),
0
though, of course, Y*t I yt for all t since O
X < 1 and g 0 together imply 0 < X* 1.
Although this is not strictly in accordance with
what is implied by the Cagan distributed lag
scheme advocated by Friedman (1957), it does
not present too serious a problem." Indeed,
neither does full justice to PIH. They both introduce the Friedman effect in a very crude
mannerand violate the basic assumptionof zero
correlation between the permanent and transitory components of income underlying PIH.'2
The implied proportionality between y*t and
Yt is in fact not possible in a realistic situation
because of the greater degree of uncertainty
and volatility associated with the property income which constitutes a part of both y and
10 Accordingto (16) and (17) A's would be unbiased if,
t(Yt-Yoo) (Yt-
2
(O)]
Y*t =
~~~~T
[T
Thisimplies
tht
)2
2
(ytil
yO)2
T
0.2
o we(yt
e
soo)
and
(1)
g)Tty*t-
Given that y* is defined according to (4), this may be
written as
(2)
y* -A y* t1 = (1 -)Yt
which using (1) would clearly imply (25).
Similarly, (22) and (23) imply that 83's would be unbiased if y were to grow according to
Yt = (1 + g)r Yt-T.
T
cr-l = /2
(1 +
(Yt-'1-
(3)
Using this in (4) we can easily write
X2
X ?
:PO)2;
and
L
1 g
(1+
g)2
T
S2
me
inco(Yt - tPoo(Ytwea-
a)
t
T
+ 182 7J4(yt - YOO)(Yt-2 -
o) +...
(Y2 - 50)]
(24)
This implies that p2 is negativele biased if
pr,im. However, the sign of bias(pa1) will depend on the sig'nsof 8, and S2.
+
82 T3(YT
-Y5o)
Nevertheless, it riay be important to mention
that LIV esti'mates of both ax'sand /3's would be
unbiased if either the measured or the permanent income were to grow steadily at a constant rate g. While thi's may seem unreasonable,
prima facie, yet, given that the permanent in-
L
/
1+ g
t
which again is exactly the same as (25).
11 Differentiating y*t as given in Friedman (1957, p. 144)
we have
(1)
dy*/dt =f Yt
using a discrete period approximation hereof, i.e., y*t P8Yt and that y*t grows steadily at a constant
Y*t-I
rate g, we get
(2)
Y*t [/3(1 + g)/g]yt
which implies that, although y*t and Yt are proportional,
hence both growing at the same rate, yet in this case y*t >
Yt, g > 0. This clearly differs from what we get by using
Koyck's formulation.
12 See Friedman and Becker (1957) p. 142 and Friedman
(1957) pp. 19-25.
PERMANENT INCOME AND HABIT PERSISTENCE HYPOTHESES
y*.13 Indeed, if the rate of discount were to
remain constant, secular growth in measured
income might imply secular growth in permanent income but not vice versa. Thus, we may
conclude that while the estimates of a's and
,B'sin the two models may become unbiased, the
assumption of secular growth in the permanent
or the measured income particularly when y*
is defined according to (4) is least desirable
from the point of view of PIH. Since this assumption does not so much restrict HPH, the
above conclusion only remains weak.
Thus, it is now clear that even if the two
models were estimated by the same approachLIVE - and they were subjected to the same
set of assumptions with regard to income, etc.,
their coefficient estimates would be differently
biased.
The Application of the Cochrane and Orcutt
Technique
To maintain the sequence in the previous section, we first consider the model (5). Since, in
this case, w's are necessarily autocorrelated,we
assume that they follow the autoregressive
scheme
Wt = Pwwt-1
(26)
+,Et
for t = t' and 0
and EEtEtp =
ce2
where EEt = 0
otherwise.
Combining (26) with (5), we have
(ct
-
pwct-1)
=
a, (yt
+
a2(Ct-1
-
pwYt-1)
-
pwCt2)
+
Et
(27)
or
(28)
where prime is used to denote the Cochraneand
Orcutt (1949) transformation of variables in
c't - aiY't +
a26-1
+
Et
(5).
It can then be noted by using (6) and (14)
that the ordinary least squares estimators of
a1 and a2 in (28) will be inconsistent. And so
will the estimator of k.
Let us now consider the model (7). As we
have already indicated above, the u's in Brown's
model, quite contrary to common belief, would
be generally autocorrelated.Empirical research
13 The proportionality between y* and y might make more
sense if we were to consider non-property income on both
sides as done in the case of LCH. See Ando-Modigliani
(1963).
101
using the model also confirms this. Therefore,
let us asume that the u's are generated by
Ut - puUt -1 +
(29)
7)t
where Eqt = 0 and E-tq,, =
o-,2
for t
-
t' and
0 otherwise. Then
c't -
/3iY't +
/2Ct
-1 +
?It
(30)
where c't ct - puCt-1 and It - Yt - puYt-i
are Cochrane and Orcutt transformations, and
variables are measured from their respective
means.The OLSestimatorof 81 and P2 in this
case will be both consistent and unbiased because cti and -t are uncorrelated.
Thus, unlike the a's in (A), the CochraneOrcutt estimates of /3's are unbiased and consistent. It should, however, be noted that in (A)
and (B) the knowledge of autocorrelation
among the disturbancesw's and u's is presumed.
Although it may be only reasonable to assume
such knowledge while comparing two theoretical models, we recognize that the results will
be affected if pu and p, are not assumed to be
known.
IV.
Concluding Remarks
First of all, we recognize that the use of
Koyck's or Cagan's distributed lag scheme to
construct an estimate of permanent income is
not compatible with the theory of PIH.
Next, even if one follows Friedman and constructs an estimate of permanent income with
the help of Koyck's scheme, PIH and HPH
represent two different models. Their underlying theories are remarkably different.
Finally, even though the Friedman model
with the application of Koyck's scheme looks
quite similar to the one suggested by Brown,
the models differ in terms of the nature of the
regressors,interpretationof the error term, and
the nonlinearity in parameters. As a result, the
sampling properties of the estimates of their
coefficients, in terms of bias and consistency,
also differ. Nonlinearity, of course, produces
bias in the estimates of structural parameters
if only a linearized version of a nonlinear model
is estimated.
Thus, we conclude that the application of
Koyck's model to PIH creates more problems
than it actually solves and that Brown's model
102
THE REVIEW OF ECONOMICS AND STATISTICS
is by no means "truly a complete anticipation Cagan, P., "The Monetary Dynamics of Hyperinflation,"
in M. Friedman (ed.), Studies in the Quantitative
of Friedman."
Theory of Money (Chicago: University of
Chicago Press, 1956), 25-117.
Cochrane, D. and G. H. Orcutt, "Application of Least
Squares Regression to Relationships Containing
The Bias of LIVE 2
Auto-correlated Error Terms," Journal of *the
American Statistical Association, 44 (1949), 32Let us write the sampling error using (8), (11), and
61.
(15), of LIVE a as'
J. S., Income Saving and the Theory of
Duesenberry,
(a - &) = (Z'X)-1 Z'wOO
Consumer Behavior (Cambridge: Harvard Uni= [I + A-(Al/2
+ B% + D% ? D)]-l
versity Press, 1949).
A-l(Z*'
+ Z**')woo
Fuller, W. A. and E. Martin, "The Effects of Autocorrelated Errors on the Statistical Estimation of
= A-1(Z*' + Z**')woo - A-'
Distributed Lag Models," Journal of Farm Eco(AY2+ B% + D% + D)A-1
nomics, XLIII (1961).
x (Z*' + Z**')woo ?+ . . .
(A.1)
Friedman, M., A Theory of Consumption Function
where A = Z*'X* is a matrix of nonstochastic elements
(Princeton: Princeton University Press for the
of order T and
National Bureau of Economic Research, 1957).
** o
, "Windfalls, the 'Horizon' and the Related
0l
D =r
OY**OO
Concepts, in the Permanent Income Hypothesis,"
D L
Y
(A.2)
in C. F. Christ et al. (eds.), Measurement in Ecois a matrix of order T in probability.2 Further,
nomics (Stanford: Stanford University Press,
A2 = Z*'X**, B /2 = Z**X*;
1963), 3-28.
Friedman, M. and G. S. Becker, "A Statistical Illusion
[
]
Dl/2
(A.3)
y** C**O
in Judging Keynesian Models," The Journal of
Xy**o; Y**oo
Economy, 65 (1957), 64-75.
Political
are the matrices of order 'vT in probability.
"Reply" (to the "Friedman-Becker Illusion"),
Now taking expectations on both sides of (A1) and
The Journal of Political Economy, 65 (1958), 545retaining terms to order 1/T, we obtain
549.
?
?
=
-A-'
E
E[(A ,2
B2,
Dy2 + D)A'
a)
Griliches, Z., "Distributed Lags: A Survey," Economet(Z*' + Z**')wOO]
(A.4)
rica, 35 (1967).
because
Holmes, J. M., "A Direct Test of Friedman's Permanent Income Theory," Journal of the American
(A.5)
EZ*'woo = 0 = EZ**'woo.
Statistical Association, 65 (1970), 1159-1162.
Further using (3), (10), (12), (13), and (14), it can
be easily verified that
, "The Independence of Permanent and Transitory Components of a Series Where the Permanent
= -A-1E(A
(A.6)
E(&-a)
%A-Z*'wOO).
Component is a Weighted Average of Measured
1 It should be noted that if A is a m X m nonsingular
Values," Journal of the American Statistical Assomatrix and B is a m X m matrix then
ciation (1971).
(A + B)-1 = (A + AA-1B)-1 = (I + A-1B)-1A-1.
"On Testing the Permanent Income Hypoth2 The first term in the expansionof the third step of (A.1)
esis," Discussion Paper No. 170, Dept. of Ecois of the order 1/T-/ the second is of the order 1T, and
nomics, State University of New York at Buffalo
so on. To determinethe order of magnitudein the proba(1972).
bility of differentfunctions,we refer to Cramer(1964), p.
Houthakker, H. S. and L. D. Taylor, Consumer Demand
122. Also see Nagar and Gupta (1968).
in *the United States: Analysis and Projections
(Cambridge: Harvard University Press, 1970).
REFERENCES
M. G. and A. Stuart, The Advanced Theory of
Kendall,
Ackley, G., Macroeconomic Theory (London: MacStatistics, vol. 1 (New York: Hafner Publishing
millan Company, 1961).
Company, 1963).
Ando, A. and F. Modigliani, "The Life-Cycle Hypothesis
of Saving: Aggregate Implications and Tests," Klein, L. R., "The Friedman-Becker Illusion," The
Journal of Political Economy, 65 (1958), 539-545.
American Economic Review, 53 (1963), 111-113.
, An Introduction to Econometrics (New Jersey:
Ball, R. J. and P. S. Drake, "The Relationship Between
Prentice Hall, 1962).
Aggregate Consumption and Wealth," International
Koyck, L. M., Distributed Lags and Investment
Economic Review, 5 (1964).
(Amsterdam: North-Holland Publishing Company,
Brown, T. M., "Habit Persistence and Lags in Con1954).
sumer Behaviour," Econometrica, 20 (1952), 355371.
Laumas, P. S., "A Test of Permanent Income HypothCramer, H., Mathematical Methods of Statistics
esis," The Journal of Political Economy, 77 (1969),
(Princeton: Princeton University Press, 1964).
857-861.
APPENDIX
0
PERMANENT INCOME AND HABIT PERSISTENCE HYPOTHESES
Lianos, T. P. and G. C. Rausser, "Approximate Distribution of Parameters in a Distributed Lag Model,"
Journal of the American Statistical Association,
67 (1972), 64-67.
Liviatan, N., "Tests of Permanent Income Hypothesis
Based on a Reinterview Savings Survey," in C.
Christ, et al. (eds.), Measurement in Economics
(Stanford: Stanford University Press, 1963a), 2966.
, "Consistent Estimation of Distributed Lags,"
International Economic Review, 44 (1963b).
Modigliani, F., "Fluctuations in the Saving-Income
Ratio: A Problem in Economic Forecasting,"
Studies in Income and Wealth, 27, National Bureau
of Economic Research (1949).
"The Life-Cycle Hypothesis of Saving, the
Demand for Wealth and the Supply of Capital,"
Social Research, 33 (1966), 160-217.
Modigliani, F. and A. Ando, "The Permanent Income
and the Life Cycle Hypothesis of Saving Behavior:
Comparison and Tests," in I. Friend and R. Jones
(eds.) Proceedings of the Conference on Consumption and Saving, Vol. 2 (Philadelphia: University
of Pennsylvania Press, 1960).
Modigliani, F. and R. Brumberg, "Utility Analysis and
the Consumption Function: An Interpretation of
Cross-Section Data," in K. Kurihara (ed.), Post
103
Keynesian Economics (New Brunswick: Rutgers
University Press, -1954).
Nagar, A. L. and Y. P. Gupta, "The Bias of Liviatan's
Consistent Estimator in a Distributed Lag Model,"
Econometrica, 36 (1968), 337-342.
Rao, P. and Z. Griliches, "Small Sample Properties of
Several Two Stage Regression Methods in the Context of Autocorrelated Errors," Journal of the
American Statistical Association? 64 (1969), 253272.
Singh, B., "A Test of the Permanent Income Hypothesis: A Comment," mimeographed, University of
Toronto, 1969.
"Maximum Likelihood Estimation of the Friedman Consumption Function," this REviEw, 57
(1975), 94-100.
Singh, B. and H. Drost, "An Alternative Approach to
the Permanent Income Hypothesis-An
International Comparison," this REviEw, 53 (1971), 326334.
Walters, A. A., An Introduction to Econometrics
(London: Macmillan Company, 1968).
Wold, H. 0. A. and L. Jureen, Demand Analysis (New
York: John Wiley & Sons, 1953).
Wright, C., "Estimating Permanent Income Hypothesis - A Note," The Journal of Political Economy,
77 (1969), 845-850.