Testing the life-cycle permanent income hypothesis using intra-year data for Sweden

Testing the life-cycle permanent income hypothesis usingintra-year data for Sweden

Kenneth Leong*, Michael McAleer

Department of Economics, University of Western Australia, Nedlands, WA 6907, Australia

Abstract

Real non-durable consumption expenditure for many countries typically exhibits substantial seasonal fluctuations. In this

paper, two seasonal models that are consistent with an extension of the rational expectations life-cycle permanent income

hypothesis are evaluated using quarterly seasonally unadjusted Swedish consumption expenditure. One model is a first-order

periodically integrated autoregressive model. Formal procedures for periodic integration are used to test this hypothesis. The

second model captures seasonal habit persistence in the form of a periodic seasonal ARIMA model. It is found that both

models fail to capture adequately the dynamics in Swedish consumption expenditure, which suggests a rejection of the rational

expectations life-cycle permanent income hypothesis. # 1999 IMACS/Elsevier Science B.V. All rights reserved.

Keywords: Life-cycle permanent income hypothesis; Rational expectations; Seasonality; Periodic models

1. Introduction

Consumption is a significant economic variable, so it is not surprising that consumer behaviour hasattracted considerable theoretical and empirical research over extended periods. One of the mostinfluential contributions to consumption in recent years has been the Euler equation approach of Hall[4], in which the life-cycle permanent income hypothesis (LC-PIH) is used to model the realconsumption of non-durables under rational expectations. Hall's result is that consumption expenditurefollows a random walk, which implies that only unexpected policy changes can affect consumption,given knowledge of last period's consumption.

An important aspect of consumption behaviour that has largely been ignored in the literature to dateis that of seasonality. If intra-year observed data are considered, consumption is typically found toexhibit substantial seasonal fluctuations. As consumption expenditure is highly seasonal, a detailedinvestigation of the nature of the inherent seasonality is warranted in the empirical modelling ofconsumption. Osborn [7] extended Hall's [4] approach to accommodate seasonality, derived a periodicanalogue of the random walk model for real non-durable consumption, and also considered a more

Mathematics and Computers in Simulation 48 (1999) 551±560

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* Corresponding author. Tel.: +61-8-9380-2896; fax: +61-8-9380-1016; e-mail: [email protected]

0378-4754/99/$20.00 # 1999 IMACS/Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 8 - 4 7 5 4 ( 9 9 ) 0 0 0 3 6 - 1

general model for consumption, namely an extension to the habit persistence model. Data for the UKwere used to show that consumption is inconsistent with the LC-PIH, but is consistent with its seasonalhabit persistence counterpart.

In this paper, quarterly seasonally unadjusted consumption data for Sweden are used to examine theempirical adequacy of the two seasonal models of consumption, namely the seasonal LC-PIH and theseasonal habit persistence model. The purpose in examining the seasonal extension of these popularmodels is to evaluate the extent to which intra-year data and seasonality can affect the analysis. Inaddition, a formal procedure for periodic integration is used to test the seasonal LC-PIH. As a failure ofthe seasonal LC-PIH (which implies a periodic random walk model) to model consumption expenditureadequately may be due to the presence of habit persistence, a seasonal habit persistence model is alsoapplied to the consumption series.

The plan of the paper is as follows. Section 2 discusses the nature of the seasonal fluctuations in theSwedish intra-year observed consumption data. Osborn's [7] extension of Hall's LC-PIH model ispresented in Section 3. Periodic autoregressive models and unit roots are discussed in Section 4, andthis new approach to testing Osborn's hypothesis is considered in Section 5. Section 6 introducesseasonal habit persistence, which is tested and evaluated using periodic seasonal ARIMA models.Concluding remarks follow in Section 7.

2. Seasonality and consumption

Quarterly seasonally unadjusted real non-durable consumption expenditure per capita for Sweden(1963Q1-1988Q1) is used in this paper to test the various economic hypotheses. It can be observed inFig. 1 that seasonally unadjusted consumption data exhibit marked seasonality, or systematic intra-yearfluctuations. As for a time trend, seasonality can be deterministic and/or stochastic in nature.

To obtain an indication of deterministic seasonality in Swedish consumption expenditure, the growthrate of consumption (approximated by logarithmic-differences) is regressed on a set of seasonal dummyvariables using the following regression (see [6]):

�ln ct � �1D1t � �2D2t � �3D3t � �4D4t � �t; (1)

Fig. 1. Logarithm of Swedish quarterly real non-durable per capita consumption expenditure.

552 K. Leong, M. McAleer / Mathematics and Computers in Simulation 48 (1999) 551±560

where Dst (s � 1,2,3,4) is a seasonal dummy variable, and �t is assumed to be a stationary and invertibleARMA process. The �R

2value for this regression is 0.81, which indicates that, after the removal of the

stochastic trend, seasonality accounts for over 80% of the variation in consumption expenditure.Seasonal patterns in consumption can also change over time. Slowly changing seasonal patterns can

reflect the presence of seasonal unit roots, or stochastic seasonality. The most commonly used test forstochastic seasonality is the HEGY test of Hylleberg et al. [5], which is based on the followingregression:

at � 1D1t � 2D2t � 3D3t � 4D4t � �t � �1bt � �2ct � �3dt � �4et � �t� (2)

In (2), t is a deterministic time trend:

at � �1ÿ L4�yt

bt � �1� L� L2 � L3�ytÿ1

ct � ÿ�1ÿ L� L2 ÿ L3�ytÿ1

dt � ÿ�1ÿ L2�ytÿ2

et � ÿ�1ÿ L2�ytÿ1

represent variables that have been transformed using specific filters that are subsets of the seasonaldifferencing filter (1ÿL4), and �t is an i.i.d. �0; �2

� � error term. Lagged values of at are added to (2) toensure that the residuals of the regression are white noise.

The HEGY hypotheses are formalised as follows:

1. H0: �1 � 0, H1: �1 < 0;2. H0: �2 � 0, H1: �2 < 0;3. H0: �3 � �4 � 0, H1: �3 6� 0 and/or �4 6� 0.

Standard t-tests are used for the first two null hypotheses, while an F-test is used for the third nullhypothesis. If the first null hypothesis is not rejected, there is a zero frequency unit root in the series yt.In the second case, the null hypothesis is consistent with a semi-annual unit root, implying that anyshocks to the variable will lead to permanent changes in the seasonal pattern of the variable at the semi-annual level. The third null hypothesis implies that the series have at least one of the two unit roots atthe annual frequency. With annual unit roots, a shock to the variable will change permanently theseasonal pattern of the variable at the annual level.

To determine the optimal lag length in the HEGY regression, the following testing approach isadopted. Starting with 8 lags of at and sequentially deleting insignificant lags when diagnostictests reveal the absence of serial correlation, the optimal lag length is the longest lag for which themodel does not appear to be misspecified. With 7 being the optimal lag of at in the HEGY regression,and diagnostic tests revealing no significant serial correlation (FAR,1±4 � 0.697 [0.596]), the estimatedtest statistics are t��̂1� � ÿ2:156, t��̂2� � ÿ1:993, and F��̂3; �̂4� � 1:718. At the 5% level ofsignificance, with 93 effective observations, all three null hypotheses are not rejected, indicating thatSwedish consumption expenditure is seasonally integrated, thereby requiring the filter (1ÿL4) forstationarity.

K. Leong, M. McAleer / Mathematics and Computers in Simulation 48 (1999) 551±560 553

3. The seasonal extension to the life-cycle permanent income hypothesis

Hall [4] used rational expectations to derive a random walk model of consumption. This implies that,since the previous value of consumption has incorporated all the information available at the time theexpectations were formed, lagged values of actual income should have no additional explanatory powerwhen lagged consumption is included. This is in marked contrast to the Keynesian consumptionfunction, where consumption is determined by contemporaneous income, among other variables.

Osborn [7] uses the following utility function to derive the seasonal extension to Hall's random walkhypothesis:

ut � c1ÿ�st

1ÿ �s

; 0 < �s < 1; s � 1; 2; 3; 4 (3)

where � is the degree of relative risk aversion.With � representing the seasonal subjective discount rate, total utility is given by

Ut �XT

i�0

�isut�i �

XT

i�0

�is

c1ÿ�st

1ÿ �s

!; 0 < � � 1: (4)

The consumer's optimisation problem is to maximise (4) subject to the budget constraint

ct � yt ÿ at � atÿ1�1� rtÿ1� (5)

where c is the volume of consumption, y is real labour and transfer income, a is the real asset level, andr is the real interest rate. Maximising total utility subject to the budget constraint leads to the marginalcondition

cÿ�s�1

t�1

cÿ�st

� 1

�s�1�1� r� (6)

which can be interpreted as the Euler equation when the utility function and the subjective discount ratevary seasonally. The real interest rate, r, is assumed to be constant in Osborn's analysis as the focus ison seasonality in the utility function.

A disturbance term, �t�1, is added to Eq. (6) to capture the inherent randomness in consumption.Taking natural logarithms yields

�s�1 ln ct�1 � �s ln ct ÿ ln1

�s�1�1� r� � �t�1

� �: (7)

If expectations are rational, �t�1 represents the response of consumers to the arrival of newinformation in period t�1, in which case the disturbance term has mean zero and is seriallyuncorrelated.

If the stochastic term in Eq. (7) is assumed to be normal and identically distributed, namely

ln1

�s�1�1� r� � �t�1

� �� N��s�1; �

2� (8)


a property of the log-normal distribution suggests the following conditional expectation (with It beingthe information set at the end of period t)

E1

�s�1�1� r� � �t�1jIt

� �� exp �s�1 � 1

2�2

� �: (9)

Evaluating the conditional expectation gives

1

�s�1�1� r� � exp �s�1 � 1

2�2

� �(10)

which, after taking logarithms and rearranging, yields

�s�1 � ln1

�s�1�1� r��

ÿ 1

2�2: (11)

Eq. (7) can be approximated by

ln ct�1 � s�1 � �s�1 ln ct � ut�1 (12)

where s�1 � �1=�s�1��ln�s�1 � ln�1� r� � 12�2�; and the disturbance term ut�1 is assumed to follow

an i.i.d. process.Note that, from Eqs. (7) and (12), �s�1 � �s=�s�1 for s � 1; 2; 3; 4, which directly implies that

�1�2�3�4 � 1. Hence, Eq. (12) is a periodically integrated autoregressive process of order 1, which canbe interpreted as the seasonal analogue to the random walk model of Hall [4].

4. Periodic autoregressive models with a unit root

In this section, the theory underlying periodic autoregressive models with a unit root is presented. Aperiodic autoregressive model of order 1, PAR(1), is given by

yt � �s � �sytÿ1 � �t; s � 1; 2; 3; 4 (13)

where the coefficients differ for each season and the use of quarterly observed data is assumed.For an analysis of non-stationarity, a multivariate representation of (13) is used, which involves

rewriting the model so that the coefficients are invariant to the seasons. By stacking the observationsinto vectors of annual observations, the (4�1) vector YT is formed as follows:

YT � �Y1;T ;Y2;T ;Y3;T ; Y4;T�0

where Ys;T is the observation in season s (s � 1; 2; 3; 4) in year T, and the annual index T � 1; 2; . . . ;N(with N � n=4).

Using this notation, the univariate PAR(1) model is represented by the following multivariate form:

�0YT � �� 1YTÿ1 � �T (14)

where � � ��1; �2; �3; �4�0 is the intercept vector, �T � ��1;T ; �2;T ; �3;T ; �4;T�0, �s;T is the observation of


the error process in season s of year T, and the � are (4�4) parameter matrices containing theperiodically-varying coefficients in the univariate PAR model.

The vector process YT is stationary if the root of the characteristic equation

j�0 ÿ �1�j � �1ÿ �1�2�3�4�� 0

is outside the unit circle. Hence, YT is stationary if j�1�2�3�4j < 1 but, if �1�2�3�4 � 1, then YT and yt

contain a unit root, so that yt is a PIAR(1) process. Formally, the null and alternative hypotheses ofwhether a PAR(1) process yt is periodically integrated of order 1 are given by

H0 : �1�2�3�4 � 1

and

H1 : j�1�2�3�4j < 1:

The unrestricted PAR(1) model can be written as

yt �X4

s�1

�sDst �X4

s�1

�sDstytÿ1 � �t: (15)

In empirical applications, the natural logarithm of yt is frequently used instead of the levels.With normality of the error process and fixed starting values, the maximum likelihood (ML)

estimator of the parameters is equivalent to the OLS estimator of the PAR(1) model. Under the unit roothypothesis, the restricted model is

yt �X4

s�1

�sDst � �1D1tytÿ1 � �2D2tytÿ1 � �3D3tytÿ1 � ��1�2�3�ÿ1D4tytÿ1 � �t: (16)

In order to test the unit root hypothesis, Boswijk and Franses [1] consider a one-sided likelihood ratiostatistic

LR� � �sign��̂1�̂2�̂3�̂4 ÿ 1�� n � ln RRSS

URSS

� �swhere RRSS and URSS denote the restricted and unrestricted residual sums of squares, respectively,and the estimated �s coefficients are the OLS estimates from the unrestricted model. Under the nullhypothesis, the asymptotic distribution of the test statistic is the same as that tabulated by Fuller [3] fornon-periodic autoregressive models.

5. Testing the seasonal life-cycle permanent income hypothesis

In this section, the Boswijk and Franses approach to test for a periodic unit root is used to evaluatethe seasonal LC-PIH for Sweden. A necessary condition for the seasonal LC-PIH to hold is if a PAR(1)model is empirically adequate for the data, while the sufficient condition is that the periodic unit rootnull hypothesis, �1�2�3�4 � 1, is not rejected at conventional significance levels.


The OLS estimates (which are equivalent to ML estimates) of the unrestricted PAR(1) model are asfollows (with standard errors in parentheses):

ln̂ ct � ÿ0:738D1t�0:097�

� 0:388D2t�0:065�

� 0:098D3t�0:078�

� 0:344D4t�0:080�

� 1:310D1t ln ctÿ1�0:044�

� 0:828D2t ln ctÿ1�0:030�

� 0:944D3t ln ctÿ1�0:360�

� 0:873D4t ln ctÿ1�0:038�

Diagnostics:

FAR;1ÿ4 � 8:317 �0:000�FARCH;1ÿ4 � 1:166 �0:331�LM�N� � 4:533 �0:104�URSS � 0:027447:

The diagnostic tests (with prob-values in parentheses) are for the absence of first-to-fourth-orderserial correlation in the errors (FAR,1±4), the absence of first-to-fourth-order ARCH errors (FARCH,1±4),and normality of the errors (LM(N)). As there is significant serial correlation in the residuals of theunrestricted PAR(1) model, it is possible that the appropriate dynamics have not been captured by thelow order PAR(1) process.

Imposing the unit root restriction and estimation by non-linear least squares gives

ln̂ ct � ÿ0:803D1t�0:087�

� 0:341D2t�0:057�

� 0:039D3t�0:068�

� 0:275D4t�0:067�

� 1:340D1t ln ctÿ1�0:040�

� 0:805D2t ln ctÿ1�0:027�

� 0:971D3t ln ctÿ1�0:031�

� 0:905D4t ln ctÿ1�0:0031�

Diagnostics:

FAR;1ÿ4 � 7:856 �0:000�FARCH;1ÿ4 � 1:390 �0:243�LM�N� � 2:347 �0:309�RRSS � 0:028109:

For the effective sample size of 103, the LR� test statistic for a periodic unit root is 1.567. As the 5%critical value is ÿ2.89, the null hypothesis of a periodic unit root cannot be rejected. However, withsignificant serial correlation in both the unrestricted and restricted PAR(1) regressions, which willaffect both the URSS and RRSS used in computing the LR� statistic, the outcome should be interpretedwith caution.

As a first-order PAR process is incompatible with Swedish consumption expenditure data, the LC-PIH is not supported empirically. An alternative specification to the LC-PIH is the habit persistencemodel. The next section evaluates the empirical adequacy of a seasonal habit persistence model for theSwedish consumption series.


6. Seasonal habit persistence and consumption

The intuition behind habit persistence is that consumption is dictated partly by habits. In times ofhigh economic growth and high income, consumption expenditure will be above its trend level, and thedesire for a high level of consumption is likely to remain in the future. As discussed in Osborn [7],incorporating seasonality into the utility function implies that consumers regard expenditure in eachquarter as a separate commodity, in which case habit persistence is also expected to be seasonal.

A generalisation of the periodically integrated model of order 1 to incorporate habit persistence isgiven by the following periodic seasonal ARIMA model:

�1ÿ �sL��1ÿ �sL4� ln ct � s � �t: (17)

Even though (17) is not derived from economic theory, the imposition of the periodic unit rootrestriction implies that (17) is a generalisation of the LC-PIH.

When the periodic unit root restriction is imposed, the seasonal habit persistence model isequivalent to

ln ct � 1D1t � 2D2t � 3D3t � 4D4t � �1D1t ln ctÿ1 � �2D2t ln ctÿ1

� �3D3t ln ctÿ1 � ��1�2�3�ÿ1D4t ln ctÿ1 � �1D1t ln ctÿ4 � �2D2t ln ctÿ4

� �3D3t ln ctÿ4 � �4D4t ln ctÿ4 ÿ �1�4D1t ln ctÿ5 ÿ �2�1D2t ln ctÿ5 ÿ �3�2D3t ln ctÿ5

ÿ ��1�2�3�ÿ1�3D4t ln ctÿ5 � �t: (18)

Table 1

Estimates of the seasonal habit persistence model for Sweden

Parameters Estimates

1 ÿ0.589 (0.129)

2 0.329 (0.079)

3 0.014 (0.075)

4 0.234 (0.081)

�1 1.141 (0.194)

�2 1.040 (0.177)

�3 0.925 (0.196)

�4 � (�1�2�3)ÿ1 0.911 (0.185)

�1 0.346 (0.105)

�2 0.173 (0.149)

�3 0.219 (0.162)

�4 0.206 (0.130)

Diagnostics Test statistics

FAR,1±4 6.324*

FARCH,1±4 2.078

LM(N) 2.441

Standard errors are in parentheses.* Significant at the 5% level. FAR,1±4 is a test for the absence of first-to-fourth-order serial correlation in the errors, FARCH,1±4

is a test for the absence of first-to-fourth-order ARCH errors, and LM(N) is a test of normality of the errors.


Table 1 presents the non-linear least squares estimates for Eq. (18), in which the coefficients measurethe extent to which consumption responds to changes in deterministic seasonality and seasonal changesin lagged consumption.

A test of the restriction �1 � �2 � �3 � �4 � 0 where �s is the coefficient of the seasonal lag(s � 1; 2; 3; 4), can be used to test the PIAR(1) model against the seasonal habit persistence model.However, the significant FAR,1±4 statistic suggests that the estimated seasonal habit persistence model isempirically inadequate for real non-durable consumption expenditure for Sweden. As there issignificant serial correlation, a test of �1 � �2 � �3 � �4 � 0 will be invalid.

The inadequacy of both the periodically integrated model and the seasonal habit persistence modelsuggests that the LC-PIH does not hold for Sweden. The same Swedish consumption series have beenanalysed in [2], where the approach was to model the seasonal fluctuations in the data to obtain anadequate empirical model. This is in contrast to the present paper, which involves testing a specificeconomic theory. Franses found that a higher-order PAR process (namely, a subset PAR(1,4) model ofthe form ln ct � �s � �1s ln ctÿ1 � �4s ln ctÿ4 � �t) provided an adequate explanation for Swedishconsumption expenditure, and that, at the 10% significance level, the null hypothesis of a periodic unitroot could not be rejected. It was also shown that a periodic cointegration model (namely, an errorcorrection model where the cointegration and the short-run adjustment parameters are allowed to varywith the seasons) of Swedish consumption expenditure and disposable income yielded an adequatemultivariate empirical model.

7. Concluding remarks

In this paper, seasonal fluctuations in real non-durable consumption for Sweden have been found tobe both deterministic and stochastic in nature. The rational expectations life-cycle permanent incomehypothesis has been augmented with seasonality and, using formal tests for periodic integration, it isshown that the Swedish consumption series are consistent with the hypothesis. However, a first-orderperiodic autoregressive model is insufficient to capture fully the consumption dynamics. Seasonal habitpersistence is then introduced in the form of a restricted periodic seasonal ARIMA model, but it is alsofound to be empirically inadequate for Sweden. This suggests that neither the rational expectations life-cycle permanent income hypothesis nor the habit persistence generalisation holds for Sweden. Theempirical findings in the paper suggest that seasonality plays an important role in Swedishconsumption, and should not be omitted in any empirical modelling of consumption expenditure.

Acknowledgements

The first author wishes to acknowledge the financial support of an Australian Postgraduate Awardand an Individual Research Grant from the Faculties of Economics & Commerce, Education and Law atthe University of Western Australia, and the second author acknowledges the financial support of theAustralian Research Council.

References

[1] H.P. Boswijk, P.H. Franses, Unit roots in periodic autoregressions, J. Time Series Anal. 17 (1997) 221±245.


[2] P.H. Franses, Periodicity and Stochastic Trends in Economic Time Series, Oxford University Press, Oxford, 1996.

[3] W.A. Fuller, Introduction to Statistical Time Series, Wiley, New York, 1976.

[4] R.E. Hall, Stochastic implications of the life cycle permanent income hypothesis: Theory and evidence, J. Political

Economy 86 (1978) 971±987.

[5] S. Hylleberg, R.F. Engle, C.W.J. Granger, B.S. Yoo, Seasonal integration and cointegration, J. Econometrics 44 (1990)

215±238.

[6] J.A. Miron, The economics of seasonal cycles, in: C.A. Sims (Ed.), Advances in Econometrics: Sixth World Congress of

the Econometric Society, Cambridge University Press, Cambridge, 1994.

[7] D.R. Osborn, Seasonality and habit persistence in a life-cycle model of consumption, J. Appl. Econometrics 3 (1988)

255±266.


Testing the life-cycle permanent income hypothesis using intra-year data for Sweden

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