CONSUMPTION, SAVING AND RATIONAL EXPECTATIONS: SOME FURTHER EVIDENCE FOR THE U.K.*

Transcription

1 The Economic Journal, 99 (March 1989)-, Printed in Great Britain CONSUMPTION, SAVING AND RATIONAL EXPECTATIONS: SOME FURTHER EVIDENCE FOR THE U.K.* Ronald MacDonald and Alan E. H. Speight The nature of the relationship of consumption to income and the appropriate functional specification of that relationship has long been a contentious issue in economic theory. In recent years this continuing debate is exemplified in the contrasting analyses of Hall (1978) and Davidson et al. (1978), the former establishing the 'forward-looking' rational expectations-permanent income hypothesis (REPI below) whilst the latter invoke a ' backward-looking' error correction mechanism in consumption determination (ECM below). However, the potential for observational equivalence in these discordant approaches is well established by Davidson and Hendry (1981). Moreover, empirical investigation has failed to discern the appropriate specification, particularly in the light of recent developments in the appraisal of economic time series concerned with issues of stationarity. On the one hand, studies indicating an 'excessive' dependence of consumption on income under REPI have typically been biased toward that finding by imposing stationarity in the variables under investigation as a result of differencing and detrending, as demonstrated by Mankiw and Shapiro (1985). On the other, the ECM formulation has enjoyed empirical success through combining non-stationary and stationary elements in explaining long-run equilibrium and short-run time series dynamics, but with (previously) inadequate statistical. foundation, and largely retrospective theoretical justification. Recently, the well-known difficulties associated with empirical investigation using non-stationary data have been circumvented by the increasingly familiar application of'cointegration' techniques associated with Granger (1981), Granger and Weiss (1983), and Engle and Granger (1987). Cointegration methods may be applied in the context of both REPI and ECM modelling, though for different purposes. A feature of cointegration analysis is that it allows the disequilibrium feedback term under an ECM to be more rigorously specified. More generally, it establishes the conditions for a valid ECM representation, in that if the properties associated with cointegration are not satisfied then the empirical representation of the ECM will be subject to spurious regression bias. Alternatively, under REPI, cointegration analysis may be used to generate a quasi-differenced variable which is stationary and appropriate for tests of excess consumption-income sensitivity. The application of cointegration techniques in the context of.the consumption ECM has recently been conducted with US data by Engle/and Granger (1987), and with UK data by Drobny and Hall (1987), whilst * We would like to thank the editor, an associate editor and two anonymous referees for their helpful comments on an earlier draft of this paper. The usual disclaimer applies. [83 ]

2 84 THE ECONOMIC JOURNAL [MARCH Campbell (1987) has exploited the advantages ofcointegration in testing REPI with US data. The purpose of this paper is to extend that investigation to UK data, so providing an empirical parallel to the body of literature concerning ECM tests with UK data. The following section details the implications of the REPI model and outlines a suitable methodology for testing this approach. Section II contains the empirical results and the paper closes with a concluding section. I. THE PERMANENT INCOME HYPOTHESIS, RATIONAL EXPECTATIONS AND A TESTING METHODOLOGY Hall (1978) demonstrated that the assumption of rational expectations applied to the permanent income hypothesis (plus a number of other assumptions such as constancy in the real interest rate, here following Flavin, 1981) implies that consumption should follow the random walk: 00 ««= «i-i+ "*«;««= 2 (i+», r w+,) (E,-E,_ 1 )jw (0 where c denotes consumption, r is the constant return to asset holdings, x t is labour income, u t is a white noise error and E ( is the rational expectations operator. This implication embodies the behavioural principle of REPI, and is easily tested by regressing the change in consumption on a subset of lagged variables uncorrelated with u t and testing for their statistical exclusion. An alternative representation of (1) in terms of savings has recently been derived by Campbell (1987) and is noted here as equation (2) (as we shall see below this derivation is especially useful for our econometric implementation of ( > ) ) J«= - E (1 +0 'E,(*t + i-*i+i-i)- ( 2 ) The intuitive explanation of this result is that saving 'anticipates' falling labour income (in the sense of an optimal predictor given available information), as consumption, being set according to permanent income, tends to be below current income when income is expected to fall. This is also in accord with the views on transitory income movements expressed by Friedman (1963). In order to implement (1) empirically we use a Bivariate Autoregressive (BVAR) model. Essentially this approach involves modelling consumption and income as a bivariate process, whilst imposing the cross equation restrictions implied by rational expectations. A crucial assumption underlying such an approach is that the series being modelled are covariance stationary. However, it is well known that most economic time series are non-stationary. To induce stationarity in their BVAR systems researchers have adopted two approaches: either simple first differencing, or removing the deterministic component with a time trend. However in the present context both these procedures are deficient. The detrending procedure can introduce spurious 'cyclical' behaviour into the residual and lead to an unwarranted rejection of the REPI (see Mankiw and Shapiro, 1985). One problem with first differencing

3 1989] EXPECTATIONS : FURTHER U.K. EVIDENCE 85 is that it does not allow the full set of restrictions implied by the REPI to be imposed on the BVAR system. 1 But more importantly from our perspective, Engle and Granger (1987) have demonstrated that if two variables, such as c and y (y = x t + ra t, where y is total disposable income and a is real assets) are cointegrated then an estimated BVAR containing Ac and Ay will be misspecified. 2 In particular, if c and y are cointegrated then no invertible moving average representation exists and thus no finite BVAR system exists for Ac and Ay. If both detrending and differencing are unsatisfactory how can (1) be implemented in a BVAR form? The answer to this question lies in the cointegration of c and y. If the data support cointegration of c and y then the way to proceed is to include the residual from the cointegration of y and c, which may be interpreted as savings, s, and a subset of Ay namely Ax. That is we propose estimating the BVAR representation of equation (2). As Campbell (1987) points out, a BVAR including s and Ax will be well behaved and allow imposition of all the restrictions implied by the REPI. If s and Ax together form a linearly nondeterministic, jointly convariance stationary process, then Wold's decomposition implies that the bivariate process has a unique, infinite order moving average representation. For a suitably chosen value of n, this may be approximated in finite samples by an zz-th order bivariate vector autoregression. This can be written: r.- fl (L) -*(L)ifAx l ] = r» 1( i L-c(L) i-rf(l} UJ UJ (3) where fl(l), b(l), c(l) and d(l) are n-th order scalar polynomials in the lag operator, L (L"z ( = z,_ ) and u ( = {v u,v 2t ) is a vector white noise process obeying E(u t u e _<) = (0, i = 0; 0,1 4= o). Alternatively, (3) can be more compactly expressed as: (I-A)X, = u which, for afirst order system simplifies further to the Markovian representation: X ( = AX M + (. (4) If we define the restricted information set at t 1, Q ( _ 1} as consisting solely of lagged values of s and Ax (whereby },_, is a subset of A the full, or 'true', information set) then agents forming their expectations in t 1 will obtain optimal predictions as: X,_ lit = A%_ 1, (5) where X t _ 1Jt is the optimal A-step ahead prediction of X ( _ 1+t made at time / 1. Projecting both sides of (2) onto Q. t _ Y we therefore have: Vi.* = e'ax ( _,; Ax ( _, k = g'a*x,_ 1, 1 In the context of a BVAR model of the term structure of interest rates, Shiller (1981) has argued that first differencing 'throws away restrictions' imposed by rational expectations. 2 A variable z is said to be integrated of order d [z ~ l{d)] if it has a stationary, invertible, nondeterministic ARMA representation after differencing d times. Two variables x and y are cointegrated of order d,b if some constant scalar a exists such that z = x ay, where 2 ~ \(d b). For a further discussion of the cointegration technique see Engle and Granger (1987).

4 86 THE ECONOMIC JOURNAL [MARCH and so the restrictions implied by the joint REPI are: e'=-s [i/(i+r)]*g'a*, (6) where e' and g' are (i x 2/z) selection vectors where the former has unity in the (n+ i) th element and zeros elsewhere and g' has unity in the first element and zeros elsewhere. In the next Section we present tests of these restrictions using a UK data base. II. EMPIRICAL RESULTS The data employed in conducting the tests were obtained from Economic Trends and Financial Statistics. The series for both total and non-durable consumption, c and nc, and total personal disposable income, y, were obtained from Economic Trends. The real disposable labour income series, x, was constructed from series in Economic Trends and Financial Statistics. 3 All series are quarterly seasonally adjusted and the availability of the labour income series dictated the starting point as 1966:1; the last observation is 1986:3. As a preliminary step in our analysis we examine whether the variables studied have unit roots. To this end we present, in Table 1, Augmented Dickey- Table 1 ADF Tests for a Unit Root in Raw Data Levels Changes c 079(1) -1059(0) nc 070(1) -489(1) y O-II (1) 1146 (o) x 016 (4) (1) Notes: c, nc, y and x are as defined in the text. ADF denotes the Augmented Dickey-Fuller t statistics calculated as the ratio of 4 to its estimated standard error in: z, = a + bx l_ 1+ EAz,_ ( + u r The null hypothesis is that the series in question is I(i). The critical values for t at, respectively, the i %, 5% and 10% levels are 3"5i> 2'89, 2-58 (see Fuller, 1976). Numbers in parenthesis after the statistics denote the number of lagged dependent variables included in the calculation. Fuller (ADF) statistics for our raw data series and first differences thereof. In no case can we reject the null hypothesis that the series follow I(i) processes at any of the normal significance levels. 3 The series for labour income was constructed as follows. Denote taxes on personal income, 7", social security contributions (comprising national insurance, national health and redundancy fund contributions), S, transfer payments (comprising national insurance benefits and other current grants to the personal sector), B, total income before tax, Y, total labour income before tax (consisting of wages, salaries and forces pay) X, and the implied consumers' expenditure deflator, P. Non-wage income was then apportioned into its labour component in the ratio X/Y, with taxes on personal income being similarly treated. In sum, the real disposable labour income series, x, was generated as: x = [X+{X/Y)(Y-X)-(X/Y)T-S+B]/P. n (-1

5 1989] EXPECTATIONS : FURTHER U.K. EVIDENCE 87 In order to set up the BVAR system we have to determine whether y and c (and nc) are cointegrated, since it is the residual from the cointegrating regression which we interpret as (quasi-)savings, s t.* We have already satisfied a necessary condition for y and c or nc to be cointegrated in that we have demonstrated that these series all follow random walks. In Table 2 our Table 2 Cointegration Regressions y, = c, R* = 097 CRDW = 1 00 DF = -522 ADF = -305 (1) Non-durable consumption y, = 957 ' ', R" = o-g8 CRDW =117 DF =-606 ADF =-326(1) Notts: R J is the coefficient of determination, CRDW is the Durbin-Watson statistic from the cointegration equation, and DF and ADF are, respectively, the Dickey-Fuller and Augmented Dickey-Fuller t ratios calculated from the residuals of the cointegrating regression. The 1 %, 5% and 10% critical values for these statistics are as follows: CRDW 0.511, and 0-322; DF -407, -337 and -303; ADF 377, 3-17 and 2-84 (see Engle and Granger, 1987). cointegrating regressions are presented for both measures of consumption. In both equations the coefficient on consumption is above unity, although it is closer to unity in the case of total consumption (given that both c and y are nonstationary series we cannot infer whether these coefficients are significantly different from unity). For non-durable consumption the null hypothesis of no cointegration is rejected at the 5 % level in terms of the CRDW, DF and ADF statistics. For total consumption the null hypothesis of no cointegration is also rejected at the 5% level with the CRDW and DF and at the 10% level with the ADF. 5 Given the residuals from our cointegrating regressions we proceeded to set up two BVAR systems: one for s and Ax and the other for s' and A* (where s is the residual from the cointegration of y and c and / is the residual from the cointegration ofy and TIC). Our initial task was to determine the order of the BVAR system. This is a somewhat sensitive issue since under-parameterisation will tend to bias the results and over-parameterisation will diminish the power of the tests. Here we followed two procedures. First, we chose n to minimise the 4 If the implicit proportionality factor in the relation of consumption to permanent income is not unity but takes some fractional value, fi, then the results presented continue to hold with s t being the quasidifference s, = y, cjp, which has the interpretation of being 'quasi-saving', in the sense of saving inclusive of that element of the available capacity for consumption which remains ' invested' in permanent income. This is the interpretation adopted in the empirical section of this paper. 5 As a matter ofinterest, following Muellbauer (1983), we also estimated our cointegrating equations over the slightly longer sample, 1955:1-1986:3 and the sub-sample corresponding to the fixed/floating exchange rate regime shift. In contrast to Muellbauer's findings, our results (available from the authors on request) are remarkably homogenous regardless of the sample chosen.

6 88 THE ECONOMIC JOURNAL [MARCH Akaike Information Criterion (AIC) (Akaike, 1973). Our second procedure for choosing n was to start with a general BVAR system and test down using a likelihood-ratio test until reducing the order by one could be rejected (the statistic was computed using the degrees of freedom correction recommended by Sims, 1980). For both definitions of consumption we started off with a BVAR system where n = 8. For both systems the AIC was minimised with n = 1; however, using a likelihood-ratio test we could reject the move from a four lag system to that with three. In testing the REPI restrictions we report results for both one and four lag models to ensure that our tests are not biased by the chosen parameterisation. Some descriptive statistics from estimating the BVAR systems are presented in Table 3. The reported Ljung-Box (1978) statistics are insignificant at the Non-durable consumption (' lag) (' lag) Non-durable consumption (4 lags) (4 lags) Table 3 Descriptive Statistics from BVAR System R; RJ Q, (24) Q 2 (24) FJ (031) (091) (0-52) (0-87) (0-98) (o-6 4 ) (0-98) (0-41) Notes: R 2 denotes the coefficient of determination, 0.(24) denotes the Ljung-Box statistic with 24 degrees of freedom (marginal significance levels in parenthesis), F denotes the marginal significance level for the Granger-Causality F statistic, a 1 subscript denotes that the equation is s on lagged Ax and lagged 1,12 subscript denotes that the equation is Ax on lagged Ax and lagged s, a 1 superscript denotes the exclusion test of lagged s = o and a 2 superscript denotes the exclusion test of Ax = o. i % level, or better, for all equations. However, given the relatively low power of the Ljung-Box test we also computed twelfth-order residual correlograms which indicated the lack of autocorrelation in all equations apart from the one lag non-durable system (spike 4 was significant at the 5% level-value, o - 22) and the one lag total consumption system (spike 5 was significant at the 5% level-value 0-24). Interestingly, Granger-causality tests indicate that saving Granger-causes changes in labour income; however, in all of the estimated regressions with Ax as the dependent variable the sign of the coefficient on the first lag of J was positive, in the one lag model, whilst the sum of coefficients in

7 1989] EXPECTATIONS'. FURTHER U.K. EVIDENCE 89 the four lag model was also positive, thus conflicting with a basic postulate of the model. 6 In Table 4 (a) we present our tests of the restrictions implied by (6), assuming no transitory income.' For both the 1 and 4 lag models the computed Wald statistics allow rejection of the model restrictions at very high significance levels. Non-durable consumption Non-durable consumption Table 4 Summary of Restrictions Tests (a) Model with no allowance for transitory income x\ '7 Xl (A) Model wi h allowance for transitory income x\ xl xl 39 '39 '373 (071) (0-49) (0-07) 108 (078) 1 08 (058) Xl (000) '4>9 (004) xl xl 1323 (0-04) (003) Notes: Where ^ is a linear Wald statistic, a 1 subscript denotes the one lag system, a 4 subscript denotes the four lag system, a 1 superscript denotes that the exclusion restrictions include a constant and a 2 superscript denotes that the exclusion restrictions are computed without a constant. Marginal significance levels are in parenthesis. Standard errors were calculated using the Hansen (1982) method of moments correction for heteroskedasticity. Interestingly, when we allow for transitory income the rejection of the joint REPI is not as strong. 8 Thus, for the one lag model reported in Table 4 (b) the joint hypothesis cannot be rejected at standard significance levels (i.e. the 1 to 5% levels). For the four lag model reported in Table ${b) the joint hypothesis cannot be rejected at the 1 % level in all cases but can be rejected at the 5 % 6 An Associate Editor has brought the following to our attention. At the individual level, saving is lower if labour income is expected to grow throughout life. At the aggregate level, where each individual saves for retirement (and dissaves in retirement), with a stationary population, aggregate saving is zero with zero productivity and income growth, but positive with economic growth. This result is reinforced with population growth. Thus savings are positively related with growth, so that savings may increase in the aggregate in anticipation of future income growth without violating the REPI. Whilst other factors, notably inflation, might influence this relationship, the possibility arises that aggregate effects may swamp individual effects which could then invalidate one of our subsidiary conclusions. Our finding of a stationary savings series perhaps runs counter to this view. ' For our measure of the real interest rate, r, we chose the average ex post real interest rate for our sample period, which was 6%; we do not believe that our results are significantly affected by the choice of this value. 8 The derivation of (1) presupposes the absence of transitory income. With the relaxation of this assumption (1) acquires a moving average (MA) error representation in addition to the 'surprise' element, u r In the presence of an MA error the exclusion restrictions are still applicable with information lagged two periods.

8 90 THE ECONOMIC JOURNAL [MARCH level for all cases except that of non-durable consumption inclusive of a constant. 9 III. SUMMARY AND CONCLUSIONS This paper examines some implications of the rational expectations-permanent income hypothesis (REPI) using a BVAR modelling framework with appropriate stationarity inducing transformations. Using aggregate UK data we find that income and consumption, both total and non-durable, follow random walks and are cointegrated in levels. The BVAR system performs well, and savings Granger-cause changes in labour income, but with a perverse sign; savings anticipate increased future labour income. Further, the exclusion restrictions derived from the BVAR system are strongly rejected when transitory consumption is denied, but this rejection is statistically diminished with allowance for (serially uncorrelated) transitory consumption. On balance our results are rather mixed. Nevertheless, the finding that savings positively Granger-cause labour income changes would seem to pose a serious challenge to strict REPI. University of Aberdeen Date of receipt of final copy: May ig88 REFERENCES Akaike, H. (1973), Information theory and an extension of the maximum likelihood principle', in (B. N. Petrov and F. L. Saki eds.), Second International Symposium on Information Theory, Budapest: Akademiai Kiado. Campbell, J. Y. (1987), 'Does saving anticipate declining labor income? An alternative test of the permanent income hypothesis', Econometrica, Vol. 55, pp and Clarida, R. H. (1987), 'Household saving and permanent income in Canada and the United Kingdom', in (Helpman el at., eds.), Economic Effects of the Government Budget, Boston: MIT Press (forthcoming). Davidson, J. E. H. and Hendry, D. F. (1981), 'Interpreting econometric evidence: the behaviour of consumers' expenditure in the U.K.', European Economic Review, Vol. 16, pp ,, Srba, F. and Yeo, S. (1978), 'Econometric modelling of the aggregate time-series relationship between consumers' expenditure and income in the U.K.', ECONOMIC JOURNAL, Vol. 88, pp Drobny, A. D. and Hall, S. G. (1987), 'An investigation of the long-run properties of aggregate non-durable consumers' expenditure in the U.K.', National Institute of Economic and Social Research, forthcoming. Engle, R. F. and Granger, C.W.J. (1987), 'Dynamic specification with equilibrium constraints: cointegration and error-correction', Econometrica, Vol. 55, pp Flavin, M. A. (1981), 'The adjustment of consumption to changing expectations about future income', Journal of Political Economy, Vol. 89, pp Friedman, M. (1963), 'Windfalls, the "horizon" and related concepts in the permanent-income hypothesis', in (C. Christ, et at.), Measurement in Economics, Stanford: Stanford University Press. Fuller, W. A. (1976), Introduction to Statistical Time Series, New York: Wiley. Granger, C. W.J. (1981), 'Some properties of time-series data and their use in econometric model specifications', Journal of Econometrics, Vol. 16, pp and Weiss, A. A. (1983), 'Time series analysis of error-correcting models', in (S. Karlin, T. Amemiya, and L. A. Goodman, eds.), Studies in Econometrics, Time Series and Multivariate Statistics, New York: Academic Press. 9 Since completing this paper, some related results presented in Campbell and Clarida (1988) have been brought to our attention. These authors also apply the cointegration approach to savings, and similarly report that on UK data saving Granger-causes changes in disposable labour income, though with significant negative sign, and are also able to reject the REPI BVAR restrictions.

9 1989] EXPECTATIONS : FURTHER U.K. EVIDENCE QI Hall, R. E. (1978), 'Stochastic implications of the life cycle-permanent income hypothesis: theory and evidence', Journal 0/Political Economy, Vol. 86, pp Hansen, L. P. (1982), 'Large sample properties of generalised method of moments estimates', Econometrica, Vol. 50, pp Ljung, G. M. and Box, G. E. P. (1978), 'On a measure of lack of fit in time series models', Biomelrika, Vol. 65, pp Mankiw, N. G. and Shapiro, M. D. (1985), 'Trends, random walks and tests of the permanent income hypothesis', Journal of Monetary Economics, Vol. 16, pp Muellbauer, J. (1983), 'Surprises in the consumption function', ECONOMIC JOURNAL, Conference Papers, Vol. 93, pp Shiller, R.J. (1981), 'Do stock prices move too much to be justified by subsequent changes in dividends?, American Economic Review, Vol. 71, pp Sims, C. A. (1980)-, 'Macroeconomics and reality', Econometrica, Vol. 48, pp