ON THE SPECIFICATION OF THE AUSTRALIAN CONSUMPTION FUNCTION*

ON THE SPECIFICATION OF THE AUSTRALIAN CONSUMPTION FUNCTION*

IMAD A. MOOSA

La Trobe University

I. INTRODUCTION

In a recent paper published in this journal, Lester (1993) estimated several versions of the Australian consumption function using quarterly seasonally unadjusted data. He found that the specifications suggested by Davidson et al. (1978) and by Hendry and von Ungern-Sternberg (1981) performed rather poorly, and proposed to use an adjusted income variable and a total wealth variable in order to tackle the failure of the estimated equations to pass the diagnostic and forecast tests. It is possible, however, that the Davidson et al. and Hendry-von Ungern-Sternberg type equations did not perform well, not only because of the definitions of the income and wealth variables, but also (perhaps more importantly) because they are misspecified in the light of the recent literature on seasonal and periodic cointegration. This may also be the case for Lester's final equation which is shown to pass all diagnostic tests. The objective of this paper is to show why these equations may be misspecified, and to propose an alternative approach to specifying and estimating the Australian consumption function.

11. COLNTEGRATION ANALYSIS AND MODEL SPECIFICATION

One of the virtues of cointegration analysis is that it provides a rule that enables us to avoid the problem of choosing between specifications in levels and first differences. In conventional econometrics, the choice is often made on an arbitrary ad hoc basis, involving specification search, and thus the choice between level and first difference specifications depends on reaching a compromise involving the t-statistics, the DW and R2. In cointegration analysis, on the other hand, if the underlying variables are Z(1), then the cointegrating regression is specified in levels while the error correction model is specified in first differences. Thus, the application of the difference operator, A , is determined by the order of integration of the underlying variables. The development of seasonal cointegration has solved the problem even further. We are now in a position to pick the appropriate filter (A, Ad, AA4, etc.) on the basis of the order of seasonal integration of the underlying variables.'

* I would like to thank Rodney Maddock and an anonymous referee for their comments on earlier versions of this paper. I am also grateful to Dianne De Freitas Braz for excellent research assistance.

I See Moosa and Al-Loughani (1994). and Osborn (1990). For more recent developments involving periodic integration see Franses and Romijn (1993).

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AUSTRALIAN ECONOMIC PAPERS DECEMBER 264

Lester’s work gives rise to two interesting problems which are worthy of examination. First, the application of the filters A 4 and M4 implies that the underlying variables are either Z(0,l) or I( 1,l) respectively; these should be testable hypotheses, not assumptions. Secondly, the possibility of cointegration at the seasonal frequencies was not considered and thus the error correction mechanism could be misspecified.

111. MODEL SPECIFICATION: A REVIEW OF LESTER’S WORK

Lester starts by replicating the following Davidson et al. (1978) type equation which is similar to the one estimated by Bladen-Hovel1 and Richards (1983)

where c is real private final consumption excluding durables, y is real disposable income, p is the consumer price index, and all the variables are measured in natural logarithms. When this equation was estimated allowing for a first order autoregressive process in the residuals, Lester found the estimated equation to be unsatisfactory for the following reasons: (i) A A o and M4p are both insignificant; (ii) the presence of serial correlation as indicated by the Ljung-Box Q* test; and (iii) c and y are not cointegrated, implying that the error correction term is not appropriate.*

While Lester correctly concluded that this specification of the Australian consumption function is inappropriate on the basis of standard diagnostic tests, one may elaborate on an important deficiency: the arbitrary choice of the filters A 4 and M,. The application of these filters may (or rather must) be justified either on a theoretical basis (e.g. intuition or the mathematical derivation of the model) or an empirical basis ( i e . the time series properties of the variables). Otherwise, any attempt to use these filters must involve specification search. The consumption-income relationship is well known and can be justified, for example, on the basis of Keynes’s absolute income hypothesis, but there is no theoretical reason to assume that the rate of change of the rate of change of income can affect the rate of change of consumption, and that this effect can be distinguished from the effect of the rate of change of income. Likewise, while the inclusion of inflation in the consumption function can be justified on the basis of relative price mistakes (Deaton, 1977) and money illusion and uncertainty (Freebairn, 1977), there is no theoretical reason why the effect cannot be captured without including the rate of change of the inflation rate. It is not surprising, therefore, that A A o and M,p turned out to be insignificant.

So much for theory. What about the time series properties of the variables? The specification of equation (1) implicitly indicates that c-Z(0,l) or Z(1,O); y-Z(0,l). Z(1,l) or Z(1,O); andp-Z(0,l) or Z(1,l). So, which one? We cannot tell unless we carry out a simple formal test such as the HEGY test (Hylleberg et al., 1990). If we are only guided by the time series properties of the variables, then it is only when the order of seasonal integration has been determined that we can choose the appropriate filter.3

2 The same equation was also found to be unsatisfactory when it was estimated over a different sample period and when it was estimated by OLS without allowing for the autoregressive process in the residuals.

3 The filters A, A, and AA, are respectively applied to variables that are integrated of orders (l,O), (0,l) and (1,l) to render them stationary. However, it is sometimes the case that seasonal differencing (applying the filter A,) is adequate to achieve stationarity in an Z(1,l) variable. This will be the case if x-Z(l.1) and A,x- 1(0,0). However, if A,x-Z(l.0) and M,x-Z(O.0) then the filter M, must be applied.

1995 ON THE SPECIFICATION OF THE AUSTRALIAN CONSUMPTION FUNCTION 265

Furthermore, while Lester correctly argued that the error correction term in the Davidson et a/. specification, c - y , is not appropriate, there are other reasons why it is not. Lester reached his conclusion by testing for cointegration between c and y while imposing the restriction (-1,l) on the cointegrating vector. But even if the restriction (-1, 1) is not imposed apriori, and if cointegration turned out to be the case, an error correction term of the form ( C - - ~ Y ) ~ - ~ is still not appropriate, not only because it is lagged four periods instead of the usual one as implied by the normal specification of Granger’s Representation Theorem. It is also inappropriate because it is derived from a cointegrating regression relating consumption to income only, while the error correction model as represented by equation ( I ) includes the seasonal difference of the price level, which casts some doubt on the specification of the long-run relationship (as represented by the cointegrating regression) and therefore the error correction term.

Let us illustrate this point by ignoring, for the time being only, seasonal unit roots and assuming that c -I( l), y - I ( 1) and p - I ( 2 ) , which is plausible. In this case the cointegrating regression is given by

and the corresponding error correction model is specified as

in which case a parsimonious specification can be reached by starting with rn = 5 and testing downwards using Hendry’s general to specific methodology.4.5 Obviously, the error correction term in (3) is strikingly different from, but more plausible than, the one in equation (1).

The error correction model represented by (3) will also be misspecified if cointegration at the seasonal frequencies is present but is not allowed for. This, naturally, will be valid only if the variables are seasonally integrated which is an implication of equation (1). It is also an aspect that needs to be examined when seasonally unadjusted data are used. This procedure will be outlined in the following section.

In an attempt to improve the specification of the consumption function, Lester estimated the following Hendry-von Ungem-Sternberg type equation

where y* is income adjusted for inflationary losses on net liquid assets, R is the real post-tax interest rate and I is net liquid assets used as a proxy for wealth. Again, Lester correctly reports several problems with this specification such as insignificant coefficients, wrong signs, and the

4 It is noteworthy that the application of the filters A and A2 in equations (2) and (3) is dictated by the order of integration of the variables.

5 If the inflation rate is not included in the cointegrating regression, i.e. it does not influence consumption in the long run, the rate of change of the inflation rate can still he included in the error correction model, but obviously it will not be part of the error correction term.

266 AUSTRALIAN ECONOMIC PAPERS DECEMBER

failure to pass diagnostic tests. However, equation (4) is subject to the same criticism as equation (1). The poor empirical performance of this equation, as pointed out by Lester, may be attributed to the same deficiencies as equation (1).

Let us, however, concentrate on the possible time series properties of the newly introduced variables, R and 1. The implication of equation (4) is that R-Z(0,l) and 1-Z(1,l). It is more likely, however, that R-Z(O,O) and that 1-Z(1,O) and possibly l -Z(2,O). The hypothesis that R-Z(O,O) has been confirmed by studies of the Fisher relationship (for example, Atkins, 1989).6 The second hypothesis is conceivable from the literature on multicointegration (for example, Granger and Lee, 1991). If c,-Z(l) and yl-Z(l) then s, -Z(O) (where s,=y, -c,) if c and y are cointegrated, and s,-Z(l) if they are not cointegrated as indicated by Lester. Since wealth is the accumulation of past saving, it follows that 1, = xlf~,-~ and so l I -Z (2) if s,-Z(l) and ll-Z(l) if s,-Z(0). It is extremely hard to conceive the idea that financial wealth can exhibit seasonal variation.

Once more, Lester tries to improve the specification of the consumption function by redefining wealth and adjusted income to include total private sector wealth except dwellings, w. Moreover, he introduces unemployment, u, as another explanatory variable. In effect the basic functional relationship underlying his parsimonious equation is

where y** is income adjusted for inflationary losses on the new measure of wealth. But then he defines the error correction term as

where a1 and a2 are cointegrating parameters. This error correction term is then used to estimate a parsimonious model of the form

which he finds to be the most appropriate as it passes all diagnostic tests. It seems, however, that this parsimonious specification was reached after extensive specification search, trying out the filters A4 and AA4 on all of the variables, including u, which is normally Z(0,l) or Z(0,O). It is not surprising then that none of the variables that were tried turned out to be significant, except A4y *1* and A4ct-l.

6 There seems to be a measurement error in R which is defined as R , = (1 -7)i,-A4p, where i is the 13 week TB rate and T is the tax rate. The calculation of the (ex post) real interest rate requires the nominal interest rate and the inflation rate to be measured over the same time period which is the holding period of the underlying asset. However, while the nominal interest rate is measured over the period extending between quarters t-1 and t , the inflation rate (as represented by A4pI) is measured over the period extending between quarters t -4 and t. Hence, a more appropriate definition of R is given by R, = (1 - ~ ) i ~ - 4 A p , .


So, let us sum up the arguments so far. On the basis of standard diagnostic tests Lester concluded that the Davidson et al. and Hendry-von Ungern-Sternberg equations are inappropriate for the Australian case. He attributed the deficiency of these equations to the definitions of the variables. Accordingly, he proceeded to estimate his preferred model in which the error correction mechanism incorporates modified income and wealth variables, since he did not find evidence for cointegration between consumption and income. The argument put forward here is that the equations may be misspecified because of the use of inappropriate filters and error correction mechanisms. This implies that any attempt to improve the specification of these equations requires these issues to be addressed first. It is possible, therefore, that a better specification can be reached by adopting a systematic modelling approach in the light of the recent literature on seasonal and periodic cointegration. This procedure is outlined in section IV.

IV. AN ALTERNATIVE MODELLING APPROACH

An alternative approach to modelling the consumption function using seasonally unadjusted data is to follow the example set by Engle et al. (1993) in modelling the Japanese consumption function. This procedure can also be supplemented by the method proposed by Franses (1993) to select between seasonal and periodic error correction models. Let us, for the sake of simplifying the exposition and following Engle et al. (1993), consider the consumption function

c, =f(y,) (8)

The first step is to determine the order of seasonal integration by testing for seasonal unit root at various frequencies as shown by Hylleberg et al. (1990). The procedure is based on the equation

where z l , z2 and z3 are transformed variables derived by applying some lag polynomials to the variable x. The test statistics for unit roots at frequencies 0, 1/2 and 1/4 are respectively given by the t-ratios of 6, and 6, and the F-statistic for 6, = 6, = 0.

If c, -Z@( l ) and yr -Zo(l) where 8 is the frequency, the cointegrating regressions at frequencies 0, 1/2 and 1/4 are respectively given by

z ~ ( c , ) = YO + Y I Z ~ ( Y ~ ) + Y ~ z ~ ( Y ~ - I ) + vr (12)

Cointegration is established at all frequencies if h,, u,, v,, are stationary, which can be tested on the basis of the auxiliary regressions

i= l


For the auxiliary regressions (13) and (14), the test statistics are the t-ratios of 4: and the critical values are tabulated in Engle and Granger (1987) and Engle and Yo0 (1987). Testing for unit root in V , is not straightforward because it contains complex roots. In this case the test statistics are the t-ratios of A, and A, and the F-statistic for A, n A, whose critical values are tabulated in Engle et al. (1993).

If cointegration is established at all possible seasonal frequencies and if seasonal differencing is adequate to achieve stationarity (i .e. A 4 c r and A4yt are stationary), then a seasonal error correction model can be specified as

where @JL) and Clq(L) are lag polynomials of orders p and q respectively.7 If p = 1 and q = 0, equation (16) will reduce to

which is directly comparable with Lester's equation (7). It is thus obvious that if c and y are cointegrated at any of the seasonal frequencies then equations (l), (4) and (7) are misspecified.

Franses (1993) has recently suggested that seasonally integrated series may give rise to periodic rather than seasonal integration. If this is the case, the error correction model would take the form

where the Djs are seasonal dummies. Equation (18) implies that there are varying cointegration relations per quarter and that the adjustment to disequilibrium errors can be time varying (Franses, 1993, p.8). Furthermore, Franses proposes a procedure for choosing between the two models. This procedure is based on applying the standard cointegration test statistics to the annual series containing the observations per quarter. The quarterly series cr and yr are decomposed into four annual series, cIlr and where t = 1,2, ..., n and j = 1,2,3,4. Since periodic cointegration, unlike seasonal cointegration, requires the elements in the corresponding seasons to be cointegrated, a model of periodic cointegration is more appropriate if cJ and yJ turn out to be cointegrated.

7If seasonal differencing is inadequate to achieve stationarity then the filter A A 4 is used such that A,c, and A4y, are replaced by M 4 c , and M,yr Thus, the choice of the filter is not arbitrary but is rather dictated by the time series properties of the variables.

1995 ON THE SPECIFICATION OF THE AUSlRALIAN CONSUMPTlON FUNCTION 269

v. AN ILLUSTRATION USING AUSTRALIAN DATA

The objective of this section is to demonstrate the points raised so far, particularly with respect to the use of the filters according to the time series properties of the variables and the specification of the error correction model. The objective is certainly not to study consumer behaviour in Australia by examining a full range of variables that impinge upon consumer expenditure.8 Given this humble objective, three variables are considered in this empirical exercise: non-durable consumption (as defined by Lester), c, disposable income, y , and consumer prices, p . The time series of quarterly seasonally unadjusted constant-price data cover the period 1959(3)- 1993(3).9

The first step in this empirical exercise is to test for unit roots at various frequencies to determine the order of integration of the variables. The results, which are presented in Table I, show the following: while y and c are Z(l,l), seasonal differencing (i .e. applying the filter A4) is adequate to achieve stationarity and hence A4c and A4y are stationary. p , on the other hand, is not seasonally integrated (as indicated by a significant t: 6, statistic), and seasonal differencing is not adequate to achieve stationarity (as indicated by an insignificant t : S , statistic for Ap). However, the filters A2 and AA4 can achieve stationarity in p , but since p -1(2) and is not seasonally integrated, the filter A2 is more appropriate.“) This result casts doubt on the validity of using the terms A4p and AA4p in equation ( 1 ) .

Having established that y and c are seasonally integrated, the next step is to examine the possibility of periodic cointegration along the lines suggested by Franses ( 1993). The quarter- specific observations of c and y are tested for cointegration using the Engle- Granger (1987) test and the Johansen (1988) Trace test. The results, which are reported in Table 11, show that the null hypothesis of no cointegration cannot be rejected for j = 1,2,3,4. Hence, the possibility of periodic cointegration is discarded.

The results of testing for seasonal cointegration are presented in Table 111. These results show that c and y are not cointegrated at the zero frequency,ll or at frequency 1/2 but are cointegrated at frequency 114 (ix. at the annual cycle). This result is plausible: while Figure 1 shows that both consumption and disposable income exhibit strong seasonal patterns, these patterns are different, and that is confirmed by Figures 2 and 3. One economic explanation for this finding is possibly the effect of Christmas on consumption and the effect of the end of the calendar year on disposable income, leading to a fourth quarter rise in both of them.’*

The objective is so because the argument put forward in this paper is that the main problem with the specifications outlined earlier (which must be tackled first) is not the definitions of the variables hut rather the ud hoc use of the filters and the specification of the error correction mechanism. For a good survey of the variables that determine Australian consumption see Lattimore ( 1994). 9 A longer sample period is used here than that used by Lester (1964(4)-1990(2)) for at least three reasons: ( i ) to avoid small sample bias; (ii) to demonstrate the structural stability of the estimated equation over as long a period as data availability permits; and (iii) to allow for the loss of observations resulting from the application of the filters and lag polynomials. “]Otherwise, there may be a risk of over differencing.

This finding confirms Lester’s scepticism of the error correction term c-y in equation ( I ) . However, his statement that ‘consumption and this measure of income do not cointegrate’ (p.323) is not strictly correct, because he only considered cointegration at the zero frequency. I? While both consumption and income decline in the first quarter following the fourth quarter rise, they move in different directions in the second quarter (income down, consumption up). In the third quarter income is up while consumption does not change significantly.


11.12

10.59

10.06

9.53 19

Figure 1

The Logarithms of Non-durable Consumption (c) and Disposable Income (y)

1 .oo

0.41

-0.18

-0.78 6 ' 12 24

Figure 2

36 45 Order of lags

Autocorrelation Function of the Log First Difference of Disposable Income

1995 27 1 ON THE SPECIFICATION OF THE AUSTRALIAN CONSUMPTION FUNCTION

1 .oo

0.46

-0.08

-0.62 12 24 36 45

Order of lags

Figure 3 Autocorrelation Function of the Log First Difference

of Non-durable Consumption

Figure 4 The Cumulative Sum of Recursive Residuals


TABLE I Testing for Seasonal Unit Root

Variable t:S, t:S, F:S, n 6, k LM(4)

Y

P C

A4Y A4c A4P A P A2P M4P

-2.37 -2.42 -0.58 -4.86 -4.84 -1.93 -1.74 -6.7 1 -6.31

-1.41 - 1.02 -5.07 -7.04 -6.14 -9.98 -5.12 -3.96

-10.28

0.82 0.17

28.51 58.81 90.71 83.52 29.07 29.68 88.07

7.33 7.98 6.22 2.65 6.69 5.77 6.34 2.06 5.79

Note: The 5% critical values are t:S,= -2.88, t:S, = - I .95, F.6, n S, = 3.08.

TABLE 11 Testing for Periodic Cointegration

ADF (m) Trace ( r = 0)

-2.93(0) -1.44(0) -1.56(0) -2.06(0)

10.2 1 5.91 9.44 7.88

Note: The 5% critical values are ADF = -3.53, Trace = 14.07.

TABLE 111 Testing for Seasonal Cointegration

Regressand Regressors t:+ t:A, t:A, F:A, n A, k LM(4)

Notes: The 5% critical values are t:+ = -3.17, t:A, = -4.14, t:A, = -2.14, F:AI n A, = 9.99. * The auxiliary regression includes intercept and seasonal dummies.


Having found cointegration at the frequency 1/4 only, the seasonal error correction model (12) can be estimated by imposing the restriction y , = y2 = 0. The parsimonious model turned out to be the following

A4c, = 0.012 + 0.559A,~,_1 + 0.1 19A4y, - 0.362~,-, + 0.232~1-3 (4.79) (8.04) (5.30) (-4.74) (2.89)

R2 = 0.67 DW=2.09 SC(4) = 4.39 FF(1) = 0.20 NO(2) = 1.44 HS (4) = 0.26 ARCH(4) = 3.19

The parsimonious model seems to be well determined: the coefficients are significant and correctly signed. It also passes the diagnostic tests for serial correlation, functional form, normality, heteroskedasticity and autoregressive conditional heteroskedasticity. Furthermore, the model is highly stable structurally as it passes the CUSUM and CUSUMSQ tests (Figures 4 and 5) and passes the Chow and predictive failure tests at four different break points, the earliest of which is 1974(4) (Table IV).

In order to demonstrate the superiority of this model to the Davidson et al. type equation ( 1 ) (with seasonal dummies), non-nested model selection tests are used with MI being the seasonal error correction model (17) adjusted to include a price variable, A*p. For this purpose six tests are used N which is the Cox (1961, 1962) test derived in Pesaran (1974); NT which is the adjusted Cox test derived in Godfrey and Pesaran (1983); J which is the Davidson and MacKinnon (1981) test; J A which is the Fisher-McAleer (1981) test; and EN which is the encompassing test proposed, inter alia, by Mizon and Richard (1986). All of these test statistics have t distribution except the EN test which has F distribution. In testing M1 versus M2, M1 is rejected in favour of M2 if the test statistic is significant. The results of these tests, which are reported in Table V, are unanimous in rejecting M2 in favour of M1 but not the other way round.

It is important to notice that the seasonal error correction model presented here and Lester’s equation (7) share the two terms A4ct-, and A4yr, but differ significantly in the specification of the error correction mechanism. Another difference is that while the former was reached after heavy specification search, the latter was attained via a systematic approach involving several steps, starting with testing for seasonal unit roots and ending with the general to specific methodology. This approach enables us to avoid specification search and the pitfall of applying the wrong filters.


TABLE IV Structural Stability Tests

Break Point Chow Predictive Failure

1989(4) 1.48 F(5,121) 0.78 F(15,l l l) 1984(4) 0.59 F(5,121) 0.67 F(35,91) 1979(4) 0.55 F(5,121) 0.68 F(55,71) 1974(4) 1.77 F(5,121) 0.85 F(75,Sl)

TABLE V Non-Nested Model Selection Tests

Test M1 vsM2 M2 vs M1

N NT W J

JA EN

-1.26 (0.21) -0.51 (0.61) -0.50 (0.62) 1.29 (0.20) 0.71 (0.48) 1.44 (0.20)

-24.87 (0.00) -18.99 (0.00) -13.63 (0.00) 10.46 (0.00) 6.28 (0.00)

23.40 (0.00)

Note: Marginal significance levels are given in parentheses.

19690) 1977(3) 198X4) 1993(3)

Figure 5 The Cumulative Sum of Squares of Recursive Residuals


V. CONCLUSION

Although Lester (1993) correctly identified the Davidson et al. and the Hendry-von Ungem- Stemberg type equations as being inappropriate for the Australian case, he attributed their empirical failure to the definitions of the variables and the omission of a wealth variable from the error correction term. The argument put forward in this paper is that the failure of these equations is due to the use of inappropriate filters (by overlooking the time series properties of the variables) and faulty error correction terms (by only allowing for cointegration at the zero frequency or the long run and not at other frequencies). A finding of this study is that Australian non-durable consumption and disposable income are cointegrated at the frequency 114 (annual cycle), implying that the consumption-income relationship should be modelled as a seasonal error correction model. This model was found to be superior to the Davidson et al. model. Moreover, it was also argued that identifying the order of integration of the variables is essential for applying the appropriate filter (and hence to avoid specification search), and for specifying an appropriate error correction term.

It will be interesting to estimate a full-fledged Australian consumption function along the lines suggested in this paper. One way to do that is to augment the seasonal error correction model presented here by the variables (with appropriate filters) tried out by Lester and those considered by Lattimore (1994).

REFERENCES

Atkins, F.J. (1989), ‘Cointegration, Error Correction and the Fisher Effect’, Applied Economics, vol. 21.

Bladen-Hovel], R.C. and Richards, G.M. (1983), ‘Inflation and Australian Savings Behaviour 1959-198 1 ’, Australian Economic Papers, December.

Cox, D.R. (1961), ‘Tests of Separate Families of Hypotheses’, in Proceedings o f the Fourth Berkeley Symposium, vol. 1, University of California Press.

Cox, D.R. (1962), ‘Further Results on Tests of Separate Families of Hypotheses’, Journal ofthe Royal Statistical Society, vol. 24.

Davidson, J.E.H., Hendry, D.F., Srba, F. and Yeo, S. (1978), ‘Econometric Modelling of the Aggregate Time Series Relationship Between Consumers’ Expenditure and Income in the United Kingdom’, Economic Journal, vol. 88.

Davidson, R. and MacKinnon, J.G. (1981), ‘Several Tests for Model Specification in the Presence of Alternative Hypotheses’, Econometrica, vol. 49.

Deaton, A. (1977), ‘Involuntary Saving Through Unanticipated Inflation’, American Economic Review, vol. 68.

Engle, R.F. and Granger, C.W.J. (1 987), ‘Cointegration and Error Correction: Representation, Estimation and Testing’, Econometrica, vol. 55.

Engle, R.F., Granger, C.W.J., HyIleberg, S. and Lee, H.S. (1993), ‘Seasonal Cointegration: the Japanese Consumption Function’, Journal of Econometrics, vol. 55.


Engle, R.F. and Yoo, B.S. (1987), ‘Forecasting and Testing in Cointegrated Systems’, Journal of Econometrics, vol. 35.

Fisher, G.R. and McAleer, M. (1981 ), ‘Alternative Procedures and Associated Tests of Significance for Non-nested Hypotheses’, Journal qf Econometrics, vol. 16.

Franses, P.H., (l993), ‘A Method to Select Between Periodic Cointegration and Seasonal Cointegration’, Economics Letters, vol. 41.

Franses, P.H. and Romijn, G. (1993), ‘Periodic Integration in Quarterly UK Macroeconomic Variables’, International Journal of Forecasting. vol. 9.

Freebairn, J.W. (1977), ‘Inflation and Stability of Household Consumption-Savings Function’, Economic Record, vol. 53.

Godfrey. L.G. and Pesaran, M.H. (1983), ‘Tests of Non-nested Regression Models: Small Sample Adjustments and Monte Carlo Evidence’, Journal of Econometrics, vol. 21.

Granger, C.W.J. and Lee, T. (1991), ‘Multicointegration’, in R.F. Engle and C.W.J. Granger (eds), Long-Run Economic Relationships: Readings in Cointegration (Oxford: Oxford University Press).

Hendry, D.F. and von Ungern-Sternberg, T. (1981), ‘Liquidity and Inflation Effects on Consumers’ Expenditure’, in A. Deaton (ed.), Essays in the Theory and Measurement of Consumer Behaviour (Cambridge: Cambridge University Press).

Hylleberg, S., Engle, R.F., Granger, C.W.J. and Yoo, B.S. (1990), ‘Seasonal Integration and Cointegration’, Journal of Econometrics, vol. 44.

Johansen, S. ( 1 988), ‘Statistical Analysis of Cointegrating Vectors’, Journal of Economic Dynamics and Control, vol. 12.

Lattimore, R. (1 994), ‘Australian Consumption and Saving’, Oxford Review of Economic Policy, vol. 10.

Lester, L.H. (1993), ‘A Parsimonious Consumption Function for Australia: 1964 to 1990’, Australian Economic Papers, December.

Mizon, G.E. and Richard, J.F. (1986), ‘The Encompassing Principle and its Application to Non- nested Hypotheses’, Econometrica, vol. 54.

Moosa, LA. and Al-Loughani, N.E. (1994), ‘Some Time Series Properties of Japanese Oil Imports’, Journal of International Economic Studies, No 8.

Osborn, D.R. (1990), ‘A Survey of Seasonality in UK Macroeconomic Variables’, International Journal of Forecasting, vol. 6.

Pesaran, M.H. (1974), ‘On the General Problem of Model Selection’, Review of Economic Studies, vol. 41.

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ON THE SPECIFICATION OF THE AUSTRALIAN CONSUMPTION FUNCTION*