Time-varying coefficients in the Almost Ideal Demand System: an empirical appraisal

Transcription

1 European Review of Agricultural Economics Vol 30 (2) (2003) pp Time-varying coefficients in the Almost Ideal Demand System: an empirical appraisal Mario Mazzocchi University of Reading, Reading, UK Received October 2001; final version received April 2003 Summary This paper provides a generalisation of the structural time series version of the Almost Ideal Demand System (AIDS) that allows for time-varying coefficients (TVC/AIDS) in the presence of cross-equation constraints. An empirical appraisal of the TVC/AIDS is made using a dynamic AIDS with trending intercept as the baseline model with a data set from the Italian Household Budget Survey ( ). The assessment is based on four criteria: adherence to theoretical constraints, statistical diagnostics on residuals, forecasting performance and economic meaningfulness. No clear evidence is found for superior performance of the TVC/AIDS, apart from improved short-term forecasts. Keywords: structural change, almost ideal demand system, theoretical restrictions, food demand, structural time series models JEL classification: D12, C51 1. Introduction Estimation of the Almost Ideal Demand System (AIDS) with time series data has often raised concern about the stability of the model s coefficients, especially when the length of the time series or the nature of the goods being analysed suggests that preferences might have changed over time. This concern has led to the development of different specifications for the AIDS, ranging from the inclusion of additional explanatory variables (demographics, advertising, health or food safety information indexes) to more flexible parameterisations of the functional form. Within the latter approach is a method for estimating a structural time series AIDS with time-varying coefficients (TVC/AIDS). This computer-intensive method, based on Harvey s (1989) structural time series model, has recently gained some popularity as a result of the progress in high-speed computing. The aim of this paper is to illustrate this model. It is extended to account for theoretical constraints, including the cross-equation restrictions on timevarying coefficients that have been neglected so far in the applied economics literature regarding this model. An empirical appraisal of the TVC/AIDS performance is based on a comparison with a dynamic AIDS model with trending intercept, conditional on a time series of food demand data from the Italian Household Budget Survey. # Oxford University Press and Foundation for the European Review of Agricultural Economics 2003

2 242 Mario Mazzocchi The rationale behind a time-varying parameterisation for the AIDS lies in the remark by Leybourne (1993) that the first-order Taylor series approximation to the true underlying demand system is only locally valid. The fixed coefficients of the AIDS represent the first derivative of the expenditure shares with respect to the explanatory variables (prices and total expenditure) evaluated at their chosen approximation point (usually the sample mean). Hence, the system is likely to perform badly when working with non-stationary time series or when the variability in prices or in total expenditure is large. There are several alternatives to the TVC/AIDS approach in the empirical literature dealing with structural change in food demand. 1 Confining our attention to the AIDS-based approaches, these alternative methods can be classified as follows: (i) allowing for a linear trend in the intercept; (ii) allowing for a linear trend in all parameters; (iii) allowing for parameters that depend on demographics or other explanatory variables. Method (i) consists of a consolidated augmentation of the intercept term and was also employed by Deaton and Muellbauer (1980). A linear trend term is often capable of portraying the pattern of structural change (Eales and Unnevehr, 1988), although the assumption of a constant trend over the whole sample could be inappropriate when a trend reversal is possible, as in long time series. Burton and Young (1992, 1996) address the problem of trend reversal by estimating the model in differences and including a shift, which involves a quadratic trend. The trending-intercept approach is usually framed in a dynamic specification, as the limits of static representation in a time series context have now been widely recognised. Method (ii) extends the trending effect to all parameters in the AIDS. Since the work by Moschini and Meilke (1989), many researchersrs have employed a regime-switching AIDS, allowing for flexible definition of the transition path between the two regimes (see, e.g. Burton and Young, 1996; Rickertsen, 1996; Mangen and Burrell, 2001). All these studies required either the a priori choice or the estimation of the breakpoints (i.e. the start and end time periods of the shift). Method (iii) links the pattern of structural change to time-varying demographic characteristics as in Moro and Sckokai (2000) or to other factors inducing structural change, such as advertising (as in Duffy, 2002) or health information indices (as in Ben Kaabia et al., 2001). This approach has the advantage of being informative about the sources of structural change; however, the required additional information might not be always available. The TVC/AIDS method discussed in this paper does not require any assumption about the pattern of structural change or any variables other than prices and income. Furthermore, all parameters in the model are allowed to change 1 Detailed critical discussion concerning the factors inducing structural change and related modelling issues are presented in Young (1996). Moschini and Moro (1996) provide an excellent review of parametric and nonparametric methods for modelling structural change in demand analysis, in which there is a burgeoning literature. There is much research focusing on structural change in meat demand (see, for example, Alston and Chalfant (1991) or the review by Smallwood et al. (1989)).

3 Time-varying coefficients in the Almost Ideal Demand System 243 without incurring econometric problems caused by over-parameterisation. Nevertheless, some assumption regarding the nature of parameter change (e.g. random walk) is still necessary and its main drawback is that time-varying coefficients may pick up measurement errors in the data or other noise usually accommodated in the AIDS disturbance term. 2 This paper shows the multivariate specification of the TVC/AIDS model (Section 2), describes the maximum-likelihood estimation method (Section 3) and contributes to the improvement of the structural time series specification by formally taking into consideration the theoretical constraints (Section 4). In Section 5 we provide an application to aggregate food consumption in Italy, over the estimation period Conditional on this data set, an empirical appraisal of the TVC/AIDS is made, using a dynamic AIDS model with trending intercept as the baseline model. Conclusions are drawn in Section The structural TVC/AIDS demand system Chavas (1983) was the first to suggest a Kalman-filter based approach to the estimation of a demand system with time-varying coefficients. The structural time series approach was originally applied to the estimation of AIDS equations by Leybourne (1993), who proposed a random walk structure for the coefficients and a maximum likelihood estimation method based upon the Kalman filter. The computational burden involved in this specification is relatively high, so attention thus far has been confined to the estimation of single equations, as in Al-Kahtani and Sofian (1995). Recently, Fraser and Moosa (2002) estimated the AIDS in a multivariate context, allowing for stochastic trend and seasonality but keeping price and expenditure coefficients constant. A more general specification of the TVC/AIDS extends the static AIDS (Deaton and Muellbauer, 1980) by allowing all the parameters to vary over time and basing the stochastic specification on Harvey s structural time series approach. Considering an n-good unrestricted AIDS model in its linear approximate form and after aggregation over households, the structural specification for each equation is w it ¼ it þ Xn xt ijt log p jt þ it log þ it þ u it ; i ¼ 1;...; n ð1þ j ¼ 1 where, for each time period t, w it represents the aggregate budget share of the ith good, p jt is the price of the jth good, x t is aggregate consumer expenditure, P t is Stone s price index, it is the time-varying intercept, it are seasonal components, u it is a random white-noise disturbance, and the price and expenditure coefficients vary over time. The model (1) is completed by a stochastic specification for the processes driving each parameter. The time-varying intercept it is specified as a local P t 2 We are grateful to an anonymous referee for pointing this out.

4 244 Mario Mazzocchi linear trend (Harvey, 1989, p. 170), i.e. a random walk with drift, where the drift ( it ) is itself a random walk: it ¼ i;t ÿ 1 þ i;t ÿ 1 þ e it it ¼ i;t ÿ 1 þ e it: The disturbances e it and e it are normally distributed white-noise processes. Intercept levels are driven by e it whereas changes in e it affect the drift t and hence the slope t, also allowing for trend reversal. Seasonality can be modelled by the following seasonal dummy specification: it ¼ÿ Xs ÿ 1 i;t ÿ j þ e it ð3þ j ¼ 1 where it is the seasonal component at time t, s is the periodicity (s ¼ 12 for monthly data, s ¼ 4 for quarterly data, etc.) and e it is a normally distributed white-noise stochastic term. The sum over one year of the seasonal effects, apart from the disturbance, is constrained to be zero. However, different seasonal specifications can be chosen (Harvey, 1989: 41 44). Provided that the explanatory variables, namely prices and total expenditure, are (weakly) exogenous, the structural model is completed by allowing the coefficients ijt and it to follow a random walk process as in Leybourne (1993). Other specifications, e.g. the autoregressive structure chosen by Chavas (1983), are also possible. 3 All the normally distributed disturbance terms are assumed to be uncorrelated with each other. 3. Estimation The TVC/AIDS can be expressed in the state-space form, which allows the derivation of the log-likelihood function through the Kalman filter, and the application of a maximisation procedure such as the EM (expectation þ maximisation) algorithm (Dempster et al., 1977). Following Shumway and Stoffer (1982), we provide here a multivariate generalisation for the estimation of the seemingly unrelated time series equations (SUTSE) of the TVC/AIDS. This allows us to extend the results by Leybourne (1993) to consider crossequation restrictions on the time-varying coefficients. The state-space form of the system is given by defining a measurement equation and a transition equation as follows: w t ¼ Z t a t þ e M t ð4þ a t ¼ Ta t ÿ 1 þ e T t ð5þ where the n 1 vector w t contains the expenditure shares, the m 1 state vector a t includes the m unknown parameters of system (1) and the n m matrix Z t contains the exogenous variables and other fixed values, so that ð2þ 3 The gain in generality by assuming an AR(1) structure is likely to be offset by a number of problems that the random walk specification overcomes.

5 Time-varying coefficients in the Almost Ideal Demand System 245 (1) is equivalent to (4), apart from the stochastic specification of the timevarying parameters. The stochastic transition patterns for the parameters are defined in the transition equation (5), which represents the relationship between the state vector a t and its lagged values, through the m m transition matrix T t, whose values are known. 4 The stochastic specification of the model is completed by the disturbance vectors e M it and e T it, each with mean zero and with covariance matrices equal to H t and Q t, respectively. The explicit multivariate state-space form for the TVC/AIDS is reported in Appendix A. Some further assumptions can considerably reduce the computational burden. Thus, we set H and Q to be time-independent and adopt a diagonal structure for Q, which implies that the errors of the transition equation are independent. Once a model is expressed in the state-space form, the Kalman filter can be applied. Essentially, it is a recursive procedure for computing the optimal estimates of the state vector at time t using all available information at time t, once some acceptable priors for the initial state vector and covariance matrix have been defined. The other procedure necessary for estimating (1) is the Kalman smoother. The Kalman smoother is a backward procedure, which starts from the state vectors computed through the Kalman filter and produces smoothed estimates. Furthermore, the Kalman filter allows us to derive the log-likelihood function as a function of the unknown parameters in the system and of the other parameters appearing in the state-space form, namely the error covariance matrices H and Q. The representations of the Kalman filter and smoother and of the log-likelihood function are reported in Appendix B. Maximum likelihood estimates can now be obtained using the EM algorithm, whose application to the estimation of stochastic coefficient models is illustrated by Shumway and Stoffer (1982) and Watson and Engle (1983). The EM algorithm is an iterative maximisation procedure that starts with the definition of the initial values for the state vector, for its covariance matrix and for H and Q. The following steps are then repeated iteratively: (i) Obtain filtered estimates of the state matrix (and its covariance matrix) through the Kalman filter. (ii) Starting from the output of step (i), obtain smoothed estimates for all time periods (including the initial values at time zero) through the Kalman smoother. (iii) Obtain new estimate of H and Q by maximising the log-likelihood function conditional to the smoothed estimates. (iv) Given the estimates of the initial state matrix, H and Q, repeat steps (i) (iii) until convergence is achieved, i.e. when the log-likelihood function (computed at step (iii)) and the parameter estimates are stable. 4 In the case presented here, there is no need to allow T to be time-varying, as we assume that the transition pattern is the same over the sample period.

6 246 Mario Mazzocchi The EM algorithm has the desirable property that each step always increases the likelihood and convergence is guaranteed (Wu, 1983). On the other hand, the limitation of the EM algorithm is that it may stop at some local maximum. This is not a trivial issue and the problem of choosing appropriate initial values, or opting for a diffuse prior, becomes crucial. 5 The fact that the EM algorithm might reach a local rather than a global optimum is another point in favour of the multivariate setting of the TVC/AIDS system, despite the additional computational effort required. In fact, under the multivariate approach, a joint likelihood function is maximised, whereas in the univariate case a likelihood function for each equation is necessary. Thus, it is likely that the joint likelihood procedure performs better than the single likelihood functions that are maximised with respect to a smaller number of parameters. 4. Imposing and testing theoretical constraints As for its static version, the TVC/AIDS described in (1) is expected to satisfy in each time-period the theoretical constraints of adding-up, homogeneity and symmetry: (adding-up) (homogeneity) X n k ¼ 1 X n j ¼ 1 kt ¼ 1 X n k ¼ 1 kt ¼ 0 X n k ¼ 1 kjt ¼ 0 X n k ¼ 1 kt ¼ 0 8t; j ð6þ kjt ¼ 0 8k; t ð7þ (symmetry) ijt ¼ jit 8i; j; t: ð8þ The fourth constraint, negativity, implies the negative semi-definiteness of the Slutsky matrix, computed in each time-period. 6 Doran and Rambaldi (1997) suggest introducing the constraints on time-varying coefficients in the measurement equation of the state-space model. Instead, we suggest a modification of both the measurement and transition matrices leading to a reduction in the computational burden. Like the linearised AIDS model, the TVC/AIDS specification allows for adding-up. This ensures that maximum likelihood estimates automatically satisfy adding-up (see Edgerton, 1995), but the maximisation procedure is 5 Starting the algorithm with a diffuse prior means setting to zero the state vector and to a large but finite number the covariance matrix. There is a rich literature on this issue, but Koopman (1997) concludes that the large covariance approximation is unsatisfactory and our experience confirms such results. Koopman suggests alternative specifications of the Kalman filter. However, in our situation a reliable prior is given by the constant coefficient estimates of the AIDS system. We therefore use the SUR estimates from a constant coefficient AIDS to start the algorithm. 6 As discussed in Moro and Sckokai (2000), the negativity condition can be imposed locally through a reparameterisation of the system equations. However, this would add further parameters to the TVC/AIDS, and local negativity has little meaning when we are interested in the behaviour of the parameters over the entire sample.

7 Time-varying coefficients in the Almost Ideal Demand System 247 affected by the singularity of the residual covariance matrix. The standard procedure to avoid the singularity problem is to impose adding-up and drop one equation from the system, as maximum likelihood estimates are invariant to the choice of the equation to be deleted from the system (see Barten, 1969; Bewley, 1986; McLaren, 1990). However, as discussed in the previous section, the EM optimisation algorithm only guarantees a local maximum of the likelihood function, conditional on the choice of the starting values. Hence, if the algorithm stops before the global maximum is reached, different estimates will be obtained according to which equation is dropped. Thus, it is always advisable, when running the EM algorithm, to check that the estimates are invariant to this choice. By imposing the homogeneity restriction, we can modify (1) as follows: w it ¼ it þ it þ Xn ÿ 1 ijt log j ¼ 1 pjt p nt þ it log xt P t þ u it : The system in (9) can be rewritten in the state-space form, as discussed in Appendix B, with a reduced and modified measurement matrix (one column is deleted and price ratios are used). Considering a system of n goods where one equation is dropped to avoid singularity, the new transition matrix has n 2 þ 12n ÿ 13 rows and columns (n ÿ 1 rows and n ÿ 1 columns are deleted), as the state vector loses one parameter per equation. The symmetry restriction can also be imposed by modifying the measurement and transition matrices. In a system of n goods (with no homogeneity restrictions and one equation dropped), we proceed by dropping from the state vector the number of coefficients equal to the binomial coefficient d ¼ð nÿ1, i.e. the parameters subject to the symmetry constraint. The measurement and transition matrices are adjusted accordingly. To illustrate this, let us consider for simplicity a homogeneity- and symmetryconstrained three-good system, where the last equation has been dropped to avoid singularity. The symmetry constraint requires that 12t ¼ 21t. Hence, we drop the element 21t from the state vector and rewrite the measurement equation as follows: " # w 1t w 2t ðnÿ1þ! 2 Þ¼2!ðnÿ3Þ! 2 p1t 1 0 log log p 3t ¼ log p2t p 3t p1t p 3t xt log P t ð9þ p2t xt log log p 3t P t

8 248 Mario Mazzocchi 2 1t 1t 1t 1t... 2t 2t 2t 2t t 12t 1;t ÿ 1 1;t ÿ 10 22t 2;t ÿ 1 2;t ÿ 10 3 " # þ em 1t e M : ð10þ 2t 7 5 From (10) it follows that the transition matrix loses the row and column that previously corresponded to the element 21t. Again, if homogeneity and symmetry are imposed together, the resulting transition matrix has n 2 þ 12n ÿ 13 ÿð nÿ1 2 Þ rows and columns. To test the validity of the homogeneity and symmetry theoretical restrictions over the whole sample, Rao s systemwise F-statistic (Shukur, 2002) may be adapted as follows: " # qs ÿ r det 1=s HR F RAO ¼ ÿ 1 ð11þ mg det H U where m is the number of restrictions per equation, g ¼ n ÿ 1 is the number of equations in the unrestricted system (one equation has been dropped), H R and H U represent respectively the restricted and unrestricted covariance matrices from the measurement equation, and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3g þ m þ 27 ðmgþ 2 ÿ 4 q ¼ T ÿ ; s ¼ 2 m 2 þ g 2 ; r ¼ mg ÿ 5 2 ÿ 1 where T is the total number of observations. Under the null hypothesis, the statistic F RAO is distributed as F ðmg; qs ÿ rþ. 5. An empirical appraisal on Italian data Having defined the multivariate TVC/AIDS model and extended it to account for theoretical constraints, the next step is to provide an empirical application

9 Time-varying coefficients in the Almost Ideal Demand System 249 to food consumption data. The model performance is assessed conditional on a specific data set by comparing the results with those obtained from a dynamic AIDS with trending intercept. The dynamic version of the linear approximate AIDS model follows the partial-adjustment form suggested by Alessie and Kapteyn (1991) and adopted by several workers (Edgerton et al., 1996; Rickertsen, 1996; Shukur, 2002). A vector of lagged expenditure shares is included in each equation: w it ¼ i þ X12 is s ts þ Xn ij w j;t ÿ 1 þ i t þ Xn xt ij log p jt þ i log þ u it s ¼ 1 j ¼ 1 j ¼ 1 P t i ¼ 1;...; n: ð12þ In system (12), the intercept of the AIDS model i is augmented to account for a lagged share effect, seasonal variation and a linear time trend t. To keep notation similar to (1), the monthly shift in the intercept is given by is,with s ¼ 1;...; 12 and the dummy s ts is equal to one when the period t falls in the sth month and zero elsewhere. The sum of the seasonal effects is constrained to zero over a 1-year period, i.e. P 12 s ¼ 1 is ¼ 0. All other variables have the same meaning as in (1). The additional constraint P n i ¼ 1 i ¼ 0 ensures the adding-up condition, and the condition P n j ¼ 1 ij ¼ 0 8i is necessary for identification (Edgerton et al., 1996). 7 Homogeneity and symmetry constraints can also be imposed and tested on the system in (12). Our TVC/AIDS and the dynamic AIDS show some further differences besides the time-varying coefficients of the former. There is no vector of lagged expenditure shares in the TVC/AIDS specification of equation (1), which stems from the static AIDS representation, and the linear trend that explicitly appears in (12) is instead embodied in the intercept it as a result of the drift it described in (2). 8 The criteria adopted for comparing the performance of the two models are the following: (i) adherence to economic theoretical restrictions; (ii) statistical assessment (model diagnostics); (iii) forecasting performance; (iv) economic meaningfulness. The data come from the Italian Household Expenditure Survey by ISTAT (the Italian National Institute of Statistics). Individual household expenditure data for the period January 1986 December 1999 were aggregated on a monthly basis through a weighted average, 9 leading to a sample of 168 observations. Nine aggregated goods were considered: bread and cereals, meat, fish, fats and oils, dairy, fruit and vegetables, other foods, beverages 7 Despite ensuring identification, this normalisation is rather arbitrary as discussed by Rickertsen et al. (2003). 8 As pointed out by an anonymous referee, these differences make the comparison less straightforward, as the two models are not nested. On the other hand, the TVC/AIDS presented here has a stricter relation with those existing in the literature. 9 The weights for aggregation across households were computed as in Deaton and Muellbauer (1980).

10 250 Mario Mazzocchi Table 1. Model selection and theoretical constraints Model specification Model selection Log-likelihood AIC F-Rao test TVC/AIDS Unconstrained ÿ13.28 Homogeneity ÿ Symmetry ÿ Homogeneity and symmetry ÿ Dynamic AIDS with trending intercept Unconstrained ÿ11.11 Homogeneity ÿ Symmetry ÿ Homogeneity and symmetry ÿ Significant at the 1 per cent level. Significant at the 5 per cent level. and non-foods. ISTAT consumer price indices were employed to build the price time series for each group of foods, and the non-food consumer price index was used as a proxy for the non-food group. The prices and total expenditure series were scaled to reduce the bias of the Stone index approximation and the nonfood equation was dropped from both models to avoid singularity. 10 Maximum likelihood estimates for the dynamic AIDS were obtained using the iterated seemingly unrelated regression (SUR) method. The EM algorithm for estimating the TVC/AIDS in its multivariate form was implemented in SAS through the SAS/IML matrix programming language. Out-ofsample forecast evaluation was based upon data for the years 2000 and Theoretical restrictions A first assessment of the performance of the two competing models could be given by analysing their ability to accommodate the constraints required by economic theory. The symmetry and homogeneity restrictions are tested separately and jointly using the Rao F-statistic (11) for both the TVC/ AIDS and the dynamic AIDS with trending intercept. Table 1 displays the likelihood value for each of the eight estimated models and the Akaike Information Criterion (AIC). The theoretical restrictions are rejected for all models. This is not an uncommon finding in applied demand analysis, but the results show that taking into account the coefficient constraints in each time period rather than at the normalisation point does not lead to a better adherence to theoretical 10 Parameter estimates for the dropped equation were retrieved through the adding-up constraint. Alternative systems were estimated by dropping other equations and final estimates were substantially equivalent.

11 Time-varying coefficients in the Almost Ideal Demand System 251 Table 2. Test for time-varying coefficients ( statistic) in the dynamic AIDS with trending intercept Equation Unrestricted Restricted Bread and cereals Meat Fish Dairy and eggs Fats and oils Fruit and vegetables Beverages Other foods Significant at the 1 per cent level. Significant at the 5 per cent level. restrictions. This evidence might be related to the aggregation problem, as restrictions that are true at the level of individuals do not necessarily hold for aggregated data Statistical performance Given the data set, a statistical appraisal of the two models is provided here according to two main issues: (i) coefficient stability in the dynamic AIDS versus random walk behaviour; (ii) other residual diagnostics from both models. The stability test for the dynamic AIDS was based on a test-statistic () first suggested by Nyblom (1989) and adapted to the AIDS case by Leybourne (1993). The null hypothesis is the constancy of all coefficients in the dynamic AIDS with trending intercept against the alternative that at least one coefficient follows a random walk. Table 2 shows the values of the statistic 11 for both the unrestricted and theory-restricted dynamic AIDS with trending intercept. It is worth noting some of the results. For the unconstrained model there is no clear evidence of random walk coefficients, apart from in the fruit and vegetables equation. However, if homogeneity and symmetry are imposed, the test shows how this induces instability in one or more coefficients, which could be interpreted as a misspecification problem. This is not sufficient to indicate that the TVC/AIDS specification should be preferred when constraints are imposed, but highlights some problems related to the imposition of theoretical restrictions when they are statistically rejected. Table 3 shows further diagnostics at the single equation level. The traditional R 2 statistic is not suitable for time-series models, as any model able 11 The 1 per cent (5 per cent) critical values of the statistics are approximately 7.7 (6.5) for the unconstrained equations and 7.5 (6.3) for the constrained one. We are grateful to Jukka Nyblom for suggesting a useful method to calculate the approximate critical values.

12 252 Mario Mazzocchi Table 3. Equation by equation diagnostics Dynamic AIDS with trend TVC/AIDS Equation R 2 DW Q a Res. Trend coeff. R 2 R 2 D DW Q a p.e.v. Res. Unconstrained model Bread and cereals ÿ Meat ÿ Fish Oils and fats ÿ Dairy and eggs ÿ Fruit and vegetables ÿ ÿ Beverages ÿ ÿ Other foods Constrained model Bread and cereals ÿ Meat ÿ ÿ Fish ÿ Oils and fats ÿ Dairy ÿ Fruit and vegetables ÿ ÿ Beverages ÿ Other foods ÿ Significant at the 1 per cent level. Significant at the 5 per cent level. a The number of lags for the Ljung-Box Q statistic was set as ln T, where T ¼ 168 is the number of observations. The Q statistic is asymptotically distributed as 2 (5). DW, Durbin Watson statistic; Res, standard error of residuals; p.e.v., prediction error variance.

13 Time-varying coefficients in the Almost Ideal Demand System 253 to pick up a time trend will return a value close to unity (Harvey, 1989: ). The most commonly employed measure to assess goodness-of-fit of time-varying parameters models is the prediction error variance (p.e.v.), i.e. the variance of the one-step-ahead prediction errors within the sample. This measure is similar to the residual standard errors in constant coefficient models. Furthermore, the goodness-of-fit can be assessed with respect to the performance of a simple random-walk-plus-drift model through the R 2 D statistic (Harvey, 1989: 268). A positive value of R 2 D indicates a better fit than the simpler model. Generally, the TVC/AIDS specification improves the goodness-of-fit statistics in both its unconstrained and constrained version. However, the values of the R 2 D statistic indicate that the full stochastic coefficient specification is almost unnecessary for the constrained system, as a simple randomwalk-plus-drift model would accommodate the data better than the TVC/ AIDS model. Again, this could be interpreted as evidence that the imposition of rejected theoretical constraints leads to a serial correlation problem that is captured by the stochastic specification of the TVC/AIDS. Such explanation is corroborated by evidence of serial correlation in the residuals. On the other hand, the unconstrained TVC/AIDS shows an improvement in terms of residuals diagnostics, especially for the bread and cereals, fish, dairy and eggs and other foods equations. In spite of this, there is insufficient evidence to claim that the TVC/AIDS as a whole outperforms the dynamic AIDS in terms of specification diagnostics Forecasting performance The forecasting performance of the competing specifications is assessed by outof-sample monthly projections computed from January 2000 to December Table 4 reports Theil s U statistics. On average, in the unconstrained version, the TVC/AIDS is the best forecaster in the short term (3 12 months), whereas over a 2-year period the two competitors show an identical performance. It is interesting to notice that the imposition of theoretical constraints appreciably improves the forecasts of the dynamic AIDS, confirming at least for this application the results obtained by Kastens and Brester (1996). The same does not hold true for the constrained TVC/AIDS, which outperforms its dynamic counterpart only in the shortest time span (3 months). In the longer term (2 years), the constrained dynamic AIDS shows forecasts that are at least as good as those from the TVC/AIDS model. This result may be also explained by the larger number of parameters in the TVC/AIDS. Overfitted models usually lead to good predictions in the short run, as a result of a good fit within the estimation sample and a slowly changing aggregate demand in the first forecast periods. On the other hand, overfitting can lead to an increased prediction error in the longer run We are grateful to an anonymous referee for pointing this out.

14 254 Mario Mazzocchi Table 4. Forecasting performances (Theil s U statistic) Forecast period: 3 months 6 months 12 months 24 months AIDS specification: TVC Dynamic TVC Dynamic TVC Dynamic TVC Dynamic Unconstrained model Bread and cereals Meat Fish Oils and fats Dairy and eggs Fruit and vegetables Beverages Other foods Overall system Constrained model Bread and cereals Meat Fish Oils and fats Dairy and eggs Fruit and vegetables Beverages Other foods Overall system The better forecasting results for the unconstrained TVC/AIDS in the short term are mainly due to good performance in the meat, fish and beverage equations. This outcome is further investigated through the evaluation of the economic meaningfulness of the results from the TVC/AIDS model Economic meaningfulness The parameter estimates for the unconstrained TVC/AIDS are reported in Appendix C. The values of parameters at the beginning of the sample (January 1987) 13 and at the end of the sample (December 1999) are shown in Table C1. Figures C1 C5 show the patterns of the time-varying coefficients. Both the TVC/AIDS model and the dynamic AIDS (see the trend coefficients in Table 3) reflect the slow but clear process of substitution between meat and fish over the considered period. The TVC/AIDS supplies some further information about the evolution of this process. The local-linear trend specification allows the identification of both the main trend (intercept 13 The first 12 months are necessary to obtain proper estimates of seasonality.

15 Time-varying coefficients in the Almost Ideal Demand System 255 component, Figure C1) and any recent pattern that might be emerging (trend component, Figure C2). A trend reversal situation is identified when the trend component switches from negative to positive in Figure C2. The meat equation shows a clearly negative trend until the end of 1996, whereas a positive trend (although rather unstable) is observed in the following years. The trend component of the fish equation is strictly positive across the whole sample, but also tends to become smaller from 1996 onwards. The ability of the TVC/AIDS to capture such relevant (but not necessarily long-lived) phenomena could explain the improved short-term forecasting performance (3 6 months) for those equations. One possible economic interpretation might be linked to the recovery in meat consumption after the negative peak of the BSE crisis at the beginning of Interestingly, in the longer term (1 2 years) the dynamic AIDS with constant trend provides better forecasts. As one might expect, the TVC/AIDS specification is more flexible in adjusting to significant shocks, but long-term shifts in preference are adequately captured by the simpler dynamic AIDS. However, there is a risk that the TVC/AIDS might capture noise in the data, as demonstrated by the results from the beverage equation. The confounding factor is the restructuring of the Italian household survey in Although the questionnaire remained unchanged for most of the foods, the beverages section was extended, leading to a significant increase in the assessment of aggregate beverage expenditure. This is clearly reflected by the sudden increase in the local linear trend (Figure C1), but also for the other time-varying coefficients where adjustments in the patterns emerge. On the one hand, this prevents us from giving a proper economic interpretation to the coefficients affected by the noise. On the other hand, this drastically improves the forecasting performance of the TVC/AIDS beverage equation compared with the dynamic AIDS, over the whole sample. So far, we have considered the advantages of having an intercept that allows for nonlinear trends or trend reversal. Some further attention should be given to the other behavioural parameters, the price and expenditure coefficients and the seasonal terms. As shown in Table C1, not all parameters show a noteworthy variability in terms of coefficient of variation. The meat ownprice coefficient exhibits a considerable variation over time, whereas for fruit and vegetables a significant variability is reflected in a smoothly trending coefficient. These values can be processed as time-varying elasticities, with the latter also subject to the influence of time-variation of shares (Figure 1). 14 Paradoxically, the time-varying coefficient resulted in a rather stable ownprice elasticity for the meat equation, compensating for the negative trend in the expenditure shares. However, the fruit and vegetables equation shows a trend towards a reduction in own-price elasticity. One possible interpretation is that the preference shift towards a more balanced diet has 14 This leads to the seasonal pattern in the elasticities. An alternative computation of the price and expenditure elasticities with seasonally adjusted shares, omitted here for brevity, highlighted the same time trends.

16 256 Mario Mazzocchi Figure 1. Own-price elasticities from estimates of the unconstrained TVC/AIDS model.

17 Time-varying coefficients in the Almost Ideal Demand System 257 Figure 2. Expenditure elasticities from estimates of the unconstrained TVC/AIDS model.

18 258 Mario Mazzocchi led the consumer to be less price-elastic for products that are perceived as healthy. To support such an evaluation, one can consider another parameter that displays some variability over time, namely the expenditure coefficient for the fats and oils equation, and the corresponding elasticity. As shown in Figure 2, there is a clear trend towards a decreasing elasticity in the expenditure elasticity for this product. At the end of the sample (December 1999), the expenditure elasticity for fats and oils was the lowest (0.56) of all considered food groups, suggesting that an increase in income is linked to a diminishing share of fats and oils in the diet. Considering the seasonal patterns for the same group, a significant reduction is also shown in the December (Christmas) peaks. This result could also reflect a change in taste as a result of health concerns, as tradition is very gradually giving way to a fat-reduced, healthier diet. Table 5 compares the Marshallian elasticities obtained from the TVC/AIDS with those of the dynamic AIDS with trending intercept. The comparisons replicate those of the forecasting performance. Elasticity estimates from the Table 5. Marshallian elasticities at the mean point (standard errors in parentheses) Unconstrained model Constrained model Price Expenditure Price Expenditure TVC/AIDS Bread and cereals ÿ0.62 (0.13) 0.70 (0.05) ÿ0.30 (0.59) 0.67 (0.05) Meat ÿ0.93 (0.17) 0.70 (0.05) ÿ1.06 (1.39) 0.77 (0.25) Fish ÿ0.53 (0.20) 0.64 (0.08) ÿ0.88 (0.94) 0.60 (0.19) Oils and fats ÿ2.40 (0.22) 0.69 (0.10) ÿ1.63 (1.63) 0.67 (0.11) Dairy ÿ0.64 (0.08) 0.69 (0.04) ÿ0.90 (0.42) 0.63 (0.11) Fruit and vegetables ÿ0.77 (0.33) 0.76 (0.12) ÿ0.72 (0.56) 0.83 (0.22) Beverages ÿ0.25 (0.42) 0.85 (0.22) ÿ0.26 (2.65) 0.83 (0.65) Other foods ÿ0.85 (0.17) 0.89 (0.10) ÿ0.54 (0.95) 0.51 (0.18) Non-foods ÿ ÿ Dynamic AIDS with trending intercept Bread and cereals ÿ0.38 (0.35) 0.73 (0.04) ÿ0.56 (0.24) 0.63 (0.04) Meat ÿ1.09 (0.40) 0.69 (0.04) ÿ1.11 (0.23) 0.59 (0.03) Fish ÿ0.52 (0.34) 0.60 (0.07) ÿ0.77 (0.22) 0.70 (0.06) Oils and fats ÿ2.57 (0.53) 0.81 (0.07) ÿ1.22 (0.44) 0.65 (0.07) Dairy ÿ0.73 (0.10) 0.70 (0.03) ÿ0.97 (0.06) 0.65 (0.03) Fruit and vegetables ÿ0.99 (0.11) 0.76 (0.05) ÿ0.82 (0.08) 0.72 (0.04) Beverages ÿ0.21 (0.37) 0.81 (0.10) ÿ0.50 (0.26) 0.80 (0.09) Other foods ÿ0.85 (0.28) 0.89 (0.06) ÿ0.91 (0.21) 0.81 (0.06) Non-foods ÿ ÿ

19 Time-varying coefficients in the Almost Ideal Demand System 259 unconstrained TVC/AIDS are generally more precise, whereas the imposition of theoretical restrictions leads to lower standard errors for the dynamic AIDS. Despite the provision of some meaningful additional information from the TVC/AIDS with respect to the simple examination of the trend coefficient in the dynamic AIDS, it is still questionable as to whether it is worth estimating the more complex model. Furthermore, a system incorporating demographic variables would probably be more informative, giving insights into the sources of structural change. 6. Concluding remarks This paper presents a generalisation of the structural time series AIDS allowing for time-varying parameters in the presence of cross-equation constraints. Conditional on a specific data set, we have carried out an empirical appraisal of the TVC/AIDS, using a dynamic AIDS with trending intercept as the baseline model. The assessment has been based on multiple criteria. In general, we have shown that the simpler dynamic AIDS performs well and there is no evidence for superiority of the TVC/AIDS. The comparison is exclusively based on the dynamic AIDS with trending intercept, but it might well be that other alternatives such as the AIDS with demographic variables or the switching AIDS with linear trends in all coefficients perform even better. The advantages of the TVC/AIDS with respect to the alternative models are that: (i) it does not require variables other than prices and income; (ii) there is no need to make assumptions about the timing and duration of structural change transition, or to search for break-points empirically; (iii) it allows for trend reversal and nonlinear trends. Empirical results also show a better forecasting performance of the TVC/AIDS in the short term (3 6 months), especially when specific sources of instability are identified. As discussed, these might not be economically related and the time-varying coefficients might actually pick up noise in the data, such as the partial restructuring of the Italian Household Budget Survey in 1997, which affected the measurement of beverages expenditure. Nevertheless, when the data set is affected by some break inside the sample, the random walk specification of the coefficients allows the model to adhere to the changing figures better and provide improved forecasts with respect to the constant coefficient models. The symmetry and homogeneity restrictions were rejected for both the TVC/ AIDS and the dynamic AIDS model with a trending intercept. Statistical diagnostics showed that imposing theoretical constraints in such a situation might lead to specification problems and could result in the improper detection of coefficient instability. However, the restricted dynamic AIDS turned out to be a better forecaster than its unconstrained counterpart in our case study, confirming previous empirical findings. Instead, the imposition of theoretical constraints on the time-varying coefficients led to a worsened performance of the TVC/AIDS according to all our criteria. This could be related to the

20 260 Mario Mazzocchi imposition of the theoretical restrictions on aggregated data and to the inability of the TVC/AIDS to account for adjustment costs, incorrect expectations and misinterpreted real prices, whereas the improvement in the dynamic AIDS might be driven by the consideration of habit persistence as suggested by Anderson and Blundell (1983). It remains an open question as to whether any of the above results can be generalised to other data, but this application shows what the advantages and limitations of a time-varying coefficient specification could be. Further research might also focus on testing the random walk specification on a parameter-by-parameter basis to reach a good balance between the advantages of the TVC/AIDS and the need for parsimony. Alternative stochastic specification for the coefficients could also be tested and the time-varying generalisation could be implemented on the dynamic AIDS. Acknowledgements This paper has substantially benefited from the constructive comments from three anonymous referees and the editor. The author is also grateful to Garth Holloway, Federico Perali and Richard Tiffin for their comments on an earlier draft, and to all his colleagues at the University of Reading for the stimulating environment they create. The usual disclaimer applies. References Alessie, R. and Kapteyn, A. (1991). Habit formation, intertemporal preferences and demographic effects in the Almost Ideal Demand System. Economic Journal 101: Al-Kahtani, S. H. and Sofian, B. E. (1995). Estimating preference change in meat demand in Saudi Arabia. Agricultural Economics 12: Alston, J. M. and Chalfant, J. A. (1991). Unstable models from incorrect forms. American Journal of Agricultural Economics 73: Anderson, G. and Blundell, R. (1983). Testing restrictions in a flexible dynamic demand system: an application to consumers expenditure in Canada. Review of Economic Studies 50: Barten, A. P. (1969). Maximum likelihood estimation of a system of demand equations. European Economic Review 1: Ben Kaabia, M., Angulo, A. M. and Gil, J. M. (2001). Health information and the demand for meat in Spain. European Review of Agricultural Economics 28: Bewley, R. (1986). Allocation Models: Specification, Estimation and Applications. Cambridge, MA: Ballinger. Burton, M. and Young, T. (1992). The structure of changing tastes for meat and fish in Great Britain. European Review of Agricultural Economics 19: Burton, M. and Young, T. (1996). The impact of BSE on the demand for beef and other meats in Great Britain. Applied Economics 28: Chavas, J. P. (1983). A study of structural change: the demand for meat. American Journal of Agricultural Economics 65: Deaton, A. and Muellbauer, J. (1980). An Almost Ideal Demand System. American Economic Review 70:

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