Sarah Gelper

Aurélie Lemmens

Christophe Croux

K U Leuven

Consumer Sentiment and Consumer Spending: Decomposing the Granger Causal Relationship in the Time Domain

Keywords: Cointegration, Consumer Sentiment, Granger Causality, VEC models Cointegration, Consumer Sentiment, Granger Causality, VEC models JEL classification: C32; E21

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In this article, we investigate the extent to which the US Consumer Sentiment Index (CSI) offers relevant and timely insights into future real consumption levels in the United States. The basic idea behind this research question is that if US consumers feel confident about the actual and future economic and financial situation, they would be more willing to increase their consumption. In contrast, pessimistic consumers could theoretically save more money and delay their spending further in time. This problem has already been investigated by several authors and the results were rather divided (see also [START_REF] Ludvigson | Consumer confidence and consumer spending[END_REF]Vuchelen, 2004). On the one hand, [START_REF] Desroches | The usefulness of consumer confidence indexes in the United States, Bank of Canada working paper[END_REF] and [START_REF] Roberts | What do sentiment surveys measure? Research Discussion Paper[END_REF] found that sentiment indicators contain little information to forecast consumption, even if [START_REF] Desroches | The usefulness of consumer confidence indexes in the United States, Bank of Canada working paper[END_REF] stressed the importance of the consumer sentiment in times of high economic and political uncertainty. On the other hand, [START_REF] Batchelor | Improving macro-economic forecasts, the role of consumer confidence[END_REF], [START_REF] Carroll | Does consumer sentiment forecast household spending? If so, why?[END_REF], [START_REF] Easaw | Evaluating consumer sentiments as predictors of UK household consumption behavior, are they accurate and useful[END_REF], [START_REF] Eppright | Aggregate consumer expectation indexes as indicators of future consumer expenditures[END_REF], [START_REF] Huth | The indexes of consumer sentiment and confidence -leading or misleading guides to future buyer behavior[END_REF], [START_REF] Kumar | Aggregate and disaggregate sector forecasting using consumer confidence measures[END_REF] and [START_REF] Souleles | Expectations, heterogeneous forecast errors, and consumption: micro evidence from the Michigan Consumer Sentiment Surveys[END_REF] all found the consumer sentiment index to be a useful leading indicator in predicting aggregate consumer expenditures. In this paper we want to study the term structure of the consumer sentiment index. Nobody, up to our knowledge, has looked at the time lag at which the consumer sentiment helps to explain consumption. Is the predictive power of the index mainly present at the very short run, or is it pertinent for larger forecasting horizons?

The research question to which extent the US consumer sentiment index offers relevant and timely insights into future real consumption levels in the United States, can be translated into a Granger causality framework. One variable is said to Granger cause the other if it helps to make a more accurate forecast of the other variable than had we only used the past of the latter as predictor. Note that Granger causality between two variables can not be interpreted as a real causal relationship but merely shows that one variable is leading the other one. The first research question becomes now "Does the US consumer sentiment index Granger cause US real consumption, and/or any of its components?". Most previous studies used Vector Autoregession (VAR) models to investigate this question. For example Utaka (2003) models Japanese consumer confidence and GDP as a VAR and finds that the confidence indicator Granger causes GDP, at least when working with monthly or quarterly data. However, US 1 F o r P e e r R e v i e w consumer confidence and consumption turn out to be cointegrated (as was also found by [START_REF] Throop | Consumer sentiment: its causes and effects[END_REF]. Consequently, a Vector Error Correcting model is more appropriate.

In a second part of the analysis, we compute a measure of Granger causality which allows to separate the Granger causal effect at different time lags. Indeed, when making the prediction for next month consumption level, not only the most recent value of the consumer index is relevant, but also past values. The importance of the consumer sentiment index is then measured at different lags. As such, the Granger causal relationship can be decomposed over the time domain.

In the third and last part of this empirical study, it will be shown that the information in present and past values of the CSI is not only of interest for making predictions at very short time horizons, but still has predictive power for larger forecasting horizons. Forecasts are again made with an Error Correcting model, taking the long run relationship between CSI and consumption into account.

Real consumption data are decomposed in three important parts, i.e. durables, nondurables and service consumption. As the consumer's decision on purchasing durables, nondurables and services are led by different motives, one may expect that the consumer sentiment index does not equally affect all consumption components. For instance, durable goods' consumption is known to be more easily delayed in time than non-durables'. Therefore, the analysis is made for each of the consumption components separately. The heterogeneity of the Granger causal effect on different components of consumer sentiment spending was also studied by [START_REF] Throop | Consumer sentiment: its causes and effects[END_REF] and [START_REF] Ludvigson | Consumer confidence and consumer spending[END_REF], but they considered a different decomposition of personal consumption.

The remainder of this paper is organized as follows. Section two covers the methodological aspect of testing for Granger causality and its decomposition over the time domain. The decomposition of the Granger causality measure for a vector autoregressive model is defined in [START_REF] Gouriéroux | Time Series and Dynamic Models[END_REF]. We extend this approach to cointegrated time series using nested Vector Error Correcting models. In the third section, the data are described. representation of the results is provided. Furthermore, we examine the forecasting power of the consumer sentiment for consumers' spending at different forecast horizons. Finally, section five concludes.

Methodology

This section describes the methodology which will be used to investigate the Granger causal relationship between U.S. consumer sentiment and real US consumer spending. Recall that time series X t Granger causes a time series Y t if the past of X t helps to forecast the future of Y t after controlling for the past of Y t .

The Vector AutoRegression (VAR) framework allows to test for Granger causality and explicitly includes the possibility of a feedback causality. For X t and Y t two stationary time series, a bivariate VAR model of order M is given by

X t = α 1 + M k=1 β 1,k X t-k + M k=1 γ 1,k Y t-k + ε x t , (2.1)
Y t = α 2 + M k=1 β 2,k Y t-k + M k=1 γ 2,k X t-k + ε y t , (2.2)
where the error terms ε x t and ε y t are assumed to be Gaussian white noise with zero mean and a constant covariance matrix. After estimating equations (2.1) and (2.2), several tests for Granger causality can be conducted. The series X t Granger causes Y t if the γ 2,k coefficients are jointly significant, while Y t Granger causes X t if the γ 1,k 's are jointly significant. If both the γ 1,k coefficients and the γ 2,k coefficients are jointly significant, there is evidence for a feedback relationship between X t and Y t .

Decomposition in the time domain

As previously mentioned, we are especially interested in the decomposition of the Granger causality of X t for Y t over different time lags. We follow a procedure proposed by [START_REF] Gouriéroux | Time Series and Dynamic Models[END_REF]

Y t = α + M k=1 β k Y t-k + M k=j+1 γ k X t-k + ε j+1 t , (2.3)
and

Y t = α + M k=1 β k Y t-k + M k=j γ k X t-k + ε j t , (2.4)
which both can be estimated by maximum likelihood for j = 1, . . . , M . Since the only difference between (2.3) and (2.4) is an additional regressor in (2.4), namely X t-j , model

(2.3) is nested in model (2.4). As a consequence, the estimated variance of the error terms in equation (2.4), σ2 j , will always be smaller than or equal to the error term variance for equation (2.3), σ2

j+1 . If the difference is large, the variable X t-j has significant power when forecasting Y t and controlling for its previous values. The measure of causality at lag j is then defined as

C j = ln σ2 j+1 σ2 j .
(2.5)

The causality measure at lag j describes the forecasting power of X t-j for Y t , after controlling for the past of X t-j and the past of Y t . As σ2 j+1 ≥ σ2 j , it follows that C j ≥ 0, resulting in a measure for the strength of X t-j in Granger causing Y t .

Once C j is computed, we would like to know whether it is significantly different from zero. The critical values of C j can be derived from its resemblance with the likelihood ratio (LR) test statistic. This test can be used to compare the nested models (2.3) and (2.4) and tests for the null hypothesis that γ j equals zero. If we call (2.4) the unrestricted model and (2.3) the restricted model, then the LR test statistic takes the form

LR = 2 log L(θ U ) -log L(θ R ) .
(2.6)

Here log L(θ U ) denotes the loglikelihood at the unrestricted model, with θ U the parameter vector collecting the estimate of all β k , all γ k and of the variance of the error terms. Analogously for the loglikelihood function at the restricted model, log L(θ R ). It is not difficult to show, e.g. as in Gouriéroux and Jasiak (2001, chapter 4) that

LR = T × C j ∼ χ 2 1 ,
where T is the number of observations. Hence, if C j is larger than the critical value χ 2 1,1-α /T (with χ 2 1,1-α the α-upper quantile of the chi-squared distribution with one degree of freedom) then the Granger causality measure at lag j is found to be significant at level α.

When summing up all C j , for j ranging from 1 to M , a total measure of causality is obtained as

C = M j=1 C j = ln σ2 M +1 σ2 1 .
(2.7)

It measures the total effect of all the lagged X t on the present Y t . Indeed, for j = M , equation (2.3) reduces to a regression with only lagged variables Y t on the right hand side, and we call this the empty model. On the other hand, for j = 1, we call equation (2.4) the full model since the whole past of X t is included. The total measure of causality C compares the error-variance in the empty and the full model and is again closely related to a likelihood ratio test for H 0 : γ 1 = γ 2 = . . . = γ M = 0 in the full model. In fact, we have that T × C equals the Granger-Wald test for Granger causality. Note that this causality measure C was initially proposed by [START_REF] Gouriéroux | Kullback causality measures[END_REF] on the basis of the Kullback Information Criterion.

An interesting summary measure that can be derived from the causality measures at different lags is the mean causality lag at which the lagged values of X t help to forecast Y t .

This measure is denoted by D and computed as the average of the lag orders weighted by the causality measures at each lag:

D = M j=1 j C j C .
(2.8)

The mean causality lag D indicates how long a value of the CSI remains pertinent. For example, if D is small, then only the most recent values of the CSI are important for predicting future consumption. On the other hand, if D is large, then the consumers' spending is reflected in a much longer series of past CSI values.

The Vector Error Correcting Model

The procedure proposed by [START_REF] Gouriéroux | Time Series and Dynamic Models[END_REF] for decomposing Granger causality over the time domain is attractive, but requires both series to be stationary. If X t and Y t are non-stationary, a VAR model applied on the series in differences could be taken. In our setting, however, it turns out that the consumer sentiment index is nonstationary but also cointegrated with total real consumption, as well as with each of its three components. As we know, two time series are cointegrated if there is a long run relationship between them. If we want to take this long run effect into account, an error correcting term has to be included in the model. This allows to separate the short run from the long run causality. Therefore, the Vector Error Correcting (VEC) model is used:

∆X t = α 1 + M k=1 β 1,k ∆X t-k + M k=1 γ 1,k ∆Y t-k + δ 1 U t-1 + ε x t
(2.9)

∆Y t = α 2 + M k=1 β 2,k ∆Y t-k + M k=1 γ 2,k ∆X t-k + δ 2 U t-1 + ε y t , (2.10)
where X t and Y t are integrated of order one and cointegrated time series. Equations (2.9) and (2.10) are the same as equations (2.1) and (2.2), but an error correction term, denoted by U t-1 , is added. This lagged error correction term is obtained from the Johansen procedure [START_REF] Johansen | Likelihood-based inference with cointegrated vector-autoregressive models[END_REF]. This procedure estimates by maximum likelihood the long run relationship as a cointegration equation

U t = aX t + bY t + c, (2.11)
with a = 1 by normalization. The error correcting term U t-1 gives the deviation from the long term equilibrium at t -1, and will lead to a short run adjustment in X t , Y t or both.

The coefficients δ 1 and δ 2 in (2.9) and (2.10) measure the speed of adjustment. If the series X t and Y t are cointegrated, then δ 1 and δ 2 are jointly significant, due to the representation theorem of [START_REF] Engle | Cointegration and error correction -representation, estimation, and testing[END_REF]. Since we want to measure Granger causality of X t for Y t , we focus on equation (2.10) in the sequel.

As explained in more detail in Miller and Russek (2001) In the cointegration framework, the method of [START_REF] Gouriéroux | Time Series and Dynamic Models[END_REF] for decomposing Granger causality needs to be modified slightly. For measuring the causality at lag j, we consider now the two models

∆Y t = α + M k=1 β k ∆Y t-k + M k=j+1 γ k ∆X t-k + δU t-(j+1) + ε j+1 t ,
(2.12)

and

∆Y t = α + M k=1 β k ∆Y t-k + M k=j γ k ∆X t-k + δU t-j + ε j t .
(2.13)

Because we want model (2.12) to take only X t-j-1 and past values into account, we include the deviation from the long run equilibrium in period t -(j + 1), that is U t-(j+1) . For the same reason, we include U t-j in equation (2.13). Equations (2.12) and (2.13) are both estimated by maximum likelihood. For the error correcting term, we use X t-j + bY t-j + ĉ as estimate for U t-j (and similarly for U t-(j+1) ), where b and ĉ are obtained by preliminary maximum likelihood estimation of the cointegration equation. The latter estimates are super consistent, i.e. converge at order 1/T . The estimates for the parameters in models (2.12) and (2.13) converge at the normal, slower, rate of 1/ √ T . Hence, when performing inference for the parameters γ k and δ in (2.12) and (2.13), the parameters a, b and c hidden in the definition of U t-j may be considered as constant and given by their preliminary estimates.

A formal proof of this argument is given in [START_REF] Toda | Vector Autoregression and Causality: A Theoretical Overview and Simulation Study[END_REF].

Again, we expect the estimated variance of ε j t , σ2 j , to be smaller than or equal to the estimated variance of ε j+1 t , σ2

j+1 . Herefore, it is sufficient to show that model (2.12) is nested in model (2.13). Rewrite model (2.12) as

∆Y t = α + M k=1 β k ∆Y t-k + M k=j+1 γ k ∆X t-k + γ j ∆X t-j + δU t-j + ε t .
(2.14)

The fourth and fifth term of the right hand side of this equation can be rewritten as

γ j ∆X t-j + δU t-j = γ j ∆X t-j + δ(aX t-j + bY t-j + c) -δ(aX t-j-1 + bY t-j-1 + c) + δ(aX t-j-1 + bY t-j-1 + c) = γ j ∆X t-j + δa∆X t-j + δb∆Y t-j + δ(aX t-j-1 + bY t-j-1 + c) = (γ j + δa)∆X t-j + δb∆Y t-j + δU t-j-1 . 7 F o r P e e r R e v i e

w

When plugging this back into the equation (2.14), we get

∆Y t = α + j-1 k=1 β k ∆Y t-k + (β j + δb)∆Y t-j + M k=j+1 β k ∆Y t-k + M k=j+1 γ k ∆X t-k + (γ j + δa)∆X t-j + δU t-j-1 + ε t . (2.15)
This transformation shows that model (2.13) can be rewritten in the form of model (2.12) plus an additional term in ∆X t-j . Hence, model (2.12) is nested in model (2.13) and is resulting from it by imposing the restriction that γ j + δa = 0.

The measure of the forecasting power of X t-j for Y t , after controlling for the past of X t-j

and the past of Y t is then defined in the same way as in (2.5),

C j = ln σ2 j+1 σ2 j ,
where the residual variances are now estimated via the VEC approach instead of the VAR model. Because equations (2.12) and (2.13) are nested, the likelihood ratio test is again appropriate for testing whether the Granger causality measure at lag j is significant. The definitions of the total Granger causality measure C and the mean Granger causality lag D, as defined in (2.7) and (2.8), are again applicable.

Data

The US consumer sentiment index is computed by the Survey Research Center at the University of Michigan 1 . It is derived from the answers to five questions asked to US consumers about (i) the financial situation of households compared to one year ago, (ii) the expected financial situation of households within one year, (iii) the expected general economic/financial situation of the country over the next twelve months, (iv) the economic (unemployment) expectations during the next five years, and (v) the appropriateness of buying major household durables at present. The consumer sentiment index is then computed as a weighted average of the relative scores (percentage of favorable answers minus percentage of negative

Results

Using the Johansen procedure, the consumer sentiment index is found to be cointegrated with total real consumption and each of its three components 5 . Therefore, a Granger-causality

Granger Causality Tests

We estimate the VEC model and test separately for short run, long run and overall Granger causality. The lag length of the VEC model was selected according to the Schwartz criterion. In terms of equation (2.10), short run causality is present if the γ 2,k 's are jointly significant, long run causality if δ 2 is significant, and overall causality if the γ 2,k 's and the δ 2 are jointly significant. Table 1 summarizes the results of the Granger causality tests for each of the four consumption series.

Insert Table 1 here. Test for no Granger causality of the consumer sentiment index for consumption and its components using the VEC model. The test statistic for the long run follows a t-distribution and for the short run and overall Granger causality an F -distribution.

Corresponding p-values are reported between parentheses.

The results in Table 1 show that there is strong evidence that all four consumption time series (total real consumption, real durables consumption, real nondurables consumption and

Time decomposition

Say we want to forecast the consumption of next month. Then, the measures of causality at lag j, C j , tell us what the relevant information is in the consumer sentiment index of j months ago. These measures are essentially comparing the models (2.12) and (2.13). The interpretation of C j is then as follows: it indicates how important the CSI measured at t -j is to forecast consumption at t, given that the outcomes of the CSI are already disposable for periods t -j -1, t -j -2, . . . In particular, if all C j 's are zero for j > j 0 , then it is sufficient to use only the last j 0 indices to make the forecast. On the other hand, if all C j 's are non significant for j < j 0 , then the most recent surveys are not adding much information.

If the latter would be the case, one could argue that it is too expensive to carry out monthly surveys, and decrease the frequency for collecting the survey data.

Figure 1 plots the measures of causality of the consumer sentiment at lag j for total real consumption and real consumption of durables, nondurables and services respectively.

Each value of C j can be compared with the critical value at the 5% level, indicated by the horizontal line. The maximum lag we considered was M = 18. Several checks confirm that the residuals with M = 18 are white noise and do not contain any forecasting information.

Moreover, it seems indeed reasonable, and it is confirmed by the graphs in Figure 1, that the consumer sentiment of more than a year and a half ago does not contain much information about today's consumption. As such, a graphical representation of the decomposition of the predictive power of the consumer sentiment index is obtained.

Insert Figure 1 Figure 1 shows that, as j grows, the C j 's tend to zero. This means that by going far enough back in time, we will reach a point where the lagged consumer sentiment index does not contain any additional information about today's consumption. For total real consumption (Figure 1a) we see that C 1 is by far the most important, followed by smaller Granger causality measures at the higher order lags. This implies that, to forecast the total real consumption in the next month, using the CSI of this month is most essential. In particular, carrying out monthly surveys is worth doing. For the 3 components of consumptions, the picture is slightly different.

First of all, for durables (Figure 1b) there are peaks at different lags and the relative contribution of the higher order lags to the total measure of Granger Causality C is more important. To forecast the consumption of durables in the next month, not only this month's consumer sentiment index is informative, but also those up to at least one quarter ago. This result is in line with our primary belief that the consumption of often expensive durables is easily postponed. For services (Figure 1d) a similar picture arises, but there the contribution of the most recent CSI to C is much more important. A possible explication for this is that "services" is a very heterogeneous consumption category, including items like recreation (where the willingness to buy will quickly be followed by actual consumption) and renting houses (being more comparable to a durable good).

For the nondurables (Figure 1c), only the Granger causality measure at lag two is significant. It turns out that the most recent value of CSI is adding much less here than for services and durables. This is at first sight surprising, but it can be explained by the fact that consumption of non-durables is already quite easy to predict from its own past. People have to fulfil their primary needs such as food, and will not change too abruptly their buying behavior for nondurables. For these goods, it is much harder for the CSI to add predictive power. This is in contrast with services, which are more difficult to predict from their own past, since their consumption is more subject to sentiments and impulsive behavior. Note that services make up about 59% of total personal consumption expenditure, compared to 12% for durables and 29% for nondurables.

Table 2 presents the mean causality lag D, together with the total causality measure for the four consumption series. We see that the consumption series are led by the consumer sentiment index with an average time lag between 4 to 6 months. While most predictive content is in the last survey value of the CSI, lagged values cannot be neglected and need to be taken along. The total Granger causality measure D is highest for the consumption of services, followed by durables and nondurables. As consumers feel more or less confident, they adapt more easily their consumption level in services (recreation, restaurants, bank services, . . .) than in goods. The higher total effect on durable goods compared to nondurables is in line with the findings of [START_REF] Throop | Consumer sentiment: its causes and effects[END_REF] and with the liquidity hypothesis of [START_REF] Mishkin | Illiquidity, consumer durable expenditure, and monetary policy[END_REF] which states that consumers who fear financial distress in the near future, and thus having low confidence, will prefer to hold their wealth in liquid form. Spending on nondurables can more difficultly be delayed in time as they are usually of primary need. These findings are in line with [START_REF] Delorme | Consumer confidence and rational expectations in the United States compared with the United Kingdom[END_REF] who also found that in the USA the confidence index has more predictive power for durables than for non-durables.

Insert Table 2 here. Mean lag and measure of the Granger causality of US consumer sentiment for real US consumption.

Forecasting power at different horizons

So far, we have only looked at one-step ahead forecasts of the consumption series. To gain insight on how consumer confidence helps to forecast consumption at larger time horizons, we will compare the R 2 of the following two equations:

∆Y t+h = α + M k=0 β k ∆Y t-k + ε t+h , (4.1)
and

∆Y t+h = α + M k=0 β k ∆Y t-k + M k=0 γ k ∆X t-k + δU t + ε t+h , (4.2)
where Y t equals consecutively the logarithm of each of the four consumption series, X t is the consumer sentiment index and U t the error correction term. Figure 2 gives the evolution of the R 2 of equations (4.1) and (4.2) as a function of the forecast horizons h for the four consumption series. Obviously, as can be seen from Figure 2, the predictive power decreases in h. The difference between the two lines in the graphs is of main interest here. The difference between the R 2 of equation (4.2) and the R 2 of equation (4.1) reveals how much is additionally explained by the present and past values of the consumer sentiment index. This difference can be interpreted as a measure for forecasting power of the CSI at horizon h.

Insert Table 3 here. The R 2 measure for a one step ahead forecast using the past of consumption only and using both the past of consumption and the past of consumer sentiment. The results for h equal to 1, the one-step ahead forecast, are presented in Table 3. In a previous study, Caroll, Fuhrer and Wilcox (1994) report that the lagged consumer sentiment index on its own explains about 14% in the variation of the growth of total personal expenditures. However, they did not control for the past values of consumption and neither did they include the long run relationship. Making a forecast with the simple univariate model (4.1) yields indeed an R 2 measure of around 15%, as can be seen from Table 3. Adding the consumer sentiment index as a predictor in a VEC model, increases the explained variance further by 5 to 10 percent. So the consumer sentiment index does serve as a valuable predictor and gives timely information of future consumption. Comparing the R 2 measures for the different components of consumption reveals that the consumer sentiment adds most additional information for future consumption of services, in comparison to the other consumption series (being consistent with the total measures of Granger causality in Table 2). A reason for this may be that consumption of services is more depending on the willingness to buy, than on the ability to buy. On the other hand, consumption of durables may depend more on ability than on willingness to buy, as was also found by Van Raaij and Gianotten (1990).

The additional predictive power of the CSI is lowest for non-durables. The reason for this, as we can infer from Table 3, is that this series is more easily predictable from its own past:

consumers tend to have their own buying patters for many non-durable consumption goods, which they are not changing too much on the very short run. The total R 2 for non-durables is, however, still fairly high.

Figure 2 gives additional insight in the predictive power of the CSI at larger time horizons.

First notice the sharp drop in R 2 after the one-step ahead forecast, especially for nondurables.

This drop in total predictive power is mainly due to a decrease of the part of the R 2 that can be attributed to past consumption only. The extra gain in R 2 given by the consumer sentiment index is, however, quite persistent. This shows that the CSI is indeed able to pick up a latent sentiment of the consumer, which will be reflected in his buying pattern over a 14 F o r P e e r R e v i e w longer time horizon. We conclude that the monthly surveys do have a longer term predictive content, for time horizons up to even one year.

Conclusion

This paper investigates whether the US consumer sentiment index offers timely information about future US consumer spending. Our contribution is threefold. First, while most previous analyses conduct a Vector Autoregression model, we find consumption and the consumer sentiment index to be cointegrated and therefore we test for Granger causality in a Vector Error Correction framework. This allows us to take into account the long run relationship between the two series. By doing so, a very strong long run Granger causality relation was found.

Secondly, we computed total Granger Causality measures and decomposed it over different time lags. A graphical representation of this decomposition, as in Figure 1, turns out to be useful. We find that for predicting the total consumption of next month, today's sentiment index is in general most informative, but further lags need to be taken into account as well, as is confirmed by an average causality lag between 4 and 6 months. Regarding the different components of Granger Causality, we find that the CSI has most predictive content for future spending on services. Somehow surprisingly, it turned out that for nondurables, the incremental predictive power of CSI is more limited than for durables and services.

Thirdly, we measured the forecasting power of the consumer sentiment index at different forecasting horizons. We conclude that the consumer sentiment remains a useful predictor of consumption for larger time horizons.

A Granger causality analysis is most often represented by just one number: the outcome of the test statistic. In this paper, a more complete analysis is carried out, by decomposing the causality in the time domain applied to consumer sentiment and consumer spending.

It became clear that such an approach yields new and interesting additional insights in the causal relationship.

The methodology outlined in this paper can be applied to other settings where the forecasting performance of a leading indicator needs to be analyzed. The R 2 measure for a one step ahead forecast using the past of consumption only and using both the past of consumption and the

  to determine the contribution of each time lag to the strength of the 3 Granger causal relationship. Consider the following pair of equations:

5

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  real services consumption) are Granger caused by the consumer sentiment index in the long run. However, concerning short run Granger causality, only services consumption seems to be Granger caused by the consumer sentiment index, with a p-value of 0.062. Short run changes in the consumers' confidence do not seem to contain much information on future consumption of durables or nondurables. The level of the CSI contained in the long run cointegration relation has a much more pronounced impact.

  here. Time decomposition of the Granger causality of CSI for (a) total consumption, (b) consumption of durables (c) consumption of nondurabels and (d) consumption of services. The dotted line gives the 5% critical value.

  Insert Figure2here. The R 2 measure of the forecasting model using only the past of consumption (2-line) and the model containing both the past of consumer sentiment and the past of consumption ( -line) versus the forecast horizon for (a) total consumption, (b) consumption of durables (c) consumption of nondurabels and (d) consumption of services.

Figure 1 :Figure 2 :

 12 Figures accompanying the article "Consumer Sentiment and

  Section four contains the empirical results. In particular, it is investigated at what time lag the consumer sentiment index offers valuable information about consumption. A graphical

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  , the null hypothesis that X t does not Granger cause Y t is rejected not only if the γ 2,k coefficients in equation (2.10) are jointly significant, but also when only δ 2 is significant. The coefficients γ 2,k give insight about the Granger causality between the short run components of the time series. If the γ 2,k coefficients are found to be significant, it implies that the changes in X t are important in predicting future Y

	predicting future Y t .
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t . The coefficient δ 2 provides evidence about the Granger causality between the deviation from the long run equilibrium and Y t . If δ 2 is significant, the level of X t is important in

  answers, plus 100) for each of the five questions. The consumer sentiment index surveyed at t will be denoted by CSI t . An augmented Dickey-Fuller test for presence of a unit root 2 indicates that the consumer sentiment index series needs to be differentiated of order one (∆CSI t = CSI t -CSI t-1 ) to become stationary. The consumer sentiment index CSI t plays the role of X t in the previous section on the methodology, and is aimed at Granger causing real consumption, denoted by Y t . The US real consumption data are collected by the US Bureau of Economic Analysis, an agency of the US Department of Commerce 3 . The data range from January 1978 to February 2004, resulting in 314 observations. These monthly time series are quantity indices, and are seasonally adjusted at annual rates by the data provider. The data measure the goods and services purchased by the persons resident in the United States. Total US real

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1 Data available at http://www.sca.isr.umich.edu/main.php 8 Page 10 of 25 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK consumption at time t is denoted by RC t and has three major components: (i) consumption of durables (RD t ) consisting largely of motor vehicles, furniture and household equipment, (ii) consumption of nondurables (RN D t ) containing, among others, expenditures on food, clothing and gasoline and (iii) consumption of services (RS t ) consisting mainly of rental housing, recreation, medical care and transportation. All four consumption series have been log-transformed and have a stochastic trend 4 .

2

  The ADF test statistic of order two (intercept in test statistic) equals -2.45 (p > 0.10), suggesting no rejection of the null of a unit root.3 Data available at http://www.bea.gov/bea/dn/nipaweb/index.asp , which is not the topic of interest in this paper. Since the time series log RC t , log RD t , log RN D t and log RS t all contain a stochastic trend and the CSI t does not, a trend variable is included in the cointegration equation (2.11).

	Submitted Manuscript
	analysis including an error correction term, as described is Section 2.2, is performed to gain
	insight into the term structure of the underlying causality between the time series. In the
	error correcting model, equation (2.10) is used to determine whether the consumer sentiment
	index is Granger causing real consumption or one of its components. Equation (2.9) examines
	the opposite causality
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	4 The ADF test statistic of order two (intercept and trend in test statistic) equals -1.78 (p > 0.10) for total
	consumption, -2.01 (p > 0.10) for consumption of durables, -1.05 (p > 0.10) for consumption of nondurables
	and -1.50 (p > 0.10) for services consumption, suggesting no rejection of the null of a unit root in all cases.
	5 Using the Johanson trace-test for cointegration (of order two, with trend in the cointegration equation and
	no deterministic trend in the VAR model), the null of no cointegration between the series of consumer sentiment

and the four consumption series is rejected at the 1% level for total real consumption and consumption of durables, and at the 5% level for nondurables and service consumption. 9 Page 11 of 25 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 1 :

 1 Test for no Granger causality: ResultsTest for no Granger causality of the consumer sentiment index for consumption and its components using the VEC model. The test statistic for the long run follows a t-distribution and for the short run and overall Granger causality an F -distribution. Corresponding p-values are reported between brackets.

	indicators. In a study of Krystalogianni Matysiak and Tsolacos (2004) more than 20 possible
	economic indicators for commercial real estate performance in the UK are investigated. Other
	examples are Lemmens, Croux and Dekimpe (2005), where the Granger causality of Business
	Tendency Surveys for national accounts series has been analyzed, and Binner, Bissoondeeal
	and Mullineux (2005) where indicators for turning points in inflation rate are studied.
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In this paper the consumer sentiment index is taken as a leading indicator for consumption, but there are other possible 15 Page 17 of 25 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK 1 Page 21 of 25 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 3 :

 3 The R 2 measure for a one step ahead forecast.Explained by Additionally explained consumption Y t the past of Y t by the past of CSI t

	Total

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UKSubmitted Manuscript

Acknowledgements: This research has been supported by the Research Fund K.U. Leuven and the Fonds voor Wetenschappelijk Onderzoek (Contract number G.0594.05).

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Submitted Manuscript


Francisco Javier

Miguel Vélez

Jesús Pérez-Mayo

Quantitative Methods, D31 -Personal Income, Wealth, and Their Distributions < D3 -Distribution < D -Microeconomics, D59 -Other < D5 -General Equilibrium and Disequilibrium < D -Microeconomics, H59 -Other < H5

Keywords: social accounting matrices, SAM multipliers, income distribution, inequality C69, D31, D59, H59, R15 social accounting matrices, SAM multipliers, income distribution, inequality

Social accounting matrices are adequate databases for the economic modelling. These matrices emphasize the role of households in the economy, and so, they usually disaggregate the household sector into several groups. This disaggregation allows social accounting matrices to be used for diverse income distribution analysis.

The objective of this work is to use the linear SAM models to study how inequality is modified by several exogenous injections of income. The set of multipliers and indicators presented is applied to the economy of Extremadura -a region situated in the southwest of Spain-. In particular, together with the accounting multipliers, two redistributed income matrices are presented to show how changes in final demand and in income transfers cause opposite effects in inequality. For contrasting these results, we also use Gini and Theil indices. Finally, a major reduction in both would result from an appropriate re-allocation of transfers.

Introduction

Inequality measurement is an important topic in the economic literature. However, National Accounting and, more precisely, Social Accounting Matrices have not been widely used as instruments for inequality analysis. This question has already exposed by [START_REF] Atkinson | Introduction: income distributions and economics[END_REF], who argued that income distribution should be integrate into economic analysis. Some attempts for solving this lack can be found in [START_REF] Rubio Sanz | SAM multipliers and inequality measurement[END_REF], where SAM multipliers and inequality measurement were put in touch in a country-level analysis. On the other hand, computable general equilibrium models have been applied to analyse the relationship between economic growth and income inequality - [START_REF] Hanson | Factor productivity and income inequality: a general equilibrium analysis[END_REF]-or the effects on income distribution of several energy taxation measures - [START_REF] Yang | Carbon reducing taxes and income inequality: general equilibrium evaluation of alternative energy taxation in Taiwan[END_REF].

In this sense, this paper was carried out in the framework of linear SAM modelling. The objective was to apply these models to the economy of Extremadura, in order to quantify and arrange the interdependence relationships, focusing on several results related to households and income inequality.

To attain this objective, together with a brief analysis of traditional SAM multipliers, three applications focused on income distribution analysis are presented. Firstly, two redistributed income matrices are computed. These matrices show the effects that exogenous inflows to either the different activity sectors or the households groups would Particularly worthy of note among the results was that increments in demand and increments in transfers had precisely the opposite effects. The former increased the inequality in income distribution between the different groups of households, while the latter reduced it. In addition, a major reduction in the two inequality indices would result from an appropriate re-allocation of transfers towards the low-income households.

The work is structured as follows. Section 2 presents the linear SAM multipliers in an abridged form, and shows the formulation required to obtain the redistributive multipliers. Section 3 is an overview of the SAM that was constructed for the Extremadura economy, and that will be used as the basis for the subsequent calculations.

Section 4 presents the results of the four applications performed. Finally, section 5 gives the principal conclusions drawn from the analyses.

Linear SAM models and redistribution matrix

Social accounting matrices (SAM) can be conceived of as a disaggregated matrix representation of the circular flow of income, allowing one to study the processes of the generation and distribution of income. These matrices are generally presented as square matrices, with a row and a column identically arranged for each agent or economic sector incorporated in the matrix. By convention, the row entries are interpreted as income, and the column entries as payments or expenditures. An important accounting constraint is that a SAM should satisfy the necessary equality between the sum of each row and the sum of its corresponding column.

Their principal application is as a basis for the construction of economic models. A first group of such models is that of the so-called linear SAM models. These allow one to determine the changes in income levels of the different agents that may be caused by It is important to note that, since it captures in a complete way the interrelationships between the different agents and sectors, this methodology is well suited to evaluating multiplicative effects. In addition, the level of disaggregation that SAMs normally incorporate enables the obtained multipliers to be presented with a high degree of detail.

To construct a SAM multiplier model, one must begin by classifying the SAM accounts into endogenous and exogenous. Traditionally, public administrations, the capital or savings/investment account, and the external sector accounts are usually considered exogenous. The accounts for the productive factors, the remaining institutional sectors, and the activity sectors are therefore considered endogenous 2 .

In the formulation of these models, one basically transforms and rewrites the SAM's accounting identities. Assume that the total number of accounts in the SAM, m, is apportioned between n endogenous and k exogenous accounts, and that the column vectors y n and y k represent their levels of production or income. Using A ij to denote submatrices of column-normalized coefficients -expenditure share-, the partitioned matrix structure of the SAM can be expressed in the following manner:

= = = = k n kk kn nk nn k n y y A A A A y y ( 1
)
Computing this matrix product:

F o r P e e r R e v i e w 5 x Ma x A I y A A I y A y A y nn k nk nn k nk n nn n = = = = = = = = = = = = + + + + = = = = 1 1 ) ( ) ( (2)
The column vector x shows the sum of exogenous injections received by each endogenous account. The matrix Ma allows one to relate exogenous injections of income with the incomes of the endogenous accounts, providing the termed accounting multipliers 3 .

The SAM multipliers analysis has traditionally focused on determining changes in absolute income levels. It is also important, however, to determine what changes the possible exogenous shocks would cause to the relative position of a given economic agent. The accounting multipliers can be used as the basis to define other measures that capture these relative effects. A good example is found in the redistributive multipliers set forth by Roland-Holst and Sancho (1992) 4 . Following these authors, one defines the relative income vector z n :

n n n y e y z ' = = = = (3),
where e' is a unitary row vector. Substituting the expression for y n , equation ( 2), and with matrix differentiation, one has: (5), where y ni is the i-th component of vector y n , and Ma .j is the j-th column of the matrix Ma. One observes that the sign of R ij depends, therefore, on the terms in brackets, i.e., it depends on the relationship between ) ( ), determining, therefore, a positive value of R ij . On the opposite, a negative value of R ij shows a worsening in its relative position.

[ [ [ [ ] ] ] ]

Social accounting matrix for the Extremadura economy

To carry out the subsequent applications, we took as the basis the only SAM built for the Extremadura economy, corresponding to the year 1990 6 . The set of accounts conforming this matrix (henceforth, SAMEXT90) is presented in figure 1.

The main statistical sources used have been a table of intersectoral flows (input-output table), the corresponding Regional Accounts, and a household's income and expenditure survey. Other more specific sources were also used to complete certain transactions of With respect to its disaggregation, firstly, there are two accounts for the labour and capital production factors, that reflect the value added generated in production and its distribution among the eleven household groups. These ones have disaggregated according to different criteria as age, activity sector or income level. Although the households' incomes basically come from the labour and capital factors, they also receive transfers from the government and the external sectors. With these incomes, the households consume, save, and make various payments to the government.

Regarding the activity sectors, their accounting structure is analogous to an input-output table structure. In particular, their cost structure (columns) reflects payments to the labour and capital factors, intermediate inputs, imports, and payments to the government (production and import taxes). Their rows reflect intermediate outputs, private consumption, public consumption, investment, and exports.

Finally, SAMEXT90 also includes an aggregate capital account, reflecting the overall equilibrium between savings and investment; a government account; and three accounts reflecting the relationships between the Extremadura economy and the three differentiated external sectors -rest of Spain, European Community, and rest of the world.

Analysis of the results

The presented applications are clearly aimed at an income distribution and inequality analysis. We first calculated the accounting multiplier matrix as an application for

showing the capacity that the different endogenous agents have to generate increments in income. The following three applications represent the main body of the study,

The accounting multipliers matrix

In our case, the usual closure assumption for SAM multipliers model is used, that is, the accounts for production factors, household groups, and activity sectors are considered as endogenous. Therefore, the corresponding accounting multipliers matrix -Ma (Ext) -is of order 30×30.

Although one could differentiate various submatrices that carry relevant information, in this section we shall restrict ourselves to analyzing the multipliers calculated as column sums of the matrix Ma(Ext), that we term diffusion effects (backward linkages in inputoutput terms). These multipliers show the overall effects of a unitary exogenous injection received by the endogenous account under consideration on all endogenous account incomes. Thus, agents or sectors with large diffusion effects generate significant knock-on effects, and they could hence be considered as priorities with respect to receiving exogenous impulses from public administrations.

These diffusion effects are given in table 1. It is clear that the greatest effects correspond to the service sectors, especially credit and insurance (account 28) and other salesoriented services (account 29), with a multiplier of approximately five m.u. per received give rise to major income expansion effects. The higher multipliers computed for these activity sectors correspond to a higher interdependence with the rest of endogenous agents. On the other hand, the industrial sector stands out for its small relative weight in the Extremadura economy, and its poor capacity for generating relevant knock-on effects as well (see accounts [18][START_REF] Reinert | Social accounts and the structure of the North American economy[END_REF][20][21][22][23][24].

With respect to the household groups, it is interesting to note that the multipliers for the low-income households are greater than those multipliers of the equivalent high-income groups. This result is due to the lower savings and payments for direct taxation, in relative terms, of low-income households. Consequently, there are also less income leakages to the exogenous part of the model, and so, producing a higher boost to the economic activity by consumption.

Redistributed income matrices: activity sectors -households, and householdshouseholds

In this second application, a more detailed analysis was made for two sets of multipliers related to households' incomes. One can define the activities-households multipliers as those that reflect how exogenous injections into the activity sectors affect household incomes. Moreover, the households-households multipliers as those that reflect how those incomes are affected when households receive exogenous inflow income transfers.

Using the formalism of section 2, in the following, we use both sets of multipliers to present their corresponding redistributive matrices 7 . The aim is to determine in relative terms for which household groups increments in final exogenous demand or in income transfers are beneficial, and for which they are detrimental. Nevertheless, to facilitate the Specifically, starting from the activities-households submatrix of Ma (Ext), one calculates its corresponding redistributed income matrix (table 2). The last row indicates the redistribution of household income over each sector of activity when there is an increase in its corresponding demand of one m.u. For example, if there is an exogenous increase in the demand for farming goods, 0.091 m.u. of household income would be redistributed: 0.001 m.u. corresponding to the first households group, 0.026 m.u. to the second, 0.004 m.u. to the sixth, and 0.06 m.u. to the seventh, while the rest of the households' incomes would undergo a relative worsening. One observes that these overall redistributive effects clearly reproduce the diffusion effects presented in table 1, because the sectors with more significant effects -in our case, a higher redistribution of household income-are again the service sectors (accounts 26-30), followed by farming and construction. The remaining activities, especially the industrial activities, present far smaller total redistributive effects.

It is more interesting to consider the values in the last column. This column represents the mean redistributive effects of a unitary increment in demand. These values are computed as a weighted mean of the row elements, using the shares of exogenous injections of each sector as weights.

One observes that the pattern of relative improvement or worsening showed by the mean effect remains, almost independently of which activity sector receives the exogenous injection. In particular, the results show a worsening in the relative position of the retiree 11 household groups (accounts 10-13). That is why their main source of income is government-paid pensions, so the effects that correspond to the distribution of factors incomes among households lack in their chain of interdependence. Similar reasons, namely, a relatively small share of incomes from labour and capital factors, determine a relative worsening for the active low-income households (accounts 3, 5, and 6).

In fact, the only household groups that benefit in relative terms are those of high income.

In particular, considering this last column, approximately 66% of the redistributed income corresponds to account nine (the fifth quintile of active non-farming households), 16% to account eight (the fourth quintile), 8% to account seven (third quintile), and 10% to account four (high-income active farming households). The results thus seem to show that exogenous increases in demand tend to widen even more the differences between low-income and high-income household groups.

Secondly, and to conclude this subsection, we shall consider the households-households multipliers and its corresponding redistributed income matrix (see table 3). The aim is to determine how the relative incomes of households are affected by transfers received by the households themselves.

In contrast to the previous table, one observes a clear predominance of negative signs.

The exogenous income transfers only improve the relative position of the household group that received them, so that there are no mutually beneficial linkages (symmetric pairs of positive elements).

Likewise, except for the elements in the main diagonal, for each household group the elements in its corresponding row are very similar, that is, irrespective of the household group that receives the exogenous injection, changes in its relative position are almost the same. On the other hand, although the results show certain homogeneity in the total It is important to note that the results given in the last column (mean redistributive effects) are in the opposite sense to those presented in table 2 9 . Specifically, the results logically show that a transfers increase reduce the differences between low and high incomes. Indeed, the lowest income groups are almost the only ones that benefit in their relative positions, especially some of the retiree household groups. The four household groups that previously benefited in relative terms (accounts 4, 7, 8, and 9 in table 2) now show a clear worsening in their relative position.

Measuring inequality after final demand and transfers changes

In this third application, we carried out two sets of simulations that were directly related to the previous redistributed income matrices. The objective of these applications is to confirm the results of the redistributed income matrices, by using the traditional income inequality analysis. Given these matrices, it is expected that, on one hand, increments in final demand increases the inequality level and, on the other one, increments in transfers reduce it. Apart from these qualitative aspects, the inequality indices we employed provide us quantitative information about the effects of growth and transfers on inequality.

Therefore, a first set of simulations was aimed at determining to what degree the levels of inequality are altered by increments in exogenous final demand, and the other the same, but by increments in exogenous income transfers received by households. The trials in both sets of simulations consisted of 10%, 20%, and 30% injections on each In this application, we considered two indices widely used in the literature: the Gini index, and the Theil index. The advantage of using this type of measure is that, since they are functions that assign a real number to each income distribution, they summarize all the information contained in the distribution in a single scalar. The two indices have quite different aggregation procedures, however, and therefore provide numerically distinct results.

The Gini index is probably the most commonly used inequality measure, due to its geometric interpretation and its relationship with the Lorenz curve. It is defined by the following expression:

= = = n i n j j i y y n G 1 1 2 | | 1 2 1 µ (6)
The Theil index is based on the concept of entropy, and indeed forms part of the general class of entropy measures [START_REF] Cowell | Measuring inequality[END_REF]. It is defined by the following expressions:

= = n i c i c c y c c n T 1 1 , 0 , 1 ) 1 ( 1 1 µ (7.a) = = = n i i c y n T 1 0 0 , ln 1 µ (7.b)
We set the parameter c to zero, to facilitate the optimization programming that will be presented in the next section. Beginning with the changes in demand (table 4), the results clearly show that stimulating the demand without making any other adjustment to the Extremadura economy leads to increased inequality 11 . Also, the changes in the two indices become greater, the larger the increment in the demand. It is important to note that these demand increases determine greater income increments for the high-income households. In this sense, due to the different sensitivity of both indices respect to changes in the distribution 12 , one observes that the changes in the Theil index become progressively greater than the corresponding changes in the Gini index.

The results for changes in transfers (table 5) are in the opposite sense to the preceding case: the transfers increments led to reductions in the inequality indices 13 . This was an expected result since the transfers are mainly received by the low-income household groups, thereby narrowing the gap between the values of their nominal income. As in the previous simulations, the different sensitivities of the two indices cause that the changes in the Theil index become progressively greater than the corresponding changes in the Gini index.

To conclude this subsection, it is important to note that the results of these simulations confirm the conclusions drawn from the redistributive effects. In particular, although the techniques of analysis are different, in both cases one observes that increments in final demand or income transfers have contrary effects on the evolution of inequality.

Transfers redistribution and inequality minimization

The importance of income transfers as a redistribution tool has been made clear by the preceding applications. Because of this importance, we propose a re-allocation of in the Extremadura economy and, besides, social and anti-poverty policies are much decentralized in Spain. Therefore, it is worth to explore the available actions for the regional government.

In particular, instead of increments in transfers, we propose a re-allocation of transfers, remaining constant its overall value 14 . We again used the linear SAM modelling framework showed in equation 2, since transfers redistribution modifies the exogenous inflows that the households receive, and consequently the endogenous incomes vary 15 .

Table 6 gives the two patterns of transfers redistribution that minimize the Gini and the Theil indices. In both cases, the only groups that should receive greater transfers than in the initial situation are the low initial incomes households (accounts 3, 5, 10, and 12), as well as the retiree urban high-income group (account 13). These would therefore be the only beneficiaries of this process of redistribution.

There are slight differences, though, according as to whether the Gini or the Theil index is used. In the former case, the reduction in inequality is smaller; also, other household groups -the second quintile of active non-farming households (account 6) and the higher-income rural retirees (account 11) would receive certain transfers, although less in quantity than the initial values. In the latter case, the sensitivity of the Theil index to changes in the low tail of the distribution leads to greater changes being proposed for the lower-income groups, which in turn leads to a greater reduction of the inequality.

Conclusions

A set of applications based on the methodological framework of SAM multipliers has been presented for Extremadura. In particular, following the computation of the 16 accounting multipliers, the three subsequent applications were focused on income distribution analysis -the fundamental objective of the work. The first of these applications presented the activities-households and households-households redistributive matrices. In the next, straightforward simulations were made to determine how changes in demand or in transfers alter the levels of inequality. In addition, the last application quantified the redistribution of the initial transfers that minimizes the inequality.

The results showed that low-income households have a greater capacity to generate increments in income than their high-income equivalents, although the greatest diffusion effects correspond to the service sectors. In addition, the accounts with the greatest diffusion effects are also those that present the greatest total redistributive effects in the activities-households and households-households redistributed income matrices. These last two matrices allow one to determine which household groups undergo a relative improvement and which a relative worsening in response to changes in demand or in transfers. The results showed increasing demand or increasing transfers to have opposite effects. In the former, increases in demand led to a relative improvement in high-income households at the cost of those of low income, thus widening the initial gap between the two. In the latter, however, the household groups that improved in relative terms in response to increases in transfers were clearly those of low income.

The simulations reported in the third application again investigated the effects on income distribution of changes in demand or in transfers. The results for the two indices used were coherent with the preceding case.

The last application showed how an appropriate redistribution of the transfers over the household groups allows the initial inequality indices to be significantly reduced. Given To conclude, we would make two final observations. First, we wish to call the attention of national and regional statistical bodies to the necessity of providing adequate statistical sources. These constitute the numerical support needed for any minimally updated economic analysis to be feasible. Second, we wish to stress the methodological potential of the analysis that has been described in the present work. SAM multipliers have allowed us to obtain important results related to the processes of income distribution and redistribution -results that would previously have been difficult to anticipate and quantify intuitively. 4 See also [START_REF] Roland-Holst | Interindustry analysis with social accounting methods[END_REF], [START_REF] Polo | Distribución de la renta en un modelo SAM de la economía española[END_REF] and [START_REF] Llop | Income distribution in a regional economy: a SAM model[END_REF]. Moreover, [START_REF] Cohen | Growth and equity effects of changing demographic structures in the Netherlands. Simulations within a social accounting matrix[END_REF] proposed a different approach for income redistribution analysis, presenting various relative distributive measures. See also, De [START_REF] De Miguel | Modelos SAM lineales y distribución de renta: una aplicación para la economía extremeña (Linear SAM models and income distribution: an application to the Extremadura economy)[END_REF]. 5 A generic element R ij of the said matrix indicates the direction and magnitude of the change in relative income of account i resulting from a unitary exogenous injection received by account j. It can be demonstrated that the different columns of this matrix R sum to zero, independently of how the distribution is made between endogenous and exogenous accounts. The process of income redistribution could therefore be regarded as a zero-sum game.

6 Due to the absence of a Regional Statistical Institute, the lack of statistical information is especially serious in Extremadura. These statistical limitations have determined the degree of detail of the SAMEXT90 matrix -it would have been appropriate to disaggregate the labour factor-and have prevented the construction of a SAM referred to a more recent period. For example, it is important to note that there is only one input-output table for the Extremadura economy, also referred to the year 1990. For more detailed information on the Extremadura matrix, see De Miguel, Manresa and[START_REF] De Miguel | Matriz de contabilidad social y multiplicadores contables: una aplicación para Extremadura[END_REF][START_REF] De Miguel | Matrices de contabilidad social y modelización de equilibrio general: una aplicación para la economía extremeña[END_REF].

7 "Elements of the matrix R are in a one-to-one correspondence with those of the original M [Ma], and the normalization of incomes can be chosen for the subgroup of endogenous institutions under study." Roland-Holst (1990, pp. 129). 8 It can be shown that the columns of this redistributed income matrix also sum to zero. 9 The weights used in this case are those of the exogenous injections received by the different households groups. 10 Although not presented here for the sake of clarity, these indices can also be calculated by differentiating between active workers (accounts 3-9 in the matrix) and retirees (accounts 10-13). It is also possible to differentiate between high and low incomes directly, i.e., applying the distinction to the active farming-linked households (accounts 1 and 2), the households of active workers in other sectors (accounts 3-7), of rural retirees (groups 8 and 9), and of urban retirees (groups 10 and 11). The results that we obtained in these cases showed the same tendency as described in the text. 11 By using a different methodological approach, [START_REF] Assane | An assessment of the growth and inequality causality relationship[END_REF] analyze the causal relationship between growth and inequality. 12 The Gini index is more sensitive to the changes in the centre of the distribution [START_REF] Sen | On economic inequality[END_REF], while the Theil index, with the parameter c set to zero, is more sensitive to changes in the extremes [START_REF] Shorrocks | The class of additively decomposable inequality measures[END_REF]. 13 Indeed, in the initial situation reflected by SAMEXT90, the incorporation of transfers leads to a major reduction in inequality. In particular, the initial Gini and Theil indices for primary incomes are 0.5837 and 0.9456, respectively, both clearly greater than the final income indices (0.4902 and 0.5131). The reduction in the Theil index is far greater because the transfers fundamentally affect the household groups situated at the lower tail of the distribution. 14 Isla, Moniche and Trujillo (2002) present a similar analysis for Andalusian economy. 15 The inequality indices were minimized by solving the corresponding optimization problem, using GAMS (General Algebraic Modelling System) software.

  have on the households' relative incomes. Secondly, Gini and Theil inequality indices are considered to how both indices change because of increases in final demand or in income transfers. Finally, since inequality decrease is an important goal of social policy, we calculated what the redistribution of initial income transfers should be to minimize both inequality indices.

  . Other indicators may be established based on these multipliers to determine the relative changes in endogenous incomes 1 .

  termed redistribution matrix, determines the ultimate distribution of relative incomes resulting from different exogenous shocks. 5 . To show and interpret the redistributive effects more closely, a generic element R ij can be expressed in the following manner:

  of the different household groups and on their relationships with the production sectors. In this sense, the second application deals with two redistributive effects matrices related to households' relative incomes. The third application emphasizes these results, by simulating increases in final demand and income transfers and assessing the changes in some inequality indices. Finally, the fourth application, in calculating the redistribution of the initial transfers required to minimize two standard inequality indices, shows the importance of transfers as a redistributive instrument.

  u. The farming (account 14) and construction (account 25) sectors also

  results, instead of the redistribution matrix R, we shall present a transformation of R consisting in pre-multiplying it by the term (e´ y n ). The elements of this new matrix reflect the value of the redistributed income, assuming the value of the endogenous accounts' initial incomes to remain constant 8 .

  household incomes, the previous diffusion effects are reproduced again, since the high-income households again show less capacity to generate significant effects than their low-income equivalents.

  for both indices are presented in tables four and five 10 .

  to reduce income inequality. Exogenous transfers are very significant

  entirety of these transfers comes from the public sector, the methodological framework used in the present work could be a valid referent in establishing social policy measures aimed at reducing inequality.

List of the accounts included in SAMEXT90

	Submitted Manuscript	Page 22 of 26
	Figure 1. Production factors	18. Chemical products
	1. Labour	19. Metal products and electrical material
	2. Capital	20. Transport material
		21. Food, beverages, and tobacco industries
	Households	22. Textiles, leather, shoes, and clothing
	3. Younger than 65, farming sector, low income	23. Paper and printing
	4. Younger than 65, farming sector, high income	24. Sundry industrial products
	5. Younger than 65, other sectors, 1st quintile	25. Construction
	6. Younger than 65, other sectors, 2nd quintile	26. Recovery and repair, trade and hostelry
	7. Younger than 65, other sectors, 3rd quintile	27. Transport and communications
	8. Younger than 65, other sectors, 4th quintile	28. Credit and insurance institutions
	9. Younger than 65, other sectors, 5th quintile	29. Other sales-oriented services
	10. 65 or older, rural, low income	30. Non-sales-oriented services
	11. 65 or older, rural, high income
	12. 65 or older, urban, low income
	13. 65 or older, urban, high income
	F o 16. Ferrous and non-ferrous minerals and metals Activity sectors 14. Agriculture 15. Energy 17. Non-metallic minerals F o r	EXOGENOUS ACCOUNTS (linear SAM model) 31. Capital account (savings/investment) 32. Government 33. External sector: rest of Spain 34. External sector: European Community 35. External sector: rest of the world
	r P P e e e r e r R R e v e v i e i w e w
	Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 1 . Accounting multipliers matrix Ma (Ext): diffusion effects

 1

		Effect	Rank		Effect	Rank
	1 Labour factor	4.442	11	16 Minerals (I)	1.466	28
	2 Capital factor	4.392	14	17 Minerals (II)	2.532	24
	3 Act-farm-low	4.486	9	18 Chemicals	1.322	29
	4 Act-farm-high	3.336	20	19 Metal prod.	2.110	26
	5 Act-nonfarm-1st quint	4.497	8	20 Transport material	1.048	30
	6 Act-nonfarm-2nd quint	4.413	12	21 Food ind.	3.291	21
	7 Act-nonfarm-3rd quint	3.909	16	22 Textiles	1.481	27
	8 Act-nonfarm-4th quint	3.425	19	23 Paper and printing	2.124	25
	9 Act-nonfarm-5th quint	2.939	23	24 Sundry ind.	2.988	22
	10 Ret-rural-low	4.677	5	25 Construction	4.449	10
	11 Ret-rural-high	3.707	17	26 Retail	4.545	7
	12 Ret-urban-low	4.393	13	27 Transport	4.557	6
	13 Ret-urban-high	3.491	18	28 Credit and insurance	5.017	1
	14 Farming	4.802	3	29 Other sales services	4.857	2
	15 Energy	4.088	15	30 Non-sales services	4.788	4
	MEAN EFFECT	3.586
	Source: the authors.

Table 3 . Redistributed income matrix: households -households Ac 3 Act-farm-low Ac 4 Act-farm-high

 3

	Ac 5	Ac 6	Ac 7	Ac 8	Ac 9
	Act-nonfarm -1 st q	Act-nonfarm -2 nd q	Act-nonfarm -3 rd q	Act-nonfarm -4 th q	Act-nonfarm -5 th q	Ac 10 Ret-rural-low	Ac 11 Ret-rural-high	Ac 12 Ret-urban-low	Ac 13 Ret-urban-high	Mean eff.

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Table 4 . Inequality indices after increments in demand Final indices after percentage increments in demand Percentage variation in indices after percentage increments in demand

 4

		Initial
		indices
			10%	20%	30%	10%	20%	30%
	Gini	0.4902	0.4957	0.5006	0.5050	1.1163	2.1281	3.0263
	Theil	0.5131	0.5278	0.5413	0.5539	2.8575	5.4963	7.9432
	Source: the authors.

Table 5 . Inequality indices after increments in transfers Final indices after percentage increments in transfers

 5

						Percentage variation in indices
		Initial indices				after percentage increments in transfers
			10%	20%	30%	10%	20%	30%
	Gini	0.4902	0.4846	0.4793	0.4742	-1.1483	-2.2330	-3.2591
	Theil	0.5131	0.4988	0.4859	0.4742	-2.7975	-5.3127	-7.5879
	Source: the authors.

Table 6 . Transfer redistribution for inequality indices minimization Minimization of the Gini index Minimization of the Theil index

 6 Note: all transfers are given in thousands of pesetas.

	Household groups	Initial TR	New TR	Change (%)	New TR	Change (%)
	3 Act-arm-low	8,173,622	12,910,599	57.95	21,681,023	165.26
	4 Act-farm-high	5,088,457	0	-100	0	-100
	5 Act-nonfarm-1st q 22,039,063 46,992,092	113.22	34,914,618	58.42
	6 Act-nonfarm-2nd q	14,691,296	7,588,633	-48.35	0	-100
	7 Act-nonfarm-3rd q 14,808,748	0	-100	0	-100
	8 Act-nonfarm-4th q 17,502,009	0	-100	0	-100
	9 Act-nonfarm-5th q 18,395,408	0	-100	0	-100
	10 Ret-rural-low	23,994,210 57,679,878	140.39	45,440,403	89.38
	11 Ret-rural-high	39,130,450 25,290,651	-35.37	0	-100
	12 Ret-urban-low	3,085,815	7,826,884	153.64	49,063,431	1489.97
	13 Ret-urban-high	10,479,320 19,099,661	82.26	26,288,922	150.86
	Gini initial	0.49
	Gini minimum	0.42
	Theil initial	0.51
	Theil minimum	0.24
	Source: the authors.

3 Page 26 of 26 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK 4 Page 27 of 26 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript

Acknowledgements

The first author acknowledges the financial support received from Ministerio de Ciencia y Tecnología (SEC2003-06080) and Generalitat de Catalunya (2004XT00095).

Regional Economics < R -Urban, Rural, and Regional Economics

Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

Submitted Manuscript