Carlo Altavilla
email: altavilla@uniparthenope.it

Paul De Grauwe
email: paul.degrauwe@econ.kuleuven.ac.be

Diebold-Mariano Granger-Newbold

FORECASTING AND COMBINING COMPETING MODELS OF EXCHANGE RATE DETERMINATION

Keywords: exchange rate, forecasting F31, C53 non-linearity, exchange rate modelling, forecasting

This paper investigates the out-of-sample forecast performance of a set of competing models of exchange rate determination. We compare standard linear models with models that characterize the relationship between exchange rate and its underlying fundamentals by nonlinear dynamics. Linear models tend to outperform at short forecast horizons especially when deviations from long-term equilibrium are small. In contrast, nonlinear models with more elaborate mean-reverting components dominate at longer horizons especially when deviations from long-term equilibrium are large. The results also suggest that combining different forecasting procedures generally produces more accurate forecasts than can be attained from a single model.

Introduction

There is an ongoing debate about exchange rate predictability in time series data. A large body of empirical literature, reviewed by [START_REF] Frankel | Empirical Research on Nominal Exchange Rates[END_REF] and [START_REF] Meese | Currency Fluctuations in the Post-Bretton Woods Era[END_REF], focuses on whether existing theoretical and econometric models of exchange rate determination represent good descriptions of the empirical data.

Nevertheless, the literature has not converged on a particular class of models capable of challenging the [START_REF] Meese | Empirical Exchange Rate Models of the Seventies[END_REF] result that structural macromodels cannot out-perform a naïve random-walk. Most of the studies conclude that monetary fundamentals such as the GDP differential, the inflation differential, the relative money growth, and the short-term interest rate differential have negligible outof-sample predictive power at least over short time horizons. However, there is some evidence that with longer time horizons the forecasting accuracy of fundamentals based exchange rate models improves (see e.g. [START_REF] Mark | Exchange rates and fundamentals: evidence on long-horizon predictability[END_REF] and Cheung et al. . Some authors have stressed that the poor forecasting performance of fundamental-based models is not related to the weak informative power of fundamentals. The superiority of random-walk forecasts is instead related to the weakness of the econometric techniques used in producing out-of-sample forecasts (see [START_REF] Taylor | Nonlinear Adjustment, Long-run Equilibrium and Exchange Rate Fundamentals[END_REF]. In a recent study, [START_REF] Sarno | Exchange rate and fundamentals: Footloose or evolving relationship?[END_REF] analyse how to optimally select the correct number of fundamentals to be used in computing the best forecasting model in each period. They show that ex-ante it is not possible to implement a procedure that is able to account for the frequent shifts occurring in the weight each fundamental has in driving exchange rate dynamics.

Recently, an important empirical literature has emerged showing strong empirical evidence that the dynamics governing the exchange rate behaviour may be nonlinear [START_REF] Taylor | Nonlinear Adjustment, Long-run Equilibrium and Exchange Rate Fundamentals[END_REF], Sarno, Taylor and Peel(2000)). In this line of research, [START_REF] Altavilla | Non-Linearities in the Relation between the Exchange Rate and its Fundamentals[END_REF], show that the exchange rate process can be modelled by a nonlinear error-correction model where deviations from the long-run equilibrium are mean-reverting but occasionally follow a nonstationary process. Nonlinearity leads to the inadequacy of the usual assumption made in the theoretical and empirical

T P 1 P T
But see the criticism of [START_REF] Faust | Exchange rate forecasting: the errors we've really made[END_REF]. The literature on exchange rate determinationT P F 2 FP T has produced mixed evidence on the out-of-sample predictive power of nonlinear modelsT P F 3 FP T . More precisely, the superiority of the Markov-switching forecast with respect to the random walk, firstly stressed by [START_REF] Engel | Long swings in the dollar: are they in the data and do markets know it?[END_REF], has recently been emphasized by [START_REF] Clarida | The out-of-sample success of term structure models as exchange rate predictors: a step beyond[END_REF]. However, while these models are found to produce an accurate representation of the in-sample exchange rate movements, they fail to consistently beat naïve random walk models in out-of-sample forecastingT P F 4 FP T

. It is then natural to produce comparative exercises on the forecasting ability of the nonlinear versus linear models of exchange rate determination.

The present study contributes to the ongoing debate regarding the possibility of correctly forecasting future exchange rate movements. The econometric evidence resulting from this kind of study can suggest which model should be adopted in order to achieve a better forecasting performance. A common characteristic of much of the existing studies is their focus on either linear or non-linear models. After this preliminary choice, selected models are then compared with a random walk process.

The contribution of the paper with respect to the existing literature is twofold.

First, we estimate a set of competing models including both linear and non-linear ones and show that the forecasting performance of competing models varies significantly across currencies, across forecast horizons and across sub-samples.

Second, we show that combining competing models of exchange rate forecasting, reflecting the relative ability each model has over different sub-samples, leads to more accurate forecasts.

The remainder of the paper proceeds as follows. Section 2 presents a set of competing model of exchange rate determination. Section 3 compares out-of-sample T P 2 P T See [START_REF] Sarno | The economics of exchange rates[END_REF] for a comprehensive discussion of competing models of exchange rate determination.

T P 3 P T See [START_REF] Engel | Can the Markov Switching Model Forecast Exchange Rates?[END_REF] on the use of Markov switching models for forecasting short-term exchange rate movements. More recently, [START_REF] Kilian | Why is it so difficult to beat the random walk forecast of exchange rates[END_REF] find some evidence on exchange rate predictability at horizon of 2 to 3 years by using ESTAR modelling. However, the power of the results decreases when the horizon is shorter.

T P 4 P T See [START_REF] Diebold | Nonparametric exchange rate prediction[END_REF][START_REF] Engel | Can the Markov Switching Model Forecast Exchange Rates?[END_REF][START_REF] Meese | Currency Fluctuations in the Post-Bretton Woods Era[END_REF]Rose, 1990, 1991.

Selecting and Estimating Competing Models

Our empirical analysis focuses on the forecasting properties over horizons from 1 quarter to 8 quarters of seven competing models which we, now, describe in detail.

These models are used to compute out-of-sample forecasts of three US dollar exchange rates: the dollar-euro, the dollar-sterling and the dollar-yen. All data used in the analysis are quarterly and were drawn from Datastream. The sample period goes from 1970:1 to 2005:3.

The first model consists of a driftless random walk process (RW). This framework remains a useful benchmark against which exchange rate models are judged. The dynamics of the model is as follows:

[1] ε - = + 1 t t t e e
where t e represents the nominal exchange rate.

The second model uses spectral analysis (SP). In general, exchange rate behaviour is expressed as a function of time. This representation may not necessarily be the most informative. In order to better analyse exchange rates evolution we combine a time domain approach with a frequency domain approach. In particular, spectral analysis might be useful in detecting regular cyclical patterns, or periodicities, in transformed exchange rate data. Significant information may be hidden in the frequency domain of the time series. Frequency-domain representations obtained through an appropriate transformation of the time series enable us to access this information.

In order to map the exchange rate from the time domain into the frequency domain we apply the Fourier transformation. Starting from the time series

( ) { } 1 T e t Δ
this transformation is based on the following equation:

[2] ( ) () ( ) ( ) ∑ = - - Δ = Δ T t T t ij t e T j
2 j T π , then ( ) T j e / 2π Δ
will be relatively large.

Starting from the moving average representation of the selected time series we follow the procedure outlined in [START_REF] Koopmans | The Spectral Analysis of Time Series[END_REF] to compute out-of sample forecasts using spectral techniques.

The third model is a four-variable vector error correction model (VECM):

[3] ε - - - = Δ = + Γ Δ +Π + ∑ 1 1 k t i ti t k t i x c x x [ ] π ′ = t t
( ) ( ) { } 1 1 ; , 1 exp t d t d e e γ μ γ μ - - - Γ = + ⎡ - -⎤ ⎣ ⎦ [6] ( ) ( ) 2 2
; , 1 exp

t d t d e e γ μ γ μ - - ⎡ ⎤ Γ = - - - ⎣ ⎦
where the first equation describes a logistic function (LSTAR) and the second one represents an exponential function (ESTAR). The specific form of the transition function is usually tested by employing a battery of tests proposed by [START_REF] Granger | Modelling Non-Linear Economic Relationships[END_REF]. However, in our case we directly estimate the model in its ESTAR form. In fact, the ESTAR model is more suitable than the LSTAR framework in analyzing the dynamic behaviour of the exchange rate deviations from equilibrium. This is so because the symmetry of the exponential transition function Γ 2 around μ implies a symmetric behaviour of the exchange rate adjustment regardless of whether the deviation form equilibrium is positive or negative. This means that the class of model used in the present study is: The specific form used in the analysis is:

[7] ( ) ( ) { } ( ) 2 *
[8] ( ) ( ) { } ( ) 4 2 * 1 1 1 1 exp t t t i ti t i e e e e μ μ γ μ α μ ε - - - = ⎡ ⎤ Δ = + Δ -+ - - - Δ -+ ⎣ ⎦ ∑ F o r P e e r R e v i e w 7
The fifth model is a univariate Markov-switching model (MS-AR) similar to the one estimated by Engel and Hamiltom (1990). In this model, the dynamics of discrete shifts follows a two-state Markov process with an AR(4) component. The lag structure has been tested with standard AIC, HQ and SC criteria. These statistics suggest an autoregressive structure of order four. The model has the form: ( ) and Valente (2005) however).. In the present study we compute combined forecasts using the methodology proposed in [START_REF] Hong | Inference on via generalized spectrum and nonlinear time series models[END_REF], [START_REF] Yang | Combining forecasting procedures: some theoretical results[END_REF] and [START_REF] Yang | Combining time series models for forecasting[END_REF]. We also estimated the model allowing for a shift in the mean of the variables. The results we obtained from the two specifications are very similar with respect to the regime classification as well as to the parameter values. As we expected, the differences between the two models mainly consist of the different pattern of the dynamic propagation of a permanent shift in regime. More precisely, in the MSIH model, the expected growth of the variables responds to a transition from one state to another in a smoother way. See Krolzig (1997) on the peculiarity of the two models.

[9] ( ) μ α μ ε - - = Δ - = Δ - + ∑ 4 1 ( ) ( )
μ μ μ > = ⎧ = ⎨ < = ⎩ 1 1 0 if 1 ( ) 0 if 2 t t
+ = = = = = ∀ ∈ ∑ 1 1 Pr | , 1 , {1, 2} M ij t
T P 8 P T See for example [START_REF] Bates | The combination of forecasts[END_REF], [START_REF] Granger | Improved Methods of Combining Forecasts[END_REF] and [START_REF] Clemen | Combining forecasts: a review and annotated bibliography[END_REF]. ( )

( ) 2 1 2 1 2 6 1 2 1 1 1 exp 2 1 exp 2 k t j j j t kt k t j j k j t e e e e σ ω σ - = - = = ⎡ ⎤ - ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ = ⎡ ⎤ - ⎢ ⎥ - ⎢ ⎥ ⎣ ⎦ ∑ ∑ ∑
where σ 2 t is the actual conditional variance of the exchange rate. The specific relationship imposed ensures that a weight attributed to a certain model at time t is larger the larger has been its ability to forecast the actual exchange rate in period t-1. Figure 1 shows the weights used in computing the combined forecast series.

Visual inspection provides useful information concerning the time-varying forecast abilities of competing models. A situation where the weights attributed to each model are very similar, as in the central part of the sample period for the 1-quarter-ahead forecast of the pound, suggests that the relative accuracy of the forecasts produced by each model may not be affected by a particular sub-sample period selected by the evaluation strategy. Moreover, the results obtained with one model are not different from the others. When, instead, weights are very dissimilar, a correct choice of the forecasting model may produce a significant improvement in terms of predictive accuracy.

The figure also suggests that there is a positive relationship between the volatility of the selected weights, i.e. the number of time each model account for the same proportion in the combing series, and the forecasting horizon.

Comparing out-of-sample forecasts

The seven models proposed above are estimated on a sub-sample of the historical data. Then the out-of sample forecasts of the competing models for alternative periods are evaluated. The forecast accuracy is measured by computing rolling forecasts. The estimation period goes from 1970:1 to 1989:4, while the forecast period goes from 1990:1 to 2005:3. This means that the first sequence of 1 to 8-quarter ahead forecast is generated starting from 1990:1. Then, the starting date of the forecast period is rolled forward one period, and another sequence of forecasts is generated. This loop is repeated until we have 62×1-quarter forecasts, down to 54×8-quarter forecasts.

Figure 2 provides a graphical summary of the performances of the competing models over the entire sample period in forecasting the three exchange rates. This figure illustrates the actual exchange rates (solid line) and the seven forecasting models (dashed lines) at 1-, -4 and 8-quarter-ahead horizon. Visual inspection suggests a better performance, in terms of forecast errors, of short-horizon exchange rates forecasts. However, the figure also illustrates higher long-horizon forecast volatility. This means that the gains (but also the losses) we can achieve by using a particular model are larger the longer is the forecasting horizon.

However, in order to assess the performance of the alternative models we have to analyse the forecast accuracy through a set of statistical measures.

Measuring forecast accuracy

The aim of this section is to examine the out-of-sample forecasting performance of alternative exchange rates models. Once each model has been estimated, the question arises as to how their performance may best be compared. In principle, forecast accuracy of competing models may be evaluated by employing various econometric procedures. Starting from the out-of-sample prediction errors, this paper

Point Forecasts Evaluation

We first use standard quantitative procedures involving forecast errors. The forecast error can be defined as:

$ + + + = -t k t k t k e x x , where ≥ 1 k and $ + t h
x represents the k-step-ahead forecast. The most widely used measure of forecast accuracy is the Root Mean Square Error (RMSE) . We can calculate it as follows:

[13]

+ + = ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ∑ 1/ 2 2 1 1 n t k i i RMSE e n
The comparison of forecasting performance based on this measure is summarized in Tables 1 and2.

Each table characterizes some periods of the three exchange rates histories. Table 1 analyses the periods 1992-2005 and 1991-1998. The four periods described in table 2 are 1992-1995, 1995-1998, 1998-2001, and 2001-2005. These tables also illustrate the relative ranking in terms of forecast errors for the 126 cases (seven models, three horizons, six sub-samples). The last column in each table reports the average ranks of the model over the three exchange rates. Table 2 provides some insights concerning the difference in the predictive power of linear and non-linear models. The first period, occurring in the early 1990s, embraces the European monetary system crises. During these years, the external value of both the U.K. pound and the euro fell while the Japanese Yen appreciated. Actual exchange rates appear to be better approximated by nonlinear models.

The second period covers the years 1995-1998. These years are characterized by a relatively stable behaviour of the two European currencies. In contrast, the yen has been depreciating against the U.S. dollar over the entire period.

The third period, ranging from the late 1990s to the early 2000s, embraces the launch of the Euro. In these years the RW forecasts appear to be much closer to the actual exchange rate.

Finally, the fourth period goes from the early 2000s to the end of the sample.

During these years the three currencies appreciated against the dollar. Contrary to the results of previous time periods, the forecast ability of the non-linear models seems to be higher than the one of the random walk.

The evidence emerging from these figures corroborates the hypothesis of having more than one state operating during the sample periods. In particular, the results suggest that forecasting performance varies significantly across currencies, across forecast horizons and across sub-samples. However, in general, linear models outperform at short forecast horizons especially when deviations from long-term equilibrium are small. In contrast, nonlinear models with more elaborate meanreverting components dominate at longer horizons especially when deviations from long-term equilibrium are large.

The RMSE provides a quantitative estimate of the forecasting ability of a specific model, allowing different models to be ranked, but does not provide a formal statistical indication of whether one model is significantly better than another. We also explicitly test the null hypothesis of no difference in the accuracy of the two . According to the null hypothesis the errors associated with the two forecasts are equally costly, on average.

If the null is rejected, the forecasting method that yields the smallest loss is preferred.

Given a series,

=1 { } T t t d
, of loss differentials, the test of forecast accuracy is based on:

[14]

= = ∑ 1 1 T t t d d T
The Diebold and Mariano (1995) (DM) parametric test is a well-known procedure for testing the null hypothesis of no difference in the accuracy of two competing forecasts. It is given by:

[15] ( ) ( )( ) 1 ( 1) 2 ( 1) 1 2 /() T T t t T t DM d l s t d d d d τ τ τ π τ - - - =-- = + ⎡ ⎤ = + - ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ∑ ∑
where ( )

/ ( ) l s t τ
is the lag windowT P F 10 FP T , ( ) s t is the truncation lag and T the number of observations. [START_REF] Harvey | Testing the Equality of Prediction Mean Squared Errors[END_REF] propose a modified version of the Diebold-Mariano test statistic in order to adjust for the possible wrong size of the original test when the forecasting horizon, k increases. The statistic they proposed is:

[16]

( )

1 1 1 2 2 1 2 1 MDM T T k T k k DM - - ⎡ ⎤ = + -+ - ⎣ ⎦
We follow the suggestion of [START_REF] Harvey | Testing the Equality of Prediction Mean Squared Errors[END_REF] in comparing the statistics with critical values from the Student's t distribution with (T-1) degrees of freedom, rather than from the standard normal distribution. This term is computed by using the Newey-West lag window.

T P 11 P T See [START_REF] Granger | Forecasting transformed series[END_REF] for a detailed description of the test. mean squared error) relative to the Model j at the 5% significance level. In contrast, pvalues no smaller then 0.95 mean that Model i generates a higher forecast error at the 5% level.

In absolute terms, if we consider the number of times each model significantly beats its competitors according to the two test statistics, we have different results depending on the forecast horizon. More precisely, at 1-quarter-ahead horizon, ESTAR models outperform the others 50% of the times. At 4-and 8-quarter horizon, combined forecasts are found to be the top-performing models. Of a total of 18 cases (six competitors for each of the three currencies), the percentage of times the combined forecast models beat the other models is higher than 60%. This evidence is in line with the results obtained with the RMSE in table 1 and2

Forecasts Encompassing

Forecast encompassing assesses whether any extra important information is contained in forecasts from rival models. A simple methodology to compare the forecast accuracy has been developed by [START_REF] Fair | Comparing Information in Forecasts from Econometric Models[END_REF]. The testing procedure in the Fair-Shiller approach is based on the following equation: On the basis of the Fair-Shiller tests and with the notable exception of the Japanese currency none of the selected models outperform their competitors at 1quarter-ahead horizon. It is interesting to note that when increasing the forecast horizon, combined models dominate the others. In general, the longer the forecast horizon, the worse the RW exchange rate forecasts become.

[17] ( ) ( ) α α α ε - - - - = + - + - + $ $ 1,
At 4-quarter-ahead horizon, combined forecasts still produce superior out-ofsample forecasting performance compared to other models. However, in these cases, other models also generate better forecasts if compared to the random walk. In fact, the t-values of the ESTAR and VEC models are above the 95% confidence threshold when compared with the RW, SP and MSAR models. From these tables, we can also conclude that, in most cases, at the 8-querter horizon, the forecast accuracy of nonlinear models significantly outperforms that of the competing models. 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 c 0.13 0.10 0.14 0.10 0.05 0.04 0.14 0.11 0.20 0.15 0.09 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 c 0.56 0.26 -0.65 -0.44 -0.48 -0.26 0.21 0.13 1.12 0.53 -1.13 -0.93 Coeff Tsta Coeff Tsta Coeff Tsta Coeff Tsta Coeff Tsta Coeff Tsta Coeff Tsta c -0.26 -0.50 -0.18 -0.33 -0.17 -0.32 0.42 0.65 -0.18 -0.30 -0.17 -0.32 α 1 0.59 1.89 0.04 0.29 -0.01 -0.06 0.94 1.56 0.01 0.04 0.07 0.19

α 2
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 c -0.26 -0.50 -0.26 -0.49 -0.29 -0.55 0.11 0.16 -0.28 -0.48 -0.27 -0.51 9.63 -6.74 -10.1 -6.78 -8.26 -7.32 -8.39 -7.92 -6.62 -1.72 -5.42 -5.50

Direction-of-change Measure of Forecast Accuracy

The third metric we use to evaluate forecast performance is the Percentage of Correct Direction Forecasts.

In exchange rate markets it is very insightful to know whether a specific forecast series generally co-moves with the actual exchange rate. In practice, sign forecast ability corresponds to evaluating whether the first differences in two series are the same.

For a given exchange rate model, we define + Δ e t j and t j + Δ as the predicted and actual direction of change in exchange rate j periods ahead, respectively. These two variables evolve as follows:

1 if

> ⎧ Δ = ⎨ - ⎩ = > ⎧ Δ = ⎨ - ⎩
Having calculated the two series, we can measure the performance of the forecast models in terms of correct directional change (CDC):

67.30%

[1] 60.74%

[ 7] [4.7] VEC(4) 64.02%

[4]

52.54%

[6] 70.57%

Comb(1-6) 51.30%

[3]

63.18%

[2]

55.91%

[2] [2.3]

Conclusions

This paper analyzed the out-of-sample forecasting performance of a set of competing models of exchange rate determination. The literature on currency forecasting has developed different quantitative frameworks to model the exchange rate process without, however, producing a consensus view of the ability of alternative models to forecast the exchange rate in the short-run.

In the empirical part of the paper we estimated, forecasted and compared different models thought to capture the dynamics of the exchange rates. The econometric evidence resulting from this kind of study can suggest which model should be adopted in order to achieve a better forecasting performance.

A set of forecast evaluation techniques was employed to judge the relative performance of seven competing models. The forecasting models include linear models (such as RW and VECM), non-linear models (such as MS-AR, MS-VECM and In general, the results suggest that the behaviour of the exchange rate is episodically unstable. This reflects the significant sub-sample instability of alternative forecasting performances. We found that the switching nature of the exchange rate process is inconsistent with a linear representation over the whole sample period..

Non-linear models characterize the exchange rate behaviour so that that even if its dynamics is time-varying, it is possible to identify periods of time where its behaviour becomes stable.

We considered three classes of statistical measures -point forecast evaluation, forecast encompassing and directional accuracy. The evidence emerging from the point forecast evaluation corroborates the hypothesis of having more than one state operating during the sample period. In particular, the results suggest that forecasting performance varies significantly across currencies, across forecast horizons and across sub-samples. In general, our results suggest that combining forecasts from many models yields more accurate forecasts than utilizing the forecast of a single model.

However, in most cases linear models outperform at short forecast horizons when deviations from long-term equilibrium are small. In contrast, nonlinear models with more elaborate mean-reverting components dominate at longer horizons especially when deviations from long-term equilibrium are large.

We also examined the predictive power of the various models by using the Fair and Schiller encompassing tests. These tests confirm that combined forecasts encompass the competing models for three selected currencies. Moreover, the results indicate that mean-reversion models outperform random walks. Out-of-sample forecast from non-linear models encompass naïve constant-change forecasts.

Finally, direction-of-change measures of forecast accuracy suggest that combining different frameworks for forecasting exchange rates generate more accurate forecasts in terms of sign forecastability. Again, the results suggest that combining the individual forecasts achieve, on average, the best performance among all the competing forecasts.

Overall, the empirical results suggest that the relative success of competing models of exchange rate forecasting mostly depend on the distance between the exchange rate 27 and its fundamentals., i.e. when this distance is not too large, linear models better approximate the exchange rate behaviour. In the limit, when exchange rates and their fundamentals coincide, the best forecasting model turns out to be the naïve random walk. In contrast, non-linear models significantly improve the forecast accuracy when the exchange rate deviates substantially from its fundamental. This means that ex-ante the degree of non-linearity a forecaster should take into account in determining the future movements of exchange rate depends on how large he judges the difference between exchange rate and fundamentals to be.

  dynamic adjustment of the exchange rate towards its long-run steady state is linear.

  the estimated models. Section 4 performs a quantitative exercise of forecast estimation and evaluation strategy. In particular, the study considers three classes of statistical measures -point forecast evaluation, forecast encompassing and directional accuracy. In Section 5, concluding remarks end the paper.

  rate forecast series obtained from model k, where k=1,…,6 represents the six models outlined above (i.e. RW, SP, MS-AR, VECM, MS-VECM, ESTAR) the combined forecast is obtained as:

  Figure 1: Weights used in Combining ForecastsEuro: 1-quarter-ahead

Figure 2 :

 2 Figure 2: Out-of-sample Point Forecasts of competing models Euro: 1-quarter-ahead

  measures -point forecasts, forecast encompassing tests and directional accuracy tests.

  using forecast encompassing tests. In particular, we use a modified version of the[START_REF] Diebold | Comparing Predictive Accuracy[END_REF] forecast comparison tests.

  .

  frequency-domain models based on spectral analysis (SP). We also proposed a forecasting method based on a weighted combination of individual forecast models.

  e

		where the frequencies range from zero to	(	( ) 1 t π -2	T	)	by increments of
	2 T π . Having transformed the series, we can look at the correlation between ( ) e t Δ
	and complex exponential (sine and cosine) functions of different frequencies. If a
	substantial proportion of the variance in	( ) e t Δ	is due to cycles of frequency
	(	)
		F o r
			P
			e
			e r
				R e v
							i e
							w
		2	/	exp		2	1	/
			1
							5

Table 1 :

 1 Comparing Forecast Accuracy

Panel A: 1992:1-2005:2 Panel B: 1992:1-1998:4

  Euro Rank Pound Rank Yen Rank Avg. Rank Euro Rank Pound Rank Yen Rank Avg. Rank When considering the full forecasting sample (table 1, panel A) the results indicate that different models are able to beat the random walk at different time horizons..

	While at the 1-step-ahead horizon the top-performing model is the ESTAR followed
	by the Comb(1-6), at 4-and 8-quarter-ahead horizons the MS-AR and MS-VECM
	models produce more accurate forecasts than the RW model. This evidence also
	emerges when analysing the first part of the sample (table 1, panel B).
		F o r
			P
				e
				e r
	1-quarter RW SP MS(2)-AR(4) MS(3)-VEC(4) ESTAR VEC(4) Comb(1-6)	4.16 [1] 4.40 [4] 5.76 [5] 6.54 [7] 4.28 [3] 6.41 [6] 4.36 [2]	3.66 [3] 3.66 [2] 5.29 [6] 5.82 [7] 3.63 [1] 5.10 [5] 3.98 [4]	5.10 [4] 5.09 [3] 6.05 [6] 6.43 [7] 5.02 [2] 5.19 [5] 4.94 [1]	R e v 1-quarter [2.7] RW 4.59 [1] [3.0] SP 4.96 [3] [5.7] MS(2)-AR(4) 6.33 [5] i e [7.0] MS(3)-VEC(4) 8.20 [7] [2.0] ESTAR 4.74 [2] [5.3] VEC(4) 6.88 [6] [2.3] Comb(1-6) 5.08 [4] w 4.84 [2] 4.81 [1] 6.84 [6] 6.97 [7] 4.91 [3] 5.11 [4] 5.12 [5]	5.01 [4] 5.31 [5] 6.20 [6] 6.46 [7] 4.99 [3] 4.91 [2] 4.93 [1]	[2.3] [3.0] [5.7] [7.0] [2.7] [4.0] [3.3]
	4-quarter					4-quarter
	RW	9.97 [3]	7.74 [2] 10.69 [3]	[2.7]	RW	9.04 [2]	8.55 [2] 11.13 [3]	[2.3]
	SP	11.60 [5]	8.93 [6] 13.03 [7]	[6.0]	SP	11.23 [4]	9.84 [5] 13.82 [7]	[5.3]
	MS(2)-AR(4)	10.97 [4]	8.45 [4] 11.93 [5]	[4.3]	MS(2)-AR(4)	11.44 [5] 10.04 [6] 13.34 [6]	[5.7]
	MS(3)-VEC(4)	17.10 [6] 10.90 [7] 11.99 [6]	[6.3]	MS(3)-VEC(4)	20.90 [6] 12.71 [7] 13.14 [5]	[6.0]
	ESTAR	9.88 [2]	7.89 [3] 11.46 [4]	[3.0]	ESTAR	9.74 [3]	9.00 [3] 11.47 [4]	[3.3]
	VEC(4)	18.10 [7]	8.79 [5] 10.13 [2]	[4.7]	VEC(4)	22.17 [7]	9.01 [4] 10.94 [2]	[4.3]
	Comb(1-6)	9.14 [1]	7.45 [1]	9.87 [1]	[1.0]	Comb(1-6)	8.95 [1]	8.43 [1] 10.56 [1]	[1.0]
	8-quarter					8-quarter
	RW	15.23 [5] 11.47 [6] 16.53 [5]	[5.3]	RW	11.59 [4] 20.62 [6] 161.55 [4]	[4.7]
	SP	15.61 [6] 11.67 [7] 17.72 [6]	[6.3]	SP	13.06 [5] 21.51 [7] 161.71 [6]	[6.0]
	MS(2)-AR(4)	10.46 [2]	8.13 [2] 11.51 [3]	[2.3]	MS(2)-AR(4)	10.82 [3] 19.66 [4] 161.02 [2]	[3.0]
	MS(3)-VEC(4)	14.29 [4]	8.57 [3] 11.29 [2]	[3.0]	MS(3)-VEC(4)	16.32 [7] 19.65 [3] 160.96 [7]	[5.7]
	ESTAR	11.41 [3] 10.89 [5] 18.00 [7]	[5.0]	ESTAR	9.95 [2] 20.45 [5] 161.63 [5]	[4.0]
	VEC(4)	17.47 [7]	9.66 [4] 16.64 [4]	[5.0]	VEC(4)	16.06 [6] 19.52 [2] 161.49 [3]	[3.7]
	Comb(1-6)	9.80 [1]	7.28 [1] 10.51 [1]	[1.0]	Comb(1-6)	9.42 [1] 18.97 [1] 160.93 [1]	[1.0]
									13

Table 2 :

 2 Comparing Forecast Accuracy Euro Rank Pound Rank Yen Rank Avg. Rank Euro Rank Pound Rank Yen Rank Avg. Rank Euro Rank Pound Rank Yen Rank Avg. Rank Euro Rank Pound Rank Yen Rank Avg. Rank

	Panel A: 1992:1-1995:1				Panel B: 1995:1 -1998:1
	1 -quarter					1-quarter
	RW	5.70 [2]	6.79 [2]	3.92 [4]	[2.7]	RW	3.26 [1]	1.79 [2]	6.10 [1]	[1.3]
	SP	5.77 [3]	6.48 [1]	3.97 [5]	[3.0]	SP	4.11 [5]	2.01 [4]	6.42 [4]	[4.3]
	MS(2)-AR(4)	8.27 [6]	9.70 [7]	5.29 [7]	[6.7]	MS(2)-AR(4)	4.04 [4]	2.03 [5]	7.37 [6]	[5.0]
	MS(3)-VEC(4)	10.37 [7]	9.33 [6]	4.25 [6]	[6.3]	MS(3)-VEC(4)	5.71 [7]	3.42 [7]	8.42 [7]	[7.0]
	ESTAR	6.10 [4]	6.95 [4]	3.69 [3]	[3.7]	ESTAR	3.07 [2]	1.85 [3]	6.26 [3]	[2.7]
	VEC(4)	7.63 [5]	7.03 [5]	3.12 [1]	[3.7]	VEC(4)	4.91 [6]	2.06 [6]	6.46 [5]	[5.7]
	Comb(1-6)	5.61 [1]	6.91 [3]	3.58 [2]	[2.0]	Comb(1-6)	3.61 [3]	1.62 [1]	6.25 [2]	[2.0]
	4-quarter					4-quarter
	RW	9.18 [1] 11.62 [3]	9.92 [5]	[3.0]	RW	9.11 [4]	4.35 [2] 12.62 [2]	[2.7]
	SP	10.80 [2] 11.98 [4] 12.04 [7]	[4.3]	SP	12.08 [5]	6.42 [5] 15.56 [5]	[5.0]
	MS(2)-AR(4)	13.45 [5] 13.46 [6]	9.19 [4]	[5.0]	MS(2)-AR(4)	8.75 [3]	5.40 [4] 17.23 [6]	[4.3]
	MS(3)-VEC(4)	20.59 [6] 16.93 [7]	7.72 [2]	[5.0]	MS(3)-VEC(4)	16.87 [7]	7.55 [7] 17.60 [7]	[7.0]
	ESTAR	11.42 [4] 12.36 [5] 10.75 [6]	[5.0]	ESTAR	7.25 [2]	4.66 [3] 12.17 [1]	[2.0]
	VEC(4)	29.45 [7] 11.30 [1]	7.10 [1]	[3.0]	VEC(4)	12.13 [6]	6.58 [6] 14.38 [4]	[5.3]
	Comb(1-6)	11.34 [3] 11.39 [2]	8.51 [3]	[2.7]	Comb(1-6)	5.40 [1]	4.07 [1] 13.00 [3]	[1.7]
	8-quarter					8-quarter
	RW	11.32 [3] 29.42 [6] 236.15 [5]	[4.7]	RW	12.23 [4]	4.29 [2] 19.11 [5]	[3.7]
	SP	11.92 [4] 30.30 [7] 236.21 [6]	[5.7]	SP	14.58 [5]	8.17 [7] 20.55 [6]	[6.0]
	MS(2)-AR(4)	13.16 [5] 28.09 [3] 235.73 [3]	[3.7]	MS(2)-AR(4)	8.31 [1]	6.14 [5] 16.46 [3]	[3.0]
	MS(3)-VEC(4)	15.46 [7] 28.27 [4] 235.70 [2]	[4.3]	MS(3)-VEC(4)	15.79 [6]	5.63 [3] 15.68 [2]	[3.7]
	ESTAR	10.15 [1] 28.98 [5] 236.29 [7]	[4.3]	ESTAR	10.42 [3]	5.98 [4] 17.88 [4]	[3.7]
	VEC(4)	14.95 [6] 27.41 [2] 235.82 [4]	[4.0]	VEC(4)	16.19 [7]	7.27 [6] 23.17 [7]	[6.7]
	Comb(1-6)	10.89 [2] 19.38 [1] 234.74 [1]	[1.3]	Comb(1-6)	8.45 [2]	3.51 [1] 13.21 [1]	[1.3]
	1 -quarter					1-quarter
	RW	4.14 [2]	2.13 [1]	6.10 [5]	[2.7]	RW	4.35 [3]	2.94 [2]	4.11 [3]	[2.7]
	SP	4.45 [4]	2.18 [2]	5.74 [1]	[2.3]	SP	4.35 [4]	2.96 [3]	4.19 [5]	[4.0]
	MS(2)-AR(4)	4.22 [3]	2.19 [3]	7.10 [6]	[4.0]	MS(2)-AR(4)	5.94 [6]	3.62 [5]	4.13 [4]	[5.0]
	MS(3)-VEC(4)	4.33 [4]	4.77 [7]	7.79 [7]	[6.0]	MS(3)-VEC(4)	5.13 [5]	3.97 [6]	4.31 [6]	[5.7]
	ESTAR	4.52 [5]	2.28 [5]	5.94 [4]	[4.7]	ESTAR	4.21 [2]	2.54 [1]	3.81 [2]	[1.7]
	VEC(4)	4.84 [6]	2.81 [6]	5.63 [2]	[4.7]	VEC(4)	7.79 [7]	6.35 [7]	4.77 [7]	[7.0]
	Comb(1-6)	3.85 [1]	2.26 [4]	5.84 [3]	[2.7]	Comb(1-6)	4.42 [1]	3.20 [4]	3.80 [1]	[2.0]
	4-quarter					4-quarter
	RW	9.73 [3]	5.26 [4] 11.54 [3]	[2.3]	RW	10.69 [5]	7.85 [4]	8.20 [3]	[4.3]
	SP	12.77 [6]	4.88 [1] 12.55 [5]	[5.3]	SP	10.68 [4]	8.93 [5] 12.31 [7]	[4.7]
	MS(2)-AR(4)	9.03 [1]	5.83 [5] 11.69 [4]	[3.0]	MS(2)-AR(4)	10.83 [6]	7.81 [3]	8.38 [5]	[6.0]
	MS(3)-VEC(4)	10.59 [4]	8.40 [4] 12.77 [7]	[5.7]	MS(3)-VEC(4)	11.44 [7]	9.78 [7]	8.33 [4]	[4.0]
	ESTAR	11.75 [5]	7.28 [6] 12.35 [6]	[4.7]	ESTAR	8.07 [1]	7.24 [1]	9.15 [6]	[4.3]
	VEC(4)	14.08 [7]	5.10 [3] 11.17 [2]	[3.7]	VEC(4)	9.78 [3]	9.71 [6]	7.35 [1]	[2.0]
	Comb(1-6)	9.22 [2]	4.90 [2] 10.14 [1]	[1.7]	Comb(1-6)	8.57 [2]	7.60 [2]	7.68 [2]	[2.0]
	8-quarter					8-quarter
	RW	15.09 [5]	7.37 [5] 16.65 [5]	[5.0]	RW	19.95 [7] 13.70 [7] 11.94 [4]	[6.0]
	SP	17.01 [7]	7.30 [4] 13.97 [4]	[5.0]	SP	19.61 [5] 13.01 [6] 16.10 [7]	[6.0]
	MS(2)-AR(4)	9.48 [1]	5.42 [2] 12.28 [3]	[2.0]	MS(2)-AR(4)	11.88 [1]	8.46 [1]	7.92 [1]	[1.0]
	MS(3)-VEC(4)	12.12 [3]	6.32 [3] 12.11 [2]	[2.7]	MS(3)-VEC(4)	12.88 [3]	9.36 [3]	8.98 [3]	[3.0]
	ESTAR	13.12 [4]	9.08 [7] 17.54 [7]	[6.0]	ESTAR	13.52 [4] 11.96 [4] 14.29 [6]	[4.7]
	VEC(4)	16.81 [6]	7.56 [6] 17.12 [6]	[6.0]	VEC(4)	19.77 [6] 12.00 [5] 13.83 [5]	[5.3]
	Comb(1-6)	9.98 [2]	4.52 [1]	9.74 [1]	[1.3]	Comb(1-6)	12.01 [2]	9.03 [2]	8.96 [2]	[2.0]
	More precisely, the accuracy of the alternative forecasts can be judged according to
	some loss function, (.) g . In analysing formal tests of the null hypothesis of equal
	forecast accuracy, we follow Diebold and Mariano (1995) and define the loss function
	as a function of the forecast errors. The loss differential is then denoted as

Panel C: 1998:1 -2001:1 Panel D: 2001:1-2005:2

Table 3

 3 to 5 report the modified DM and the Granger and NewboldT P F 11

	FP T	test

T P 10 P T

Table 3 :

 3

				Diebold-Mariano			Granger-Newbold
			Euro	Pound	Yen	Euro	Pound	Yen
	Model i	Model j	Test Stat p-value Test Stat p-value Test Stat p-value Test Stat p-value Test Stat p-value Test Stat p-value

Table 4 :

 4

	Model i	Model j	Test Stat p-value Test Stat p-value Test Stat p-value Test Stat p-value Test Stat p-value Test Stat p-value
	RW	SP

Table 5 :

 5

				Diebold-Mariano			Granger-Newbold
			Euro	Pound	Yen	Euro	Pound	Yen
	Model i	Model j	Test Stat p-value Test Stat p-value Test Stat p-value Test Stat p-value Test Stat p-value Test Stat p-value

Table 6 :

 6 Dollar/Euro -Fair and Shiller encompassing test

	1-quarter-ahead
	m1
	m2
		Coeff Tsta Coeff Tsta Coeff Tsta Coeff Tsta Coeff Tsta Coeff Tsta Coeff Tsta
	c α 1 α 2 c α 1 α 2 c α 1 α 2 c α 1 α 2 c α 1 α 2 c α 1 α 2 c α 1 α 2	-0.03 -0.05 -0.07 -0.12 -0.02 -0.04 0.03 0.05 0.04 0.07 0.04 0.08 0.24 0.71 0.09 0.63 0.15 1.54 0.44 1.54 0.11 1.05 0.42 1.63 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.05 -0.08 -0.02 -0.04 0.04 0.07 0.04 0.06 0.06 0.10 0.05 0.32 0.15 1.35 0.59 1.44 0.09 0.83 0.58 1.58 0.18 0.47 0.01 0.04 -0.24 -0.51 0.13 0.35 -0.30 -0.62 -0.07 -0.12 -0.05 -0.08 -0.03 -0.05 0.00 0.00 0.24 0.71 0.04 0.07 0.03 0.05 0.02 0.03 0.11 0.19 0.00 0.00 0.18 0.47 0.26 1.65 0.44 1.39 0.10 1.00 0.59 1.65 0.09 0.63 0.05 0.32 -0.20 -0.87 0.00 0.02 0.08 0.56 -0.14 -0.69 -0.02 -0.04 -0.02 -0.04 0.04 0.07 0.02 0.04 0.04 0.06 0.03 0.04 0.00 0.00 0.01 0.04 -0.20 -0.87 0.31 0.96 0.07 0.71 0.29 0.63 0.15 1.54 0.15 1.35 0.26 1.65 0.11 0.96 0.14 1.32 0.06 0.36 0.03 0.05 0.04 0.07 0.03 0.05 0.02 0.04 0.07 0.12 0.05 0.09 0.00 0.00 -0.24 -0.51 0.00 0.02 0.11 0.96 0.06 0.55 0.28 0.77 0.44 1.54 0.59 1.44 0.44 1.39 0.31 0.96 0.38 1.24 0.23 0.56 0.04 0.07 0.04 0.06 0.02 0.03 0.04 0.06 0.07 0.12 0.05 0.09 0.00 0.00 0.13 0.35 0.08 0.56 0.14 1.32 0.38 1.24 0.39 1.24 0.11 1.05 0.09 0.83 0.10 1.00 0.07 0.71 0.06 0.55 0.02 0.14 0.04 0.08 0.06 0.10 0.11 0.19 0.03 0.04 0.05 0.09 0.05 0.09 0.00 0.00 -0.30 -0.62 -0.14 -0.69 0.06 0.36 0.23 0.56 0.02 0.14 0.42 1.63 0.58 1.58 0.59 1.65 0.29 0.63 0.28 0.77 0.39 1.24
	4-quarter-ahead
	m1
	m2
		Coeff Tsta Coeff Tsta Coeff Tsta Coeff Tsta Coeff Tsta Coeff Tsta Coeff Tsta
	c α 1 α 2	0.13 0.10 0.19 0.14 0.06 0.05 0.15 0.12 0.25 0.18 0.09 0.07 -0.11 -0.43 0.06 0.24 0.08 0.95 0.48 1.89 -0.05 -0.53 0.81 3.40

Table 7 :

 7 Dollar/Pound -Fair and Shiller encompassing test

	α 1 α 2 c α 1 α 2 c α 1 α 2 c α 1 α 2 c α 1 α 2 c α 1 α 2	0.00 0.00 0.31 0.78 -0.81 -0.55 -0.65 -0.44 0.00 0.00 0.26 0.95 1.17 7.76 1.17 7.72 -0.69 -0.38 -0.48 -0.26 -0.70 -0.48 1.17 7.72 0.57 4.70 1.40 6.67 0.29 1.80 1.66 10.64 0.26 0.95 0.36 1.08 -0.13 -0.43 0.18 0.46 0.06 0.25 -0.70 -0.48 -0.53 -0.43 -0.29 -0.20 -1.18 -0.98 -0.18 -1.04 0.87 4.81 0.23 2.11 1.47 5.16 1.40 5.30 0.84 5.80 1.14 7.79 0.18 0.78 -0.40 -0.28 -0.02 -0.01 -1.05 -0.91 0.00 0.00 0.36 1.08 1.40 5.30 1.15 5.98 0.30 2.32 2.01 8.63 0.57 4.66 0.57 4.70 -0.18 -1.04 0.37 3.76 0.57 4.86 -0.24 -1.95 0.29 0.18 0.21 0.13 -0.53 -0.43 -0.40 -0.28 -0.79 -0.54 -0.92 -0.79 0.00 0.00 -0.13 -0.43 0.84 5.80 0.37 3.76 -0.48 -3.17 1.36 6.68 1.38 6.80 1.40 6.67 0.87 4.81 1.15 5.98 1.95 7.53 0.44 2.16 1.04 0.49 1.12 0.53 -0.29 -0.20 -0.02 -0.01 -0.79 -0.54 -1.39 -1.13 0.00 0.00 0.18 0.46 1.14 7.79 0.57 4.86 1.95 7.53 1.71 10.28 0.30 1.93 0.29 1.80 0.23 2.11 0.30 2.32 -0.48 -3.17 -0.08 -0.81 -1.17 -0.98 -1.13 -0.93 -1.18 -0.98 -1.05 -0.91 -0.92 -0.79 -1.39 -1.13 0.00 0.00 0.06 0.25 0.18 0.78 -0.24 -1.95 0.44 2.16 -0.08 -0.81 1.66 10.83 1.66 10.64 1.47 5.16 2.01 8.63 1.36 6.68 1.71 10.28

Table 8 :

 8 Dollar/Yen -Fair and Shiller encompassing test

	α 1 α 2 c α 1 α 2	0.00 0.00 0.24 1.08 0.65 5.93 0.68 6.77 0.36 0.85 1.84 9.27 1.80 8.95 1.33 7.61 1.48 9.59 1.64 5.27 -1.85 -2.55 -1.53 -1.82 -1.62 -2.24 -1.92 -2.74 1.39 0.90 -5.42 -5.50 0.00 0.00 -0.13 -0.77 -0.32 -1.80 -0.36 -2.23 0.56 2.34 0.77 4.63 2.00 14.52 2.03 14.00 2.43 8.88 2.44 10.28 1.76 10.61 1.49 9.30	1.49 9.30 0.77 4.63

Table 9

 9 reports the proportions of forecasts that correctly predict the actual directions of the dollar-euro, dollar-pound and dollar-yen. If a model has a CDC greater than 50% this means that it performs better than a random toss of a coin. The results suggest that in most cases the number of times the direction of change of both the actual and forecasted series agrees is greater than what would be expected by chance. Combined models outperform the other models over all forecasting horizons.ESTAR models are the second top performing models over both 1 and 4 quartersahead. At 8-step-ahead, with the exception of the COMB, MS-VECM models collect a higher number of agreements than the other five competitors. In general, non-linear models generate more accurate forecasts in terms of direction-of-change than their linear competitors.

	F o r
	P
	e
	e r
	R e v
	i e
	w

Table 9 :

 9 Percentage of quarters where forecasts detect the correct direction of change

		Euro	Rank	Pound Rank	Yen	Rank Avg. Rank
	1-quarter-ahead
	RW	64.02%	[5]	50.90%	[7]	64.02%	[5]	[5.7]
	SP	59.10%	[7]	59.10%	[4]	65.66%	[4]	[5.0]
	MS(2)-AR(4)	62.38%	[6]

Page 27 of 29 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK

	Submitted Manuscript
	F o r F o r
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	e r e r
	R e v R e v
	i e i e
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Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript


Paul Dolan

Aki Tsuchiya
email: a.tsuchiya@sheffield.ac.uk

Determining the parameters in a social welfare function using stated preference data: an application to health

Keywords: social welfare function, preference elicitation, equality-efficiency trade-off JEL Classification: I10 social welfare function, preference elicitation, equality -efficiency trade-off

Determining the parameters in a social welfare function using stated preference data: an application to health

I. INTRODUCTION

An important consideration when establishing priorities in the public sector is the amount of benefit generated by alternative allocations. As a result, there has been considerable research effort devoted to developing technologies that allow the benefits from a range of public services to be measured and subsequently valued. If benefits were the only consideration, then the objectives of public policy could be defined in terms of the maximisation of these benefits. However, policy-makers, as well as the general public, are also likely to be concerned with how benefits are distributed.

Although standard economic models assume that people do not care about inequalities, there is increasing interest in the economics literature in people's preferences regarding fairness [START_REF] Rabin | Incorporating fairness into game theory and economics[END_REF][START_REF] Fehr | A theory of fairness, competition and cooperation[END_REF]. These models focus on self-centred inequality aversion in the sense that people care only about their own outcomes relative to those of other people, but they do not care about inequality amongst other people per se. This has important implications for how social welfare functions (SWFs) are constructed and how preferences are aggregated (for example, see [START_REF] Quesada | Positional independence in preference aggregation[END_REF]. In this paper, we are concerned with an individual's distributional preferences as they relate to the treatment of other people. Specifically, our inquiry is into whether a SWF can be constructed from people's other-regarding, or social, preferences [START_REF] Menzel | How should what economists call 'social values' be measured?[END_REF].

The SWF is, in principle, a powerful device for determining how best to balance these competing objectives of efficiency and equality. However, in practice, there has hitherto been only limited success in developing a SWF that is operationally useful. There have been some attempts to estimate the parameters of SWFs from the stated preferences of individuals. For example, [START_REF] Amiel | Thinking about Inequality[END_REF] have asked respondents to choose between different distributions of income across population groups. Similar attempts have been made to estimate the SWF for health, using relatively small samples of students [START_REF] Dolan | The measurement of preferences over the distribution of benefits: the importance of reference points[END_REF]. In this context, a policy that maximises population health might be of relatively less benefit to less healthy groups, or a policy that reduces inequalities might forego the opportunity to improve the health of the relatively healthy. This paper demonstrates how the stated preferences of a sample of the general public can be used to estimate the parameters of a SWF in the domain of health. The issues addressed in this paper are of real policy concern in many countries, such as the United Kingdom, Australia and New Zealand, that have put into place policies that seek to improve overall population health and reduce health inequalities (Department of Health, 1999;[START_REF] Rice | Capitation and Risk Adjustment in Health Care Financing: An International Progress Report[END_REF].

In order for the SWF approach to be operationally useful, two main questions need to be answered: 1) what type of SWF is to be employed; and 2) how is the shape of the SWF to be determined? Sections 2 and 3 deal with each of these questions in turn and Section 4 presents the design of an empirical study that elicited the public's preferences over two health programmes, one that maximises health and one that reduces inequalities in health between particular population subgroups. Section 5 shows how these data can be used to derive a set of relative weights to be given to a unit health gain to people from different population subgroups and hence to estimate the shape of the SWF. Section 6 discusses the implications of the results.

II. DEFINING THE SWF

The SWF in economics textbooks are welfarist in the sense that they rely on an individual's subjective assessment of her own well-being and, as such, they are concerned with the distribution of individual utility. In the few empirical studies that have sought to empirically estimate the parameters in a SWF, economists have used more readily quantifiable proxies for utility, such as in income [START_REF] Amiel | Thinking about Inequality[END_REF]. Health economics have a tradition of expressing utility in health-related terms e.g. in terms of quality-adjusted life years (QALYs), where the quality adjustment weight -the 'Q' in the QALY -is typically defined in terms of dimensions of a health state classification system (see for example [START_REF] Drummond | Methods for the Economic Evaluation of Health Care Programmes[END_REF]. Whilst health (and income for that matter) represents only a sub-set of the determinants of utility, it more readily allows for interpersonal comparisons and may suffer from fewer of the ethical problems with comparisons of utility identified by [START_REF] Sen | Inequality reexamined[END_REF]. From a public policy perspective, citizens and policy-makers may well prefer to focus on the distribution of health across society rather than on the distribution of utility (see for example, [START_REF] Dolan | Distributing Health Care: Economics and Ethical Issues[END_REF][START_REF] Tsuchiya | Chapter 20 Social choice in health and health care[END_REF].

In short, there are good normative and practical reasons for defining a health-related SWF in terms of the different levels of health experienced by different groups [START_REF] Dolan | The measurement of individual utility and social welfare[END_REF]. In this paper, differences in health are represented as differences in average life. When the analytical objective is that of searching for a more equal distribution of health (rather than utility or health-related utility), a SWF in terms of health is potentially more useful in a policy context since life is more readily interpersonally comparable than utilities (see [START_REF] Olsen | Theories of justice and their implications for priority setting in health care[END_REF].

In this study, we assume that health-related social welfare is a function of: a) the average levels of health of different groups within a given population; and b) the inequalities in health that exist between those groups. Of course, differences in health exist within any population sub-group, as well as between groups. In principle, the SWF could be estimated across groups of any size, including across individuals, but in practice it would be impossible to get reliable health data at such a micro level.

There are a number of functional forms that this SWF can take. In order to deal with varying degrees of inequality, an additive SWF with convexity to the origin to allow for inequality aversion has been widely used in the literature [START_REF] Atkinson | On the measurement of inequality[END_REF][START_REF] Little | Project appraisal and planning for developing countries[END_REF][START_REF] Layard | Allowing for income distribution[END_REF]. Let us start with a SWF with a constant elasticity of substitution (CES):

[ ] [ ] , 1 1 r r b r a H H W - - -+ = β α , 0 , > b a H H , 1 = + β α , 1 - ≥ r , 0 ≠ r [1]
where W is the health-related social welfare and H a and H b are the average levels of health of groups of equal size. (The function is, of course, generalisable to more than two groups, and to groups of different sizes.) The nature of the SWF, and the resulting iso-welfare curves, is determined by r and α. The parameter r measures the degree of aversion to inequality, as represented by the convexity of the iso-welfare curves. If r = -1, social welfare is equal to the sum of individual health and there is no aversion to inequality. This utilitarian-type SWF results in iso-welfare contours that are straight lines with a gradient of -1. If r > -1, there is aversion to inequality, or diminishing marginal rate of social substitution (MRSS) between the health of the two groups: along a given iso-welfare curve, the greater the inequalities in health between the two groups, the greater is the weight given to the worse-off group relative to the better-off group. In the extreme, r

approaches ∞ and all that matters is the health of the worse-off group. This results in a Rawlsian-type SWF with right-angled iso-welfare curves.

The parameters α and β determine the rate at which the welfare of subgroups a and b enter the social welfare calculus. It might be argued, for example, that relatively less weight should be given to those who considered more responsible for their poor health [START_REF] Le Grand | Equity and choice[END_REF]Schokkaeart and Devooght, 2003). In the literature on SWFs, however, it is common to assume 'anonymity', which implies that both individuals and groups are equally deserving of any given gain in well-being [START_REF] Musgrave | The Theory of Public Finance[END_REF][START_REF] Harsanyi | Morality and the theory of rational behaviour[END_REF][START_REF] Boadway | Welfare Economics[END_REF]. In this paper, we will also make this anonymity assumption i.e. that α = β = 0.5.

The objective of this study is to derive the implied weights to be given to a unit health gain to one group relative to another. This is represented as the MRSS along the relevant iso-welfare curve. As such, reference is made only to contours of the SWF, and not to the level of social welfare implied by these contours. In this respect, the CES SWF is equivalent to the Atkinson SWF [START_REF] Atkinson | On the measurement of inequality[END_REF] that was first proposed to address income distributions and has been applied to the health context [START_REF] Wagstaff | QALYs and the equity-efficiency tradeoff[END_REF]. The CES specification is chosen as the baseline specification because it is individualistic, additive, non-decreasing (or monotonic), strictly concave, exhibits constant relative inequality aversion (or scale independence or homotheticity) and, with α = β, it also satisfies anonymity, and thus, satisfies all the conventional requirements of a SWF.

We will also look at two alternative SWF specifications by way of sensitivity analysis.

These are derived from the generic form:

F o r P e e r R e v i e w 5 [ ] λ κ b a b a H H c H H W - - + = ) ( 2 , where 0 > κ , 1 ≥ κ λ , 0 ≥ c , [2]
which provides a family of SWFs that are increasing in total health and decreasing in inequality in health [START_REF] Abásolo | Exploring social welfare functions and violation of monotonicity: an example fro inequalities in health[END_REF].

When κ = λ = 2, the contours become hyperbolic and when κ = 1 and λ = 2, they become parabolic. The parabolic and hyperbolic specifications are individualistic, additive, inequality averse and symmetric. The parabolic specification satisfies constant relative inequality aversion, while the hyperbolic specification satisfies constant absolute inequality aversion instead. The main characteristic of these two specifications is that beyond a given level of inequality, social welfare ceases to be non-decreasing in composite health.

III. ESTIMATING THE MRSS GIVEN A SWF SPECIFICATION

The question now is how do we identify the MRS, given a SWF specification? One way is to elicit the preferences of the general public over stylised questions specifically designed to allow us to identify points on the same social welfare contour. [START_REF] Williams | Intergenerational equity: an exploration of the 'fair innings' argument[END_REF] suggests that respondents could be presented with the current unequal distribution of health and then asked to think about an equal distribution of health that makes them indifferent between the two distributions, which corresponds to the concept of equally distributed equivalent income by [START_REF] Atkinson | On the measurement of inequality[END_REF]. In this way, the general format of the questions would be similar to those used in empirical studies that have attempted to measure the degree of inequality aversion in relation to income distributions [START_REF] Amiel | Thinking about Inequality[END_REF]Amiel et al, 1999). However, whilst it is possible to take income from one person and transfer it to another, it is not possible to redistribute health in the same way.

Therefore, it seems more appropriate to design the empirical study in terms of the distribution of gains in health from an initial position.

Figure 1 shows the basis of the questions. The initial situation (I) is presented to respondents together with a programme (A) that will benefit both groups by the same amount. They are then presented with an alternative programme (B) that targets the benefit on the worse-off group. The aim then is to determine, in an iterative way, how much Programme B would have to benefit the worse-off group For the baseline CES specification, the value of r can be obtained by using standard spreadsheet procedures (e.g. the "goal seek" tool in MS Excel), by looking for the value of r that makes W [1] identical at two points, X and Y. Alternatively, for a mathematical solution, see [START_REF] Abásolo | Exploring social welfare functions and violation of monotonicity: an example fro inequalities in health[END_REF]. The weight implied to the less advantaged group a relative to group b is calculated from the MRSS:

) 1 ( r a b a b H H dH dH +       = - .
Provided r > -1, MRSS increases exponentially with the extent of the equalityefficiency trade-off (since the iso-welfare contour in Figure 1 is convex), and so the mean of any group of values would give greater relative weight to the preferences of those most concerned about equality. This makes it difficult to account for the strength of each individual's preferences in the overall preferences of a group. For this reason, we will concentrate our analysis and interpretation on the median. Use of the median is also consistent with the median voter rule, which has been used to model public policy choices [START_REF] Mueller | Public Choice[END_REF]. Furthermore, the relationship between the number of years traded off and the level of inequality aversion implies that the mean of the former will not correspond to the mean of the latter, thus making the median a more attractive summary measure to use.

For the hyperbolic and the parabolic specifications, suppose H a (X), H b (X) and H a (Y), H b (Y) represent two points on the same indifference curve. By solving

W [2] (X)=W [2] (Y) for C: [ ] [ ] [ ] [ ] β β α α ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( Y H Y H X H X H Y H Y H X H X H C b a b a b a b a - - - + - + = ,
the relative weight for the less advantaged group at point (H a ,H b ) will be given by: Any one respondent could be asked to adopt a number of different perspectives when answering questions of the kind used in this study (see [START_REF] Dolan | An inquiry into the different perspectives that can be used to elicit preferences to inform prioritysetting[END_REF]. In this study, we asked respondents to adopt a citizen-type perspective, where they are not explicitly asked to think about being in either group. This is the perspective adopted by [START_REF] Amiel | Thinking about Inequality[END_REF] (1955) and [START_REF] Etzioni | The case for a multiple utility conception[END_REF]. We therefore collected information on a range of background characteristics in order to examine the extent to which self-interest might be playing a part in responses.

[ ] [ ] [ ] [ ] ) 1 ( ) 1 ( ) 1 ( ) 1 ( - - - - - + + - - + = - β α β α β α β α b a b a b a b a a b H H C H H H H C H H dH dH .

IV. THE QUESTIONNAIRE DESIGN

Differences in health in this study, as noted in Section 2, have been defined in terms of average life expectancy. The most obvious differences in mortality in the UK exist between the social classes (Acheson 1998). Of the six social classes often used in British surveys, we employ data concerning the top and the bottom classes, which highlights the extent of the prevailing inequalities and has the advantage that the fraction of the population in each of these classes is roughly the same (about 7% in each case). On average, people in the highest social class (such as doctors and other professionals) live five years longer than those in the lowest social class (unskilled manual workers such as cleaners.

Scenarios with population subgroups other than social class are also used.

Differences of the same magnitude (five years) in average life expectancy exist between women and men. This means that by presenting separate respondents with identical questions regarding life expectancy, but relating them to differences by sex instead of by social class, it is possible to test whether the degree of inequality aversion is a function of the groups across which the inequalities exist. To further test the sensitivity of inequality aversion, other respondents were presented with the same life expectancy differences across groups that were simply defined as the 'healthiest 20%' and the 'unhealthiest 20%' of the population. The questionnaire was administered during a face-to-face interview, which gave the interviewer the opportunity to assess the respondent's understanding of the task and provided the respondent with the opportunity to ask any clarificatory questions.

The interview began with a brief description of the task and an explanation of the population sub-group used. The questionnaire was developed through in-depth interviews and extensive piloting, during which time it emerged that the clearest way in which to represent the health of the two groups was in the form of graphical representations, as shown in the Appendix. Respondents were first asked to make a discrete choice between Programme A (that benefits both groups by the same amount) and Programme B (that targets the same amount of overall benefit on the worse-off group). They were told that the two groups were of approximately equal size and that the two Programmes would cost the same.

For those respondents who chose Programme A, it was assumed that, since they were unwilling to target the worse off group when overall benefits were the same, they would also be unwilling to target the worse off group when overall benefits were reduced, and so no further sub-questions were asked. Those respondents who chose Programme B were presented with a series of pairwise choices in which the benefits from choosing B were gradually reduced. This order was chosen to make the trade-off between efficiency and equality as transparent as possible and because it was felt that it would be cognitively less demanding for respondents than a random order that would have required them to 'jump around' between different trade-offs. Note that respondents were not provided with the opportunity to state that they were indifferent between the two Programmes. This option was in the pilot interviews but was never chosen and in fact caused confusion.

The interviews were carried out in two rounds using different respondents. In the first round around half the respondents were given the social class scenario and the other half were given the sex scenario. In the second round around half the respondents were given the social class scenario and the other half were given the quintiles scenario. The response categories presented in the two rounds (independently of scenario) are shown in Table 1. Respondents in the first round who initially chose Programme B were presented with six additional pairwise For those respondents who initially chose to target on Programme B but then switched at some point to Programme A, their point of indifference has been taken

to be half-way between the last point at which they chose B and the first point at which they chose A. The first columns of Table 2 present the implied points of indifference, their associated inequality aversion parameters depending on SWF specification, and their corresponding implied relative weights to the worst off group at the initial point, given the options in Table 1. The precise trade-offs made by those who choose not to target and by those who always choose to target are indeterminate, and so, strictly speaking, inequality aversion can only be calculated for those respondents who switch from Programme B to Programme A at some point. Having said this, for those who chose A in the initial pairwise comparison, we have assumed that they are inequality neutral (although we cannot rule out the possibility that some respondents may have favoured increased inequality). For those who always chose B, we have assumed that they are indifferent at the implied point presented on the last row, but again we cannot be sure.

As can be seen, the choice functional form for the SWF does not have much effect on the relative weight given to the two groups except in the highly inequality averse region. And even in this range, it is more important to identify the preferences accurately than it is to identify the correct functional form to represent those preferences. Under extreme inequality aversion, where a reduction in inequality is preferred even when it entails loss in the health of the better off so that the monotonicity principle is violated, the CES specification can no longer accommodate such preferences. However, the hyperbolic and the parabolic specifications can, and the relative weights across these two functional forms are similar to each other. were male (compared to 47% for the region as a whole); 50% were aged under 45 (compared to 50%); 67% had children (compared to 66%); 54% were employed (compared to 56%); and 60% had the minimum level of education (compared to 61%). This paper is based on the life expectancy question, which appeared at the beginning of the interview and was answered by all 130 respondents.

The results are summarised in Table 2 where the last five columns present the distribution of responses. The numbers of respondents were 29 and 37 for the social class scenario across the two rounds, 31 for the sex scenario (in round one)

and 33 for the quintiles scenario (in round two). Since the implied trade-offs that respondents made between the social classes did not differ across the rounds (Mann Whitney U Test, p>0.05), pooled responses are also reported. Using these pooled results, the median respondent is indifferent between people in the highest and lowest social classes living on average to be 80 and 75, respectively (i.e. the outcome for choosing Programme A), and these groups living to be 78 and 75.5, respectively (i.e. the outcome for Programme B from the median respondent). Depending on the SWF specification, this implies that a marginal health gain to the lowest social class is valued 6.8 to 9.9 times more than a marginal health gain to the highest social class. This is also the median response when the sub-groups are defined in terms of the healthiest and unhealthiest quintiles of the population. However, when identical data are presented but the sub-groups are defined by sex, the median preference is to favour no targeting of men at all, thus implying that a marginal health gain to men and to women are equally valued. The responses were not related to any of the personal characteristics (using the χ 2 test, p>0.05). This study has sought to determine the shape of a health-related SWF from people's stated preferences over various equality -efficiency trade-offs. While a CES was used as the baseline specification, similar results are derived from the hyperbolic and the parabolic specifications.

Overall, the results seem plausible, suggesting that there is aversion to inequalities in life expectancy, but its extent is sensitive to the groups across which the inequalities exist. However, the study also raises a number of methodological issues that warrant further discussion. In the first part of each question, the information regarding the size of the health gains of the two Programmes was easy to understand and, in the second part of each question, the implications of choices were made clear through changes in the size of the bars on the graph. Nevertheless, to facilitate this visual representation, the scales on the graphs did not start at zero (see the Appendix), and this could have led some respondents to perceive that the relative difference between the two groups was larger than it really was.

In general, it has been shown that very subtle changes in the framing of a question can sometimes have a dramatic effect on responses (for an excellent review, see [START_REF] Rabin | Psychology and economics[END_REF]. This study was designed to minimise the effects of certain framing effects but it is impossible to remove every potential bias. For example, we were aware of the evidence from other studies that suggests that respondents might be reluctant to give all the benefit to one individual or group (see, for example, Cuadras-Morato et al 2001). We went further, though, and asked respondents who chose not to target if they would have targeted if there had instead been a one-year benefit to the better-off group (and hence a three-year benefit to the worse-off group). None of these respondents chose to revise their answers.

It is now well established that respondents may give greater weight to the losses of one group as compared to an equivalent gain to the other group [START_REF] Schweitzer | Multiple reference points, framing, and the status-quo bias in health-care financing decisions[END_REF].

Therefore, the questions were designed so that neither Programme in the two questions involved any losses, and so that neither Programme was presented as representing the status quo. However, it is possible that loss aversion may also be present when considering potential as well as actual losses from a particular reference point [START_REF] Dolan | The measurement of preferences over the distribution of benefits: the importance of reference points[END_REF]. Therefore, if some respondents adopted the potential gains available to both groups in Programme A as their reference point, then Programme B would involve a 'loss' to the better-off group. It would be interesting, and policy relevant, to test with further research how sensitive the degree of inequality aversion is to variation in the initial situation.

There is a status quo bias of a different kind that might have made respondents more inclined to stick with Programme B if they chose it initially. This relates to the fact that respondents were always presented with response categories in the same order; that is, Programmes A and B start out being equally effective and then B becomes incrementally less effective. This ordering was chosen to make the equality -efficiency trade-off as transparent as possible and was informed by the results from the pilot interviews which suggested that the trade-off questions would have been cognitively too difficult if the ordering of the response categories was randomised. However, there is the possibility of a status quo bias whereby some respondents get 'locked into' choosing B throughout (see Samuelsen and Zeckhauser 1988). On the other hand, there is some limited evidence that shows there may be a 'left hand side' bias: when respondents are asked to choose between two options laid out next to each other, the default choice is the option on the left hand side, and the right hand side option will be chosen only when it is significantly more preferable than the default option on the left hand side (see, for example, [START_REF] Mcintosh | Eliciting preferences for hernia repair: A comparison of strength of preference modelling and discrete choice experiments[END_REF]. Thus, there are two potential biases working in opposite directions.

Despite these concerns about the data, we believe that this study represents an advance in terms of both the methodology used and the implications for future research that seeks to enhance the policy usefulness of stated preference data. It suggests that people are willing to forego overall health in order to reduce differences in average life expectancy between the social classes. On the other hand, differences in the average life expectancies of men and women did not seem to matter much at all, with the median respondent unwilling to sacrifice any overall gains in life expectancy in order to target men. [START_REF] Tsuchiya | A "Fair Innings" Between the Sexes: Are Men Being Treated Inequitably?[END_REF] tries In conclusion, this study has demonstrated that, using carefully designed questionnaire instruments, the SWF can develop from being a theoretical construct to becoming a potentially powerful practical policy tool. A survey instrument can be designed that elicits meaningful trade-off responses from the general population that can then be used to determine the shape of the SWF. We therefore believe that the study indicates a promising new avenue of economic enquiry that is relevant to important policy questions in health care and other areas of public policy.

  be considered equally as valuable as Programme A. Once indifference between Programmes A and B has been established, the MRSS of the SWF can be calculated.

  response categories in the second round of interviews were revised in the light of the distribution of responses from the first round, resulting in only four additional pairwise choices in the second round. In addition to some of the response categories in the first round being largely redundant, this allowed us to test whether respondents were following a particular pattern of responses e.g. choosing the middle option.

  interview a broadly representative sample of the general population, every 8 th person on the electoral register in three wards in York, UK, was contacted and invited to participate, for which they would receive £15. Out of a total of 1,500 letters of invitation, 467 people (31%) agreed to take part. To ensure representativeness, 140 respondents were selected for interview based on information on a broad range of characteristics obtained from their reply slips. In total, 130 individuals were interviewed. The interviews took place at the University of York and lasted for about an hour. The achieved sample was broadly representative of the population of the Yorkshire and Humberside region: 48%

  some of the reasons for the very different attitudes towards health inequalities by sex as compared to those by social class.

Figure

  Figure 1: The SWF and the life expectancy questions

  in their empirical studies on income inequalities. To us, and as famously emphasised by Rousseau (1762), there is a legitimate distinction between a person's self-regarding preferences based on her own self-interest and her society-regarding preferences which reflect her views about what society should look like. The distinction has more recently received attention -and support -from a number of economists and political scientists, including Harsanyi

Table 1 :

 1 The response optionsThe initial situation is one in which group a (the worst-off group) live to be 73 and group b (the best-off group) live to be 78. The numbers in the Table show average increases in life expectancy per group depending on the Programmes chosen for each of the pairwise choices.

	1 st round of interviews			2 nd round of interviews
	Programme A	Programme B	Programme A	Programme B
	Group a Group b	Group a Group b Group a Group b Group a Group b
	+2	+2	+4	+0	+2	+2	+4	+0
	+2	+2	+3.5	+0	-	-	-	-
	+2	+2	+3	+0	+2	+2	+3	+0
	+2	+2	+2.5	+0	-	-	-	-
	+2	+2	+2	+0	+2	+2	+2	+0
	+2	+2	+1.5	+0	+2	+2	+1.5	+0
	+2	+2	+1	+0	+2	+2	+1	+0
	"-" indicates where a response category was not offered to respondents

Page 21 of 23 Editorial Office, Dept of Economics, Warwick University, Coventry CV4 7AL, UK Submitted Manuscript F o r P e e r R e v i e w 21Table 2 :

 2 The three SWF specifications and the results Inequality aversion parameter is r for the CES, C for the hyperbolic and the parabolic 2. Relative weight at initial point is the implied equality weight given to group b relative to group a at the initial point where life expectancy for groups a and b are 73 and 78 respectively.3. Median respondent in bold; "-" indicates where a response category was not offered to respondents

	response options	as inferred from	Indifference points	parameter 1	aversion	inequality	CES	at initial point 2	relative weight	parameter 1	aversion	hyperbolic inequality at initial point 2	relative weight	parameter 1	aversion	parabolic inequality at initial point 2	relative weight	(n= 29)	Round 1	Social class 3 Round 2 (n=37)	(n=66)	Pooled	Sex 3 (n=31)	Quintiles 3 (n=33)
	(75,80) ~ (77,78)	-1.0		1.00	0.00	1.00	0.00	1.00		9	15	24	20	12
	(75,80) ~ (76.75,78)		2.3			1.25	3.30	1.25	0.01	1.24		0	-	0	1	-
	(75,80) ~ (76.5,78)		5.9			1.58	6.80	1.58	0.02	1.56		-	5	5	-	2
	(75,80) ~ (76.25,78) (75,80) ~ (75.75,78) (75,80) ~ (75.50,78) (75,80) ~ (75.25,78) (75,80) ~ (74.75,78) unspecified unspecified 9.8 2.05 20.1 4.04 27.9 6.80 41.2 16.38 (75,80) ~ (74.25,78) unspecified unspecified (75,80) ~ (73.5,78) unspecified unspecified Notes o r P e e r 10.57 19.36 24.68 30.94 47.96 77.25 225.84 1. F R	2.08 4.57 9.94 -83.12 -4.40 -2.28 -1.31	0.03 0.06 0.08 0.10 0.16 0.25 0.74	2.04 4.36 9.00 -559.00 -4.58 -2.32 -1.31	0 2 -11 4 0 3	--11 -5 0 1	0 2 11 11 9 0 4	0 1 -5 3 -1	--14 -3 1 1
											e
												v i e
													w

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

ACKNOWLEDGEMENTS

We are indebted to Rebecca Shaw who conducted the interviews. The Economic and Social Research Council (Award No: L128251050) supported Rebecca Shaw and the fieldwork. We would like to thank John Cairns, Angus Deaton, Atsushi Higuchi, Daniel Kahneman, Peter Lambert, Chris McCabe, Jan Abel Olsen, and the anonymous referee for Applied Economics for their helpful comments. The paper has also benefited from the comments made by participants at a range of seminars, including those held at Princeton, Toronto, East Anglia and Oslo. We are grateful to all the respondents who agreed to take part. Last but not least, special thanks are due to Peter Smith and Alan Williams. The usual disclaimers apply.

Appendix: Example of the questions across social class

As you might know, average life expectancy differs by social class.

Whilst actual life expectancy varies between individuals, on average, people in social class 1 live to be 78 and in social class 5 they live to be 73.

Imagine that you are asked to choose between two programmes which will increase average life expectancy. Both programmes cost the same.

In the two graphs below the light grey part shows average life expectancy, and the dark grey part shows the increase in life expectancy. There is a separate graph for each of the programmes. The presentation of the choices was of the same kind as that illustrated above

As you can see, Programme