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Nonstationary term premia and cointegration of the term structure
Economics Letters 80 (2003) 409–413 www.elsevier.com / locate / econbase
Nonstationary term premia and cointegration of the term structure Kai Carstensen* Kiel Institute for World Economics, Duesternbrooker Weg 120, D-24105 Kiel, Germany Received 17 June 2002; accepted 25 March 2003
Abstract This paper proposes a model of the term structure with nonstationary term premia which exhibit a factor structure. This explains the common empirical finding of a cointegrating rank smaller than the one predicted by the rational expectations hypothesis of the term structure. An application to German interest rate data yields easily interpretable results. 2003 Elsevier B.V. All rights reserved. Keywords: Interest rates; Nonstationary factors JEL classification: E43
1. Introduction The analysis of the cointegration implications of the rational expectations hypothesis of the term structure (REHTS) has been one of the first applications of the cointegration methodology requiring cointegrating rank N 2 1 in a set of N interest rates. Empirically, this is supported at the short end of the maturity spectrum (Hall et al., 1992) but generally rejected at the long end (Shea, 1992; Zhang, 1993; Wolters, 1995) where too small a cointegrating rank is found. This implies that there is more than a single common nonstationary trend which drives the term structure. Moreover, given rank N 2 1 the REHTS implies that the cointegrating vectors are made up by the N 2 1 linearly independent spreads. If, however, the rank is smaller than N 2 1 it not clear how to identify and, then, interpret the cointegrating vectors. This paper offers an interpretation from a factor-theoretic perspective. In Section 2 a simple theoretical model of the term structure is presented which allows
* Tel.: 1491431-8814-258; fax: 149-431-8814-525. E-mail address: [email protected] (K. Carstensen). 0165-1765 / 03 / $ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016 / S0165-1765(03)00139-3
K. Carstensen / Economics Letters 80 (2003) 409–413
410
for nonstationary factors driving the term premia. As an empirical example, the German term structure is analysed in Section 3. Section 4 concludes.
2. A model of nonstationary term premia As demonstrated by Tzavalis and Wickens (1997), the REHTS predicts that the spread between a n-period yield R (n) and a 1-period yield r t is determined by a term premium T (n) and the expected t t future changes of the short yield
O (1 2 i /n)E [Dr
n21 (n)
(n)
R t 2 rt 5 T t 1
t
t1i
].
(1)
i 51
If interest rates are integrated processes, their rationally expected changes must be stationary. Consequently, the only reason for a possible nonstationary spread in (1) and thus an additional stochastic trend in the term structure can be a nonstationary term premium as argued by Evans and Lewis (1994). Paralleling the literature on continuous equilibrium models of the term structure (Dai and Singleton, 2000), we assume the term premium is a linear combination of J , N non-redundant nonstationary factors fjt , j 5 1, . . . , J and a stationary remainder term w t(n)
Of J
T
(n) t
5
(n) j jt
f 1 w (n) t
n 5 2, . . . , N.
(2)
j51
(n)
Substituting (2) into (1) and subsuming the stationary parts within 2 u t
Of
yields
J
(n)
R t 2 rt 2
(n) j jt
(n)
f 5 2 ut ,
(3)
j51
where the right-hand side is stationary by definition and, thus, the left-hand side describes a cointegrating relation. Collecting all N 2 1 cointegrating relations in matrix form yields
f1 N 21 , 2 IN21gR t 1 F Ft 5 u t ,
(4)
where 1 N 21 denotes a N 2 1 3 1 vector of ones, R t 5 (r t , R t(2) , . . . , R t(N ) )9, F is the N 2 1 3 J matrix of factor loadings, Ft 5 ( f1t , . . . , fJt )9 and u t 5 (u t(2) , . . . , u t(N ) )9. Since the factors Ft are unobservable they can be eliminated by solving (4) for Ft and substitute it back which yields B9R t 5 (IN 21 2 F (F 9F )21 F 9)u t
(5)
with B9 5 (IN21 2 F (F 9F )21 F 9)f1 N 21 , 2 IN 21g. Still, B9R t is cointegrating but with rank N 2 J 2 1 because the cointegrating space is projected into the space spanned by the observable variables R t . As a consequence, performing a cointegration analysis for a set of N yields may lead to an estimated cointegrating rank of N 2 J 2 1 because there are J nonstationary factors driving the term premia. While B itself is difficult to interpret, it is straightforward to find the null space b' of B9 which is given by the factor weights
K. Carstensen / Economics Letters 80 (2003) 409–413
b' 5 [1 N
1 1 F f]5 : 1
3
0 f 1( 2 ) : ) f (N 1
... ...
411
0 f J(2 ) : , ) f (N J
4
...
(6)
where F f 5 (0 913 J , F 9)9. The first vector of the null space 1 N is the only one if term premia are stationary (J 5 0). It represents a factor that has the same loading on any interest rate and may thus be labeled the level factor as, e.g., proposed by Litterman and Scheinkman (1991) and Knez et al. (1994). Note that it implies linear homogeneity of the cointegrating vectors, a property which can be statistically tested. Concerning the remaining J nonstationary factors, additional assumptions are needed for full identification as discussed below.
3. The German term structure It is found by Wolters (1995, 1998) and Nautz and Wolters (1996) that the German term structure contains more than one nonstationary factor. We replicate their analysis using yields of German government bonds 1 with time to maturity of n 5 1, 2, 3, . . . , 10 years. We find that the yields show strong signs of non-normality, especially excess kurtosis and conditional heteroskedasticity. Hence, we supplement conventional unit root tests by robust alternatives. The results support the common view that the yields are integrated of order one.2 In the next step, we perform two cointegration tests, see Table 1. In addition to the Gaussian trace (LR) test of Johansen (1988), the pseudo likelihood ratio (PLR) test of Lucas (1997) based on a t distribution with 5 degrees of freedom is used. Both tests indicate the existence of cointegrating rank 8 and, thus, two nonstationary factors. Consequently, one of these factors affects the yields via nonstationary term premia (J 5 1). Table 1 Cointegration tests H0 :r
0
1
2
3
4
5
6
7
8
9
LR LR 95 PLR PLR 95
939.1 251.3 793.6 179.1
634.6 208.4 584.3 152.9
458.7 169.5 438.8 123.1
333.7 134.7 308.3 100.1
220.1 103.8 228.9 78.85
127.2 76.96 186.2 58.14
67.93 54.09 69.54 41.21
36.04 35.19 32.35 28.47
13.36 20.25 14.11 16.69
6.41 9.17 2.66 7.17
Notes: All cointegration tests are performed with one lagged difference in the VECM and a constant restricted to the cointegration space. LR is the Gaussian (Johansen) trace test with 95% critical values taken from MacKinnon et al. (1999). PLR is the pseudo-likelihood ratio test statistic based on a t(5) distribution with 95% critical values PLR 95 obtained from simulation of the limit distribution as proposed by Lucas (1997).
1
We use monthly yield data from January 1974 to May 2000 of hypothetical zero bonds which are estimated by the German Bundesbank from the observed interest rates of German quoted Bundesanleihen, Bundesoligationen and Bundesschatzanweisungen. 2 The results can be obtained from the author upon request.
K. Carstensen / Economics Letters 80 (2003) 409–413
412
Table 2 Factor loadings in the 2-factor model R t( 4 )
R t( 5 )
R t( 6 )
R t( 7 )
R t( 8 )
R t( 9 )
R t( 10 )
Unrestricted estimation 1 0.044 0.058 (1.087) (1.116) 0 0.597 0.793 (2.755) (4.578)
0.057 (1.115) 0.773 (5.857)
0.048 (1.101) 0.659 (6.785)
0.038 (1.082) 0.515 (7.477)
0.027 (1.060) 0.370 (8.004)
0.017 (1.039) 0.234 (8.415)
0.008 (1.019) 0.111 (8.739)
1
Homogeneity imposed 1 1 1 0 0.371 0.495 (1.548) (2.970)
1 0.486 (4.256)
1 0.418 (5.380)
1 0.329 (6.344)
1 0.238 (7.168)
1 0.151 (7.872)
1 0.072 (8.477)
1 9
rt
R (t 2 )
R t( 3 )
9
Notes: Asymptotic standard errors are given in brackets below the estimates.
Given cointegrating rank 8, we first estimate the (unidentified) cointegration space by means of the quasi maximum likelihood method based on a t(5) distribution as proposed by Lucas (1997). Next, we calculate the null vectors b' which are unidentified because any vectors b˜ ' 5 b' k, with k being a regular (J 1 1) 3 (J 1 1) matrix, would span the same space. Therefore, we need J 1 1 5 2 identifying restrictions per vector. In accordance with the REHTS, the first factor shall be interpreted as a level factor. To achieve this, we restrict the loading on the shortest and the longest yield to be equal and arbitrarily fix them to 1. To identify the second factor we follow Litterman and Scheinkman (1991) and Knez et al. (1994) who find a slope factor to be important for explaining the term structure. Since this factor acts via the term premium it does not affect the shortest yield. Additionally, we arbitrarily normalize the weight f (10) to N 2 1. 1 The estimation results are displayed in Table 2. Without imposing linear homogeneity on the cointegration space the weights of the first factor are near to the theoretically predicted values of one. Imposing homogeneity implies one restriction on each cointegrating vector and yields a likelihood ratio test statistic of 13.5, which is well below the critical value of x 20.95 (8) 5 15.5. Linear homogeneity can thus be accepted. The weights estimated under homogeneity are given in the lower part of Table 2. For the second factor they are monotonically increasing which suggests that it is in fact sensible to interpret it as a slope factor.
4. Conclusion In this paper, it is shown how one can interpret the results of a cointegration analysis of the term structure if a cointegrating rank smaller than the theoretical value N 2 1 is found which is an empirically relevant case. It is demonstrated that the rank is reduced by the number of nonstationary factors which drive the term premia and the space orthogonal to the cointegration space contains the factor weights. It may thus be called the factor space. It might be easier to find restrictions which identify the factor space instead of the cointegration space because there is some a priori knowledge on how the factors affect specific yields.
K. Carstensen / Economics Letters 80 (2003) 409–413
413
An application to the German term structure leads to sensible results. In addition to a nonstationary level factor that governs the overall interest rate level, it is found that the term premium is driven by another nonstationary factor which can be interpreted as a slope factor.
Acknowledgements Thanks to Gerd Hansen and Marc Paolella for helpful comments
References Dai, Q., Singleton, K.J., 2000. Specification analysis of affine term structure models. Journal of Finance 55, 1943–1978. Evans, M.D.D., Lewis, K.K., 1994. Do stationary risk premia explain it all? Journal of Monetary Economics 33, 285–318. Hall, A.D., Anderson, H.M., Granger, C.W.J., 1992. A cointegration analysis of treasury bill yields. Review of Economics and Statistics 74, 116–126. Johansen, S., 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12, 231–254. Knez, P.J., Litterman, R., Scheinkman, J., 1994. Explorations into factors explaining money market returns. Journal of Finance 49, 1861–1882. Litterman, R., Scheinkman, J., 1991. Common factors affecting bond returns. Journal of Fixed Income 1, 54–61. Lucas, A., 1997. Cointegration testing using pseudolikelihood ratio tests. Econometric Theory 13, 149–169. MacKinnon, J.G., Haug, A.A., Michelis, L., 1999. Numerical distribution functions of likelihood ratio tests for cointegration. Journal of Applied Econometrics 14, 563–577. ¨ Nautz, D., Wolters, J., 1996. Die Entwicklung langfristiger Kreditzinssatze: Eine empirische Analyse. Kredit und Kapital 29, 481–510. Shea, G.S., 1992. Benchmarking the expectations hypothesis of the interest-rate term structure: an analysis of cointegration vectors. Journal of Business & Economic Statistics 10, 347–366. Tzavalis, E., Wickens, M.R., 1997. Explaining the failures of the term spread models of the rational expectations hypothesis of the term structure. Journal of Money, Credit, and Banking 29, 364–380. Wolters, J., 1995. On the term structure of interest rates—empirical results for Germany. Statistical Papers 36, 193–214. Wolters, J., 1998. Untersuchung der Renditestruktur am deutschen Kapitalmarkt 1970–1996. In: Baltensperger, E. (Ed.), ¨ auf Finanz- und Devisenmarkten. ¨ Spekulation, Preisbildung und Volatilitat Duncker and Humblot, Berlin, pp. 129–152. Zhang, H., 1993. Treasury yield curves and cointegration. Applied Economics 25, 361–367.
Nonstationary term premia and cointegration of the term structure Kai Carstensen* Kiel Institute for World Economics, Duesternbrooker Weg 120, D-24105 Kiel, Germany Received 17 June 2002; accepted 25 March 2003
Abstract This paper proposes a model of the term structure with nonstationary term premia which exhibit a factor structure. This explains the common empirical finding of a cointegrating rank smaller than the one predicted by the rational expectations hypothesis of the term structure. An application to German interest rate data yields easily interpretable results. 2003 Elsevier B.V. All rights reserved. Keywords: Interest rates; Nonstationary factors JEL classification: E43
1. Introduction The analysis of the cointegration implications of the rational expectations hypothesis of the term structure (REHTS) has been one of the first applications of the cointegration methodology requiring cointegrating rank N 2 1 in a set of N interest rates. Empirically, this is supported at the short end of the maturity spectrum (Hall et al., 1992) but generally rejected at the long end (Shea, 1992; Zhang, 1993; Wolters, 1995) where too small a cointegrating rank is found. This implies that there is more than a single common nonstationary trend which drives the term structure. Moreover, given rank N 2 1 the REHTS implies that the cointegrating vectors are made up by the N 2 1 linearly independent spreads. If, however, the rank is smaller than N 2 1 it not clear how to identify and, then, interpret the cointegrating vectors. This paper offers an interpretation from a factor-theoretic perspective. In Section 2 a simple theoretical model of the term structure is presented which allows
* Tel.: 1491431-8814-258; fax: 149-431-8814-525. E-mail address: [email protected] (K. Carstensen). 0165-1765 / 03 / $ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016 / S0165-1765(03)00139-3
K. Carstensen / Economics Letters 80 (2003) 409–413
410
for nonstationary factors driving the term premia. As an empirical example, the German term structure is analysed in Section 3. Section 4 concludes.
2. A model of nonstationary term premia As demonstrated by Tzavalis and Wickens (1997), the REHTS predicts that the spread between a n-period yield R (n) and a 1-period yield r t is determined by a term premium T (n) and the expected t t future changes of the short yield
O (1 2 i /n)E [Dr
n21 (n)
(n)
R t 2 rt 5 T t 1
t
t1i
].
(1)
i 51
If interest rates are integrated processes, their rationally expected changes must be stationary. Consequently, the only reason for a possible nonstationary spread in (1) and thus an additional stochastic trend in the term structure can be a nonstationary term premium as argued by Evans and Lewis (1994). Paralleling the literature on continuous equilibrium models of the term structure (Dai and Singleton, 2000), we assume the term premium is a linear combination of J , N non-redundant nonstationary factors fjt , j 5 1, . . . , J and a stationary remainder term w t(n)
Of J
T
(n) t
5
(n) j jt
f 1 w (n) t
n 5 2, . . . , N.
(2)
j51
(n)
Substituting (2) into (1) and subsuming the stationary parts within 2 u t
Of
yields
J
(n)
R t 2 rt 2
(n) j jt
(n)
f 5 2 ut ,
(3)
j51
where the right-hand side is stationary by definition and, thus, the left-hand side describes a cointegrating relation. Collecting all N 2 1 cointegrating relations in matrix form yields
f1 N 21 , 2 IN21gR t 1 F Ft 5 u t ,
(4)
where 1 N 21 denotes a N 2 1 3 1 vector of ones, R t 5 (r t , R t(2) , . . . , R t(N ) )9, F is the N 2 1 3 J matrix of factor loadings, Ft 5 ( f1t , . . . , fJt )9 and u t 5 (u t(2) , . . . , u t(N ) )9. Since the factors Ft are unobservable they can be eliminated by solving (4) for Ft and substitute it back which yields B9R t 5 (IN 21 2 F (F 9F )21 F 9)u t
(5)
with B9 5 (IN21 2 F (F 9F )21 F 9)f1 N 21 , 2 IN 21g. Still, B9R t is cointegrating but with rank N 2 J 2 1 because the cointegrating space is projected into the space spanned by the observable variables R t . As a consequence, performing a cointegration analysis for a set of N yields may lead to an estimated cointegrating rank of N 2 J 2 1 because there are J nonstationary factors driving the term premia. While B itself is difficult to interpret, it is straightforward to find the null space b' of B9 which is given by the factor weights
K. Carstensen / Economics Letters 80 (2003) 409–413
b' 5 [1 N
1 1 F f]5 : 1
3
0 f 1( 2 ) : ) f (N 1
... ...
411
0 f J(2 ) : , ) f (N J
4
...
(6)
where F f 5 (0 913 J , F 9)9. The first vector of the null space 1 N is the only one if term premia are stationary (J 5 0). It represents a factor that has the same loading on any interest rate and may thus be labeled the level factor as, e.g., proposed by Litterman and Scheinkman (1991) and Knez et al. (1994). Note that it implies linear homogeneity of the cointegrating vectors, a property which can be statistically tested. Concerning the remaining J nonstationary factors, additional assumptions are needed for full identification as discussed below.
3. The German term structure It is found by Wolters (1995, 1998) and Nautz and Wolters (1996) that the German term structure contains more than one nonstationary factor. We replicate their analysis using yields of German government bonds 1 with time to maturity of n 5 1, 2, 3, . . . , 10 years. We find that the yields show strong signs of non-normality, especially excess kurtosis and conditional heteroskedasticity. Hence, we supplement conventional unit root tests by robust alternatives. The results support the common view that the yields are integrated of order one.2 In the next step, we perform two cointegration tests, see Table 1. In addition to the Gaussian trace (LR) test of Johansen (1988), the pseudo likelihood ratio (PLR) test of Lucas (1997) based on a t distribution with 5 degrees of freedom is used. Both tests indicate the existence of cointegrating rank 8 and, thus, two nonstationary factors. Consequently, one of these factors affects the yields via nonstationary term premia (J 5 1). Table 1 Cointegration tests H0 :r
0
1
2
3
4
5
6
7
8
9
LR LR 95 PLR PLR 95
939.1 251.3 793.6 179.1
634.6 208.4 584.3 152.9
458.7 169.5 438.8 123.1
333.7 134.7 308.3 100.1
220.1 103.8 228.9 78.85
127.2 76.96 186.2 58.14
67.93 54.09 69.54 41.21
36.04 35.19 32.35 28.47
13.36 20.25 14.11 16.69
6.41 9.17 2.66 7.17
Notes: All cointegration tests are performed with one lagged difference in the VECM and a constant restricted to the cointegration space. LR is the Gaussian (Johansen) trace test with 95% critical values taken from MacKinnon et al. (1999). PLR is the pseudo-likelihood ratio test statistic based on a t(5) distribution with 95% critical values PLR 95 obtained from simulation of the limit distribution as proposed by Lucas (1997).
1
We use monthly yield data from January 1974 to May 2000 of hypothetical zero bonds which are estimated by the German Bundesbank from the observed interest rates of German quoted Bundesanleihen, Bundesoligationen and Bundesschatzanweisungen. 2 The results can be obtained from the author upon request.
K. Carstensen / Economics Letters 80 (2003) 409–413
412
Table 2 Factor loadings in the 2-factor model R t( 4 )
R t( 5 )
R t( 6 )
R t( 7 )
R t( 8 )
R t( 9 )
R t( 10 )
Unrestricted estimation 1 0.044 0.058 (1.087) (1.116) 0 0.597 0.793 (2.755) (4.578)
0.057 (1.115) 0.773 (5.857)
0.048 (1.101) 0.659 (6.785)
0.038 (1.082) 0.515 (7.477)
0.027 (1.060) 0.370 (8.004)
0.017 (1.039) 0.234 (8.415)
0.008 (1.019) 0.111 (8.739)
1
Homogeneity imposed 1 1 1 0 0.371 0.495 (1.548) (2.970)
1 0.486 (4.256)
1 0.418 (5.380)
1 0.329 (6.344)
1 0.238 (7.168)
1 0.151 (7.872)
1 0.072 (8.477)
1 9
rt
R (t 2 )
R t( 3 )
9
Notes: Asymptotic standard errors are given in brackets below the estimates.
Given cointegrating rank 8, we first estimate the (unidentified) cointegration space by means of the quasi maximum likelihood method based on a t(5) distribution as proposed by Lucas (1997). Next, we calculate the null vectors b' which are unidentified because any vectors b˜ ' 5 b' k, with k being a regular (J 1 1) 3 (J 1 1) matrix, would span the same space. Therefore, we need J 1 1 5 2 identifying restrictions per vector. In accordance with the REHTS, the first factor shall be interpreted as a level factor. To achieve this, we restrict the loading on the shortest and the longest yield to be equal and arbitrarily fix them to 1. To identify the second factor we follow Litterman and Scheinkman (1991) and Knez et al. (1994) who find a slope factor to be important for explaining the term structure. Since this factor acts via the term premium it does not affect the shortest yield. Additionally, we arbitrarily normalize the weight f (10) to N 2 1. 1 The estimation results are displayed in Table 2. Without imposing linear homogeneity on the cointegration space the weights of the first factor are near to the theoretically predicted values of one. Imposing homogeneity implies one restriction on each cointegrating vector and yields a likelihood ratio test statistic of 13.5, which is well below the critical value of x 20.95 (8) 5 15.5. Linear homogeneity can thus be accepted. The weights estimated under homogeneity are given in the lower part of Table 2. For the second factor they are monotonically increasing which suggests that it is in fact sensible to interpret it as a slope factor.
4. Conclusion In this paper, it is shown how one can interpret the results of a cointegration analysis of the term structure if a cointegrating rank smaller than the theoretical value N 2 1 is found which is an empirically relevant case. It is demonstrated that the rank is reduced by the number of nonstationary factors which drive the term premia and the space orthogonal to the cointegration space contains the factor weights. It may thus be called the factor space. It might be easier to find restrictions which identify the factor space instead of the cointegration space because there is some a priori knowledge on how the factors affect specific yields.
K. Carstensen / Economics Letters 80 (2003) 409–413
413
An application to the German term structure leads to sensible results. In addition to a nonstationary level factor that governs the overall interest rate level, it is found that the term premium is driven by another nonstationary factor which can be interpreted as a slope factor.
Acknowledgements Thanks to Gerd Hansen and Marc Paolella for helpful comments
References Dai, Q., Singleton, K.J., 2000. Specification analysis of affine term structure models. Journal of Finance 55, 1943–1978. Evans, M.D.D., Lewis, K.K., 1994. Do stationary risk premia explain it all? Journal of Monetary Economics 33, 285–318. Hall, A.D., Anderson, H.M., Granger, C.W.J., 1992. A cointegration analysis of treasury bill yields. Review of Economics and Statistics 74, 116–126. Johansen, S., 1988. Statistical analysis of cointegration vectors. Journal of Economic Dynamics and Control 12, 231–254. Knez, P.J., Litterman, R., Scheinkman, J., 1994. Explorations into factors explaining money market returns. Journal of Finance 49, 1861–1882. Litterman, R., Scheinkman, J., 1991. Common factors affecting bond returns. Journal of Fixed Income 1, 54–61. Lucas, A., 1997. Cointegration testing using pseudolikelihood ratio tests. Econometric Theory 13, 149–169. MacKinnon, J.G., Haug, A.A., Michelis, L., 1999. Numerical distribution functions of likelihood ratio tests for cointegration. Journal of Applied Econometrics 14, 563–577. ¨ Nautz, D., Wolters, J., 1996. Die Entwicklung langfristiger Kreditzinssatze: Eine empirische Analyse. Kredit und Kapital 29, 481–510. Shea, G.S., 1992. Benchmarking the expectations hypothesis of the interest-rate term structure: an analysis of cointegration vectors. Journal of Business & Economic Statistics 10, 347–366. Tzavalis, E., Wickens, M.R., 1997. Explaining the failures of the term spread models of the rational expectations hypothesis of the term structure. Journal of Money, Credit, and Banking 29, 364–380. Wolters, J., 1995. On the term structure of interest rates—empirical results for Germany. Statistical Papers 36, 193–214. Wolters, J., 1998. Untersuchung der Renditestruktur am deutschen Kapitalmarkt 1970–1996. In: Baltensperger, E. (Ed.), ¨ auf Finanz- und Devisenmarkten. ¨ Spekulation, Preisbildung und Volatilitat Duncker and Humblot, Berlin, pp. 129–152. Zhang, H., 1993. Treasury yield curves and cointegration. Applied Economics 25, 361–367.