INTRODUCTION Liquidity Preference hypothesis (LPH) implies ex ante return on government securities is a monotonically increasing function of time to maturity. Underlying intuition is that longer term bonds are more risky and hence have a higher expected return. No formal test the LPH with respect to conditioning information has been performed.
THEORY R E t Xtt, jm + tt, + j = 1. 1 Pt, j = = E t M tt, j, j + tt, + 1 ( j) ( Ytt, + j) Pt+ 1, j 1 Et+ 1[ Mt+ 1, t+ j] = =. P E[ M ] t, j t tt, + j * * ( ) ( ) Et rtt, + 1 j ytt, + 1 cov t Mtt, + 1, Mt+ 1, t+ j. j 1 1 y y = E y y ( ) tt, + j tt, + 1 t t+ it, ++ i 1 tt, + 1 j i= 1 1 Et j j 1 i= 1 cov t+ i 1 * * ( Mt+ i 1, t+ i, M t+ i, t+ j),
EXISTING EVIDENCE Direct tests of LPH Fama (84) McCulloch (87) Richardson, Richardson and Smith (92) Evidence seems to suggest monotonicity. However-unconditional tests. More interesting question is whether expected returns are monotonic conditional on all available information. Informal evidence that they are not Fama (86) Stambaugh (88) Klemkosky and Pilotte (92)
PROBLEMS IN TESTING #1: Model implies inequality restrictions on the parameters to be estimated. Liquidity preference hypothesis implies j -month term premium exceeds the j 1-month term premium (for all j ). How do we test these restrictions? Look to recent econometrics literature (see Gourieroux, Holly and Monfort (1982), Kodde and Palm (1986), Wolak (1989a, 1989b, 1991)).
#2: Conditional expected returns are unobservable to the econometrician Liquidity preference hypothesis holds at each point in time t, conditional on available information. Importance: 1. Unconditional tests can lack power (ignore information). 2. Investigate model s implications with respect to current information. How do we account for this? Appeal to recent tests for conditional asset pricing models (see H-S (1982, 1983), G-F (1985), K-S (1986), etc )
SUMMARY Combining these disciplines leads to test methodology which: 1. Requires weak assumptions on underlying processes. 2. Requires little knowledge of conditional moments. 3. Has widespread applications:
TEST METHODOLOGY Define H τ t + 1 as the continuously compounded return from t to t + 1 on a bill with maturity τ at date t. The term premium in the return on a bill with maturity τ is defined as P E[ H H ], τt+ 1 t τt+ 1 lt+ 1 where ( ) Nt+ 1 2t+ 1 P P... P τ = 2,..., N + 1 N+ 1 t+ 1 under the null model. Thus we have a model which implies [ ] (1) E H H t τt+ 1 τ 1t+ 1 0
Model (1) implies [ ] E H H D t τt+ 1 τ 1t+ 1 = t 0 where D t is defined as this difference. Observations: Econometrician has less information than do agents. Restrict ourselves to z + t, which is nonnegative.
Because z + t is nonnegative E ( H H ) z = D z 0 + + t τt+ 1 τ 1t+ 1 t t t (2) Rearranging (2), and applying the law of iterated expectations, + ( τt+ 1 τ 1t+ 1) t θ 0 Dz + = (3) E H H z + where θ E Dt zt 0. Dz + = (4) Observations: (3) provides a set of orthogonality conditions to + estimate θ Dz + (all we need observe are Hτt 1, Hτ 1t 1, Zt not the unknown D t ). + + and Note that θ Dz + is the mean of the product of observables. Null model implies vector of inequality restrictions θ Dz + 0.
ADVANTAGES 1. The econometrician does not require a model for conditional expectations. 2. The econometrican can often point to the interesting instrumental variables, but may not know how they enter the model. (Note here there is no assumed functional form). 3. Restrictions given in (4) can be tested using the technology developed recently in the inequality testing literature. Joint tests: - Correlation across estimators of vector Dz θ + (e.g. F verus t tests). - Take account of serial correlation.
TEST STATISTIC Write the restriction given in (3) and (4) as a system of N moment conditions: ( ) E H H z = θ + τt+ 1 τ 1t+ 1 1t Dz ( ) E H H z = θ + 1 + τt+ 1 τ 1t + 1 Nt Dz + N H 0 : θ 0 = 1,..., N (5) versus H A Dz + i N : θ R. Dz + i i
STEP 1 Estimate the same means of the product of the observable variables. In particular, ˆ 1 = H H z = 1,..., N. T + θ ( t 1 1t 1) Dz + τ + τ + it i i T t= 1 The vector ˆDz θ + is asymptotically normal with mean θ Dz + and variance-covariance matrix Ω.
STEP 2 Derive estimates under the null restriction by minimizing deviations from the unrestricted model; ˆ ( ) ' 1 min ( θ + θ + ) Ω θ + θ +, θ Dz Dz Dz Dz Dz + subject to θ 0. ˆ Dz + Let θ Dz + be the solution to this quadratic program.
STEP 3 Generate statistic for testing the null hypothesis, e.g. test how close the restricted estimates θ Dz + are to the unrestricted estimates ˆ θ Dz + Under the null, the difference. should be small. ( ) ' ˆ ˆ 1 ( ˆ ) W T θ + θ + Ω θ + θ + (6) Dz Dz Dz Dz
STEP 4 Final step is to calculate W ' asymptotic distribution. Wolak (1989a) and others show that the statistic is now distributed as a weighted sum of chi-squared variables with different degrees of freedom. Ωˆ,,, (7) N 2 Pr χk c w N N k k = 0 T where c R + is the critical value for a given size and the ˆ weigh w( N, N k, Ω T ) is the probability that θ Dz + has exactly N k positive elements.
INTUITIVE EXAMPLE Suppose the estimators ˆDz θ + have asymptotically zero covariance. Consider two restrictions, i.e. θ 0 Dz + and 1 θ + > 0. Dz2 1. Consider the weight ( 2, 2, T ) 2. Consider the weight ( T ) 3. Consider the weight ( T ) w Ω w= χ ˆ 1 2 4, 0 w 2,1, Ω w=, χ. ˆ 1 2 2 1 w 2,0, Ω w=, χ. ˆ 1 2 4 2 Therefore, for a fixed size α, the critical value in this case solves the equation 1 1 1 α = Pr χ c + Pr χ c + Pr χ c 4 2 4 2 2 2 ( 0 ) ( 1 ) ( 2 )
OBSERVATIONS Generalizes to larger sets of restrictions and to nonzero covariance among the estimators. Weights have a closed form solution for a small number of restrictions ( N 5) (see Kudo (1963) for exact calculations). For larger restrictions, Koddle and Palm (1986, Table 1, page 1246) provide upper and lower bound critical values which do not require calculation of the weights. If between these bounds, Wolak (1989b) describes an approximate method for calculating the weights based on a monte carlo simulation.
FAMA REPLICATION Table 1 replicates (1986). Period: November 71-July 84. Holding period returns are: H 1t + 1 H 3t + 2 H 6t + 3 H 11t + 6 Leads to restriction ( + + ) ( + + ) ( + + ) E H11, t 6 H6, t 3 zt θ 1 = 0 E H6, t 3 H3, t 2 zt θ 2 = 0 E H3, t 2 H1, t 1 zt θ 3 = 0, with the restriction 0 θ i i
Panel A Instrument = Nonmonotonicity of forward rate term structure Results F6,11 F3,6 F1,3 F0,1 We cannot reject null of LPH P-value is.382 for the 0/1 instrument and.301 for the informative instruments. Why? 1. Auto-correlated errors 2. Cross-correlation across premiums. 3. Distribution of joint test. Illustrates different conclusions one can reach from formal inequality restrictions tests.
Panel B Now partition information into finer elements. z1 F1,3 F0,1 = 1if < and 0 otherwise z2 F3,6 F1,3 = 1if < and 0 otherwise z3 F6,11 F3,6 = 1if < and 0 otherwise Leads to restriction ( + + ) ( + + ) ( + + ) E H11, t 6 H6, t 3 z1 t θ 1 = 0 E H6, t 3 H3, t 2 z2t θ 2 = 0 E H3, t 2 H1 t 1 z3t θ 3 = 0, with the restriction 0 θ i i
RESULTS We can reject null of LPH P-value is.011 for the 0/1 instrument and.012 for the informative instruments. negative term premia seen to be associated with particular non monotonic forward rates e.g. any non monotonicity implies θ 2 =.170 but non monotonicity of forward rate of corresponding maturity implies θ 2 =.659.
TEST OF LPH ACROSS ALL MATURITIES 1 month return on Govt. bonds of following maturity 2-6 months 7-12 months 13-36 months 37-60 months 61-120 months 120+ months Sample period January 1972-December 1994
INSTRUMENTS Non monotonic yield curve 1. Uninformative 2. Informative Downward sloping yield curve 1. Uninformative 2. Informative
RESULTS Summary statistics on instruments and term premiums Formal tests
CONCLUSIONS Provide formal tests for whether conditional expected returns on bonds are increasing with maturity of bond Document and test properties of holding period return across the entire maturity spectrum Document nonmonotonicities related to the shape of the yield curve