Improvement in Finite Sample Properties of the Hansen-Jagannathan Distance Test Yu Ren Department of Economics Queen’s University Katsumi Shimotsu Department of Economics Queen’s University May 22, 2006 0-0 Motivation • Hansen and Jagannathan (1997) propose the HJ-distance to evaluate stochastic discount factor. It is the quadratic form of the pricing error weighted by the second moment of the portfolios returns. • Jagannthan and Wang (1996) develop a specification test of the HJ-distance based on linear asset pricing models . • Ahn and Gadarowski (2004) find that the the HJ-distance test overrejects correct models too severely in commonly used sample size. 1 Table1.Rejection frequencies of the HJ-distance test Number of Observations T =160 T =330 T =700 1% 5.8 3.3 1.1 5% 15.1 10.6 7.1 10% 23.9 18.9 12.8 1% 99.8 53.7 13.8 5% 100.0 76.0 30.8 10% 100.0 84.3 44.6 Fama-French Model 25 Portfolios 100 Portfolios 2 Basic Setting • N portfolios with T observations • Rt : t-th period returns, N × 1 • Yt : K × 1 factor vector including a 1 • mt = Yt0 δ linear combination of factors • mt : a stochastic discount factor proxy 3 Hansen-Jagannathan Distance • Define wt (δ) = Rt Yt0 δ − 1N , G = E[Rt Rt0 ] p • HJ(δ) = E[wt (δ)]0 G−1 E[wt (δ)] • This distance is equal to the least squares distances between the stochastic discount factors proxies and the family of stochastic discount factors that price correctly. 4 Sample Estimation of the HJ-Distance • Jagannathan and Wang (1996) use sample moments to estimate the HJ-distance q HJT (δ) = wT (δ)0 G−1 T wT (δ), • wT (δ) and GT are sample analogue of E(wt (δ)) and G PT −1 0 • DT = T R Y t t t=1 PT −1 • wT (δ) = T t=1 wt (δ) = DT δ − 1N PT −1 0 • GT = T R R t t t=1 5 Estimation of SDF and null Hypothesis q • δT = arg min wT (δ)0 G−1 T wT (δ) • We can test the hypothesis that the SDF, Yt0 δT , prices the portfolios returns correctly. • H0 : E[Rt (Yt0 δT )] = 1N 6 Asymptotical Distribution of the HJ-Distance Asymptotic distribution of the HJ-distance is: d T [HJT (δT )]2 → NX −K λj υj j=1 where υ1 ,. . . υN −K are independent χ2 (1) random variables, and λ1 ,. . . λN −K are nonzero eigenvalues of the following matrix: ψ = S 1/2 G−1/2 [IN − (G−1/2 )0 D(D0 G−1 D)−1 D0 G−1/2 ](G−1/2 )0 (S 1/2 )0 7 p-value of the HJ-distance PN −K • The distribution of j=1 λj υj depends on ψ. PN −K • We simulate the distribution of j=1 λj υj by randomly drawing N − K random variables from χ2 (1) distribution M times. • Obtain an empirical p-value by PM PN −K −1 2) p=M I( λ v ≥ T [HJ (δ )] i ij T T j=1 i=1 8 Fama-French Three-Factor Model Rit = α + X1t β1i + X2t β2i + X3t β3i + eit • Fama and French (1993) • Rit : Fama-French 25 (or 100)portfolio returns. • Xit are Fama-French factors. 9 Monte Carlo Simulation • Collect time series of monthly returns for Fama-French portfolios and Fama-French factors. • Simulate the factors from normal distribution with the mean and the covariance equal to the sample mean and the sample covariance matrix derived from the actual data of Fama-French factors. • Calibrate Fama-French portfolio returns according to Fama-French three factor model. • Do the HJ-distance test based on the simulated factors and calibrated returns. • Repeat the above process for 1000 times. 10 Table1.Rejection frequencies of the HJ-distance test Number of Observations T =160 T =330 T =700 1% 5.8 3.3 1.1 5% 15.1 10.6 7.1 10% 23.9 18.9 12.8 1% 99.8 53.7 13.8 5% 100.0 76.0 30.8 10% 100.0 84.3 44.6 Fama-French Model 25 Portfolios 100 Portfolios 11 Possible Reason for the Over-rejection GT = T −1 T X d t ) + E(R b t )0 E(R b t) Rt Rt0 = Cov(R t=1 • E(Rt ) can be estimated accurately by the sample mean for the sample size of our interest. • The sample covariance matrix can be a very inaccurate estimate of Cov(Rt ) when N/T is not negligible. • Inverting the sample moment as the weighting matrix can amplify the inaccuracy. 12 Table3. Rejection frequencies with the exact weighting matrix G Number of Observations T =160 T =330 T =700 1% 1.0 1.2 0.7 5% 4.9 5.7 5.0 10% 9.9 11.8 11.9 1% 1.1 1.6 1.3 5% 7.4 7.4 5.8 10% 16.8 14.7 13.6 Fama-French Model 25 Portfolios 100 Portfolios 13 Covariance Matrix based on a Structure Model • True covariance matrix of returns: Σ • Asset pricing model (Structure model): R = Xβ + ² • Covariance matrix of returns implied by the structure model: Φ = β 0 Cov(Xt )β + ∆ • Estimated covariance matrix based on the structure model: \ +D F = b0 Cov(X)b • Sample covariance: S 14 Improve Estimation of G by Shrinkage • Shrinkage Estimation: combine sample covariance and structure model covariance matrix with an optimal weight. • Sample covariance S has small bias, but it has a large estimation error. • Covariance matrix F based on a structure model has a large bias, but small estimation error. • Shrinkage estimator balances these two with an optimal weight α 15 Shrinkage Estimation • Loss function: L(α) = kαF + (1 − α)S − Σk2 • k · k is Frobenius norm: for any N × N matrix Z, PN PN 2 2 2 kzk = T race(Z ) = i=1 j=1 zij • Minimize the expected loss function: minα E[L(α)] 16 Optimal α √ √ √ PN PN PN PN i=1 j=1 V ar( T sij ) − i=1 j=1 Cov( T fij , T sij ) ∗ α = PN PN √ √ PN PN 2 i=1 j=1 V ar( T fij − T sij ) + T i=1 j=1 (φij − σij ) • sij is the typical element of sample covariance S • fij is the typical element of estimated structure based covariance matrix F • φij is the typical element of structure based covariance matrix Φ • σij is the typical element of the true covariance matrix Σ 17 Consistent Estimators of the Optimal Weight Assumption 1. Individual stock returns are independent and identically distributed over time. Assumption 2. Stock returns have finite fourth moments. Use consistent estimators to replace the numerator and the denominator in the formula of α∗ 18 Shrinkage Estimator Theorem 1. Define an estimate of the optimal shrinkage constant as PN PN PN PN i=1 j=1 pij − i=1 j=1 rij α ˆ= PN PN i=1 j=1 (gij + T cij ) ˆ is: Σ ˆ =α The shrinkage covariance matrix Σ ˆ F + (1 − α ˆ )S. 19 Table 5. Rejection frequencies with shrinkage estimation of G Number of Observations T =160 T =330 T =700 1% 0.9 1.1 0.5 5% 5.3 6.0 4.8 10% 10.7 11.6 10.7 1% 4.9 2.2 1.7 5% 23.9 11.5 7.3 10% 39.9 22.1 15.0 Fama-French Model 25 Portfolios 100 Portfolios 20 Compare Table 1. with Table 5. Number of Observations T =160 T =330 T =700 Fama-French Model: 25 Portfolios Table 1. 1% 5.8 3.3 1.1 5% 15.1 10.6 7.1 10% 23.9 18.9 12.8 1% 0.9 1.1 0.5 5% 5.3 6.0 4.8 10% 10.7 11.6 10.7 Table 5.(with Shrinkage) 21 Compare Table 1. with Table 5. (continued) Number of Observations T =160 T =330 T =700 Fama-French Model: 100 Portfolios 100 Portfolios 1% 99.8 53.7 13.8 5% 100.0 76.0 30.8 10% 100.0 84.3 44.6 1% 4.9 2.2 1.7 5% 23.9 11.5 7.3 10% 39.9 22.1 15.0 100 Portfolios 22 The Density Functions for α ˆ in Fama-French 25 Portfolios Model 20 density T=160 15 10 5 0 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.8 0.85 0.9 0.95 1 0.8 0.85 0.9 0.95 1 20 density T=330 15 10 5 0 0.65 0.7 0.75 20 density T=700 15 10 5 0 0.65 0.7 0.75 23 The Density Functions for α ˆ in Fama-French 100 Portfolios Model 15 density T=160 10 5 0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.7 0.75 0.8 0.85 0.9 0.95 1 0.7 0.75 0.8 0.85 0.9 0.95 1 15 density T=330 10 5 0 0.6 0.65 15 density T=700 10 5 0 0.6 0.65 24 Premium-Labor Three-Factor Model Rit = α + X1t β1i + X2t β2i + X3t β3i + eit • Jagannathan and Wang (1996) • Rit is constructed in the same way with Fama-French model, but ONLY includes the stocks of nonfinanical firms listed in NYSE, AMEX. • Xit are Premium-Labor factors. 25 Compare Table 1. with Table 5. Number of Observations T =160 T =330 T =700 Premium-Labor Model: 25 Portfolios Table 1. 1% 14.9 11.3 9.2 5% 31.9 26.0 19.5 10% 42.7 34.4 29.0 1% 4.7 5.9 6.4 5% 16.4 17.0 16.2 10% 26.3 26.9 23.8 Table 5.(with Shrinkage) 26 Compare Table 1. with Table 5. (continued) Number of Observations T =160 T =330 T =700 Premium-Labor Model: 100 Portfolios 100 Portfolios 1% 99.7 79.1 36.8 5% 99.9 90.1 59.1 10% 99.9 94.3 69.6 1% 16.6 14.9 11.1 5% 41.5 35.2 27.0 10% 58.3 50.4 38.9 100 Portfolios 27 The Density Functions for α ˆ in Premium-Labor 25 Portfolios Model 15 density T=160 10 5 0 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.8 0.85 0.9 0.95 1 0.8 0.85 0.9 0.95 1 15 density T=330 10 5 0 0.65 0.7 0.75 15 density T=700 10 5 0 0.65 0.7 0.75 28 The Density Functions for α ˆ in Premium-Labor 100 Portfolios Model 20 density T=160 15 10 5 0 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0.75 0.8 0.85 0.9 0.95 1 0.75 0.8 0.85 0.9 0.95 1 25 T=330 density 20 15 10 5 0 0.65 0.7 25 T=700 density 20 15 10 5 0 0.65 0.7 29
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