The Slow Convergence of OLS Estimators of α, β and Portfolio. β and Portfolio Weights under Long Memory Stochastic Volatility

Transcription

1 The Slow Convergence of OLS Estimators of α, β and Portfolio Weights under Long Memory Stochastic Volatility New York University Stern School of Business June 21, 2018

2 Introduction Bivariate long memory stochastic volatility model The Long Memory Stochastic Volatility Model (LMSV) was introduced by Breidt, Crato and de Lima (1998), and Harvey (1998). The model yields long memory in squares of stock returns (or other positive powers of absolute returns), as is found in actual data. (See, for example, Ding, Granger and Engle, 1993). What consequences would long memory in volatility have on financial econometrics? The question has not been fully explored. We will consider a bivariate LMSV model and examine the implications of long memory in volatility on estimation of the regression intercept and slope (α, β) and on the optimal portfolio weights.

3 Bivariate long memory stochastic volatility model Bivariate long memory stochastic volatility model Bivariate LMSV model x t = µ x + e h 1,t 2 ɛ1,t y t = µ y + e h 2,t 2 ɛ2,t Market and Stock Excess Returns (CAPM) Returns on two stocks (Portfolio) Assumptions h i,t = k=0 ψ i,ke i,t k, with ψ i,j c i j d i 1, for i = 1, 2. {e 1,t, e 2,t } is i.i.d with zero mean, var(e i,t ) = σ 2 e i <, ρ e = cov(e 1,t, e 2,t ). {ɛ 1,t, ɛ 2,t } is i.i.d with zero mean, ρ ɛ = cov(ɛ 1,t, ɛ 2,t ). {h 1,t, h 2,t } and {ɛ 1,t, ɛ 2,t } are mutually independent. µ x and µ y are population means for {x t} and {y t}. d i = Memory Parameters, in (0, 1/2).

4 Capital Asset Pricing Model The slow convergence rates of estimated regression coefficients Theoretical results for ˆα and ˆβ Capital Asset Pricing Model E(r i,t ) r f = β [E(r m,t ) r f ] where r i,t the return for i th asset or portfolio; r f risk free rate; r m,t the return of the market (index). The estimation of β is typically obtained by running a regression: r i,t r f = α + β (r m,t r f ) + ε t

5 The slow convergence rates of estimated regression coefficients Theoretical results for ˆα and ˆβ Test of α and β in CAPM Define β = cov(x t, y t )/var(x t ), α = E[y t βx t ]. Test whether there is significant abnormal return/test the performance of the fund manager H 0 : α = 0 Test whether the individual risk premium is equal to the market risk premium H 0 : β = 1 Could the general feeling that β is time-varying (or that α is nonzero) be due simply to unexpectedly large estimation error?

6 The rate of convergence of ˆβ The slow convergence rates of estimated regression coefficients Theoretical results for ˆα and ˆβ The logarithm of the standard deviation of ˆβ (σ True ( ˆβ)) versus the logarithm of the sample size. the slope of the red line (d 1 = d 2 = 0.4) is the slope of the blue line (d 1 = d 2 = 0) is log standard errors d i = 0.4 d i = log sample size

7 The slow convergence rates of estimated regression coefficients Theoretical results for ˆα and ˆβ Table: Ratio of True to Estimated Squared Standard Errors, σ2 True( β) ˆσ 2 ( β) Sample Size, T d 1 = 0.1, d 2 = d 1 = 0.2, d 2 = d 1 = 0.3, d 2 = d 1 = 0.4, d 2 = The ratio σ2 True( β) is increasing with d and T. ˆσ 2 ( β) σ 2 True( β): the true squared standard error of ˆβ based on simulation. ˆσ 2 ( β): the traditional squared estimated standard error of ˆβ

8 The slow convergence rates of estimated regression coefficients Theoretical results for ˆα and ˆβ Coverage rate of ˆβ ± 2σ True ( ˆβ) Sample size, T d 1 = 0.1, d 2 = % 95.2% 94.2% 95.6% 94.8% 95.4% d 1 = 0.2, d 2 = % 95.4% 95.6% 95.2% 93.8% 94.4% d 1 = 0.3, d 2 = % 95.0% 95.8% 95.2% 94.0% 94.2% d 1 = 0.4, d 2 = % 95.0% 94.8% 96.0% 95.0% 95.0% Coverage rate of ˆβ ± 2ˆσ( ˆβ) Sample size, T d 1 = 0.1, d 2 = % 92.2% 92.8% 92.6% 92.0% 91.8% d 1 = 0.2, d 2 = % 86.4% 86.6% 81.6% 79.6% 61.6% d 1 = 0.3, d 2 = % 65.0% 57.8% 50.2% 42.4% 23.6% d 1 = 0.4, d 2 = % 32.8% 26.4% 17.4% 13.2% 4.0% The coverage rate of the confidence interval based on the traditional standard error ˆσ( ˆβ) decreases with the increase of the sample size T and the long memory parameters.

9 Theorem 1 The slow convergence rates of estimated regression coefficients Theoretical results for ˆα and ˆβ The OLS estimator ˆβ obtained by regressing {y t } T t=1 on {x t} T t=1 with intercept satisfies T 1 2 d ( ˆβ β) where d = max(d 1, d 2 ) and D N(0, σ 2 Asy. ( ˆβ)) σasy. 2 ( ˆβ) T = β 2 lim var[t 1 2 d ( h 2,t T 2 h 1,t 2 )]. t=1 ˆβ has slow convergence rate.

10 Theorem 2 The slow convergence rates of estimated regression coefficients Theoretical results for ˆα and ˆβ For the OLS estimator ˆα, T 1 2 d (ˆα α) T 1 2 (ˆα α) where σ 2 = var(y t ) + β 2 var(x t ). D N(0, µ 2 xσ 2 Asy. ( ˆβ)), µ x 0, D N(0, σ 2 ), µ x = 0, ˆα has slow convergence rate if µ x 0, but not if µ x = 0.

11 The slow convergence rates of estimated regression coefficients Theoretical results for ˆα and ˆβ The rate of convergence of ˆα when µ x 0 The logarithm of the true standard error of ˆα (σ True (ˆα)) versus the logarithm of T. µ x = µ y = 0.05 the slope of the red line (d 1 = d 2 = 0.4) is the slope of the blue line (d 1 = d 2 = 0) is log standard errors d i = 0.4 d i = log sample size

12 The slow convergence rates of estimated regression coefficients Theoretical results for ˆα and ˆβ The rate of convergence of ˆα when µ x 0, µ x = 0 The logarithm of the true standard error of ˆα (σ True (ˆα)) versus the logarithm of T d 1 = d 2 = 0.4 the slope of the blue line (µ x = µ y = 0.05) is the slope of the red line (µ x = µ y = 0) is log standard errors µ x = 0 µ x = log sample size

13 The slow convergence rates of estimated regression coefficients Theoretical results for ˆα and ˆβ Table: σ2 True ( α) ˆσ 2 ( α) for different sample sizes(d 1 = d 2 = 0.4). Sample Size µ x = 0, µ y = µ x = , µ y = µ x = , µ y = µ x = 0.05, µ y = σtrue 2 ( α): the true squared standard error of ˆα based on simulation. ˆσ 2 ( α): the traditional squared estimated standard error of ˆα

14 The slow convergence rates of estimated regression coefficients Theoretical results for ˆα and ˆβ Coverage rate of ˆα ± 2σ True (ˆα) Sample size µ x = 0, µ y = % 94.8% 94.4% 94.8% 95.0% 94.4% µ x = , µ y = % 94.8% 95.0% 95.8% 95.0% 95.0% µ x = , µ y = % 94.8% 94.8% 96.0% 95.0% 95.0% µ x = 0.05, µ y = % 94.8% 94.8% 96.0% 95.0% 95.0% Coverage rate of ˆα ± 2ˆσ(ˆα) Sample size µ x = 0, µ y = % 93.8% 94.2% 94.6% 95.0% 93.8% µ x = , µ y = % 33.4% 28.0% 18.8% 14.4% 4.0% µ x = , µ y = % 33.0% 26.8% 17.4% 13.4% 4.0% µ x = 0.05, µ y = % 33.0% 27.0% 17.2% 13.4% 4.0%

15 Proposition 1 The slow convergence rates of estimated regression coefficients Theoretical results for ˆα and ˆβ T σ(ˆα) T σ( ˆβ) P σ ε σ x, P σ ε 1 + µ2 x σx 2, where ε t = y t (α + βx t ) and σ 2 ε = var(ε t ). The usual standard errors, obtained from the regression output, are too optimistic, converging to zero as 1/ T.

16 Minimum Variance Portfolio Consider two assets (A and B) with returns {x t } and {y t }. The weight for asset A of w m x = σ 2 y σ xy σ 2 x + σ 2 y 2σ xy minimizes the variance of the portfolio return var(r p ) = w 2 σ 2 x + (1 w) 2 σ 2 y + 2w(1 w)σ xy, where the portfolio weights are w, 1 w on Assets A, B.

17 Table: The estimated convergence rate for ŵ m x. Convergence Rate case d 1 d 2 σ 2 e 1 σ 2 e 2 σ 2 ɛ 1 σ 2 ɛ 2 ŵ m x Theoretical e e e e e e e e

18 Theorem 4: Estimated weight in min-variance portfolio Under suitable regularity conditions (see working paper), T 1/2 d (ŵ m x w m x ) D N(0, C), C > 0.

19 Proposition 2 Let σ xy = cov(x t, y t ), ˆσ xy = 1 T T t=1 (x t x)(y t ȳ). Then where T ( 1 2 d) (ˆσ xy σ xy ) D N(0, σ 2 ) T σ 2 = β 2 σx 2 lim var(t 1 2 d ( h 1,t T 2 + h 2,t 2 )). t=1 Slow convergence of sample covariance is inherited by the estimated portfolio weight.

20 Minimum variance portfolio weight n = 1320 n = case1 n = 1320 n = case n = 1320 n = case3 n = 1320 n = case4.

21 Table: Excess Variance, var(r p )/var m (r p ) 1 case Median Mean Max var(r p ): the variance obtained by using the estimated weight (ŵ m x ) var m (r p ): the variance obtained by using the true weight (w m x ) Actual variance of the portfolio may be far from the minimum

22 Ordinary t-stat for α, β typically diverges under the null hypothesis. We focus here on β. The problem: the traditional estimated standard error converges to zero faster than ˆβ β 0 does. Possible Remedy: Estimate the true standard error. Unfortunately, the SE depends on nuisance parameters which would also need to be estimated.

23 Alternative Approach: 1. Use self-normalization to construct a test statistic that converges in distribution under the null hypothesis for all values of the nuisance parameter d. 2. Use subsampling to obtain critical values for the test. This approach has been used in the long-memory context by, e.g., Jach, McElroy and Politis (2012, 2016), and Bai, Taqqu and Zhang (2016).

24 Consider testing H 0 : β = β 0. Since β = cov(x t, y t ) var(x t ), the parameter θ = cov(x t, y t ) β 0 var(x t ) is zero under H 0. We estimate θ by [ T ˆθ T ] T = T 1 (x t x)(y t ȳ) β 0 (x t x) 2 t=1 t=1.

25 Self-Normalization: The self-normalized test statistic is where S t,b = D 2 T,B = 1 T G T,β = T t=1 ˆθ T T 1 D T,B, ( S t,b t T S T,B) 2, t B k, and B k = (x k x)(y k ȳ) ˆβ(x k x) 2. k=1 Under H 0, the denominator of G T,β converges in distribution at the same rate as the numerator, so that G T,β has a limiting distribution.

26 Subsampling: Note that ˆθ T θ T 1 D T,B has a limiting distribution for all values of θ and d. We estimate the quantiles of this limiting distribution using subsampling (with overlapping subsamples of size b). This yields asymptotically correct critical values for the test. We do not need to estimate any nuisance parameters (such as d) to perform the test. On each subsample of the data we compute a replicate of the self-normalized test statistic, centered by an estimate of θ. These replicates have the same limit distribution as above.

27 Simulations: we compare the size of the self-normalized subsampling-based test and the ordinary t-test for testing the null hypothesis H 0 : β = β 0 versus the two-tailed alternative hypothesis H A : β β 0. We used µ x = µ y = 0, var(ɛ 1,t ) = var(ɛ 2,t ) = 1, ρ ɛ1,ɛ 2 = 0.2; {h 1,t } and {h 2,t } were mutually independent ARFIMA(0, d, 0) processes with d = d x = d y = 0.37 and equal innovation variances of These parameter values are calibrated to empirical studies of S&P returns (see, for example, Deo, Hurvich and Lu, 2005).

28 Since the test statistic diverges under H A, the subsampling procedure will yield a consistent test. Here, since we are simply studying size, we take β 0 to be the true value of β (β = ) corresponding to the given data generating mechanism. We considered two significance levels (5% and 1%), two sample sizes (T = 4000, T = 6000), and three values of b (200, 400, 600). We used 5000 simulated replications.

29 Table: Sizes of self-normalized subsampling-based test and ordinary t-test of H 0 : β = β 0 versus H A : β β 0. Size of subsample is b. 5% 1% T = 4000 T = 6000 T = 4000 T = 6000 b = b = b = t-test Both tests are severely oversized, but the self-normalized subsampling-based test is typically far less oversized than the ordinary t-test. The size of the t-test increases with T, as predicted by our theoretical results. The size of the self-normalized subsampling-based test increases with b, for a given T and nominal significance level.

30 The OLS-estimator ˆβ in the regression under long memory volatility has slower convergence rate than T. The larger the long memory parameter, the slower the convergence rate. The same results apply to OLS-estimator ˆα when the population mean of the regressor is not equal to zero. The traditional standard errors of ˆα and ˆβ are too optimistic. Therefore, the corresponding t-statistics diverge under the null hypothesis and the confidence intervals are too narrow.

31 Self-normalized subsampling-based inference for α, β can yield asymptotically correct size (though we don t prove this), and in simulations is less oversized than the ordinary t-test.

32 The estimated minimum variance portfolio weights have the same slow convergence rate as above. The actual variance of the portfolio may be far from the minimum when using the estimated weights for the minimum variance portfolio.

33 Future Work Our theoretical results on slow convergence in estimation of α, β and portfolio weights continue to hold for a bivariate pure-jump log price model in continuous time. In this model, transactions are triggered by Cox processes driven by fractional Gaussian noise. Estimation can be done in discrete or continuous time. More work is needed to get good finite-sample performance for self-normalized subsampling-based inference (perhaps by using a different normalizing statistic) and to establish the asymptotic validity of this procedure under our bivariate LMSV model.