Testing for Weak Form Efficiency of Stock Markets

Testing for Weak Form Efficiency of Stock Markets Jonathan B. Hill 1 Kaiji Motegi 2 1 University of North Carolina at Chapel Hill 2 Kobe University The 3rd Annual International Conference on Applied Econometrics in Hawaii September 10-11, 2017 Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 1 / 22

Introduction Testing for Weak Form Efficiency of Stock Markets = Testing for Unpredictability of Stock Returns (cf. Fama, 1970) x IID x MDS White Noise x x Many applications cf. Lim and Brooks (2009) Only few applications REASON: It is hard to establish a formal white noise test. BREAKTHROUGH: Shao s (2010, 2011) dependent wild bootstrap. This paper tests for the white noise hypothesis of stock returns, using the dependent wild bootstrap. A rejection of the white noise hypothesis might serve as helpful information for arbitragers, because it indicates the presence of non-zero autocorrelations at some lags. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 2 / 22

Introduction Dependent Wild Bootstrap (Shao, 2010; Shao, 2011) Adaptive Market Hypothesis (cf. Lo, 2004; Lo, 2005) Full Sample Shao (2010): Temperature Shao (2011): Stock returns Rolling Window New! PERIODICITY IN CONFIDENCE BANDS REASON: Fixed block size produces similar bootstrapped autocorrelations in every windows. REMEDY: Randomizinga block size across bootstrap samples and windows. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 3 / 22

Introduction Daily S&P 500 White noise hypothesis is often rejected during Iraq War and the subprime crisis. We observe significantly negative autocorrelations during crisis periods. cf. Fama and French (1988) Campbell et al. (1993) Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 4 / 22

Table of Contents 1) Introduction 2) Dependent Wild Bootstrap 3) Stock Price Data (S&P 500) 4) Empirical Results 5) Conclusions Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 5 / 22

Review of Dependent Wild Bootstrap Consider full sample analysis as a benchmark. Consider covariance stationary {y 1,..., y n }. Assume E[y t ] = 0 for notational simplicity. Population quantities: γ(0) = E[y 2 t ], γ(h) = E[y t y t h ], ρ(h) = γ(h) γ(0). Sample quantities: ˆγ n (0) = 1 n n yt 2, ˆγ n (h) = 1 n t=1 n t=h+1 y t y t h, ˆρ n (h) = ˆγ n(h) ˆγ n (0). Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 6 / 22

Review of Dependent Wild Bootstrap White noise hypothesis is that ρ(h) = 0 for all h 1. Naïve approach: Box and Pierce s (1970) asymptotic χ 2 test n L ˆρ 2 n(h) d χ 2 L. h=1 This is not a white noise test but an IID test, because the asymptotic χ 2 property requires an IID assumption. Example: Box-Pierce test rejects GARCH, a well-known example of non-iid MDS, with probability 1 asymptotically. We do not want to reject GARCH, because it is white noise. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 7 / 22

Review of Dependent Wild Bootstrap A formal white noise test requires Shao s (2010, 2011) dependent wild bootstrap. To describe how it works, fix lag length h and consider testing for ρ(h) = 0. How can we construct a confidence band for ˆρ n (h)? Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 8 / 22

Review of Dependent Wild Bootstrap 1 Set a block size b n (typically b n = n). 2 Generate iid {ξ 1, ξ 2,..., ξ n/bn }. Define an auxiliary variable: ω = [ ξ 1,..., ξ 1, ξ 2,..., ξ 2,..., ξ n/bn,..., ξ n/bn ]. }{{}}{{}}{{} b n terms b n terms b n terms 3 Compute a bootstrapped autocorrelation: ˆρ (dw) n (h) = 1 ˆγ n (0) 1 n n t=h+1 ω t [y t y t h ˆγ n (h)]. 4 Repeat Steps 2-3 M times and sort ˆρ (dw) n,(1) (h) < < ˆρ(dw) n,(m) (h). 5 The 95% confidence band is C(h) = [ˆρ (dw) n,(0.025m) (h), ˆρ(dw) n,(0.975m) (h)]. 6 If ˆρ n (h) C(h), then we do not reject ρ(h) = 0. If ˆρ n (h) / C(h), then we reject ρ(h) = 0. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 9 / 22

Review of Dependent Wild Bootstrap Block 1 Block 2 Block 3 Block 20 y 1 y 2, y 2 y 3 Generate i.i.d. random numbers ξ 1,, ξ 20 ~ N(0, 1) y 3 y 4, y 4 y 5, y 5 y 6 y 6 y 7, y 7 y 8, y 8 y 9 y 57y 58, y 58y 59, y 59y 60 ξ 1 ξ 2 ξ 3 ξ 20 Compute bootstrapped autocorrelation Repeat M times Construct confidence band Examle: h = 1; n = 60; b n = 3. Preserved dependence within each block. No dependence across different blocks. Asymptotically correct size and consistency under weak dependence (e.g. GARCH, bilinear). Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 10 / 22

Hidden Pitfall: Periodic Confidence Bands Now consider rolling window analysis. Suppose that window size is n = 60 and block size is b n = 3. In window #1 (y 1,..., y 60 ), we have [ y 1 y 2, y 2 y 3, y 3 y 4, y 4 y 5, y 5 y 6, y 6 y 7, y 7 y 8, y 8 y 9,..., y 57 y 58, y 58 y 59, y 59 y 60 ]. }{{}}{{}}{{}}{{} Block 1 ( ξ 1 ) Block 2 ( ξ 2 ) Block 3 ( ξ 3 ) Block 20 ( ξ 20 ) In window #4 (y 4,..., y 63 ), we have [ y 4 y 5, y 5 y 6, y 6 y 7, y 7 y 8, y 8 y 9,..., y 57 y 58, y 58 y 59, y 59 y 60, y 60 y 61, y 61 y 62, y 62 y 63 ]. }{{}}{{}}{{}}{{} Block 20 ( ξ Block 1 ( ξ 1 ) Block 2 ( ξ 2 ) Block 19 ( ξ 19 ) 20 ) Similar bootstrapped autocorrelations appear in windows #1, #4, #7,.... = Periodicity with b n = 3 cycles. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 11 / 22

Hidden Pitfall: Periodic Confidence Bands 0.5 0-0.5 2 4 6 8 10 12 y 1,..., y 71 i.i.d. N(0, 1). Window size is n = 60. There are 71 60 + 1 = 12 windows. Block size is b n = 3. We plot ˆρ n (1) and 95% confidence band for each window. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 12 / 22

Remedy: Randomized Block Size Block size is b n = [c n]. Conventional choice that c = 1 produces periodicity. We propose to draw c U(1 δ, 1 + δ) independently across rolling windows and bootstrap samples. Randomness across windows removes periodicity. Randomness across bootstrap samples reduces the volatility of confidence bands. We naïvely choose δ = 0.5 (i.e. c U(0.5, 1.5)). Open question: optimal choice of δ. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 13 / 22

Remedy: Randomized Block Size Illustrative Example: y 1, y 2,..., y 400 i.i.d. N(0, 1). Window size is n = 240. There are 400 240 + 1 = 161 windows. Block size is b n = [c n] = [c 240] = [c 15.5]. We choose either c = 1 or c U(0.5, 1.5). We plot ˆρ n (1) and 95% confidence band for each window. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 14 / 22

Remedy: Randomized Block Size 0.2 0.2 0.1 0.1 0 0-0.1-0.1-0.2 50 100 150-0.2 50 100 150 a. c = 1 b. c U When c = 1, confidence bands have clear periodicity. When c U(0.5, 1.5), the periodicity evaporates dramatically. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 15 / 22

Stock Price Data: S&P 500 300 0.2 200 Iraq Shanghai 0.1 Iraq 0 100 0 Subprime AA+ Jan05 Jan10 Jan15-0.1-0.2 Subprime AA+ Shanghai Jan05 Jan10 Jan15 a. Level b. Log Return Daily data of S&P 500. January 1, 2003 October 29, 2015 (3,230 days). Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 16 / 22

Autocorrelations at Lag 1 0.5 0.5 0 0-0.5 Jan05 Jan10-0.5 Iraq Subprime Jan05 Jan10 AA+ a. c = 1 b. c U Window size is n = 240 (roughly a year). Block size is b n = [c n] = [c 15.5]. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 17 / 22

Cramér-von Mises White Noise Test White noise requires that ρ(h) = 0 for all h 1. Following Shao (2011), we use the Cramér-von Mises statistic: { π n 1 } 2 C n = n ˆγ n (h)ψ h (λ) dλ, ψ h (λ) = sin(hλ) 0 hπ. h=1 Bootstrapped p-values are computed based on the dependent wild bootstrap (with a randomized block size). Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 18 / 22

P-Values of Cramér-von Mises Test 1 0.5 0 Jan05 Jan10 P-values over rolling windows. S&P has significant autocorrelations during Iraq War and the subprime mortgage crisis. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 19 / 22

Conclusions 1. Outline Test for white noise hypothesis of stock returns Perform rolling window analysis with dependent wild bootstrap 2. Contributions Find that a fixed block size results in periodic confidence bands Reveal that the periodicity stems from repeated block structures Propose randomizing a block size to remove the periodicity 3. Empirical Finding White noise hypothesis is rejectedfor S&P during crisis periods -- Significantly negative autocorrelations at lag 1 Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 20 / 22

References Box, G. E. P. and D. A. Pierce (1970). Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models. Journal of the American Statistical Association, 65, 1509-1526. Campbell, J. Y., S. J. Grossman, and J. Wang (1993). Trading Volume and Serial Correlation in Stock Returns. Quarterly Journal of Economics, 108, 905-939. Fama, E. F. (1970). Efficient Capital Markets: A Review of Theory and Empirical Work. Journal of Finance, 25, 383-417. Fama, E. F. and French, K. R. (1988). Permanent and Temporary Components of Stock Prices. Journal of Political Economy, 96, 246-273. Lim, K.-P. and R. Brooks (2009). The Evolution of Stock Market Efficiency over Time: A Survey of the Empirical Literature. Journal of Economic Surveys, 25, 69-108. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 21 / 22

References Lo, A. W. (2004). The Adaptive Market Hypothesis: Market Efficiency from an Evolutionary Perspective. Journal of Portfolio Management, 30, 15-29. Lo, A. W. (2005). Reconciling Efficiency Markets with Behavioral Finance: The Adaptive Market Hypothesis. Journal of Investment Consulting, 7, 21-44. Shao, X. (2010). The Dependent Wild Bootstrap. Journal of the American Statistical Association, 105, 218-235. Shao, X. (2011). A Bootstrapped-Assisted Spectral Test of White Noise under Unknown Dependence. Journal of Econometrics, 162, 213-224. Hill & Motegi (UNC & Kobe U.) Weak Form Efficiency of Stock Markets September 10-11, 2017 22 / 22