Expecting the Unexpected: Uniform Quantile Regression Bands with an application to Investor Sentiments

Transcription

1 Expecting the Unexpected: Uniform Bands with an application to Investor Sentiments Boston University November 16, 2016

2 Econometric Analysis of Heterogeneity in Financial Markets Using s Chapter 1: Expecting the Unexpected: Uniform Quantile Regression Bands with an Application to Investor Sentiments Chapter 2: Heterogeneity in Development Funding for Micro enterprises: Evidence from Sri Lanka Field Experiments Chapter 3: Housing Price Volatility, Hot Money and the Capital Account in China (with Kevin P. Gallagher)

3 Motivation Classical Linear (Koenker&Bassett, 1978) Q Y X (τ x) = β(τ) x, τ (0, 1) A flexible model for conditional distributions; Characterize the impact on distributional features beyond the mean; Applications: wage/income inequality (labor); determinants of infant weights (health); VaR (finance)...

4 Motivation Outline Introduction Motivation Literature Model Set up Inference Example Application Simulation Conclusion Summary Outlook

5 Motivation Motivation (QR) Uniform confidence bands for QR estimators in a time series setting Serially correlated error terms Empirical Question Compare differences in QR estimators at different quantiles of the conditional distribution.

6 Motivation Example Relationship between realized returns and investor sentiments using quantile regressions; Investor Sentiments: weekly survey; stock returns: daily data; Q: Why interested in regression quantile uniformity? A: Compare QR coefficients across quantiles of conditional distributions.

7 Literature Literature Review Classical : Chesher (2003), Koenker (2005), Su & Xiao (2008), Chernozhukov & Fernandez-Val (2010), Härdle & Song (2010), Chernozhukov et al (2013). Dynamic : Koenker & Xiao (2006), Qu (2008), Escanciano & Velasco (2010), Wu & Zhou (2014). Investor Sentiments: Fama & French (1992), Greenwood & Shleifer (2014), Shiller (2014), Ni et al (2015), Nardo (2015).

8 Literature Related Research Qu (2008): testing for structural changes in regression quantiles, Journal of Econometrics. Greenwood & Shleifer (2014): comparing financial survey data on expectations with expected returns,review of Financial Studies. Wu & Zhou (2014): quantile structural change testing for linear models with a wide class of non-stationary regressors and errors.

9 Literature Results Develop uniform confidence bands for QR estimator over quantiles under the time series setting with serially correlated error terms; QR estimates of investor sentiments on realized stock returns vary at different quantiles of the return distribution; At higher quantiles in the stock returns, more optimistic predictions tends to turn into lower future returns.

10 Literature Uniform QR Bands What is uniform confidence bands? To identify confidence sets for which coverage of QR estimator converges to asymptotic size independent of quantiles. Why needed? Better than pointwise confidence bands in order to compare difference in QR at different quantiles. What s difficult? To develop the uniform QR inference in a time series setting and apply for investor sentments and stock returns.

11 Literature Contributions Develop a method for uniform inference on conditional quantiles with serially correlated error terms; Analyze the financial survey weekly data using quantile regressions; Detect a changing pattern in the relationship between investor expectations and realized S&P500 returns.

12 Set up Model Q yt x t (τ x t ) = β(τ) x t Denote (y t, x t ) time series sequences of random variables. Assume that the τth conditional quantile function of y t given x t is linear in x t. y t : realized returns and x t : investor sentiments.

13 Set up Quantile regression estimators ˆβ(τ) can be estimated by min β(τ) T ρ τ (y t β(τ) x t ), t=1 where ρ τ (u) = u(1(u < 0) τ). (Koenker, 2005) Denote the full sample sub-gradient as S T (τ, β(τ)) = T 1/2 where φ τ (u) = 1(u 0) τ T t=1 x t φ τ (y t β(τ) x t ),

14 Set up Assumptions A1. (y t, x t ) t=1,...,t is a strictly stationary and ergodic time series such that T 1/2 max 1 t T x t = o p (1). A2. Let F (. x t ) = F t (.) denote the conditional distribution function of y t given x t. F t (.) has continuous Lebesgue density f t (.) = f(. x t ) uniformly bounded away from 0 and over quantiles τ. f t (y 1 ) f t (y 2 ) C y 1 y 2 with E[C x t 2 ] < and C is a constant.

15 Set up Assumptions(continued): A3. E[φ τ (y t β 0 (τ) x t ) x t ] = 0, a.s. for some unique β 0 (τ) B R p, where β 0 (τ) is an interior point of the compact set B for each τ. A4. There exist a random variable A and a constant 0 k 1 < 1/2 such that T 1 T t=1 x t A T T k 1 a.s. In addition, sup E(A k 2 ) C < for some k 2 > 2.

16 Set up Assumptions(continued): A5. There exist k 3 k 4 > 1, (k 3 1)/(k 4 1) > 1 + 2k 1 and C 3 <, C 4 < such that for T > 1, T 1 T t=1 E[T 1 [E(x tx t )] k 3 C 3 T t=1 (x tx t ) k 4 ] C 4 If E(x tx t ) 2 C 5 <, t, we can take k 3 = 2, k 4 = 3/2. A4-5 ensure stochastic equi-continuity of the process S T (τ, β(τ)) over τ. A5 is used for the tightness of certain sequential weighted empirical process.

17 Set up Assumptions(continued): A6.1 T 1 T t=1 x tx t = J 0 + o p (1) where J 0 is a finite, symmetric and positive definite matrix. A6.2 T 1 T t=1 f t(β 0 (τ) x t )x t x t = H 0 + o p (1) holds uniformly over τ, where H 0 is a finite, symmetric and positive definite matrix for each τ. A6 provides general assumptions to facilitate the derivation of the asymptotic distribution of the QR estimator over quantiles.

18 Set up Serially Correlated Error Terms Notations For a random variable X, let X q := (E X q ) 1/q be its L q norm. For a d-dimensional random vector v = (v 1,..., v d ), v = v 2 i, let v q = v 2 i q. v L q if v q <. Let filtrations i = (..., η i 1, η i ) and (j) i = (..., η j 1, η j, η j+1,..., η i ), (j i), where ({η i } i=, {η j } j= ) are i.i.d. random variables.

19 Set up Serially Correlated Error Terms Error Terms Denote the error term in the quantile regression model by e t (τ) = y t β ( τ) x t. Denote e t (τ) = G τ (i, t,l ). If G τ (i,.), i = t/t is a smooth function in i, e t (τ) changes smoothly over (0,1]. t,l = {..., ɛ 0,l, ɛ 1,l,..., ɛ t,l }. For l s, {ɛ l } and {ɛ s } are independent i.i.d. r.v s.

20 Set up Serially Correlated Error Terms Assumption S1 Assume the error term e t (τ) = G τ (i, t,l ) satisfies that i, j (0, 1), (G τ (i, t,l ) G τ (j, t,l ))/ i j v C for some constant v > 1. The dependence measure of the error term in L v norm satisfies G τ (i, t,l ) G τ (i, t,l ) v Mχ l where M is sufficient large constant and χ (0, 1). Serial Correlation Assumption S1 requires that the process e t to be short range dependent with exponentially decaying dependence measures. It covers a broad class of serially correlated error terms, such as invertible ARMA process.

21 Inference Lemma 1 Under Assumption 1-5, S1 and for large T, T ( ˆβ(τ) β0 (τ))h 0 (τ) [S T (τ, ˆβ(τ)) S T (τ, β 0 (τ))] = o p (1) uniformly in τ. ˆβ(τ) is the QR estimate of β0 (τ) using the full sample. T H 0 (τ) = lim T 1 f t (β 0 (τ) x t )x t x t T t=1.

22 Inference Proposition 1 Based on Wu and Zhou(2014), under assumption S1, on a richer probability space, there exists a d-dimensional zero-mean Gaussian process U τ (t), with covariance function γ(t, s) = min(t,s) 0 Σ 2 τ (r)dr, for i = t/t (0, 1], t = 1,..., T, Σ 2 τ (i) = k Cov(G τ (i, 0 ), G τ (i, k )), s.t. S T (τ, β 0 (τ)) U τ = o p (T 1/4 log 2 T ). (1) where S T (τ, β 0 (τ)) = T 1/2 T t=1 x tφ τ (y t β(τ) x t ).

23 Inference Theorem 1 Suppose assumptions A1-5 and S1 holds, we have the following weak convergence in distribution: T ˆβT (τ) β 0 (τ) T H 1 0 (τ)u τ (2)

24 Inference Bootstrapping Based on Wu and Zhou (2014), ˆΛ cn (n, τ) converges to H 0 in probability and Ψ m are uniformly consistent estimate of U τ (t). Ψ m = ˆΛ cn (n, τ) = n i=1 φ(ê i (τ)/c n )x i x i nc n (3) n (m(n m + 1)) 1/2 (ˆω i,m m n ˆω 1,n)R i, (4) i=1 where ω j,m = j+m 1 r=j φ τ (ê r (τ))x r and {R i } n i=1 are i.i.d. standard normals.

25 Inference Remark Inference: i.i.d. case Under Assumption 1-6 and for large T, T ( ˆβ(τ) β0 (τ)) H 0 (τ) 1 J 1/2 0 U τ where denotes weak convergence in the Skorokhod space of τ and U τ is a Kiefer process with E[U τ ] = 0 and E[U(τ 1 )U(τ 2 )] = (τ 1 τ 2 τ 1 τ 2 )I p. Recall H 0 (τ) = T 1 f t (β 0 (τ) x t )x t x t + o p (1) uniformly in τ [ω, 1 ω] and ω (0, 1/2).

26 Inference Corollary Under Assumption 1-6, the uniform confidence bands Ĉα(τ) for the QR estimator: lim T P ( ˆβ(τ) Ĉα(τ), τ (0, 1)) = α, Ĉ α (τ) = [ ˆβ(τ) : T ( ˆβ(τ) β 0 (τ))/ˆσ(τ) U α ], For the scalar estimator ˆβ(τ), Ĉ α (τ) = [ ˆβ(τ) ˆσ(τ)U α, ˆβ(τ) + ˆσ(τ)U α ] where ˆσ(τ) is the estimated uniform standard deviation of the QR estimator and U α is the (1 α) percentile of sup τ U τ.

27 Inference Algorithm Uniform Bands for Quantile Estimators Step 1: Obtain the consistent QR estimators. Step 2: Estimate the variance of the QR estimator. Step 3: Simulate the critical value and calculate the uniform confidence bands.

28 Application Empirical Research Questions Investors Sentiments How do investors expectations on the stock market relate to the real S&P 500 stock market performance? Do such correlations vary across the conditional distribution of realized returns of the stock market? Compare coefficients of dynamics quantile regression of financial time series. Learn about the asymptotic properties of the QR estimators.

29 Application Investor Expectation Measures Who are these potential investors taking the survey? American Association of Individual Investors (AAII): Investor Sentiment Survey Individuals are polled from the ranks of the AAII membership on a weekly basis(july 24, Feb 20, 2014).Only one vote per member is accepted in each weekly voting period. Question: Do you feel the direction of the stock market over the next six months will be up (bullish), no change (neutral) or down (bearish)? bull-bear spread= % bullish - % bearish

30 Application Model Is investor expectation consistent with realized returns of stock markets and expected returns? Q Rt+k E t (τ E t ) = α(τ) + β(τ)e t, (5) Denote E t as the bull-bear spread, an instrumental measure of investors expectation from AAII survey. Denote R t+k as the k-time ahead realized stock market return based on the daily S&P500 index. R t+k = log P t+k log P t+k 1

31 Application Uniform Bands at 90% nominal level QR Coefficients of Investor Sentiments spread over τ [0.05, 0.95]

32 Application

33 Application Interpret the results There is pronounced heterogeneity in the slope parameter of the quantile regression. This coefficient is slightly positive at lower quantiles while significantly negative at higher quantiles. The negative relationship suggests that, at such quantiles, more optimistic predictions tends to turn into lower future returns, a puzzling phenomenon that awaits further study.

34 Simulation Simulation Procedure Step 1: Select m and c n based on minimum volatility (MV) method, first advocated in Politis et.al (1999); Step 2: Generate B times{ψ m } for sample size T; Step 3: Calculate E b (τ) = ˆΛ 1 c n (T, τ)ψ m (τ) for every given τ; Step 4: Let E b = sup τ E b (τ). Let E (1) E (2)... E (B) be the order statistics of E b. Then E (1 α)b is the level α critical values for the QR estimator coefficient uniform confidence bands.

35 Simulation Simulation Data Generating Process y t = 1 + x t + u t x t is χ 2 (3)/3 Case 1: u t is i.i.d.n(0, 1). Case 2: u t = 0.5u t 1 + ɛ t and ɛ is N(0,1). Calculated the simulated coverage probabilities.

36 Simulation Simulation Results

37 Simulation Simple Linear Quantile Structure Change Test Test investors expectation and real returns changes across quantiles. H( ˆβ(τ)) = (X X) 1/2 T t=1 x tφ τ (y t+1 x t ˆβ(τ)) SQ τ = sup λ [0,1] (τ(1 τ)) 1/2 (H λ ( ˆβ(τ)) H 1 ( ˆβ(τ))). = max( z 1,..., z k ) for a generic vector z = (z 1,..., z k ) Based on Qu(2008), under certain assumptions, [λn] i=1 x t[φ τ (y t+1 x ˆβ(τ, t λ)) φ τ (y t+1 x tβ 0 )] = λnh 0 ( ˆβ(τ, λ) β 0 ) + no p (0) H 0 = plim n n 1 n i=1 f i(f 1 (τ))x i x i i

38 Simulation Robustness Test There is no structural break for each quantile from 0.2 to 0.8. The difference between the two quantiles doesn t indicate a structure change so that the comparison holds for the uniform bands. Quantile spread return Rm return Rd Critical values # of changes Table : Analysis based on a single conditional quantile function

39 Summary Expecting the Unexpected Correlations between investors sentiment and realized returns vary across the distribution of real returns. Investors sentiment is different from the expected returns, especially in the tails of the realized return distribution. Investors response to extreme situations asymmetrically.

40 Summary Empirical Results Distributional Evidence in Investor Expectations Investor sentiments are negatively correlated with realized stock market returns during expansions. There are asymmetry responses in high and low quantiles of stock returns. More optimistic predictions tends to turn into lower future returns at higher quantiles.

41 Summary Conclusion Information of investment behaviors can be reflected from investor sentiments. QR coefficient of realized SP500 index returns and investor sentiments: downward sloping from positive to negative. Investment behavior interpretation Investors are slightly risk averse in the crises and expect worse. Investors can hardly expect to reach the peaks/extremes. Investors show pessimistic when stock market is in its expansions.

42 Outlook Future Challenges Study the volatility of stock market with respect to investor expectation variation Self-selection issue in investor survey data: whether investors are more likely to be optimistic/pessimistic and report good/bad expectations. Dynamics of investor sentiment: the speed for inefficiency to be resolved in weekly updated data. Rational expectation tests in quantiles. Lags choices and determinants of investor expectations. Dynamics of predictability in overall stock returns.