# Introduction to Quantile Regression

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1 Introduction to Quantile Regression CHUNG-MING KUAN Department of Finance & CRETA National Taiwan University June 13, 2011 C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

2 Lecture Outline 1 Introduction 2 Quantile Regression Quantiles Quantile Regression (QR) Method QR Models 3 Algebraic Properties Equivariance Gooness of Fit 4 Asymptotic Properties Heuristics QR Estimator as a GMM Estimator Asymptotic Distribution C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

3 Lecture Outline (cont d) 5 Estimation of Asymptotic Covariance Matrix 6 Hypothesis Testing Wald Tests Likelihood Ratio Tests 7 Treatment Effect Average Treatment Effect (ATE) Quantile Treatment Effect (QTE) 8 Empirical Applications Return-Volume Relations Effects of NHI on Saving C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

4 Introduction The behavior of a random variable is governed by its distribution. Moment or summary measures: Location measures: mean, median Dispersion measures: variance, range Other moments: skewness, kurtosis, etc. Quantiles: quartiles, deciles, percentiles Except in some special cases, a distribution can not be completely characterized by its moments or by a few qunatiles. Mean and median characterize the average and center of y but may provide little info about the tails. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

5 Conventional Methods For the behavior of y conditional on x, consider regression y t = x tβ + e t. Least squares (LS): Legendre (1805) Minimizing T t=1 (y t x tβ) 2 to obtain ˆβ T. x ˆβ T approximates the conditional mean of y given x. Least absolute deviation (LAD): Boscovich (1755) Minimizing T t=1 y t x tβ to obtain ˇβ T. x ˇβT approximates the conditional median of y given x. Both the LS and LAD methods provide only partial description of the conditional distribution of y. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

6 Mosteller F. and J. Tukey, Data Analysis and Regression: What the regression curve does is (to) give a grand summary for the averages of the distributions corresponding to the set of xs. We could go further and compute several different regression curves corresponding to the various percentage points of the distributions and thus get a more complete picture of the set. Ordinarily this is not done, and so regression often gives a rather incomplete picture. Just as the mean gives an incomplete picture of a single distribution, so the regression curve gives a correspondingly incomplete picture for a set of distributions. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

7 Quantiles The θ th (0 < θ < 1) quantile of F Y is q Y (θ) := F 1 Y (θ) = inf{y : F Y (y) θ}. q Y (θ) is an order statistic, and it can also be obtained by minimizing an asymmetric (linear) loss function: θ y q df Y (y) + (1 θ) y>q y<q y q df Y (y). The first order condition of this minimization problem is 0 = θ df Y (y) + (1 θ) df Y (y) y>q y<q = θ[1 F Y (q)] + (1 θ)f Y (q) = θ + F Y (q). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

8 Sample Quantiles The sample counterpart of the asymmetric linear loss function is 1 T T ρ θ (y t q) = 1 [θ y T t q + (1 θ) t=1 t:y t q t:y t<q where ρ θ (u) = (θ 1 {u<0} )u is known as the check function. ] y t q, Given θ, minimizing this function yields the θ th sample quantile of y. Key point: Other than sorting the data, a sample quantile can also be found via an optimization program. Given various θ values, we can compute a collection of sample quantiles, from which the distribution can be traced out. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

9 ρ θ θ =0.8 θ =0.5 θ =0.2 Figure: Check function ρ θ (u) = (θ 1 {u<0} )u. u C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

10 Quantile Regression (QR) Method Koenker and Basset (1978) Given y t = x tβ + e t, the θ th QR estimator ˆβ(θ) minimizes V T (β; θ) = 1 T T ρ θ (y t x tβ) t=1 where ρ θ (e) = (θ 1 {e<0} )e. For θ = 0.5, V T is symmetric, and ˆβ(0.5) is the LAD estimator. x ˆβ(θ) approximates the θ th conditional quantile function Q y x (θ), with ˆβ i (θ) the estimated marginal effect of the i th regressor on Q y x (θ). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

11 Finding the Solution to V T Difficulties in estimation: The QR estimator ˆβ(θ) does not have a closed form. V T is not everywhere differentiable, so that standard numerical algorithms do not work. A minimizer of V T (β; θ) is such that the directional derivatives at that point are non-negative in all directions w: d dδ V T (β + δw; θ) = 1 δ=0 T T ψθ (y t x tβ, x tw)x tw, t=1 ψ θ (a, b) = θ 1 {a<0} if a 0, ψ θ (a, b) = θ 1 {b<0} if a = 0. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

12 Let b be the point such that y t = x tb for t = 1,..., k. This is a minimizer of V k because the directional derivative is 1 k k ( ) θ 1{ x t x w<0} t w, t=1 which must be non-negative for any w. Thus, b a basic solution to the minimization of V T. Other basic solutions: b(κ) = X(κ) 1 y(κ), each yielding a perfect fit of k observations. The desired estimator ˆβ(θ) can be obtained by searching among those basic solutions, for which a linear programming algorithm will do. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

13 Linear Programming y = Xβ + e can be expressed as y = X(β + β ) + (e + e ) = Az, where A = [X, X, I T, I T ] and z = [ β +, β, e +, e ], with β + and β the positive and negative parts of β, respectively. Let c = [0, 0, θ1, (1 θ)1 ]. Minimizing V T (β; θ) with respect to β is equivalent to the following linear program: 1 min z T c z, s.t. y = Az, z 0. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

14 Some Remarks ˆβ(θ) is also the QMLE based on an asymmetric Laplace density: f θ (e) = θ(1 θ) exp[ ρ θ (e)]. Due to linear loss function, ˆβ(θ) is more robust to outliers than the LS estimator. The estimated θ th quantile regression hyperplane must interpolate k observations in the sample. (Why?) QR is not the same as the regressions based on split samples because every quantile regression utilizes all sample data (with different weights). Thus, QR also avoids the sample selection problem arising from sample splitting. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

15 QR: Location Shift Model DGP: y t = x tβ o + ε t = β 0 + x tβ 1 + ε t, where ε t are i.i.d. with the common distribution function F ε. The θ-th quantile function of y is Q y x (θ) = β 0 + x β 1 + Fε 1 (θ), and hence quantile functions differ only by the intercept term and are a vertical displacement of one another. The model can also be expressed as y t = [β 0 + Fε 1 (θ)] + x } {{ } tβ 1 + ε t,θ, β 0 (θ) where Q εθ x(θ) = 0. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

16 QR: Location-Scale Shift Model DGP: y t = x tβ o + (x tγ o )ε t, where ε t are i.i.d. with the df F ε. The θ th quantile function of y is Q y x (θ) = x tβ o + (x tγ o )Fε 1 (θ), and hence quantile functions differ not only by the intercept but also the slope term. The model can also be expressed as y t = x t[β o + γ o Fε 1 (θ)] + ε } {{ } t,θ, β(θ) where Q εθ x(θ) = 0. The QR estimator ˆβ(θ) converges to β(θ), and x ˆβ(θ) approximates the θ th quantile function of y given x, Q y x (θ). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

17 Algebraic Properties: Equivariance Let ˆβ(θ) be the QR estimator of the quantile regression of y t on x t. Scale equivariance: For y t = c y t, let ˆβ (θ) be the QR estimator of the quantile regression of y t on x t. For c > 0, ˆβ (θ) = c ˆβ(θ). For c < 0, ˆβ (1 θ) = c ˆβ(θ). ˆβ (0.5) = c ˆβ(0.5), regardless of the sign of c. Shift equivariance: For y t = y t + x tγ, let ˆβ (θ) be the QR estimator of the quantile regression of y t on x t. Then, ˆβ (θ) = ˆβ(θ) + γ. Equivariance to reparameterization of design: Given X = XA for some nonsingular A, ˆβ (θ) = A 1 ˆβ(θ). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

18 Equivariance to monotonic transformations: For a nondecreasing function h, IP{y a} = IP{h(y) h(a)}, so that Q h(y) x (θ) = h ( Q y x (θ) ). Note that the expectation operator does not have this property because IE[h(y)] h(ie(y)) in general, except when h is linear. Example: If x β is the θ th conditional quantile of ln y, then exp(x β) is the θ th conditional quantile of y. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

19 Goodness of Fit Specification: y t = x 1t β 1 + x 2t β 2 + e t. A measure of the relative contribution of additional regressors x 2t is 1 V T (ˆβ1 (θ), ˆβ 2 (θ); θ ) V T ( β1 (θ), 0; θ ), where V T ( β 1 (θ), 0; θ ) is computed under the constraint β 2 = 0. A measure of the goodness-of-fit of a specification is thus R 1 (θ) = 1 V T (ˆβ(θ); θ ) ). V T (ˆq(θ), 0; θ where ˆq(θ) is the sample quantile and V T (ˆq(θ), 0; θ ) is obtained from the model with the constant term only. Clearly, 0 < R 1 (θ) < 1. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

20 Asymptotic Properties: Heuristics Ignoring y t = q, the FOC of minimizing T 1 T t=1 ρ θ(y t q) is g T (q) := 1 T T (1 {yt<q} θ). t=1 Clearly, g T (q) is non-decreasing in q (why?), so that ˆq(θ) > q iff g T (q) < 0. Thus, We have IP [ T (ˆq(θ) q(θ) > c ] = IP [ g T ( q(θ) + c/ T ) < 0 ]. IE [g T ( q(θ) + ( var [g T q(θ) + c )] ( = F q(θ) + T c ) θ f (q(θ)) c T T c )] = 1 T T F (1 F ) 1 θ(1 θ). T C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

21 Setting λ 2 = θ(1 θ)/f 2 (q(θ)), IP [ T (ˆq(θ) q(θ) ) > c ] [ ( ) gt q(θ) + c/ T = IP θ(1 θ)/t < 0 [ gt ( q(θ) + c/ T ) = IP c θ(1 θ)/t λ < c λ [ ( ) gt q(θ) + c/ T f (q(θ))c/ T = IP θ(1 θ)/t ] ] < c λ ] D 1 Φ(c/λ), by a CLT. This implies T (ˆq(θ) q(θ) ) D N (0, λ 2 ). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

22 GMM Estimation Given q moment conditions IE[m(w t ; β o )] = 0, β o (k 1) is exactly identified if q = k and over-identified if q > k. When β o is exactly identified, the GMM estimator ˆβ of β o solves T 1 T t=1 m(w t; β) = 0. Asymptotic Distribution of the GMM Estimator Given the GMM estimator ˆβ of β o, T (ˆβ β o ) A N ( 0, G 1 o Σ o G 1 o with Σ o = IE[m(w t ; β o )m(w t ; β o ) ], and 1 T T IP β m(w t ; β o ) G o := IE[ β m(w t ; β o )]. t=1 ), C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

23 QR Estimator as a GMM Estimator The QR estimator ˆβ(θ) satisfies the asymptotic FOC : 1 T T t=1 ϕ θ (y t x t ˆβ(θ)) := 1 T T t=1 The (approximate) estimating function is thus 1 T T ( ) x t θ 1{yt x t. β<0} t=1 The expectation of the estimating function is x t ( θ 1{yt x t ˆβ(θ)<0}) = oip (1). IE { x t [ θ IE ( 1{yt x t β<0} x t )]} = IE { xt [ θ Fy x (x tβ) ]}. When β is evaluated at β(θ), F y x (x tβ) must be θ so that the moment conditions are IE [ ϕ θ ( yt x tβ(θ) )] = 0. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

24 Asymptotic Distribution When integration and differentiation can be interchanged, G(β) = IE [ β ϕ θ (y t x tβ) ] = β IE { x t [ θ Fy x (x tβ) ]} = IE [ x t x tf y x (x tβ) ]. Then, G(β(θ)) = IE [ x t x tf eθ x(0) ]. 1 {yt x t β(θ)<0} is Bernoulli with mean θ and variance θ(1 θ), so that ( Σ(β) = IE x t x t IE [( ) 2 ] ) θ 1 {yt x t β<0} xt. Then, Σ(β(θ)) = θ(1 θ) IE(x t x t) =: θ(1 θ)m xx. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

25 Asymptotic Normality of the QR Estimator T [ˆβ(θ) β(θ) ] D N (0, θ(1 θ)g ( β(θ) ) 1 Mxx G ( β(θ) ) 1 ), where M xx = IE(x t x t) and G(β(θ)) = IE [ x t x tf eθ x(0) ]. Conditional heterogeneity is characterized by the conditional density f eθ x(0) in G(β(θ)), which is not limited to heteroskedasticity. If f eθ x(0) = f eθ (0), i.e., conditional homogeneity, T [ˆβ(θ) β(θ) ] ( D N 0, ) θ(1 θ) [f eθ (0)] 2 M 1 xx. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

26 Estimation of Asymptotic Covariance Matrix Consistent estimation of D(β(θ)) = G ( β(θ) ) 1 Mxx G ( β(θ) ) 1. Estimation of M xx : M T = T 1 T t=1 x tx t. Digression: Differentiating both sides of F (F 1 (θ)) = θ: df 1 (θ) dθ = 1 f (F 1 =: s(θ), (θ)) differentiating a quantile function yields a sparsity function. Estimating the sparsity function: Using a difference quotient of empirical quantiles ŝ T (θ) = [ F 1 T (θ + h 1 T ) F T (θ h T ) ] /(2h T ). 1 F T (θ): Letting ê (i) be the i th order statistic of QR residuals ê t, F 1 T (τ) = ê (i), τ [(i 1)/T, i/t ). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

27 Hendricks and Koenker (1991): Estimating f e(θ) x (0) in G(β(θ)) by ˆf t = x t 2h T [ˆβ(θ + h T ) ˆβ(θ h T ) ], and estimating G by ĜT = 1 T T t=1 ˆf t x t x t. Powell (1991): Estimating G(β(θ)) by ĜT = 1 2Tc T T 1 { êt(θ) <ct }x t x t, t=1 where c T 0 and T 1/2 c T as T. STATA: Bootstrap C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

28 Standard Wald Test H 0 : Rβ(θ) = r, where R is q k and r is q 1. T [ˆβ(θ) β(θ) ] D ( ) N 0, θ(1 θ)d(β(θ)). Under the null, T R (ˆβ(θ) β(θ) ) = T ( Rˆβ(θ) r ) where Γ(β(θ)) = RD(β(θ))R. The Null Distribution of the Wald Test D N ( 0, θ(1 θ)γ(β(θ)) ), W T (θ) = T [ Rˆβ(θ) r ] Γ(θ) 1 [ Rˆβ(θ) r ] /[θ(1 θ)] D χ 2 (q), where Γ(θ) = R D(θ)R, with D(θ) a consistent estimator of D(β(θ)). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

29 Sup-Wald Test H 0 : Rβ(θ) = r for all θ S (0, 1). The Brownian bridge: B q (θ) d = [θ(1 θ)] 1/2 N (0, I q ), and hence Γ(θ) 1/2 T [ Rˆβ(θ) r ] Thus, W T (θ) D Bq (θ). D B q (θ)/ θ(1 θ) 2, uniformly in θ. The Null Distribution of the Sup-Wald Test sup θ S B q (θ) 2 W T (θ) sup, θ(1 θ) θ S where S is a compact set in (0, 1). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

30 To test Rβ(θ) = r, θ [a, b], set a = θ 1 <... < θ n = b and compute sup -W T = sup W T (θ i ). i=1,...,n Koenker and Machado (1999): [a, b] = [ɛ, 1 ɛ] with ɛ small. For s = θ/(1 θ), B(θ)/ θ(1 θ) = d W (s)/ s, so that IP sup B q (θ) 2 { } < c θ(1 θ) = IP W sup q (s) 2 < c, s θ [a,b] with s 1 = a/(1 a), s 2 = b/(1 b). s [1,s 2 /s 1 ] Some critical values were tabulated in DeLong (1981) and Andrews (1993); the other can be obtained via simulations. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

31 Likelihood Ratio Tests Let ˆβ(θ) and β(θ) be the constrained and unconstrained estimators and ˆV T (θ) = V T (ˆβ(θ); θ) and Ṽ T (θ) = V T ( β(θ); θ) be the corresponding objective functions. Given the asymmetric Laplace density: f θ (u) = θ(1 θ) exp[ ρ θ (u)], the log-likelihood is T L T (β; θ) = T log(θ(1 θ)) ρ θ (y t x tβ). t=1 2 times the log-likelihood ratio is 2 [ L T (ˆβ(θ); θ) L T ( β(θ); θ) ] = 2 [ Ṽ T (θ) ˆV T (θ) ]. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

32 Koenker and Machado (1999): LR T (θ) = 2[ Ṽ T (θ) ˆV T (θ) ] θ(1 θ)[f eθ (0)] 1 D χ 2 (q). This test is also known as the quantile ρ test. Koenker and Bassett (1982): For median regression, LR T (0.5) = 8[ Ṽ T (0.5) ˆV T (0.5) ] because f e0.5 (0) = 1/4. [f e0.5 (0)] 1 = 2 [ Ṽ T (0.5) ˆV T (0.5) ], C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

33 Average Treatment Effect (ATE) Evaluating the impact of a treatment (program, policy, intervention). Let D be the binary indicator of treatment and X be covariates. Y 1 (Y 0 ) is the potential outcome when an agent is (is not) exposed to the treatment. The observed outcome is Y = DY 1 + (1 D)Y 0. We observe only one potential outcome (Y 1i or Y 0i ) and hence can not identify the individual treatment effect, Y 1i Y 0i. We may estimate the ATE: IE(Y 1 Y 0 ). Under conditional independence: (Y 1, Y 0 ) D X, IE(Y D = 1, X ) IE(Y D = 0, X ) = IE(Y 1 Y 0 X ), so that the ATE is IE(Y 1 Y 0 ) = IE[IE(Y 1 Y 0 X )]. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

34 Using the sample counterpart of IE(Y D = 1, X ) IE(Y D = 0, X ) we have ÂTE = 1 N [ˆµ1 (X N i ) ˆµ 0 (X i ) ]. i=1 For the dummy-variable regression: Y i = α + D i γ + X i β +e } {{ } i, i = 1,..., n, µ D the LS estimate of γ is ÂTE. Other estimators: Kernel matching, nearest neighbor matching, propensity score matching (based on p(x) = IP(D = 1 X = x)), etc. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

35 Quantile Treatment Effect (QTE) Let F 0 and F 1 be, resp., the distributions of control and treatment responses. Let (η) be the horizontal shift from F 0 to F 1 : F 0 (η) = F 1 (η + (η)). Then, (η) = F 1 1 (F 0 (η)) η, and the θ th QTE is, for F 0 (η) = θ, QTE(θ) = F 1 1 (θ) F 1 0 (θ) = q Y1 (θ) q Y0 (θ), the difference between the quantiles of two distributions. We may apply the QR method to Y i = α + D i γ + X i β + e i, the resulting QR estimate ˆγ(θ) is the estimated θ th QTE. Other: A weighting estimator based on the propensity score. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

36 Difference in Differences The impact of a program (policy) may be observed after certain period of time. To identify the true treatment effect, the potential change due to time (other factors) must be excluded first. Define the following dummy variables: (i) D i,τ = 1 if the i th individual receives the treatment; (ii) D i,a = 1 if the i th individual s in the post-program period; (iii) D i,aτ = D i,τ D i,a. Model: Y i = α + α 1 D i,τ + α 2 D i,a + α 3 D i,aτ + X i β + e i. For the treatment group in pre- and post-program periods, the time effect is α 2 + α 3. For the control group in pre- and post-program periods, the time effect is α 2. The treatment effect is the difference between these two effects: α 3. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

37 Empirical Study: Return-Volume Relations Granger non-causality is defined in terms of distribution. Existing tests focus on its implications: Non-causality in mean (linear model): Granger (1969, 1980). Non-causality in variance: Granger et al. (1986), Cheung and Ng (1996). Nonlinear causality: Hiemstra and Jones (1994). Non-causality in some quantiles: Lee and Yang (2006), Hong et al. (2006). Empirical studies on return-volume relations are typically based on least-squares regressions and find that volume does not Granger cause return. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

38 Notions of Granger Non-Causality x does not Granger cause y in distribution if F yt (η (Y, X ) t 1 ) = F yt (η Y t 1 ), η R. x does not Granger cause y in mean if IE[y t (Y, X ) t 1 ] = IE(y t Y t 1 ). x does not Granger cause y in all quantiles if Q yt (τ (Y, X ) t 1 ) = Q yt (τ Y t 1 ), τ [0, 1]; cf. Lee and Yang (2006), Hong et al. (2006). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

39 Notions of Granger Non-Causality x does not Granger cause y in distribution if F yt (η (Y, X ) t 1 ) = F yt (η Y t 1 ), η R. x does not Granger cause y in mean if IE[y t (Y, X ) t 1 ] = IE(y t Y t 1 ). x does not Granger cause y in all quantiles if Q yt (τ (Y, X ) t 1 ) = Q yt (τ Y t 1 ), τ [0, 1]; cf. Lee and Yang (2006), Hong et al. (2006). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

40 Notions of Granger Non-Causality x does not Granger cause y in distribution if F yt (η (Y, X ) t 1 ) = F yt (η Y t 1 ), η R. x does not Granger cause y in mean if IE[y t (Y, X ) t 1 ] = IE(y t Y t 1 ). x does not Granger cause y in all quantiles if Q yt (τ (Y, X ) t 1 ) = Q yt (τ Y t 1 ), τ [0, 1]; cf. Lee and Yang (2006), Hong et al. (2006). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

41 Causal Relations between Return and Volume Chuang, Kuan, and Lin (2009, JBF): 2 stock market indices: NYSE and S&P 500, from Jan 1990 to June 30, 2006 with 4135 and 4161 obs. Model: r t = a(τ)+b(τ) t ( t ) 2+ q q T +c(τ) α T j (τ)r t j + β j (τ) ln v t j +e t, where β j (τ) represents the quantile causal effect of ln v t j on r t, cf. Gallant, Rossi, and Tauchen (1992). Null hypothesis: β j (τ) = 0 for all τ in (0, 1), i.e., Granger non-causality in quantiles. A sup-wald test will do. j=1 j=1 C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

42 S&P 500: β 1 (τ) S&P 500: β 2 (τ) Figure: QR and LS estimates of the causal effects of log volume on return. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

43 NYSE: β 1 (τ) NYSE: β 2 (τ) NYSE: β 3 (τ) C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

44 A Summary There is no causality in mean, but there are significant and heterogeneous causal effects of volume on return quantiles. Causal effects have opposite signs at the two sides of dist. Causal effects are stronger at more extreme quantiles. The pairwise causal effects are symmetric about the median. With log volume (return) on the vertical (horizontal) axis, causal relations exhibit symmetric V shapes across quantiles, cf. Karpoff (1987), Gallant et al. (1992), and Blume et al. (1994). This implies that return dispersion (volatility) increases with volume. These results remain valid when the model includes rt j 2, and they are robust in different sample periods. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

45 NYSE: β 1 (τ) S&P 500: β 1 (τ) Figure: QR and LS estimates of the causal effects of log volume on return: C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

46 NYSE: β 1 (τ) S&P 500: β 1 (τ) Figure: QR and LS estimates of the causal effects of log volume on return: C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

47 Empirical Study: Effects of NHI on Saving There were 3 major health insurance programs in Taiwan: Labor Insurance (1950), Government Employees Insurance (1958), and Farmers Health Insurance (1985). Government Employees Insurance (GEI): Covers the employees (including retirees) in the public sector; the coverage was extended to spouses in 1982, to parents in 1989, and to children in Labor Insurance: Covers only the employees in the private sector whose ages are 15 60, but not their spouse, parents, and children. The NHI launched in 1995 covers every citizen in Taiwan without discrimination, and its coverage is the same as that of GEI. We want to examine the effect of the enforcement of NHI on the precautionary saving in Taiwan. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

48 Data from the SFIE The data are taken from the Survey of Family Income and Expenditure (SFIE). This nation-wide survey is not panel and uses new sampling every year. The sample size is about 16,000 each year in the early 90 s and drops to about 14,000 from the mid-90 s. The modules in the SFIE include individual socio-demographic and socio-economic characteristics along with income and outlays of each income earner within the household. The SFIE provides details on each household member s income sources and job attributes that are crucial for estimation of permanent income and identification of treatment/control groups. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

49 We employ 10 waves of SFIE from 1990 through 2000 (excluding 1995), with and as the periods before and after the enforcement of NHI. Our sample includes only the households with non-farm heads aged years. There are 68,738 and 58,355 observations for the periods and , respectively. While the average household real income increases moderately from NTD 0.97 million during to more than NTD 1.2 million during , the corresponding average saving rates decrease by 50%, from 0.23 during to 0.12 during C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

50 Treatment and Control Groups Groups are partitioned according to the working status of household head and his/her spouse. 1 Case 1: 2 Case 2: Control: At least one of head and spouse in the public sector; Treatment: Neither head nor spouse in the public sector. Strict control: Head and spouse in the public sector; Quasi control: One of head and spouse in the public sector; Treatment 1: Head and spouse in the private sector; Treatment 2: One of head and spouse in the private sector. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

51 Unconditional Group Means and ATE Sample Obs. Mean S.D. Obs. Mean S.D. Difference Full Sample 68, , (0.001) Control 13, , (0.003) Treatment 55, , (0.002) Strict Control 2, , (0.006) Quasi Control 11, , (0.003) Treatment 1 24, , (0.002) Treatment 2 30, , (0.003) C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

52 Comparison with Chou, Liu, & Hammitt (2003) 1 Data: CLH03 include 1995 but we do not. CLH03 exclude the sample with negative saving but we do not. Note that such sample was 11.8% in but jumped to 27.2% in , leading to an average of 18.9% of the entire sample. 2 Dependent variable: CLH03 uses ln(y C), but we approximate the saving rate by ln Y ln C. 3 Covariates: We partition some covariates (income and head age) into groups and study the groupwise ATE and QTEs. 4 Model: CLH03 make pairwise comparisons, but we estimate a model with multiple treatment groups. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

53 Empirical Models 1 Control vs. one treatment group: S i = α + X i β + D i γ + (NHI i )δ + [(NHI i ) D i ]ζ + e i. 2 Control vs. multiple treatment groups: S i = α + X i β D i (j)γ j + (NHI i )δ j=0 2 [(NHI i ) D i (j)]ζ j + e i. j=0 where D(j) are the indicators of the j-th treatment group (j = 0 for quasi-control group, j = 1 and 2 for treatment 1 and 2 groups). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

54 1 To explore possible heterogeneity of the treatment effects, we partition the data into 5 permanent income groups (based on income quintiles) and 5 age groups (20 29,..., 60 69). We estimate the following model: S i = α + X i β D i K i (j)γ j + j=1 5 (NHI i K i (j))δ j j=1 5 [(NHI i ) D i K i (j)]ζ j + e i, j=1 where D is the binary indicator of the treatment group, and K(j), j = 1,..., 5, are the indicators of 5 income groups or 5 age groups. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

55 Empirical Result theta Figure: DID estimates of the ATE and QTE: 1 treatment group. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

56 Empirical Result theta theta theta Figure: DID estimates of the ATE and QTE: Quasi control (left), treatment 1 (middle), and treatment 2 (right). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

57 Empirical Result Yq1 Yq2 Yq3 Yq4 Yq Figure: DID estimates of the ATE and QTE: 5 income groups (left) and 5 age groups (right). C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

58 A Summary The NHI has significantly negative (crowd-out) effect on precautionary saving, and such impact (QTE) is stronger for higher savers (magnitude increasing with saving quantile). The estimated conditional ATEs are quite close to the unconditional ATEs. For example, for the treatment 1 and treatment 2 groups, the estimated ATEs are 4.3% and 3.5% (vs. unconditional ATEs: 4.3% and 4.1%). For the quasi control group, the ATE is insignificant. The QTE patterns for treatment 1 and 2 groups are opposite to those of CLH03. The negative impact is the largest for the highest income group and the oldest (age 60 69) group, cf. CLH04. C.-M. Kuan (Finance & CREAT, NTU) Intro. to Quantile Regression June 13, / 56

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