Quantile Cointegration in the Autoregressive Distributed-Lag ...

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orders, as also confirmed by Pesaran and Shin (1998). This implies that the QARDL orders selected by BIC may be consistently higher for some quantiles than for other quantiles. Our simulations are conducted across three quantile levels, τ 1 = 0.25, τ 2 = 0.50 and τ 3 = 0.75 as follows: For the long-run parameters, B ∗ (τ ) = [β ∗ (0.25), β ∗ (0.5), β ∗ (0.75)] ′ and the the short-run parameters, Γ ∗ (τ ) := [γ ∗ (0.25), γ ∗ (0.5), γ ∗ (0.75)] ′ , we use (j) ˜W n (B) and (j) ˜W n (Γ) to test the hypotheses: H (j) 0 (B) : S jB ∗ (τ ) = s j versus H (j) 1 (B) : S jB ∗ (τ ) ≠ s j and H (j) 0 (Γ) : S jΓ ∗ (τ ) = s j versus H (j) 1 (Γ) : S j Γ ∗ (τ ) ≠ s j for j = 1, 2, 3, 4. Next, for Φ ∗ (τ ) := [φ 1∗ (0.25), . . . , φ p∗ (0.25), φ 1∗ (0.50), . . . , φ p∗ (0.50), φ 1∗ (0.75), . . . , φ p∗ (0.75)] ′ (j) , we use ˜W n (Φ) to test the hypotheses: ˜H(j) 0 (Φ) : ˜Sj Φ ∗ (τ ) = s j versus ˜H (j) 1 (Φ) : ˜S j Φ ∗ (τ ) ≠ s j , for j = 1, 2, 3, 4, where ˜S 1 = ( 1, 0 ′ p−1 , −1, 0′ p−1 , 0, p−1) 0′ , ˜S2 = (0, 0 ′ p−1 , 1, 0 ′ p−1 , −1, 0′ p−1 ), ˜S 3 = ( ′. 1, 0 ′ p−1 , 0, 0′ p−1 , −1, p−1) 0′ , and ˜S4 = (˜S′ 1 , ˜S 2) ′ In principle, if the different lag orders are selected based on BIC, the null hypotheses can be different for the short-run parameters (of course, the same for the long-run parameter). Nevertheless, the essence of inference is intact when testing the coefficients on the first lagged dependent variable. For example, ˜W (1) n (Φ) tests φ 1∗ (0.25) = φ 1∗ (0.50), and (4) ˜W n (Φ) tests φ 1∗ (0.25) = φ 1∗ (0.50) = φ 1∗ (0.75). Under the DGP condition in (23), we expect the empirical rejection rates of the Wald test statistics to be close to the nominal level. For j = 1, 2, and 3, and (j) ˜W n (B), (j) (j) ˜W n (Φ), and ˜W n (Γ) follow X1 2 (4) ; and ˜W n (B), (4) ˜W n (Φ), (4) ˜W n (Γ) follow X2 2 . We consider sample sizes of 50, 200, 400, 600, 800, 1, 000, and 2, 000, and we let σ ∗ = 0.00, 0.10, 0.20, 0.30, and 0.40. Notice that, as σ ∗ increases, {U t } becomes more serially correlated; and thus, the larger size distortion is expected for σ ∗ = 0.4. We compute the Wald test statistics using the same methodology as in the previous subsection and present our simulation results in Tables 7, 8, and 9. For brevity, we examine the empirical rejection rates at 5% significance only. >> The overall simulation results can be summarised as follows. First, the sizes of the Wald test statistics all tend to the nominal level as the sample size increases, implying that the Wald tests are asymptotically well approximated by the claimed null distributions even in the presence of MA(1) errors. Second, in order to examine the asymptotic validity at other nominal levels, we also draw the EPDFs and ECDFs of the Wald tests for σ ∗ = 0.4 and n = 2, 000. Figures 3 and 4 show that the claimed asymptotic null distributions well approximate the empirical distribution functions. Third, as expected, we note that there are somewhat non-negligible size distortions especially in the small sample (say, n = 50) as σ ∗ increases. >> 23
5 Empirical Application In a classic study on dividend policy, based on interviews with 28 managers, Lintner (1956) observes that firms gradually adjust dividends in response to changes in earnings. Lintner also observes that firms are reluctant to make dividend changes that have to be reversed in the near future. An important implication of this finding is that managers make dividend adjustments in response to unanticipated and non-transitory changes in their firms’ earnings in order to attain a long-run target payout ratio. Empirical research generally supports Lintner’s partial adjustment framework at both the firm and aggregate levels, e.g., Fama and Babiak (1968), Marsh and Merton (1987), Garrett and Priestley (2000), von Eije and Megginson (2008) and Andres et al. (2009). In the literature, however, the estimates of adjustment speed are widely different. Fama and Babiak (1968) and Aivazian et al. (2006) estimate speeds 0.37 and 0.24, respectively. Using the US real estate investment trusts data from 1992-2003, Hanyunga and Stephens (2009) report widely different estimates of the speed of adjustment: 0.028 and 0.04 at a quarterly frequency and 0.379 and 0.371 at an annual frequency by the panel OLS and Tobit methods, respectively. These large differences are mainly due to the well-known small-T bias in dynamic AR regression (e.g., Hurwicz, 1950; Nickell, 1981). Recently, Brav et al. (2005) surveyed 384 financial executives to determine the factors that drive dividend and share repurchase decisions. The most important findings are as follows: executives try to avoid reducing dividends per share (93.8% agreed) and aims to maintain a smooth dividend stream (89.6%); they are reluctant to make changes that will be reversed (77.9%) because there are negative consequences to cutting dividends (88.1%). These responses are mostly consistent with Lintner’s observations, although the link between dividends and earnings has weakened in the subsequent 50 years. With the new survey evidence in mind, Leary and Michaely (2011) address an important issue of why firms smooth dividends by studying the cross-sectional differences in the behavior of dividend smoothing and providing empirical evidence against alternative explanations. Their main findings suggest that dividend smoothing is most common in the U.S. in large and mature firms that are not financially constrained, possess ce low levels of asymmetric information, and are readily susceptible to agency conflicts. Furthermore, they document that dividend smoothing has steadily increased over the past century, even before firms began to repurchase shares in the mid-1980s. Chen et al. (2012) also show that aggregate dividends are dramatically more smoothed in the post-world war II period (1946-2006) than in the prewar period (1871-1945). In particular, they report that, during the post-world war II period, dividends adjust to the earnings target at a speed about one-fourth that of the prewar period. This implies that dividend growth is unpredictable for the firms that have most smoothed their dividends but predictable for the firms that have least smoothed their dividends. By a simulation analysis, 24
Page 1 and 2: Quantile Cointegration in the Autor
Page 3 and 4: manner to Xiao’s (2009) quantile
Page 5 and 6: 2 Quantile Autoregressive Distribut
Page 7 and 8: where ⎡ Γ ∗ (τ) := ⎢ ⎣ µ
Page 9 and 10: ganelli (2004) for the value at ris
Page 11 and 12: Ω (r, ˜r, τ) as its minimal cond
Page 13 and 14: Using this identity, we show in the
Page 15 and 16: test statistic: ( ) ′ W n (φ) :=
Page 17 and 18: (ii) For each r, τ ∈ (0, 1), we
Page 19 and 20: ( ) ′ W n (Γ) := n S˜Γ n (τ )
Page 21 and 22: To estimate the relevant statistics
Page 23: Two remarks are in order. First, th
Page 27 and 28: We find that D t and E t are non-st
Page 29 and 30: on the current location of the divi
Page 31 and 32: of the four parameters are displaye
Page 33 and 34: statistics using rolling estimation
Page 35 and 36: a spurious relationship at other qu
Page 37 and 38: (v) First, we consider the case wit
Page 39 and 40: [ (iii) Notice that D −1 H G′ K
Page 41 and 42: tions A, B, and C of Xiao (2009). T
Page 43 and 44: ecause (˜ς n (τ) − ς ∗ (τ)
Page 45 and 46: asymptotic covariance matrix betwee
Page 47 and 48: Denis, D.J. and I. Osobov (2008):
Page 49 and 50: Machado, J. (1993): “Robust Selec
Page 51 and 52: W n (1) (B) Level \ Sample Size 50
Page 53 and 54: W n (1) (Γ) Level \ Sample Size 50
Page 55 and 56: W n (1) (Φ) Level \ Sample Size 50
Page 57 and 58: ˜W n (1) (B) σ ∗ \ Sample Size
Page 59 and 60: ˜W n (1) (Γ) σ ∗ \ Sample Size
Page 61 and 62: ECDFs of W n (1) (B), W n (2) (B),
Page 63 and 64: (1) (2) (3) (1) (2) ECDFs of ˜W n
Page 65 and 66: ECM Parameter (ζ ∗ (τ)) Long-Ru
Page 67 and 68: p-values of W n (1) (β) Tests p-va

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