Non-IVX benchmarks: Bonferroni Q, ARM, hybrid t, empirical likelihood • ivx

library(ivx)

Five non-IVX procedures are included as the benchmarks the IVX literature compares against: the Bonferroni Q-test of Campbell & Yogo (2006), cy_test(); the augmented regression method of Amihud, Hurvich & Wang (2009), arm(); the control-function regression of Elliott (2011), elliott_cf(); the hybrid switching t-test of Harvey, Leybourne & Taylor (2021), hlt_test(); and the unified empirical likelihood test of Liu, Yang, Cai & Peng (2019), el_test().

Bonferroni Q-test

cy_test() is the Campbell & Yogo (2006) procedure, the feasible version of the sup-bound / Bonferroni idea of Cavanagh, Elliott & Stock (1995). If $\rho$ were known, the UMP conditional test of $\beta = 0$ is the t-ratio of the regression augmented with $x_t - \rho x_{t-1}$ (the “Q-test”), whose estimate \[ \hat\beta(\rho) = \frac{\sum_t x^\mu_{t-1}\big(y_t - \tfrac{\sigma_{ue}}{\sigma_e\omega}(x_t - \rho x_{t-1})\big) - \tfrac{T}{2}\tfrac{\sigma_{ue}}{\sigma_e\omega}(\omega^2 - \sigma_v^2)}{\sum_t x^{\mu 2}_{t-1}} \] (their eq. 25, with the AR($p$) correction of Appendix A) has standard error $\sigma_u\sqrt{1-\delta^2}/(\sum x^{\mu 2}_{t-1})^{1/2}$. Since $\rho$ is not consistently estimable, a confidence interval for it is obtained by inverting the DF-GLS statistic (Stock, 1991; Elliott, Rothenberg & Stock, 1996) and the Bonferroni interval for $\beta$ runs from $\hat\beta(\bar\rho) - 1.645\,se$ to $\hat\beta(\underline\rho) + 1.645\,se$. The levels of the interval for $\rho$ come from the paper’s Table 2, which tightens the plain Bonferroni bound so that the one-sided test has size exactly 5% for some $c$.

The DF-GLS null quantiles as a function of $c$ are simulated once (data-raw/dfgls-quantiles.R, $c \in [-100, 10]$, 20 000 replications of a 600-step OU process) and shipped as internal data; the 5% quantile at $c = 0$ reproduces the $-1.95$ of Elliott et al. (1996).

cy_test(Ret ~ DP, data = kms)
#> 
#> Call:
#> cy_test(formula = Ret ~ DP, data = kms)
#> 
#> Bonferroni Q-test (Campbell & Yogo, 2006)
#> 
#> delta = -0.972, DF-GLS = -1.468 (p = 2), CI for c at levels (0.055, 0.082): [-9.319, 1.045], rho: [0.991, 1.001]
#> OLS slope = 0.006193; Q-estimates at the ends of the rho interval: 0.009076, 0.0004903
#> 90% Bonferroni confidence interval for beta: [-0.000973, 0.01054]
#> 5% one-sided Q-tests: H1 beta > 0 do not reject H0; H1 beta < 0 do not reject H0
cy_test(Ret ~ EP, data = kms)
#> 
#> Call:
#> cy_test(formula = Ret ~ EP, data = kms)
#> 
#> Bonferroni Q-test (Campbell & Yogo, 2006)
#> 
#> delta = -0.7909, DF-GLS = -3.014 (p = 3), CI for c at levels (0.065, 0.17): [-29.24, -11.74], rho: [0.9717, 0.9886]
#> OLS slope = 0.008694; Q-estimates at the ends of the rho interval: 0.02198, 0.01469
#> 90% Bonferroni confidence interval for beta: [0.01056, 0.02611]
#> 5% one-sided Q-tests: H1 beta > 0 reject H0; H1 beta < 0 do not reject H0

In a Monte Carlo with $T = 200$ and 500 replications, the right-tailed 5% Q-test rejects a true null 4.4–5.8% of the time and the left-tailed one 1.2–3.6% for $\delta \in \{-0.9, -0.5\}$ and $c \in \{0, -20\}$ — the asymmetry the paper describes (Section 3.4: left-tailed probability “can be as small as 1.2%”).

Augmented regression method

arm() is reduced-bias OLS for stationary persistent predictors (Demetrescu & Rodrigues, 2022, build ivx_ra() on the same augmentation idea).

Idea

With $y_t = \alpha + \beta' x_{t-1} + u_t$ and $x_t = \mu + \Phi x_{t-1} + v_t$, write $u_t = \phi' v_t + e_t$ with $e_t$ independent of $x$. OLS of $y_t$ on $x_{t-1}$ is biased because $u_t$ is correlated with $v_t$ and hence with the future of the regressor (Stambaugh, 1999). Regressing $y_t$ on $x_{t-1}$ and $v_t$ removes it — if $v_t$ were known. Using OLS VAR residuals does not help (they are orthogonal to $x_{t-1}$ by construction), so the VAR coefficient matrix is first bias-corrected with the Nicholls & Pope (1988) expansion \[ E[\hat\Phi - \Phi] = -\tfrac{1}{n}\,\Sigma_v\Big[(I-\Phi')^{-1} + \Phi'(I-\Phi'^2)^{-1} + \sum_{\lambda \in \mathrm{Spec}(\Phi)} \lambda (I-\lambda\Phi')^{-1}\Big]\Gamma_0^{-1}, \] iterated up to iter times (Kendall’s $1 + 3\phi$ in the scalar case), and the corrected residuals $v^c_t = x_t - \hat\mu^c - \hat\Phi^c x_{t-1}$ are the control variables.

The OLS covariance of the augmented regression is wrong for $\hat\beta^c$ because $v^c_t$ is estimated. The paper’s estimator (eqs 7–8) adds the uncertainty of $\hat\Phi^c$: \[ \widehat{\mathrm{cov}}[\hat\beta^c] = (\hat\phi'\hat\Sigma_v\hat\phi)\,(X'X)^{-1}_{xx} + \hat\sigma_e^2 \Big[\tfrac{\sum_t r_{it} r_{jt}}{\sum_t r_{it}^2 \sum_t r_{jt}^2}\Big]_{ij}, \] with $r_{jt}$ the residual of $x_{j,t-1}$ on the other regressors of the augmented regression. summary() reports $t$-ratios and the joint Wald test from this matrix.

m <- arm(Ret ~ DP + TBL, data = kms)
summary(m)
#> 
#> Call:
#> arm(formula = Ret ~ DP + TBL, data = kms)
#> 
#> Coefficients:
#>      Estimate Std. Error t value Wald Ind Pr(> chi)
#> DP   0.002050   0.003797   0.540    0.292     0.589
#> TBL -0.046509   0.055999  -0.831    0.690     0.406
#> 
#> Joint Wald statistic:  1.068 on 2 DF, p-value 0.5862
#> Multiple R-squared:  0.02415,    Adjusted R-squared:  0.0194
m$Phi      # bias-corrected VAR(1) coefficients
#>                DP           TBL
#> DP   9.965559e-01 -0.0001402752
#> TBL -5.682961e-05  0.9964614118
m$Phi_ols  # OLS
#>               DP        TBL
#> DP   0.992685362 0.02492259
#> TBL -0.000183203 0.99299626

Check

With $\phi = 0.9$, $\mathrm{corr}(u, v) = -0.9$, $n = 100$ and $\beta = 0$ the OLS slope has mean bias about $0.037$ ($\approx -\rho(1+3\phi)/n$) over 400 replications; arm() reduces it to $0.003$, with $\hat\Phi^c$ averaging $0.894$ against an OLS mean of $0.857$. In the paper’s two-predictor Case 1 the 5% $t$-tests reject 7–8% at $n = 100$, in line with the 6–10% ($n = 50$) and 5–7% ($n = 200$) the paper reports.

Caveats

Assumes stationary predictors: the correction diverges as the largest eigenvalue of $\Phi$ approaches one, and the iteration stops if the corrected VAR becomes non-stationary. For near-unit-root predictors use ivx() or ivx_ra().
VAR(1) only, short horizon only, no bootstrap.

Control-function regression

Elliott (2011) shows that if stationary covariates $z_t$ are available that are contemporaneously correlated with the shocks to both the predictor and the response, the Wald test of $\beta = 0$ in \[ y_t = \alpha + \beta' x_{t-1} + \gamma' Z_t + \tilde u_t, \qquad Z_t = (z_t', z_{t-1}', \dots, z_{t-q}')', \] is asymptotically $\chi^2$ whatever the persistence of $x_t$, because the covariates “orthogonalise” the innovations (his Theorem 2 versus the Elliott–Stock 1994 distribution of Theorem 1). The covariates are a modelling choice, not a data construction — arm() and ivx_ra() are the feasible versions where the control variable is built from the predictor’s own innovations. elliott_cf() runs the regression, reports the Wald test with Eicker-White standard errors and the correlation that remains between the regression residuals and the predictor innovations, which should be near zero.

elliott_cf(Ret ~ DP, ~ TBL, data = kms, lags = 1)
#> 
#> Call:
#> elliott_cf(formula = Ret ~ DP, covariates = ~TBL, data = kms, 
#>     lags = 1)
#> 
#> Control-function predictive regression (Elliott, 2011), 1 covariate lag(s)
#> 
#>    Estimate Std. Error t value Pr(>|t|)
#> DP 0.005605   0.005102   1.099    0.272
#> (Eicker-White standard errors)
#> 
#> Wald statistic: 1.207 on 1 DF, p-value 0.2719
#> Remaining innovation correlation: DP -0.977

With a covariate that does absorb the correlation ($z_t = v_t + $ noise, $\delta = -0.9$, $c = 0$, $T = 200$) the 5% test rejects a true null 4% of the time against 23% for plain OLS (500 replications).

Hybrid t-test

hlt_test() implements $T_{hyb}$ of Harvey, Leybourne & Taylor (2021) for a single predictor. It uses only regression t-ratios: the standard one, $T$, and $\tilde T$, in which the predictor is quasi-GLS demeaned with $\bar c = 7$ as in Elliott et al. (1996). Under strong persistence their null distributions depend on $c$ and on $\rho_{xy}$, so the paper tabulates conservative critical values — the maximum over $c$ of the $\alpha$-level quantile — as response surfaces in $\hat\rho_{xy}$ (their Table 1), and switches:

if the ADF normalised-bias statistic (lag length by MBIC) is below $-4\sqrt{T}$ the predictor is weakly persistent: $T$ with a normal critical value;
otherwise, for an upper-tail test, $T$ with $cv(\hat\rho_{xy})$ when $\hat\rho_{xy} > -0.1$ and $\tilde T$ with $\tilde{cv}(\hat\rho_{xy})$ when $\hat\rho_{xy} < -0.1$ (quasi-GLS demeaning pays off only when the endogeneity is strong); lower-tail tests mirror this.

hlt_test(Ret ~ DP, data = kms)
#> 
#> Call:
#> hlt_test(formula = Ret ~ DP, data = kms)
#> 
#> Hybrid predictability test (Harvey, Leybourne & Taylor, 2021)
#> 
#> Selected test: T~_con (quasi-GLS-demeaned t, conservative critical value)
#> statistic = 1.298, 5% critical value = 1.944 (alternative: beta > 0): do not reject
#> slope = 0.006172, t =  1.63, t (quasi-GLS) = 1.298, ADF = -5.97 (p = 21, cutoff -128.6), rho_xy = -0.9485
hlt_test(Ret ~ TBL, data = kms, alternative = "less", level = 0.1)
#> 
#> Call:
#> hlt_test(formula = Ret ~ TBL, data = kms, alternative = "less", 
#>     level = 0.1)
#> 
#> Hybrid predictability test (Harvey, Leybourne & Taylor, 2021)
#> 
#> Selected test: T_con (OLS-demeaned t, conservative critical value)
#> statistic = -1.403, 10% critical value = -1.306 (alternative: beta < 0): reject
#> slope = -0.07836, t = -1.403, t (quasi-GLS) = -1.384, ADF = -6.707 (p = 6, cutoff -128.6), rho_xy = -0.05541

In the paper’s design ($T = 200$, upper-tail 5% test) the empirical size in 1000 replications is 0.07/0.04/0.05 for $\rho_{xy} = -0.9$ and $\rho = 1, 0.95, 0.5$, and 0.00/0.02/0.04 for $\rho_{xy} = 0.9$ — the “undersize for positive $\rho_{xy}$ in the more persistent cases” the paper notes is the price of the conservative critical values. The test is one-sided by construction and has no p-value; the object reports the selected test, its statistic and critical value.

Unified empirical likelihood test

el_test() implements Liu, Yang, Cai & Peng (2019). Their model adds the lagged difference of the predictor, \[ Y_t = \alpha + \beta_1 \Delta X_{t-1} + \beta_2 X_{t-2} + U_t, \] so that $Y_t$ can be stationary whether or not $X_t$ is. The intercept is removed by differencing at lag $m = \lfloor n/2 floor$ (Zhu, Cai & Peng, 2014), and the empirical likelihood is built on the two score equations, the second weighted by $1/\sqrt{1 + \tilde X_{t-2}^2}$ so that its sample variance converges whatever the persistence. The profile EL ratios for $\beta_2 = 0$ (no predictability), $\beta_1 = 0$ and the joint null are then $\chi^2(1)$, $\chi^2(1)$, $\chi^2(2)$ with no tuning parameter and no persistence classification (Theorem 2). The price is efficiency: only $m - 2$ differenced observations enter, and the paper reports that the test on $\beta_1$ is oversized at $n = 200$.

el_test(Ret ~ DP, data = kms)
#> 
#> Call:
#> el_test(formula = Ret ~ DP, data = kms)
#> 
#> Unified empirical likelihood test (Liu, Yang, Cai & Peng, 2019), m = 516
#> 
#>                  Estimate EL ratio    df Pr(> chi)
#> beta1 (dX[t-1]) -0.133221    4.456 1.000    0.0348
#> beta2 (X[t-2])   0.008196    2.404 1.000    0.1210
#> joint                        7.250 2.000    0.0266

The EL dual problem is solved by Newton’s method with Owen’s (2001) pseudo-logarithm; the profiles are minimised over the nuisance coefficient by BFGS from the OLS start.

The Monte Carlo figures quoted in this vignette are reproduced by research/replication/extensions-2026/size-checks.R in the source repository.

References

Amihud, Y., Hurvich, C. M., & Wang, Y. (2009). Multiple-predictor regressions: Hypothesis testing. Review of Financial Studies, 22(1), 413–434.
Amihud, Y., & Hurvich, C. M. (2004). Predictive regressions: A reduced-bias estimation method. Journal of Financial and Quantitative Analysis, 39(4), 813–841.
Campbell, J. Y., & Yogo, M. (2006). Efficient tests of stock return predictability. Journal of Financial Economics, 81(1), 27–60.
Cavanagh, C. L., Elliott, G., & Stock, J. H. (1995). Inference in models with nearly integrated regressors. Econometric Theory, 11(5), 1131–1147.
Elliott, G. (2011). A control function approach for testing the usefulness of trending variables in predictive regressions and econometric models. Journal of Econometrics, 164(1), 79–91.
Elliott, G., Rothenberg, T. J., & Stock, J. H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813–836.
Harvey, D. I., Leybourne, S. J., & Taylor, A. M. R. (2021). Simple tests for stock return predictability with good size and power properties. Journal of Econometrics, 224(1), 198–214.
Liu, X., Yang, B., Cai, Z., & Peng, L. (2019). A unified test for predictability of asset returns regardless of properties of predicting variables. Journal of Econometrics, 208(1), 141–159.
Nicholls, D. F., & Pope, A. L. (1988). Bias in the estimation of multivariate autoregressions. Australian Journal of Statistics, 30A, 296–309.
Owen, A. B. (2001). Empirical Likelihood. Chapman & Hall.
Stock, J. H. (1991). Confidence intervals for the largest autoregressive root in U.S. macroeconomic time series. Journal of Monetary Economics, 28(3), 435–459.
Stambaugh, R. F. (1999). Predictive regressions. Journal of Financial Economics, 54(3), 375–421.