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Bonferroni Q-Test of Campbell and Yogo (2006)

Source: R/cy-test.R
cy_test.Rd

cy_test implements the Bonferroni Q-test of Campbell and Yogo (2006), the standard feasible version of the sup-bound / Bonferroni approach of Cavanagh, Elliott and Stock (1995). For a single predictor \(x_t = \gamma + \rho x_{t-1} + v_t\) (AR(p) dynamics allowed, Appendix A), the procedure is:

  1. estimate the innovation correlation \(\delta\) between the predictive-regression residual and the ADF innovation of the predictor (lag length by BIC, \(p \in [1, p_{max}]\));

  2. compute the DF-GLS statistic of Elliott, Rothenberg and Stock (1996) and invert its local-to-unity null distribution (Stock, 1991) into a confidence interval \([\underline c, \bar c]\) for \(c = T(\rho - 1)\), with lower and upper levels \(\underline\alpha_1(\delta)\), \(\bar\alpha_1(\delta)\) from the paper's Table 2, which refine the Bonferroni bound so that the one-sided test has size 5\

  3. for \(\rho\) at each end of the interval compute the Q-estimate \(\hat\beta(\rho)\) of eq. (25) - OLS of \(y_t\) on the demeaned \(x_{t-1}\) after subtracting \((\sigma_{ue}/\sigma_e\omega)(x_t - \rho x_{t-1})\), with the Phillips-Perron-type correction \(\tfrac{T}{2}(\sigma_{ue}/\sigma_e\omega)(\omega^2 - \sigma_v^2)\) when \(p > 1\) - and its standard error \(\sigma_u (1-\delta^2)^{1/2}/(\sum x^{\mu 2}_{t-1})^{1/2}\);

  4. the 90\ \([\hat\beta(\bar\rho) - 1.645\,se,\; \hat\beta(\underline\rho) + 1.645\,se]\) (eq. 17); the 5\ bound is positive and \(\beta \ge 0\) if the upper bound is negative.

The DF-GLS null quantiles are tabulated by simulation for \(c \in [-100, 10]\) (see data-raw/dfgls-quantiles.R); a statistic outside the tabulated range is clamped to the boundary, which for very negative values (a clearly stationary predictor) makes the interval for \(c\) start at \(-100\). Table 2 is given for \(\delta < 0\); for \(\hat\delta > 0\) the predictor is sign-flipped, which flips \(\beta\) and the alternative, and the results are mapped back.

Usage

cy_test(formula, data, lag_max = NULL, na.action)

cy_test_fit(y, x, lag_max = NULL)

# S3 method for class 'cy_test'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

Arguments

formula

an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted.

data

n optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which lm is called.

lag_max

maximum ADF lag order for the BIC search; the default is \(\lfloor 12 (T/100)^{1/4} \rfloor\) lagged differences.

na.action

a function which indicates what should happen when the data contain NAs. The default is set by the na.action setting of options, and is na.fail if that is unset. The ‘factory-fresh’ default is na.omit. Another possible value is NULL, no action. Value na.exclude can be useful.

y

response vector.

x

an object of class "cy_test".

digits

minimal number of significant digits.

...

unused.

Value

an object of class "cy_test": a list with ci (the 90\ interval for \(\beta\)), reject (named logical: greater, less), estimate (OLS slope), beta_rho (\(\hat\beta\) at \(\underline\rho\) and \(\bar\rho\)), se, delta, dfgls, c_ci, rho_ci, alpha1, lag and n.

References

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., Rothenberg, T. J., & Stock, J. H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813-836.

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.

Examples

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
#>