Hybrid t-Test for Return Predictability (Harvey, Leybourne & Taylor)

hlt_test implements the double-switching hybrid procedure \(T_{hyb}\) of Harvey, Leybourne and Taylor (2021, Section 3.3) for a single predictor. Two regression t-ratios are used: the standard OLS t-ratio \(T\) (eq. 5) and the variant \(\tilde T\) in which the predictor is quasi-GLS demeaned with \(\bar c = 7\) (eq. 7). Under weak persistence the standard t-ratio is compared with normal critical values; under strong persistence the limiting null distributions depend on the local-to-unity parameter and on the innovation correlation \(\rho_{xy}\), and the tests are run with the paper's asymptotically conservative critical values (maximised over \(c\)) obtained from the response surfaces in its Table 1.

Usage

hlt_test(
  formula,
  data,
  alternative = c("greater", "less"),
  level = 0.05,
  lag_max = NULL,
  na.action
)

hlt_test_fit(y, x, alternative = "greater", level = 0.05, lag_max = NULL)

# S3 method for class 'hlt_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.
alternative: direction of the one-sided test.
level: significance level; one of 0.1, 0.05, 0.025, 0.01 (the levels for which response surfaces are available).
lag_max: maximum ADF lag order; the default is the paper's \(\lfloor 12 (T/100)^{1/4} \rfloor\).
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 "hlt_test".
digits: minimal number of significant digits.
...: unused.

Value

an object of class "hlt_test": a list with the selected test ("T_N", "T_con" or "T~_con"), its statistic, cv and reject indicator, plus t, t_gls (both t-ratios), adf, adf_lag, rho_xy and estimate (the OLS slope).

Details

The procedure is: (1) if the ADF normalised-bias statistic \(T\hat\rho/(1 - \sum_i \hat\psi_i)\) from an ADF regression with lag length chosen by the MBIC of Ng and Perron (2001) is below \(-4\sqrt{T}\), the predictor is treated as weakly persistent and the standard test \(T_N\) is used; (2) otherwise, for an upper-tail test, \(T\) with critical value \(cv(\hat\rho_{xy})\) if \(\hat\rho_{xy} > -0.1\) and \(\tilde T\) with \(\tilde{cv}(\hat\rho_{xy})\) if \(\hat\rho_{xy} < -0.1\) (mirrored for lower-tail tests), where \(\hat\rho_{xy}\) is the correlation of the ADF residuals with the predictive-regression residuals.

References

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.

Ng, S., & Perron, P. (2001). Lag length selection and the construction of unit root tests with good size and power. Econometrica, 69(6), 1519-1554.

Examples

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
#>