Unified Empirical Likelihood Test for Predictability (Liu, Yang, Cai & Peng)

el_test implements the unified empirical likelihood (EL) test of Liu, Yang, Cai and Peng (2019, Section 2.2) for the predictive regression with unknown intercept $$Y_t = \alpha + \beta_1 \Delta X_{t-1} + \beta_2 X_{t-2} + U_t,$$ where the lagged difference of the predictor is included so that the response can be stationary when the predictor is not. The intercept is removed by differencing at lag $m = \lfloor n/2 \rfloor$ ($\tilde Y_t = Y_{t+m} - Y_t$, $\tilde X_t = X_{t+m} - X_t$), and the EL function is built from the score equations $\tilde Z_{t1} = \tilde e_t \Delta\tilde X_{t-1}$ and $\tilde Z_{t2} = \tilde e_t \tilde X_{t-2}/\sqrt{1 + \tilde X_{t-2}^2}$, $t = 3, \dots, m$, with $\tilde e_t$ the model error. The weight on the second equation makes its sample variance converge whether the predictor is stationary, nearly integrated or a unit root, so that the profile EL ratios for $H_0: \beta_2 = 0$ (no predictability), $H_0: \beta_1 = 0$ and the joint null are $\chi^2(1)$, $\chi^2(1)$ and $\chi^2(2)$ without knowing the persistence (Theorem 2). Single predictor only.

Usage

el_test(formula, data, na.action)

el_test_fit(y, x)

# S3 method for class 'el_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.
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 "el_test".
digits: minimal number of significant digits.
...: unused.

Value

an object of class "el_test": a list with the EL ratio statistics stat (named beta2, beta1, joint), their p.value, the unconstrained EL estimates estimate, the OLS estimates ols of $(\beta_1, \beta_2)$ on the differenced data, and m.

References

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.

Owen, A. B. (2001). Empirical Likelihood. Chapman & Hall.

Examples

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
#>