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Control-Function Predictability Test of Elliott (2011)

Source: R/elliott-cf.R
elliott_cf.Rd

elliott_cf runs the augmented predictive regression of Elliott (2011, eq. 9), $$y_t = \alpha + \beta' x_{t-1} + \gamma' Z_t + \tilde u_t,\qquad Z_t = (z_t', z_{t-1}', \dots, z_{t-q}')',$$ where \(z_t\) are user-supplied stationary covariates that are contemporaneously correlated with the innovations of the persistent predictors \(x_t\) and of the response. If they absorb that correlation (the "orthogonalising" condition, Section 5 of the paper), the Wald test of \(\beta = 0\) has a standard chi-square limit whatever the persistence of \(x_t\) (Theorem 2); without them it has the non-standard Elliott-Stock (1994) distribution (Theorem 1). Unlike arm() and ivx_ra(), which build the control variable from the data, the covariates here come from the user

  • the paper's example is predicting returns with the dividend-price ratio using contemporaneous price-related variables.

Usage

elliott_cf(formula, covariates, data, lags = 0, robust = TRUE, na.action)

elliott_cf_fit(y, x, z, lags = 0, robust = TRUE)

# S3 method for class 'elliott_cf'
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.

covariates

a one-sided formula giving the orthogonalising covariates \(z_t\) (contemporaneous with the response).

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.

lags

number of lags \(q\) of the covariates to include.

robust

logical; if TRUE (default) Eicker-White standard errors are used.

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 "elliott_cf".

z

matrix of covariates.

digits

minimal number of significant digits.

...

unused.

Value

an object of class "elliott_cf": a list with coefficients (on the predictors), se, tstat, Wald, df, p.value, gamma (coefficients on the covariates and their lags), rho_resid (remaining innovation correlation per predictor) and the underlying lm fit.

Details

The remaining innovation correlation after augmentation is returned as a diagnostic: it is the correlation between the regression residuals and the residuals of an AR(1) of each predictor on the same covariates, and should be close to zero for the test to be reliable.

References

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., & Stock, J. H. (1994). Inference in time series regression when the order of integration of a regressor is unknown. Econometric Theory, 10(3-4), 672-700.

Examples

# the T-bill rate as covariate for the dividend-price ratio (illustration only)
elliott_cf(Ret ~ DP, ~ TBL, data = kms)
#> 
#> Call:
#> elliott_cf(formula = Ret ~ DP, covariates = ~TBL, data = kms)
#> 
#> Control-function predictive regression (Elliott, 2011), 0 covariate lag(s)
#> 
#>    Estimate Std. Error t value Pr(>|t|)
#> DP 0.005664   0.005099   1.111    0.267
#> (Eicker-White standard errors)
#> 
#> Wald statistic: 1.234 on 1 DF, p-value 0.2667
#> Remaining innovation correlation: DP -0.977 
#>