Fitting Residual-Augmented IVX Models

ivx_ra implements the residual-augmented (bias-reduced) IVX estimator of Demetrescu and Rodrigues (2022). An autoregression of order p is fitted to the predictors, the predictive regression is augmented with its residuals (in the spirit of Amihud and Hurvich, 2004), and the slope on the lagged predictors is estimated by IVX. Inference uses the heteroskedasticity-robust standard errors of the paper (eq. 9 and 14), which are valid whether the predictors are stationary or near-integrated.

Usage

ivx_ra(
  formula,
  data,
  ar = "auto",
  ar_ic = c("aic", "bic"),
  ar_max = 5,
  horizon = 1,
  beta = 0.95,
  cz = 1,
  na.action,
  contrasts = NULL,
  model = TRUE,
  x = FALSE,
  y = FALSE,
  ...
)

# S3 method for class 'ivx_ra'
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.
ar: order of the autoregression fitted to the predictors: "auto" selects it by ar_ic in levels (as recommended in the paper), or a positive integer for a fixed order.
ar_ic: information criterion for ar = "auto".
ar_max: maximum order considered when ar = "auto".
horizon: forecast horizon \(h\); see Details.
beta, cz: tuning parameters of the IVX instrument \(z_t = \sum_{j=0}^{t-1} (1 - c_z/n^\beta)^j \Delta x_{t-j}\). Defaults (beta = 0.95, cz = 1) follow Kostakis et al. (2015).
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.
contrasts: an optional list. See the contrasts.arg of model.matrix.default.
model: logical. If TRUE the model.frame of the fit is returned.
x: an object of class "ivx_ra".
y: logical. If TRUE the response of the fit is returned.
...: additional arguments to be passed to the low level regression fitting functions (see lm).
digits: the number of significant digits to use when printing.

Value

an object of class c("ivx_ra", "ivx"); the usual ivx methods (summary, vcov, delta, ...) apply. Additional components: gamma (coefficients on the augmentation residuals) and ar_order.

Details

For horizon > 1 the estimator is the transformed-regression residual-augmented IVX of Demetrescu, Rodrigues and Taylor (2023), eqs (4.9), (4.11) and (5.5)-(5.7): the single-period response is regressed on the \(h\)-period transformed instrument \(z_t^{trf,(h)} = \sum_{i=\max(1,t-h+1)}^{\min(t,T-h)} z_i\) (eq. 4.4), which accounts for the overlap of the long-horizon regression without HAC estimation. At horizon = 1 it coincides with the short-horizon estimator. The coefficients estimate the \(h\)-period slope \(\beta_h\); fitted values and residuals are those of the transformed (non-overlapping) regression, and the Kostakis et al. (2015) intercept correction is applied only at horizon = 1.

The autoregression of the predictors is fitted without an intercept and its residuals are demeaned before augmentation, which is the paper's preferred \(\tilde t_{ivx}^{\mu_0}\) statistic (Sections 4-5); the standard errors include the finite-sample correction of Kostakis et al. (2015) as in the paper's simulations.

References

Demetrescu, M., & Rodrigues, P. M. M. (2022). Residual-augmented IVX predictive regression. Journal of Econometrics, 227(2), 429-460.

Demetrescu, M., Rodrigues, P. M. M., & Taylor, A. M. R. (2023). Transformed regression-based long-horizon predictability tests. Journal of Econometrics, 237(2), 105316.

Amihud, Y., & Hurvich, C. M. (2004). Predictive regressions: A reduced-bias estimation method. Journal of Financial and Quantitative Analysis, 39(4), 813-841.

Examples

ivx_ra(Ret ~ DP, data = kms)
#> 
#> Call:
#> ivx_ra(formula = Ret ~ DP, data = kms)
#> 
#> Residual-augmented IVX, AR order p = 5 (aic)
#> 
#> Coefficients:
#>        DP  
#> -0.002383  
#> 

summary(ivx_ra(Ret ~ DP + TBL, data = kms, ar = 2))
#> 
#> Call:
#> ivx_ra(formula = Ret ~ DP + TBL, data = kms, ar = 2)
#> 
#> Coefficients:
#>      Estimate Std. Error t value Wald Ind Pr(> chi)
#> DP  -0.002924   0.005016  -0.583    0.340      0.56
#> TBL -0.051281   0.054856  -0.935    0.874      0.35
#> (Eicker-White standard errors)
#> 
#> Joint Wald statistic:  1.654 on 2 DF, p-value 0.4374
#> Multiple R-squared:  0.02494,	Adjusted R-squared:  0.02304
#> 

# long horizon (Demetrescu, Rodrigues & Taylor, 2023)
ivx_ra(Ret ~ DP, data = kms, horizon = 12)
#> 
#> Call:
#> ivx_ra(formula = Ret ~ DP, data = kms, horizon = 12)
#> 
#> Residual-augmented IVX, AR order p = 5 (aic)
#> 
#> Coefficients:
#>       DP  
#> -0.01775  
#>