IV Predictability Tests with Combined Instruments

ivx_iv implements the instrumental-variable tests of Breitung and Demetrescu (2015): the predictive regression is estimated by 2SLS with Eicker-White standard errors (their eq. 12) using instruments that are less persistent than the predictor. Two families are available. Type-I instruments are transformations of the predictor itself: the fractional difference \(\Delta_+^{d} x_{t-1}\) ("frac") and the long difference \(x_{t-1} - x_{t-1-k_T}\) ("diff"), which keep power when the predictor is stationary. Type-II instruments are deterministic and correlate with a near-integrated predictor only: the sine function \(\sin(\pi t/T)\) ("sin"). Their 2SLS combination ("comb", the paper's IVcomb and the authors' recommendation) is asymptotically dominated by whichever instrument is informative, so the squared t-ratio (and the Wald statistic with several predictors) is chi-square whatever the persistence of the predictor (Theorems 3 and 7).

Usage

ivx_iv(
  formula,
  data,
  instruments = c("comb", "sin", "frac", "diff"),
  d = 0.5,
  kappa = 0.2,
  eta = 0.85,
  na.action,
  contrasts = NULL,
  model = TRUE,
  x = FALSE,
  y = FALSE,
  ...
)

# S3 method for class 'ivx_iv'
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.
instruments: instrument set; see Details.
d: order of the fractional difference for "frac", in (0, 1/2]; the paper uses 1/2.
kappa, eta: the long-difference lag is \(k_T = \lfloor \kappa T^\eta \rfloor\) (paper: 0.2 and 0.85), truncated to \(t - 1\).
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_iv".
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_iv", "ivx") so the ivx methods apply; instruments holds the instrument matrix aligned with the regressors.

Details

With \(K\) predictors each type-I instrument is built per predictor and the sine instruments use frequencies \(\sin(k\pi t/T)\), \(k = 1, \dots, K\), so that the instrument vector stays linearly independent (Assumption 5). The IVX instrument of Kostakis et al. (2015) is the paper's "mild integration" type-I case and is available through ivx().

References

Breitung, J., & Demetrescu, M. (2015). Instrumental variable and variable addition based inference in predictive regressions. Journal of Econometrics, 187(1), 358-375.

Examples

ivx_iv(Ret ~ DP, data = kms)
#> 
#> Call:
#> ivx_iv(formula = Ret ~ DP, data = kms)
#> 
#> 2SLS with instruments: sin1, frac_DP
#> 
#> Coefficients:
#>     DP  
#> 0.0101  
#> 
summary(ivx_iv(Ret ~ DP + TBL, data = kms, instruments = "frac"))
#> 
#> Call:
#> ivx_iv(formula = Ret ~ DP + TBL, data = kms, instruments = "frac")
#> 
#> Coefficients:
#>       Estimate Std. Error t value Wald Ind Pr(> chi)
#> DP  -0.0455575  0.2025258  -0.225    0.051     0.822
#> TBL -0.0009177  0.8732213  -0.001    0.000     0.999
#> (Eicker-White standard errors)
#> 
#> Joint Wald statistic:  3.117 on 2 DF, p-value 0.2104
#> Multiple R-squared:  0.8671,	Adjusted R-squared:  0.8667
#>