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Fitting IVX Quantile Predictive Regressions

Source: R/ivx-qr.R
ivx_qr.Rd

ivx_qr implements the IVX-QR predictability test of Lee (2016): the \(\tau\)-quantile of the response is regressed on the IVX-filtered predictors (Section 3.3, Proposition 3.2), giving a test of \(H_0: \beta_\tau = 0\) with a standard chi-square limit whatever the persistence of the predictors. Requires the quantreg package.

Usage

ivx_qr(
  formula,
  data,
  tau = 0.5,
  beta = 0.95,
  cz = 5,
  na.action,
  contrasts = NULL,
  model = TRUE,
  x = FALSE,
  y = FALSE,
  ...
)

# S3 method for class 'ivx_qr'
print(x, digits = max(3L, getOption("digits") - 3L), ...)

# S3 method for class 'ivx_qr'
summary(object, ...)

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.

tau

quantile level(s) in (0, 1). A vector fits one model per level.

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_qr".

y

logical. If TRUE the response of the fit is returned.

...

further arguments passed to quantreg::rq().

digits

the number of significant digits to use when printing.

object

an object of class "ivx_qr".

Value

For a single tau, an object of class c("ivx_qr", "ivx") (so summary(), vcov() etc. apply) with components tau, rho_tau, sparsity (\(\hat f_u(0)\)) and rq (the underlying quantreg::rq fit). For several tau, a list of such objects named by tau.

Details

The instrument is \(z_t = (1 - c_z/n^\beta) z_{t-1} + \Delta x_t\). Lee (2016) normalises \(c_z = 5\) and picks \(\beta\) from a look-up table indexed by the estimated QR endogeneity \(\hat\rho(\tau) = -\mathrm{corr}(1\{\hat u_t < 0\}, \hat u_{x,t})\), which is returned as rho_tau so the rule can be applied by the user; the default beta = 0.95 follows Kostakis et al. (2015). The sparsity \(f_u(0)\) is estimated by a Gaussian kernel with Silverman's bandwidth (footnote 4 of the paper).

References

Lee, J. H. (2016). Predictive quantile regression with persistent covariates: IVX-QR approach. Journal of Econometrics, 192(1), 105-118.

Examples

if (requireNamespace("quantreg", quietly = TRUE)) {
  summary(ivx_qr(Ret ~ DP, data = kms, tau = 0.5))
  ivx_qr(Ret ~ DP + TBL, data = kms, tau = c(0.1, 0.5, 0.9))
}
#> $`0.1`
#> 
#> Call:
#> ivx_qr(formula = Ret ~ DP + TBL, data = kms, tau = 0.1)
#> 
#> IVX-QR at tau = 0.1
#> 
#> Coefficients:
#>       DP       TBL  
#> -0.02831   0.18938  
#> 
#> 
#> $`0.5`
#> 
#> Call:
#> ivx_qr(formula = Ret ~ DP + TBL, data = kms, tau = 0.5)
#> 
#> IVX-QR at tau = 0.5
#> 
#> Coefficients:
#>        DP        TBL  
#>  0.008204  -0.171321  
#> 
#> 
#> $`0.9`
#> 
#> Call:
#> ivx_qr(formula = Ret ~ DP + TBL, data = kms, tau = 0.9)
#> 
#> IVX-QR at tau = 0.9
#> 
#> Coefficients:
#>       DP       TBL  
#>  0.02674  -0.43021  
#> 
#>