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Fitting Systems of IVX Predictive Regressions

Source: R/ivx-sys.R
ivx_sys.Rd

ivx_sys estimates a system of predictive regressions \(y_t = \mu + A x_{t-1} + \varepsilon_t\) with an \(m\)-vector response and \(r\) predictors of arbitrary persistence by IVX, and tests linear restrictions on \(A\) with the IVX-Wald statistic of Kostakis, Magdalinos and Stamatogiannis (2023), eqs (15) and (23), whose covariance has the Kronecker form \((Z'X)^{-1} \otimes I_m\) around \(Z(K)'Z(K) \otimes \hat\Sigma - n \bar z \bar z' \otimes \hat\Sigma_{FM}\). Long horizons are handled as in ivx(). For \(m = 1\) the results coincide with ivx().

Usage

ivx_sys(
  formula,
  data,
  horizon,
  na.action,
  contrasts = NULL,
  model = TRUE,
  x = FALSE,
  y = FALSE,
  beta = 0.95,
  cz = 1,
  bandwidth = NULL,
  ...
)

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

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

Arguments

formula

a formula whose left-hand side is a matrix, e.g. cbind(y1, y2) ~ x1 + x2.

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.

horizon

is the horizon (default horizon = 1 corresponds to a short-horizon regression). For horizon > 1 the estimator and modified Wald statistic are those of Kostakis et al. (2023), eqs (15) and (23): K-period sums of the response and of the lagged predictors, a single-lag instrument in the signal matrix and the K-period sum of the instrument in the covariance.

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_sys".

y

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

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).

bandwidth

Newey-West bandwidth for the long-run covariance estimate. The default NULL uses \(n^{1/3}\) as in Kostakis et al. (2015).

...

additional arguments to be passed to the low level regression fitting functions (see lm).

digits

the number of significant digits to use when printing.

object

an object of class "ivx_sys".

Value

an object of class "ivx_sys" with the coefficient matrix coefficients (responses in rows, predictors in columns), matching se, tstat and Wald_Ind matrices, the joint Wald statistic Wald_Joint (\(\chi^2(mr)\)), the per-equation Wald statistics Wald_Eq (\(\chi^2(r)\)), and vcov of vec(coefficients) (column-major, names response:predictor).

References

Kostakis, A., Magdalinos, T., & Stamatogiannis, M. P. (2023). Taking stock of long-horizon predictability tests: Are factor returns predictable? Journal of Econometrics, 237(2), 105380.

Examples

ivx_sys(cbind(Ret, DE) ~ DP + TBL, data = kms)
#> 
#> Call:
#> ivx_sys(formula = cbind(Ret, DE) ~ DP + TBL, data = kms, horizon = 1)
#> 
#> Coefficients (responses in rows):
#>      DP         TBL      
#> Ret   0.006145  -0.080717
#> DE    0.311083  -4.014921
#> 

summary(ivx_sys(cbind(Ret, DE) ~ DP + TBL, data = kms, horizon = 4))
#> 
#> Call:
#> ivx_sys(formula = cbind(Ret, DE) ~ DP + TBL, data = kms, horizon = 4)
#> 
#> Coefficients:
#>          Estimate Std. Error t value Wald Ind Pr(> chi)    
#> Ret:DP   0.006579   0.004601   1.430    2.045     0.153    
#> DE:DP    0.316547   0.019563  16.181  261.819    <2e-16 ***
#> Ret:TBL -0.073549   0.058238  -1.263    1.595     0.207    
#> DE:TBL  -3.979948   0.279933 -14.217  202.137    <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Equation Wald statistics on 2 DF:
#>      Wald p-value
#> Ret 3.527  0.1715
#> DE  488.3  <2e-16
#> 
#> Joint Wald statistic:  493.7 on 4 DF, p-value < 2.2e-16 
#>