Test whether lags of the regressors are missing (pre-determinedness)

Experimental. A variable-addition LM test in the spirit of Remark 9 of Perron and González-Coya (2026): FGLS needs the regressors to be pre-determined (uncorrelated with current and future innovations), which can fail when lags of the regressors belong in the model. The steps are (a) the Durbin regression (AR order by BIC) giving \(\hat\beta^D\); (b) the residuals \(\hat u^D = y - X\hat\beta^D\), demeaned; (c) for \(l = 1, \dots,\) max_lag, the statistic \(LM_l = n (SSR_0 - SSR_1)/SSR_0\), asymptotically \(\chi^2\) with as many degrees of freedom as added lags, from the auxiliary regressions without and with lags \(1..l\) of the stochastic regressors.

Usage

test_predetermined(
  formula,
  data,
  max_lag = 4L,
  ar_order = NULL,
  kmax = NULL,
  regressors = NULL,
  alpha = 0.05,
  scale = c("innovations", "errors"),
  na.action = stats::na.fail,
  time = NULL,
  allow_gaps = FALSE
)

# S3 method for class 'fgls_predetermined'
print(x, ...)

Arguments

formula: A model formula, as in stats::lm(). Include an intercept unless the model truly has none.
data: A data frame, a list, a ts/mts object, or a 'zoo'/'xts' object containing the variables in the formula. Rows must be in time order.
max_lag: Largest number of lags added.
ar_order: Fixed AR order of the Durbin regression; NULL (default) selects it by BIC up to kmax.
kmax: Largest AR order considered: NULL (default, \(\lfloor 12 (T/200)^{1/4} \rfloor\)), an integer, or "theory" (\(\lfloor (T/\ln T)^{1/3} \rfloor\)). It is capped so that the Durbin regression uses at most half of the sample as regressors.
regressors: Names of the regressors whose lags are tested (default: all non-constant regressors).
alpha: Level used for the advice.
scale: "innovations" (default) or "errors".
na.action: stats::na.fail() (default) stops on missing values; stats::na.omit() or stats::na.exclude() drop them. For time series only leading and trailing missing values can be dropped. With na.exclude, residuals() and fitted() are padded with NA to the rows selected by subset, as for stats::lm().
time: Time index for data frames: the name of a Date, POSIXct or numeric column of data, or a vector with one value per row. It is used to check that rows are sorted and have no gaps, and must have no missing values in the sample. Convert character dates first, e.g. with as.Date() or zoo::as.yearmon().
allow_gaps: Accept a time index with gaps, or a subset that skips rows (lags are then positional), with a warning.
x: A "fgls_predetermined" object.
...: Not used.

Value

An object of class "fgls_predetermined": a list with

table: a data frame with columns lags, LM, df and p.value, one row per number of added lags;
ar_order: the AR order of the Durbin regression;
regressors: the names of the regressors whose lags were tested;
alpha: the level used for the advice;
rejected: TRUE if some test rejects at level alpha;
advice: a sentence summarizing the result;
scale: the scale used, "innovations" or "errors".

The print method returns x invisibly.

Details

With scale = "innovations" (default) the auxiliary regression is run on the innovation scale: \(\hat u^D\) and all regressors are filtered with the Durbin AR filter and lagged \(\hat u^D\) enter as nuisance regressors, which keeps the \(\chi^2\) approximation valid with pre-determined but non-exogenous regressors. scale = "errors" uses \(\hat u^D\) and the raw regressors (the literal reading of Remark 9).

References

Perron, P. and González-Coya, E. (2026). Feasible GLS for time series regression. Journal of Econometrics, 258, 106346. doi:10.1016/j.jeconom.2026.106346

González-Coya, E. and Perron, P. (2024). Estimation in the presence of heteroskedasticity of unknown form: a Lasso-based approach. Journal of Econometric Methods, 13(1), 29–48. doi:10.1515/jem-2023-0007

Romano, J. P. and Wolf, M. (2017). Resurrecting weighted least squares. Journal of Econometrics, 197(1), 1–19. doi:10.1016/j.jeconom.2016.10.003

Examples

sim <- simulate_serial(300, seed = 1)
test_predetermined(y ~ x, data = sim)
#> Warning: test_predetermined() is experimental; its interface and details may change.
#> Pre-determinedness LM test (PG26 Remark 9, experimental)
#> Durbin AR order: 1; regressors tested: x; scale: innovations
#> 
#>  lags      LM df p.value
#>     1 0.05382  1  0.8165
#>     2 0.94706  2  0.6228
#>     3 1.16658  3  0.7610
#>     4 1.18574  4  0.8804
#> 
#> No rejection at 5%: no evidence of missing regressor lags.