The bounds test asks whether a long-run level relationship exists
between y and X without first having to decide whether
the regressors are I(0) or I(1). It is run on the conditional
error-correction form
$$\Delta y_t = c_0 [+ c_1 t] + \pi_{yy} y_{t-1}
+ \sum_j \pi_{yx,j} x_{j,t-1} + \sum_{i=1}^{p-1} \psi_i \Delta y_{t-i}
+ \sum_j \sum_{i=0}^{q-1} \beta_{ji} \Delta x_{j,t-i} + u_t,$$
in which the lagged levels carry the long-run information and
the differences absorb the short-run dynamics.
Arguments
- y
Numeric vector, the dependent series.
- X
Numeric matrix of regressors; a plain vector is read as a single regressor.
- p
Lag order of
y: \(p-1\) lagged differences enter.- q
Lag order of each regressor: \(q-1\) lagged differences of each \(x\) enter, alongside the contemporaneous difference.
- case
Deterministic specification, 3 or 5.
Value
Named list with f_statistic, t_statistic,
df_num, df_den, n_used, k,
f_pvalue_iid, pi_yy, case, p, q,
method.
Details
Two statistics are reported. f_statistic is the Wald/F statistic
for the joint null \(\pi_{yy} = 0\) and \(\pi_{yx,j} = 0\) for
every \(j\) — no level relationship — which is the bounds test
proper. t_statistic is the t-ratio for \(\pi_{yy} = 0\) alone,
the complementary test on the speed of adjustment.
Neither statistic has a standard null distribution: under the null both limits depend on whether the regressors are I(0) or I(1), which is why the test is read against a pair of critical values — a lower bound assuming all regressors are I(0) and an upper bound assuming all are I(1). A statistic above the upper bound rejects, one below the lower bound does not, and one between the two is inconclusive.
The critical value bounds are not returned. They are tabulated
in Tables CI(i)–CI(v) of Pesaran, Shin & Smith (2001), one panel per
deterministic case, and no accessible copy of those tables could be
obtained when this function was written — the article is paywalled
and no authoritative open restatement was found. Reproducing them from
memory would be inventing numbers, so the statistics are returned for
the caller to compare against the published table for their
case and number of regressors. f_pvalue_iid is a
diagnostic only: it is the p-value from the ordinary F distribution,
which is not the null distribution of this test and is always
too small.
case follows the paper's numbering of the deterministic terms:
2 restricted intercept and no trend, 3 unrestricted intercept and no
trend, 4 unrestricted intercept and restricted trend, 5 unrestricted
intercept and unrestricted trend. Cases 3 and 5 are fitted here; cases
2 and 4 place the deterministic term inside the error-correction term
and are rejected rather than silently fitted as case 3.
Mirrors morie.fn.ardlmd on the Python side.