The name is absent

1(#) specifies the number of initial ordinates to be removed from the regression for roblpr. Some researchers have found that
such exclusion improves the properties of tests based on log-periodogram regressions. The default value of 1 is zero.

j (#) specifies that the log periodogram employed in roblpr is to be computed as an average of adjacent ordinates. The default
value of j is 1, so that no averaging is performed. If j is 2, the number of ordinates is halved; with a j of 3, divided by
three, and so on. When j is greater than 1, the value of powers should be set large enough so that the averaged ordinates
are sufficient in number.

constraints (uumlitt) specifies the constraint numbers of the linear constraints to be applied during estimation in roblpr. The
default is to perform unconstrained estimation. This option allows the imposition of linear constraints prior to estimation
of the pooled coefficient vector. For instance, if varlist contains prices, dividends, and returns, and your prior (or previous
findings) states that prices’ and dividends’ order of integration is indistinguishable, one might impose that constraint to
improve the power of the F test provided by roblpr. You would specify the constraints prior to the roblpr command
and then provide the list of constraints in the constraints option to roblpr.

Technical note on constraints. When constraints are imposed it is difficult to identify the number of numerator degrees of
freedom in the test for equality of d coefficients reported at the bottom of roblpr’s output. Since constraints can be of
any general form and it is possible to specify constraints that are not unique, roblpr determines the degrees of freedom
from the rank of the matrix used to compute the Wald statistic. Determining that matrix rank from a numerical standpoint
can be problematic, in which case roblpr may overstate the number of constraints being tested and thereby incorrectly
compute the numerator degrees of freedom for the test. This rarely has a meaningful impact on the statistical test, but you
may wish to test only the unconstrained coefficients if the computed degrees of freedom are wrong.

For example, after the final example below, we could perform the test by typing test ftap == ftaret. In this case, the
degrees of freedom were correct, so we needn’t have gone to the trouble.

Description

The model of an autoregressive fractionally integrated moving average process of a time series of order (p, d, q), denoted
by ARFIMA(p, d, q), with mean μ, may be written using operator notation as

Φ(Z)(1 - L')^d {y_t -μ)= Θ(L~)e_t, e_t ~ i.i.d.(0,σ_e²)

where L is the backward-shift operator,

Φ(Z) = 1 - φ_lL----- φ_pL^p

θ(ɪ) = 1 + i?i-L H----+ 0,₇∕A, and (1 — Z)^d is the fractional differencing operator defined by
with Γ(∙) denoting the gamma (generalized factorial) function. The parameter d is allowed to assume any real value. The
arbitrary restriction of d to integer values gives rise to the standard autoregressive integrated moving average (ARIMA) model.
The stochastic process y_t is both stationary and invertible if all roots of Φ(Z) and Θ(Z) lie outside the unit circle and ∣d∣ < 0.5.
The process is nonstationary for d ≥ 0.5, as it possesses infinite variance; for example, see Granger and Joyeux (1980).

Assuming that d ∈ [0,0.5), Hosking (1981) showed that the autocorrelation function, p(∙), of an ARFIMA process is
proportional to k^2d~¹ as к —> ∞. Consequently, the autocorrelations of the arfima process decay hyperbolically to zero as
fc-÷∞in contrast to the faster, geometric decay of a stationary ARMA process. For d ∈ (0,0.5), ∑j=~_n ∣p(J)∣ diverges as
n —> ∞, and the arfima process is said to exhibit long memory, or long-range positive dependence. The process is said to
exhibit intermediate memory (anti-persistence), or long-range negative dependence, for d E ( —0.5,0). The process exhibits short
memory for d = 0, corresponding to stationary and invertible ARMA modeling. For d ∈ [0.5,1) the process is mean reverting,
even though it is not covariance stationary, as there is no long-run impact of an innovation on future values of the process.

If a series exhibits long memory, it is neither stationary (ʃ(θ)) nor is it a unit root (ʃ(l)) process; it is an ʃ(d) process,
with d a real number. A series exhibiting long memory, or persistence, has an autocorrelation function that damps hyperbolically,
more slowly than the geometric damping exhibited by “short memory” (ARMA) processes. Thus, it may be predictable at long
horizons. Long memory models originated in hydrology and have been widely applied in economics and finance. An excellent
survey of long memory models is given by Baillie (1996).

There are two approaches to the estimation of an ARFIMA (p, m g) model: exact maximum likelihood estimation, as
proposed by Sowell (1992), and semiparametric approaches, as described in this insert. Sowell’s approach requires specification
of the p and v values, and estimation of the full arfima model conditional on those choices. This involves all the attendant

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