The name is absent

with associated periodogram ordinates I_v (λ_s) = υ_x (λ_s) r⅛ (λ_s)* (Phillips 1999b, 9). He notes that both υ_x (λ_s) and, thus, I_v (λ_s)
are observable functions of the data. The log-periodogram regression is now the regression of log‰ (λ_s) on a_s = log ∣1 — e^lλ- ∣.
Defining a = m~¹ ∑Xι ^αs and x_s = a_s — ð, the modified estimate of the long-memory parameter becomes

∑Xι^l°g^(λ_s)                                 ∕_fix

2∑χ_ι⅛

Phillips proves that, with appropriate assumptions on the distribution of ⅞, the distribution of d follows

in distribution, so d has the same limiting distribution at d = 1 as does the GPH estimator in the stationary case so d is consistent
for values of d around unity. A semiparametric test statistic for a unit root against a fractional alternative is then based upon the

statistic Phiips 1999a, 10)
with critical values from the standard normal distribution. This test is consistent against both d < 1 and d > 1 fractional
aternatives.

roblpr computes the Robinson (1995) multivariate semiparametric estimate of the long memory (fractional integration)
parameters, d(g), of a set of G time series, y{g), g = 1,G with G ≥ 1. When applied to a set of time series, the d{g) parameter
for each series is estimated from a single log-periodogram regression which allows the intercept and slope to differ for each
series. One of the innovations of Robinson’s estimator is that it is not restricted to using a small fraction of the ordinates of the
empirical periodogram of the series, that is, the reasonable values of power need not exclude a sizable fraction of the original
sampe size. The estimator also allows for the removal of one or more initial ordinates and for the averaging of the periodogram
over adjacent frequencies. The rationale for using non-default values of either of these options is presented in Robinson (1995).

Robinson (1995) proposes an alternative log-periodogram regression estimator which he claims provides “modestly superior
asymptotic efficiency to J(0)”, (J(O) being the Geweke and Porter-Hudak estimator) Robinson (1995, 1052). Robinson’s
formuation of the log-periodogram regression also allows for the formulation of a multivariate model, providing justification for
tests that different time series share a common differencing parameter. Normality of the underlying time series is assumed, but
Robinson caims that other conditions underlying his derivation are milder than those conjectured by GPH.

We present here Robinson’s multivariate formulation, which applies to a single time series as well. Let X_t represent a
G-dimensional vector with g^th element X_gt,g = 1,... ,G. Assume that X_t has a spectral density matrix ./'Z_7r eŋʌ/ (λ) dw. with
eg,he element denoted as ∕₅⅛ (λ). The gth diagonal element, f_gg (λ), is the power spectral density of X_gt. For 0 < C_g < ∞
and — 1∕2<d₅<1∕2, assume that f_gg (λ) ~ C_gλ~^2da as λ —> 0+ for g = 1,..., G. The periodogram of X_gt is then denoted

Without averaging the periodogram over adjacent frequencies nor omission of I initial frequencies from the regression, we may
define Y_gk = log∕₅ (λ⅛). The least squares estimates of c = (eɪ,... ,cg)^z and d = (dɪ,... ,⅛)^z are given by

= vec{Y'Z(Z'Z)~¹}

where Z = (Z₁,...,Z_m)', Z_k = (1,-21ogλ_fe)^z, Y = (Y₁,...,Y_β), and Y_g = (Y_g,₁,. ..,Y_g,_m)' for m periodogram
ordinates. Standard errors for d_g and for a test of the restriction that two or more of the d_g are equal may be derived from the
estimated covariance matrix of the least squares coefficients. The standard errors for the estimated parameters are derived from
a pooed estimate of the variance in the multivariate case, so that their interval             estimates differ from those of their univariate

counterparts. Modifications to this derivation when the frequency-averaging (J) or omission of initial frequencies (1) options are
seected may be found in Robinson (1995).

Examples

Data from Terence Mills’ Econometric Analysis of Financial Time Series on UK FTA All Share stock returns (ftaret) and
dividends (ftadiv) are analyzed.

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