Estimate long memory in a set of timeseries via Log-Periodogram Regression
roblpr varname [if exp] [in range] [, powers(numlist) l(#) j(#) constraints(numlist)]
roblpr is for use with time-series data. You must tsset your data before using roblpr; see help tsset. roblpr supports the by prefix, which may be used to operate on each time series in a panel. Alternatively, the if qualifier may be used to specify a single time series in a panel.
roblpr computes the Robinson (1995) multivariate semiparametric estimate of the long memory (fractional integration) parameters, d(g), of a set of timeseries, y(g), g=1,G. G may be one. If a series exhibits long memory, it is neither stationary (I) nor is it a unit root (I) process; it is an I(d) process, with d a real number. When applied to a set of timeseries, 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. The standard errors for the estimated parameters are derived from a pooled estimate of the variance in the multivariate case, so that their interval estimates differ from those of their univariate counterparts.
A choice must be made of the number of harmonic ordinates to be included in the spectral regression. 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. The estimator also allows for the removal of one or more initial ordinates, and for the averaging of the periodogram over adjacent frequencies.
Results of the estimation are returned after the command; type return list to see the returned results. If more than one series is specified in varlist, an F-test of the equality of the d(g) coefficients is performed, and those values are returned as well.
powers(numlist) specifies the number of ordinates (nord) entering the log-periodogram regression. Each value in numlist is taken as the value p in nord = N^p. The default value of p, following Robinson, is 0.9. More than one p may be specified in numlist only if there is a single variable in the varlist.
l(#) specifies the number of initial ordinates to be excluded from the log-periodogram regression. Some researchers have found that such exclusion improves the properties of tests based on log-periodogram regressions. The default value of l is zero.
j(#) specifies that the log-periodogram is to be computed from sums of adjacent periodogram ordinates. The default value of j is 1, so that no averaging is performed. With j=2, the number of ordinates is halved; with j=3, divided by three; and so on.
powers(constraints) specifies the constraint numbers of the linear constraints to be applied during estimation; see help constraint. 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 the 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. One 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 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.
. use http://fmwww.bc.edu/ec-p/data/Mills2d/fta.dta
. roblpr ftap
. roblpr ftap ftadiv
. roblpr ftap ftadiv ftaret, j(2)
. roblpr ftap, power(0.7 0.75:0.9)
. roblpr ftap, j(2)
. roblpr D.ftap, j(2) l(1)
. constraint define 1 ftap = ftadiv . roblpr ftap ftadiv ftaret, c(1)
Christopher F. Baum, Boston College, USA firstname.lastname@example.org
Vince Wiggins, StataCorp LP email@example.com
References Robinson, P.M. Log-Periodogram Regression of Time Series with Long Range Dependence. Annals of Statistics 23:3, 1995, 1048-1072.
On-line: regress, time, tsset, ac, corrgram; gphudak (if installed), modlpr (if installed)