Open Access
A simple test for the equality of integration orders
(2012) Hualde Bilbao, Javier; Economía; Ekonomia
A necessary condition for two time series to be nontrivially cointegrated is the equality of their respective integration orders. Thus, it is standard practice to test for order homogeneity prior to testing for cointegration. Tests for the equality of integration orders are particular cases of more general tests of linear restrictions among memory parameters of different time series, for which asymptotic theory has been developed in parametric and semiparametric settings. However, most tests have been developed in stationary and invertible settings, and, more importantly, many of them are invalid when the observables are cointegrated, because they usually involve inversion of an asymptotically singular matrix. We propose a general testing procedure which does not suffer from this serious drawback, and, in addition, it is very simple to compute, it covers the stationary/nonstationary and invertible/noninvertible ranges, and, as we show in a Monte Carlo experiment, it works well in finite samples.
Open Access
Estimation of the cointegrating rank in fractional cointegration
(2012) Hualde Bilbao, Javier; Economía; Ekonomia
This paper proposes an estimator of the cointegrating rank of a potentially cointegrated multivariate fractional process. Our setting is very flexible, allowing the individual observable processes to have different integration orders. The proposed method is automatic and can be also employed to infer the dimensions of possible cointegrating subspaces, which are characterized by special directions in the cointegrating space which generate cointegrating errors with smaller integration orders, increasing the “achievement” of the cointegration analysis. A Monte Carlo experiment of finite sample performance and an empirical analysis are included.
Open Access
Fixed bandwidth inference for fractional cointegration
(Wiley, 2019) Hualde Bilbao, Javier; Iacone, Fabrizio; Economía; Ekonomia
In a fractional cointegration setting we derive the fixed bandwidth limiting theory of a class of estimators of the cointegrating parameter which are constructed as ratios of weighted periodogram averages. These estimators offer improved limiting properties over those of more standard approaches like ordinary least squares or narrow band least squares estimation. These advantages have been justified by means of traditional asymptotic theory and here we explore whether these improvements still hold when considering the alternative fixed bandwidth theory and, more importantly, whether this latter approach provides a more accurate approximation to the sampling distribution of the corresponding test statistics. This appears to be relevant, especially in view of the typical oversizing displayed by Wald statistics when confronted to the standard limiting theory. A Monte Carlo study of finite-sample behaviour is included.
Open Access
Fixed bandwidth asymptotics for the studentized mean of fractionally integrated processes
(Elsevier, 2017) Hualde Bilbao, Javier; Iacone, Fabrizio; Economía; Ekonomia
We consider inference for the mean of a general stationary process based on standardizing the sample mean by a frequency domain estimator of the long run variance. Here, the main novelty is that we consider alternative asymptotics in which the bandwidth is kept fixed. This does not yield a consistent estimator of the long run variance, but, for the weakly dependent case, the studentized sample mean has a Student- limit distribution, which, for any given bandwidth, appears to be more precise than the traditional Gaussian limit. When data are fractionally integrated, the fixed bandwidth limit distribution of the studentized mean is not standard, and we derive critical values for various bandwidths. By a Monte Carlo experiment of finite sample performance we find that this asymptotic result provides a better approximation than other proposals like the test statistic based on the Memory Autocorrelation Consistent (MAC) estimator of the variance of the sample mean.
Open Access
Small‐b and fixed‐b asymptotics for weighted covariance estimation in fractional cointegration
(Wiley, 2015) Hualde Bilbao, Javier; Iacone, Fabrizio; Economía; Ekonomia
In a standard cointegrating framework, Phillips (1991) introduced the weighted covariance (WC) estimator of cointegrating parameters. Later, Marinucci (2000) applied this estimator to fractional circumstances and, like Phillips (1991), analysed the so-called small-b asymptotic approximation to its sampling distribution. Recently, an alternative limiting theory (ﬁxed-b asymptotics) has been successfully employed to approximate sampling distributions. With the purpose of comparing both approaches, we derive here the ﬁxed-b limit of WC estimators in a fractional setting, ﬁlling also some gaps in the traditional (small-b) theory. We also provide some Monte Carlo evidence that suggests that the ﬁxed-b limit is more accurate.
Open Access
A novel test of economic convergence in time series
(Springer, 2025-01-03) Hualde Bilbao, Javier; Olmo, José; Economía; Ekonomia; Institute for Advanced Research in Business and Economics - INARBE
This paper proposes a novel test for the hypothesis of economic convergence. We extend the standard definition of convergence based on the parity condition and say that two economies converge if the time series of economic output are positively cointegrated and cotrended. With this definition in place, our main contribution is to propose a test of positive cointegration that does not require estimation of the cointegrating relationship, but is able to differentiate between positive and negative cointegration. Once the possibility of positive cointegration is established in a first stage, we test for cotrending in a second stage. Our sequential proposal enjoys an excellent performance in small samples due to the fast convergence of our novel test statistic under positive cointegration. This is illustrated in a simulation exercise where we report clear evidence showing the outperformance of our proposed method compared to existing methods in the related literature that test for economic convergence using cointegration methods. The results are particularly strong for sample sizes between 25 and 50 observations. The empirical application testing for economic convergence between the G7 group of countries over the period 1990–2022 confirms these findings.
Open Access
Revisiting inflation in the euro area allowing for long memory
(Elsevier, 2017) Hualde Bilbao, Javier; Iacone, Fabrizio; Economía; Ekonomia
We analyse inflation and inflation differentials in the euro area allowing for long memory and a new type of limiting theory denoted fixed-bandwidth. Our results differ from those based on standard normal asymptotics and the short memory assumption, and we also find that the inflation differentials between 'core' and 'peripheral' countries are strongly persistent. 'Core' economies appear to have less persistent differentials and may be more integrated, while 'peripheral' countries with high inflation may find themselves under competitive pressure for a long time.
Open Access
Truncated sum-of-squares estimation of fractional time series models with generalized power law trend
(Institute of Mathematical Statistics, 2022) Hualde Bilbao, Javier; Nielsen, Morten Ørregaard; Economía; Ekonomia
We consider truncated (or conditional) sum-of-squares estimation of a parametric fractional time series model with an additive deterministic structure. The latter consists of both a drift term and a generalized power law trend. The memory parameter of the stochastic component and the power parameter of the deterministic trend component are both considered unknown real numbers to be estimated and belonging to arbitrarily large compact sets. Thus, our model captures different forms of nonstationarity and noninvertibility as well as a very flexible deterministic specification. As in related settings, the proof of consistency (which is a prerequisite for proving asymptotic normality) is challenging due to non-uniform convergence of the objective function over a large admissible parameter space and due to the competition between stochastic and deterministic components. As expected, parameter estimates related to the deterministic component are shown to be consistent and asymptotically normal only for parts of the parameter space depending on the relative strength of the stochastic and deterministic components. In contrast, we establish consistency and asymptotic normality of parameter estimates related to the stochastic component for the entire parameter space. Furthermore, the asymptotic distribution of the latter estimates is unaffected by the presence of the deterministic component, even when this is not consistently estimable. We also include Monte Carlo simulations to illustrate our results.
Open Access
Estimation of long-run parameters in unbalanced cointegration
(Elsevier, 2014) Hualde Bilbao, Javier; Economía; Ekonomia
This paper analyses the asymptotic properties of nonlinear least squares estimators of the long run parameters in a bivariate unbalanced cointegration framework. Unbalanced cointegration refers to the situation where the integration orders of the observables are different, but their corresponding balanced versions (with equal integration orders after filtering) are cointegrated in the usual sense. Within this setting, the long run linkage between the observables is driven by both the cointegrating parameter and the difference between the integration orders of the observables, which we consider to be unknown. Our results reveal three noticeable features. First, superconsistent (faster than √ n-consistent) estimators of the difference between memory parameters are achievable. Next, the joint limiting distribution of the estimators of both parameters is singular, and, finally, a modified version of the ‘‘Type II’’ fractional Brownian motion arises in the limiting theory. A Monte Carlo experiment and the discussion of an economic example are included.

Hualde Bilbao, Javier

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