TESTS OF UNIT ROOTS AND COINTEGRATION HYPOTHESES IN ECONOMETRIC MODELS*

A number of statistical procedures for testing the unit roots hypotheses and the cointegration hypothesis have been proposed by statisticians and econometricians. This paper unifies many of the previous studies on unit roots tests and cointegration tests in the framework of a multivariate regression model and develops some new test statistics. We give a convenient quadratic representation of the limiting distributions of test statistics using stochastic integrals with respect to Brownian motions. The test procedures in this paper include the statistics for testing the unit root, the double unit roots, the seasonal unit roots, and the cointegrating relations for special cases. We also discuss some useful generalizations of unit roots tests and cointegration tests for empirical studies.     

Testing for Seasonal Fractional Roots in German Real Output

Abstract. This paper examines the seasonal structure of German real GNP per capita by using a version of Robinson's (1994) tests which is suitable in the context of seasonality. This method has several advantages over alternative approaches when testing for seasonal unit roots. First, unlike standard tests, which are nested in AR alternatives, it is embedded in fractional alternatives. Second, it allows testing at the zero frequency and at each of the seasonal frequencies separately. Third, it makes it possible to test for different orders of integration at each of the frequencies simultaneously. The empirical analysis suggests that the real output series may have a unit root at the zero frequency, and fractional rather than unit roots at the seasonal ones. This is in contrast to the findings reported by Lutkepohl et al. (1999) in their study on German money demand, and shows the importance of modelling the seasonal features of the data in alternative ways.     

Testing fractional integration with monthly data

Seasonal roots can help to explain the seasonal fluctuations in macroeconomic time series. In this paper we concentrate on monthly data and look at different versions of Robinson’s (1994) tests for testing unit roots and other fractionally integrated hypotheses when the root is located at zero and/or at the seasonal frequencies. A Monte Carlo experiment is carried out to check the power of these tests against different fractional alternatives, and an empirical application, using Spanish monthly data for the consumer price index, is also carried out in the article.     

Testing for PPP: Should we use panel methods?

A common finding in the empirical literature on the validity of purchasing power parity (PPP) is that it holds when tested for in panel data, but not in univariate (i.e. country-specific) analysis. The usual explanation for this mismatch is that panel tests for unit roots are more powerful than their univariate counterparts. In this paper we suggest an alternative explanation. Existing panel methods assume that cross-unit cointegrating relationships, that would tie the units of the panel together, are not present. Using simulations, we show that if this important underlying assumption of panel unit root tests is violated, the empirical size of the tests is substantially higher than the nominal level, and the null hypothesis of a unit root is rejected too often even when it is true. More generally, this finding warns against the automatic use of panel methods for testing for unit roots in macroeconomic time series.First version received: November 2001/Final version received : October 2003     

Testing for seasonal integration and cointegration: The Austrian consumption income relationship

This paper analyzes the nature of seasonal fluctuations in quarterly observations for Austrian consumption and income data. We begin with univariate tests of the order of integration and then move on to tests of cointegration. Seasonally adjusted as well as raw data are used in these tests. in univariate tests, the outcome for seasonally adjusted and raw data is in line. The unit roots at the zero frequency found in the seasonally adjusted series are also present in the raw data. In bivariate tests, the results for seasonally adjusted and raw data differ. While we find cointegration at the zero frequency between consumption and income for seasonally adjusted series, this hypothesis is generally rejected for the raw data.     

Bayesian analysis of periodic unit roots in the presence of a break

A Bayesian testing approach for a periodic unit root in quarterly and monthly data is presented. Further a Bayesian test is introduced to test for unit roots at (non)seasonal spectral frequencies. All procedures admit one structural break in the periodic trend function, where the occurrence of a break and the associated timing are treated as additional model parameters. A Bayesian model averaging (BMA) approach is proposed and power functions of the tests are computed. Overall the results indicate that the BMA periodic unit root test exhibits favourable test properties even in small samples. In an empirical application the presented testing procedures are used to test for (non)seasonal forms of unemployment persistence among OECD countries.     

Trade Balance and Exchange Rate: Unit Roots, Co-integration and Long Memory in the US and the UK

This paper deals with the relationship between the balance of trade and the exchange rate in the US/UK case. Many authors have studied this issue for many countries, but despite the intensive research, there is still no agreement about the effectiveness of currency devaluation to increase a country's balance of trade. We first analyse the relationship between the two variables using unit roots and co-integration methods, and the results are ambiguous. We try a new approach based on fractional integration. The unit root hypothesis is rejected in case of the trade balance in favour of smaller orders of integration, while this hypothesis is not rejected for the exchange rate. Thus, the two series do not possess the same order of integration. We sort this problem out by taking the exchange rate as an exogenous variable, and including it in a regression model where the residuals might follow a fractionally integrated model.     

Unit roots and persistence in the nominal interest rate: a confirmatory analysis applied to the OECD

Abstract.  This paper investigates the stochastic properties of long-term and short-term nominal interest rates for the OECD over the post-war era. For that purpose, we employ univariate unit root tests as well as panel unit root and stationarity tests that explicitly allow for cross-sectional dependence. Overall, we find overwhelming evidence that the nominal interest rate contains a unit root, which may be driven by a stochastic common factor. The computation of half-lives through impulse-response functions also points to a high degree of persistence. This has important implications for the cointegration analysis of the Fisher equation, the uncovered interest parity, and the term structure.     

New panel results on cointegration of international health expenditure and GDP

This article tests for existence of cointegration between health expenditure and GDP using data from 25 OECD countries for the period 19607ndash;1997. The empirical modelling is based on a heterogeneous bivariate vector error correction panel model that allows for trending data as well as intercepts and trends in the cointegrating relations. Univariate country-by-country and panel unit root tests generally fail to reject the null of a unit root in the health expenditure and GDP variables. Country-by-country results based on the Johansen multivariate likelihood-based inference indicate somewhat mixed results on country-specific cointegration with a rank of one found for 12 countries and a rank of zero for the remaining 13 countries. Application of a new panel test for cointegration rank with higher power than the individual tests indicates that health expenditure and GDP are cointegrated around linear trends.     

Exchange rates dynamics revisited: a panel data test of the fractional integration order

We test the possibility that exchange rates from nine developed countries have a unit root against the alternate possibility that they are fractionally integrated. Theoretically, exchange rates are only expected to follow a random walk under restrictive assumptions. However, most traditional unit root tests cannot reject a unit root in exchange rates, and time series tests that allow for fractional integration have given inconclusive results. To increase the power of the test of the integration order we develop two panel data tests of the fractional integration order. Monte Carlo simulations show that these tests are correctly sized and have relatively high power compared to other similar tests. Moreover, our empirical results show that we can reject a unit root in exchange rates with a high probability, but the integration order is close to one. This indicates that exchange rates are mean-reverting, although the reversion is slow, resulting in long swings.     

A managerial model of investment: overlooked alternatives

The long-run purchasing power parity (PPP) hypothesis is tested for nine bilateral sterling exchange rates, using recently developed techniques on cointegration and seasonal integration. The empirical findings show that none of the exchange rates and relative prices contain seasonal unit roots, but all have an autoregressive unit root. The cointegration tests overwhelmingly reject the PPP hypothesis as a long-run equilibrium condition for all countries concerned.     

New tests for stationarity and parity reversion: Evidence on New Zealand real exchange rates

The present paper discusses the stochastic stationarity of New Zealand exchange rates in light of new time series methods and new tests. The question of whether the real exchange rates have a unit root or are mean reverting is set in the general framework of fractionally integrated models. The estimates sustain the claim that New Zealand real exchange series are not stationary. However, it is shown that nonstationarity is compatible with parity reversion in the framework of fractional unit-root models.     

COMMON STOCHASTIC TRENDS: EVIDENCE FROM THE LONDON METAL EXCHANGE

Recent evidence from cointegration theory points towards the efficiency of the London Metal Exchange. We show that on theoretical grounds this evidence could be misleading. We also conduct multivariate and univariate unit roots tests on prices of three different metals, namely: copper, lead and zinc. The price data are seasonal and unadjusted quarterly data from the London Metal Exchange and they cover the period from 1972.1–1987.4. The evidence presented here supports the presence of common stochastic trends in metal price movements.     

Do Financial and Institutional Variables Enhance the Impact of Remittances on Economic Growth in Latin America and the Caribbean? A Panel Cointegration Analysis

Using recently developed panel unit root and panel cointegration tests and the Fully-Modified OLS methodology (FMOLS), this paper estimates the impact of remittances on the economic growth of selected upper and lower income Latin American & Caribbean (LAC) countries over the 1990–2007 period. Despite the large flow of remittances to the region, there have been relatively few empirical studies assessing the impact of remittances on economic growth in LAC. Panel unit root tests suggest that several of the macro variables included in the model exhibit unit roots, yet, at the same time, Pedroni’s panel cointegration test determined that there is a cointegrating relationship among the variables in the estimated model. The FMOLS estimates suggest that remittances have a positive and significant effect on economic growth in both groups of countries. The estimates also indicate that both the degree of economic freedom and credit provided by the banking system have a positive and significant effect on economic growth in upper (middle) income LAC countries. The sign of the interaction term between remittances and the credit (and EFI) variables suggest that remittances act as substitutes for these variables. Finally, the effect of remittances on both sets of countries is stronger in the presence of a financial (credit) variable.     

中国宏观经济和金融总量结构变化及因果关系研究

宏观经济和金融总量是否平稳是研究总量动态特征以及总量之间关系的前提。本文在考虑经济中结构变化的基础上对中国宏观经济和金融总量的时序列是具有单位根的非平稳还是分段趋势平稳进行了研究,结果发现在检验的10个总量中,有6个,即实际GDP、人均实际GDP、就业、实际银行信贷、实际储蓄负债和实际固定投资等总量的时序列是围绕着1个或2个结构断点的分段趋势平稳。分段趋势平稳的结论对于政策主导下的长期经济发展战略和短期经济稳定措施是否有效,以及总量之间关系的研究具有重要的启示。在单位根检验结果的基础上,本文还对消除趋势后的分段趋势平稳总量之间的因果关系进行了分析。     

Modeling Mexican agricultural production

Results of time series tests (including unit root and deterministic and stochastic cointegration tests) imply that a mixture of differenced and cointegrated model specifications are warranted for econometric models of Mexican agricultural supplies and input demands. Test results are sensitive to choice of functional form and the set of regressors. For example, share equations should be estimated using differenced data, but output supply and input demand equations generally should not. Generalized Leontief and quadratic functional forms are preferred over the translog. Symmetry and curvature of a restricted profit function are rejected. Short-run output supplies and input demands are generally inelastic. First version received: February 1998/Final version received: November 1998     

An extension of the asymmetric causality tests for dealing with deterministic trend components

This article extends the asymmetric causality tests, as developed by Hatemi-J (2012), for dealing with deterministic trend parts. It is shown how integrated variables up to three degrees with deterministic trend parts can be transformed into positive and negative cumulative partial components. These cumulative components can be used for implementing the asymmetric causality tests based on a Wald test statistic that is shown to follow a chi-square distribution asymptotically. Each solution is expressed as a proposition and a mathematic proof is provided for each underlying proposition. This issue is important because most economic or financial variables seem to be characterized by both stochastic as well as deterministic trend parts. An empirical application is provided in order to show how the oil prices and the exchange rates as integrated variables with drift and trend can be transformed into cumulative partial sums of positive and negative components. The conducted causality tests reveal that allowing for asymmetry has important repercussions for the underlying causal inference between these two variables.     

A reappraisal of the evidence on PPP: a systematic investigation into MA roots in panel unit root tests and their implications

Panel unit root tests of real exchange rates—as opposed to univariate tests—usually reject non-stationarity. These tests, however, could be biased if the real exchange rate contained MA roots. Indeed, two independent arguments claim that the real exchange rate, being a sum of a stationary and a non-stationary component, is possibly an ARIMA (1, 1, 1) process. Monte Carlo simulations show how systematic changes in the parameters of the components, of the test equation and of the correlation matrix affect the size of first and second-generation panel unit root tests. Two components of the real exchange rate—the real exchange rate of a single good and a weighted sum of relative prices—are constructed from the data for a panel of countries. Computation of the relevant parameters reveals that panel unit root tests of the real exchange rate are severely oversized, usually much more so than simple augmented Dickey-Fuller tests. Thus, the evidence for purchasing power parity from first and second-generation panel unit root tests may be merely due to extreme size biases.     

On the (mis)specification of seasonality and its consequences: An empirical investigation with US data

It is well known that mis-specification of a trend leads to spurious cycles in detrended data (see, e.g., Nelson and Kang (1981)). Seasonal-adjustment procedures make assumptions, either implicitly or explicitly, about roots on the unit circle both at the zero and seasonal frequencies. Consequently, seasonal-adjustment procedures may produce spurious seasonal variation and other statistically undesirable effects. In this paper we document for a large class of widely used US quarterly macroeconomic series the effects of competing seasonal-adjustment procedures on the univariate time-series properties of the adjusted series. We also investigate which procedures are most appropriate given the properties of the data. Overall, we find very significant differences and evidence that several U.S. macroeconomic time series contain a mixture of deterministic and stochastic seasonal components.     

Seasonal fluctuations and equilibrium models of exchange rate

Most of the evidence on dynamic equilibrium exchange rate models is based on seasonally adjusted consumption data. Equilibrium models have not worked well in explaining the actual exchange rate. However, the use of seasonally adjusted data might be responsible for the spurious rejection of the model. This article presents a new equilibrium model for the exchange rates that incorporates seasonal preferences. The fit of the model to the data is evaluated for five industrialized countries using seasonally unadjusted data. Our findings indicate that a model with seasonal preferences can generate monthly time series of the exchange rate without seasonality even when the variables that theoretically determine the exchange rate show clear seasonal behaviours. Further, the model can generate theoretical exchange rates with the same order of integration than actual exchange rates, and in some cases, with the same stochastic trend.