Why Frequency Matters for Unit Root Testing

Chapter 2 used the standard AR model for the inference on unit roots. However, one may contemplate testing for a unit root using different model specifications, and doing so might result in different inferential results. This chapter introduces inferential procedures for a unit root that use model specifications that depart from the standard AR model.

The existing empirical studies on the effectiveness of RBI intervention in the exchange rate market are based on single equation/GARCH models. Recognizing the interdependence between exchange rate returns, conditional volatility of exchange rate and RBI intervention, this study uses a simultaneous equations framework to examine the efficacy of intervention by RBI. Additionally, we control for the impact of macro-economic fundamental variables and micro-structure variable in our model. The econometric methodology used in the study includes two steps. In the first step, DF-GLS, Ng-Perron and, Lee and Strazicich unit root tests are used to examine the stationarity of the variables. In the second step, the simultaneous equations model is estimated using the generalized method of moments (GMM)-IV. This study is based on secondary data at monthly frequency, spreading from 1996:08 to 2016:10. The main findings of the study are as follows. The results of unit root tests suggest that all variables included in the analysis are stationary. The system GMM-IV estimates of the model indicate that exchange rate returns are influenced by RBI intervention, output growth differential, money supply growth differential, stock market returns, capital flows, EUR-USD exchange rate returns, change in trade balance differential and order flow. Additionally, the estimates of exchange rate volatility equation suggest that RBI intervention significantly decreases exchange rate volatility. Further, we find that volatility in capital flows has significant and positive impact on exchange rate volatility. The estimates of the central bank intervention equation suggests that exchange rate returns have a significant and negative impact on RBI intervention in foreign exchange rate market. We also find evidence for the presence of asymmetry in RBI intervention. The key implications of the study indicate that RBI intervention is effective in stabilizing the Indian exchange rate market in the face of both domestic and external shocks. Furthermore, exchange rate returns are significantly affected by RBI intervention, macro-economic variables and micro-structure variable. The findings of this study may be useful for formulating India's monetary policy.

The purpose of this study is to examine the Efficiency Market Hypothesis (EMH) from the perspective of the Algerian exchange rate market. We apply different tests of dependence, long memory, volatility clustering and unit root tests over the three main Algerian exchange rate returns series vis–à-vis the US Dollar, the Euro, and the British Pound. Empirical findings suggest that combined Autoregressive Moving Average (ARMA)-Fractionally Integrated Generalized Autoregressive Conditional Heteroskedastic (FIGARCH) models were the most appropriate to represent the behavior of exchange rate returns. We also compare the predictive qualities of the estimated models and the Random Walk (RW) in terms of out-of-sample forecasting. The results are held to imply the rejection of the EMH in the Algerian exchange rate market. Therefore, the exchange rates fluctuations can be predicted, which may help public authorities intervene in the exchange market and assess the consequences of different economic policies.

Many financial time series are nonstationary and are modeled as ARIMA processes; they are integrated processes (I(n)) which can be made stationary (I(0)) via differencing n times. I(1) processes have a unit root in the autoregressive polynomial. Using OLS with unit root processes often leads to spurious results; a cointegration analysis should be used instead. Unit root tests (URT) decrease spurious cointegration. The Augmented Dickey Fuller (ADF) URT fails to reject a false null hypothesis of a unit root under the presence of structural changes in intercept and/or linear trend. The Zivot and Andrews (ZA) (1992) URT was designed for unknown breaks, but not under the null hypothesis. Lee and Strazicich (2003) argued the ZA URT was biased towards stationarity with breaks and proposed a new URT with breaks in the null. When an ARMA(p,q) process with trend and/or drift that is to be tested for unit roots and has changepoints in trend and/or intercept two approaches that can be taken: One approach is to use a unit root test that is robust to changepoints. In this paper we consider two of these URT's, the Lee-Strazicich URT and the Hybrid Bai-Perron ZA URT(Herranz, 2016.) The other approach we consider is to remove the deterministic components with changepoints using the Bai-Perron breakpoint detection method (1998, 2003), and then use a standard unit root test such as ADF in each segment. This approach does not assume that the entire time series being tested is all I(1) or I(0), as is the case with standard unit root tests. Performances of the tests were compared under various scenarios involving changepoints via simulation studies. Another type of model for breaks, the Self-Exciting-Threshold-Autoregressive (SETAR) model is also discussed.

Weekly series of agricultural prices usually exhibit seasonal variations and the stationarity of these variations should be taken into account to analyse price relationships. However, unit root tests at seasonal frequencies are unlikely to have good power properties. Furthermore, movements in actual price series are often not as expected when unit roots are present. Therefore, stationarity tests at seasonal frequencies also need to be applied. In this paper, a procedure to test for the null hypothesis of stationarity at seasonal frequencies was extended to the weekly case. Once critical values were obtained by simulation exercises, unit root and stationarity tests were applied to weekly retail prices of different agricultural commodities in Spain. The most relevant finding was that many unit roots that seasonal unit root tests failed to reject did not seem to be present from the results of seasonal stationarity tests, whereas seasonal unit root tests led to the rejection of some unit roots that seemed to be present according to the results of seasonal stationarity tests. In conclusion, unit root tests should be complemented with stationarity tests before making decisions about the behaviour of seasonal patterns.

Testing for unit roots in flow data sampled at different frequencies

The efficient market hypothesis explains the random walk hypothesis suggesting that stock prices are independent of each other, hence, it is impossible to earn abnormal profits. The positive effect of a well-functioning and highly efficient stock market on the performance of an economy motivated the Philippine Stock Exchange to pursue massive modernization initiatives. This research provides evidence of the existence of random walk in the Philippine stock market employing the Augmented Dickey-Fuller (1981) and Phillips-Perron (1988) unit root tests, the Lo-MacKinlay's (1988) conventional variance ratio test, and Chow-Denning's (1993) simple multiple variance ratio test. Results of the ADF and PP unit root tests confirm the necessary condition for a random walk. The Chow-Denning (1993) maximum /z/ statistic and the Wald test statistic as in Richardson and Smith (1991) for the joint hypotheses and the Lo and MacKinlay (1988) individual statistics variance ratio test generally accepted the null hypothesis of a random walk. That is, the unit root and variance ratio tests consistently indicate that the null hypothesis of random walk cannot be rejected. The existence of a random walk in weak-form efficiency can be attributed to market liquidity as a result of continuous development and modernization of the Philippine equity market.

Since threshold autoregressive models were discovered, many unit root tests have been developed to test the unit root null hypothesis when considering regime change. On the other hand, Sollis, J Time Ser Anal 25:409–417, 2004, indicates that a threshold unit root test could be combined with some smooth transition logistic functions which are introduced by Leybourne et al., J Time Ser Anal 19:83–97, 1998. This paper investigates whether the Caner and Hansen, Econometrica 69:1555–1596, 2001, unit root test could be expanded with smooth transition functions and demonstrates the performance of this new unit root testing process with Monte Carlo simulations. Simulation results for finite sample properties show reasonable empirical size and power values. Also, the proposed unit root testing procedure is used to test unit root null hypothesis for industrial production indices of United States of America and Turkey.KeywordsThreshold modelSmooth transitionUnit rootMonte Carlo simulations

There is a theoretical case for real exchange rates to be stationary, but conventional unit root tests generally find nonstationarity in most economic data expressed in nominal terms; exchange rates in particular. Perron (1989) questioned the latter interpretation on the basis that the presence of a unit root may be a manifestation of not allowing for structural change - a finding reaffirmed later by Zivot and Andrews (1992) and Clemente, Montanes, and Reyes (1998) when single and double sudden and gradual endogenous breakpoints are accounted for in unit root tests. This paper considers testing for structural breaks and unit roots - in the presence of structural shifts - in the univariate data generating process (DGP) of the key nominal foreign exchange rates of the South African rand. Additionally, the connexions between the timing of the structural shifts and important economic and noneconomic events are explored. The key findings show overwhelming support for structural shifts in the DGP and nonstationarity across all exchange rates examined - even after accounting for structural change. The convergence of the t-statistics for the yen/rand towards their critical values when at most two breaks are allowed for in the unit root tests is also a significant discovery; suggesting that stationarity is a possibility contingent on the sample period selected and the true number of breaks incorporated in the tests.

Nonstationarity in real exchange rates using unit root tests with a level shift at unknown time

The paper empirically examines the validity of purchasing power parity (PPP) in Pakistan. For this purpose, unit root properties of real exchange rates (RERs) of Pakistan against its 21 major trading partners are examined using Fourier ADF (FADF) and Fourier KPSS (FKPSS) unit root tests for the period 1983Q1 to 2014Q4. Fourier unit root tests are used as they consider multiple temporary structural breaks and nonlinearity of the data. FADF test rejects the null hypothesis of unit root (non-stationary) in three RER series, while FKPSS test rejects the null hypothesis of stationary in nine RER series. Thus, FADF unit root test provide support for PPP hypothesis in three exchange rate series, while FKPSS test validates PPP theory in 12 exchange rate series. For robustness analysis, PPP equation is also estimated using regression analysis. The regression results show that PPP hypothesis is valid for nine exchange rate series. These findings suggest that PPP hypothesis partially holds in Pakistan.

Purchasing Power Parity and the European single currency: Some new evidence

Cointegrating Tests of Purchasing Power Parity Using ZNZ Patterned VECM

Several recent studies have found evidence of purchasing power parity by using long time series of data to combat the low power of unit root tests. The ability to reject unit roots in real exchange rates, however, only implies that PPP holds in the absence of structural change. We test for multiple structural changes in six real exchange rates, and find strong evidence against both the unit root and the no-structural-change hypotheses. We call this result quasi purchasing power parity. When structural change is accounted for, real exchange rates display mean reversion that is much faster than previously calculated. © 1998 John Wiley & Sons, Ltd.

PurposeIn the Iranian economy, investing in the housing market has been very important and beneficial for investors and households, because of inflationary environment, low real interest rates, underdeveloped financial and tax systems and economic sanctions. Hence, prediction of house prices is the main concern of housing market agents in the economy. The purpose of this paper is to test the stationary properties of Iran's provinces to improve the prediction of future housing prices.Design/methodology/approachIn this paper, the authors have tested the stationary properties of 20 Iran’s province centers over the period from 1993 to 2017 using a novel Fourier quantile unit root test and conventional ordinary/generalized least squares (O/GLS) linear unit root/stationary tests.FindingsAccording to conventional O/GLS linear unit root/stationary tests, most of the house prices series exhibit random walk behavior, whereas by applying the Fourier quantile unit root test, the null hypothesis of unit root is rejected for 15 out of 20 series. Other results indicated that house prices of cities responded differently to positive and negative shocks.Originality/valuePrevious studies only addressed conventional OLS or GLS linear unit root or stationary tests, but novel Fourier quantile unit root test was not used. New results were obtained based on this unit root test, that, as a priori knowledge, will help benefiting from the positive effects, or avoiding being victimized by the negative effects.