Choice of Spectral Density Estimator in Ng-Perron Test: A Comparative Analysis

Malik, Muhammad Irfan; Atiq-ur-Rehman Article Choice of Spectral Density Estimator in Ng-Perron Test: A Comparative Analysis International Econometric Review (IER) Provided in Cooperation with: Econometric Research Association (ERA), Ankara Suggested Citation: Malik, Muhammad Irfan; Atiq-ur-Rehman (2015) : Choice of Spectral Density Estimator in Ng-Perron Test: A Comparative Analysis, International Econometric Review (IER), ISSN 1308-8815, Econometric Research Association (ERA), Ankara, Vol. 7, Iss. 2, pp. 51-63, https://doi.org/10.33818/ier.278040 This Version is available at: https://hdl.handle.net/10419/238816 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ International Econometric Review (IER) 51 Choice of Spectral Density Estimator in Ng-Perron Test: A Comparative Analysis Muhammad Irfan Malik and Atiq-ur-Rehman International Islamic University Islamabad and International Islamic University Islamabad ABSTRACT Ng and Perron (2001) designed a unit root test, which incorporates the properties of DFGLS and Phillips Perron test. Ng and Perron claim that the test performs exceptionally well especially in the presence of a negative moving average. However, the performance of the test depends heavily on the choice of the spectral density estimators used in the construction of the test. Various estimators for spectral density exist in the literature; each have a crucial impact on the output of test, however there is no clarity on which of these estimators gives the optimal size and power properties. This study aims to evaluate the performance of the Ng-Perron for different choices of spectral density estimators in the presence of a negative and positive moving average using Monte Carlo simulations. The results for large samples show that: (a) in the presence of a positive moving average, testing with the kernel based estimator gives good effective power and no size distortion, and (b) in the presence of a negative moving average, the autoregressive estimator gives better effective power, however, huge size distortion is observed in several specifications of the data-generating process. Key words: Ng-Perron Test, Monte Carlo, Spectral Density, Unit Root Testing JEL Classifications: C01, C15, C63 1. INTRODUCTION Unit root testing is a well-known and one of most debated issues in econometrics. There are lots of economic and econometric implications of the existence of a unit root in time series data, including the incidence of spurious regression (Atiq-ur-Rehman, 2011; Libanio, 2005). Due to its importance, many tests and testing procedures were developed for testing for a unit root. However, the size and power properties of unit root tests have always been subject to debate. In many economic time series models, errors may have heterogeneity and temporal dependence of unknown forms. This is the main source of size and power distortion of unit root tests. In order to draw more accurate inferences from estimates of parameters, constructing unit root tests based on long run variance (LRV) estimates has become important. LRV estimates take serial correlation and heterogeneity into account. The key to constructing an LRV is to estimate the spectral density (SD hereafter) at zero frequency. There are two main types of SD estimators: (1) autoregressive estimator of spectral density, (2) kernel based estimator of spectral density. However, literature does not provide any information about the relative performance of these estimators of spectral density. Many of existing tests for unit root, including the Ng-Perron test, use an estimator of spectral density at Muhammad Irfan Malik, PhD. Scholar (Econometrics), International Institute of Islamic Economics, International Islamic University Islamabad, (email: irfan[email protected]m), Tel: +92 321 6150274. Atiq-ur-Rehman, Assistant Professor (Econometrics), International Institute of Islamic Economics, International Islamic University Islamabad. Malik and Rehman-Choice of Spectral Density Estimator in Ng-Perron Test: A Comparative Analysis 52 zero frequency. Ng and Perron (2001) have developed a new suit of tests, which according to them outperforms the other tests, especially in case of a negative moving average process. The output of the test is also affected critically by the choice of spectral density estimator, and the literature does not provide any guide in this regard. Ng and Perron (2001) do not discuss the effect of the choice of spectral density estimator and thus leave practitioners without guidance regarding the choice of an estimator of spectral density. This study aims to investigate the properties of Ng-Perron test for different choices of SD estimators using Monte Carlo simulations. We examine the size distortion and effective power of the test, with both autoregressive (AR) estimator and kernel based (KB) estimators of spectral density, in the presence of a negative and positive moving average. The remainder of the paper is organized as follows: In Section 2 we discuss the Ng-Perron test and various estimators of spectral density. Section 3 consists of our Monte Carlo design. Section 4 explains the results. Section 5 provides details of detecting the sign of a moving average. Section 6 presents some concluding remarks. 2. EFFECT OF SPECTRAL DENSITY ESTIMATOR ON OUTPUT OF NG-PERRON TEST: A REAL DATA ILLUSTRATION Like other tests, the output of Ng-Perron test depends crucially on the choice of the spectral density estimator, and the final decision may be quite contradictory for two different choices of the density. This fact is illustrated below with the help of a real data example. We apply the Ng-Perron test on log GDP of UK from 1951-2007. Table 2.1 provides the outputs of Ng-Perron test with both estimators of spectral density estimators. Spectral Density Estimator Ng-Perron tests With Drift With Drift and Trend AR Estimator -8.25* -25.25* KB estimator (Parzen Kernel) 1.54* -3.81* Critical Value 5% Critical Value -8.1 -17.3 Table 2.1 Output of Ng-Perron test with AR and KB estimator for log UK GDP data Notes: * 5% level of significance According to the results in Table 2.1, the Ng-Perron test statistics is below the critical value for the autoregressive estimator of spectral density; hence, the unit root hypothesis should be considered rejected. On the other hand, for Kernel based estimator of spectral density, the Ng- Perron test statistics is far above the critical value; thus, the null of unit root could not be rejected even at a loose significance level. Therefore, the person applying the unit root test may be confused in the choice of result. In response to this ambiguity, we designed our study to compare the size and power properties of Ng-Perron test so that a practitioner may get some guidance on the selection of an optimal spectral density estimator. 3. COMPUTATION OF NG-PERRON TEST In this section we discuss the Ng-Perron unit root test and different estimators of spectral density at zero frequency. International Econometric Review (IER) 53 3.1. Test Statistics Dufour and King (1991) and Elliott et al. (1996) found that local GLS detrending of the data yields significant power gains. Phillips and Perron (1987) found that use of SD could improve the performance of the test. Ng and Perron (2001) combine GLS detrending with SD to design a new test. The proposed test consists of a suite of four tests, namely MZa, MZt, MSB, and MPT. The four test statistics proposed by Ng-Perron are:   k fyT MZ t a2 )0( ˆ ) ~ (21    MSBMZMZ at * 2/1 )0( ˆ        f k MSB            1 212 0 212 )0( ˆ ) ~ ()1( )0( ˆ ) ~ ( t t t t dwhen f yTckc dwhen f yTckc MPT where represent drift and drift and trend in DGP, and - . The symbol indicates the estimate of spectral density at frequency zero. 3.2. Spectral Density at Frequency zero The spectral density at frequency zero represent the heteroskedasticity and autocorrelated corrected (HAC) standard error. There are many ways to estimate the spectral density, which can be divided into two types: (a) autoregressive spectral density, and (b) kernel based spectral density, which can be further subdivided into four types. This hierarchy is summarized in the following Figure 3.1: Figure 3.1 Summary of Spectral Density Hierarchy The computational details of these estimators are as under: SD Estimator Autoregressive Parzen Kernel Bartlett Kernel Quadratic Spectral Kernel Tucky-Hanning Kernel Kernel Based Malik and Rehman-Choice of Spectral Density Estimator in Ng-Perron Test: A Comparative Analysis 54 3.2.1. Autoregressive (AR) Estimator of Spectral Density Autoregressive estimator of spectral density was proposed by Stock (1990; see also Stock, 1994; Perron and Ng, 1998). This estimator, based on the estimation of parametric model, is identical to the equation of the ADF test equation. After having GLS detrending series estimate the regression equation given below: tl T llttt yyy      ˆ ~~~ 1 1 (3.1) Autoregressive estimator of spectral density is: 2 2 ))1( ˆ 1( ˆ )0( ˆ    f where (1) = and – – . (1) is the sum of coefficients of lags of . Here – – and represent the variance of residuals ( ) from the equation (3.3). 3.2.2. Kernel Based (KB) Estimator of Spectral Density Non parametric kernel based estimator of spectral density was proposed by Phillips (1987) and then restructured by Phillips and Perron (1988). Kernel based estimator of spectral density is the weighted sum of auto covariance, in which weights are decided by the kernel and bandwidth parameter. Estimating the equation using GLS detrended series, ttt yy   1 ~~ (3.2) The kernel based estimator given as: )/()( ˆ )0( ˆ )( ljKjf lT lTj      (3.3) jT j jT tjtt      1 ˆˆ )( ˆ   where l is bandwidth parameter, which act as a truncation lag in the covariance weighting, and K is the kernel function, which can be estimated in multiple ways listed below. (j) is jth order auto covariance of residual from equation (3.2). For the estimation of the kernel estimator of spectral density we consider the following kernels: 1. Bartlett Kernel 2. Parzen Kernel International Econometric Review (IER) 55 3. Quadratic Spectral Kernel – 4. Tukey-Hanning Kernel where x = for all kernels. Asymptotically, all of these kernels are equivalent (Andrews, 1991). The computational details are given below. Ng and Perron point out that these four tests are equivalent in terms of size and power. Throughout our discussion, MZa is taken as representative of these four. 4. MONTE CARLO EXPERIMENT In order to compare the performance of Ng-Perron test with AR and KB estimator of spectral density, we perform extensive Monte Carlo experiments, which is given in Figure 4.2 below. Figure 4.2 Flow Chart of Monte Carlo Experiment Generate Series Apply Ng-Perron test with Autoregressive Estimator Parzen Kernel Bartlett Kernel Quadratic Spectral Kernel Tucky-Hanning Kernel Apply Ng-Perron test with Kernel Based Estimator Compute Size/ Power Compute Size/ Power Chose Autoregressive and Moving Average Coefficient Select DGP Compare Results Malik and Rehman-Choice of Spectral Density Estimator in Ng-Perron Test: A Comparative Analysis 56 Every step of the above mentioned Monte Carlo experiment is summarized as under: 4.1. Data Generating Process The following forms of the data generating process were used to conduct the Monte Carlo experiment: DGP-I yt = α + ut ut = ρut–1 + δe t–1 + et, DGP-II yt = α + + ut ut = ρut–1 + δe t–1 + et, DGP-I resembles an ARMA process with an intercept but no trend, whereas DGP-II resembles ARMA with drift and trend, where t = , ,…,T. 4.1.1. Autoregressive Coefficient Setting the autoregressive coefficient ρ = 1 will generate a unit series, which could be used to compute the power of Ng-Perron test, whereas setting Rho < 1 generates stationary series, which can be used to compute the power of the test. The following values of Rho were used for the Monte Carlo experiment: 0.99, 0.98, 0.95, 0.90, 0.85, 0.80, and 0.70. 4.1.2. Moving Average Coefficient The aim of this study was to evaluate the performance of the Ng-Perron test both for positive and negative moving average processes. The following values were used in the experiment: -0.80, -0.60, -0.40, -0.20, 0, 0.2, 0.4, 0.6, and 0.8 4.1.3. Calculating Size and Size Distortion Ng and Perron provide a set of asymptotic critical values for their test. The test statistics calculated on the series generated under the null were compared with these critical values in order to calculate the actual size of the test. The size distortion is the difference between the actual size and nominal level of significance. 4.1.4. Calculating Power and Effective Power The power of the test was computed by applying unit root tests to series generated with a stationary root. The probability of rejection of the null is the power of the test. However, for several data generating processes, heavy size distortion was observed. Since it is not reasonable to compare the power of two tests with different sizes, we have used the effective power of the tests for comparison. The effective power was calculated as follows: Effective Power for a DGP = Actual Power at Rho < 1 – Actual Size for Rho = 1. 5. MONTE CARLO RESULTS This section illustrates the equivalence of KB estimators. There are four choices of kernels in this study whose computational details are given in Section 3. The figures below summarize the size and power of Ng-Perron test for different choice of kernels. The figures show that the power curves remains same of various choices of kernels. The experiment was repeated for a var y DGP’ a d m a u b a d. A a y a k does not significantly affect the size and power of test, therefore there is no need of International Econometric Review (IER) 57 summarizing the simulations for all four kernels. Only one of these kernels will be sufficient to observe the behavior of the remaining ones. Figure 5.3 The Size and Power of MZa when Sample Size is 150 and δ = 0.4 with DGP-I Figure 5.4 The Size and Power of MZa when Sample Size is 80, δ = -0.4 with DGP-I For the comparison of size and power of the test with AR estimator and KB estimator, we use the Parzen kernel as a representative for these four kernels. 5.2. Effective Power versus Power Our results indicate that the size of the test is not stable, rendering comparison of the power meaningless. For a more meaningful comparison, we compare the distortion in size and the effective power of the test. Size distortion is the difference between the observed size and theoretical size (here 5%) of the test; effective power is defined as the difference between the empirical power and empirical size of the test. 5.3. Performance of Test with DGP-I Both KB and AR estimators of spectral density are equivalent mathematically at zero lag length/ lag truncation for any data generating process. We discuss the performance of the test 0 1 2 3 4 5 6 1 2 3 4 5 Perecntage Lag Truncation Size Bartlett Parzen Quadratic-Spectral Tucky-Hanning 0 10 20 30 40 50 60 1 2 3 4 5 Percentage Lag Truncation Power for Rho=0.95 Bartlett Parzen Quadratic-Spectral Tucky-Hanning 0 5 10 15 20 25 30 35 40 45 1 2 3 4 5 Percentage Lag Truncation Size Bartlett Parzen Quadratic-Spectral Tucky-Hanning 0 10 20 30 40 50 60 70 80 90 1 2 3 4 5 Percentage Lag Truncation Power for Rho=0.95 Bartlett Parzen Quadratic-Spectral Tucky-Hanning Malik and Rehman-Choice of Spectral Density Estimator in Ng-Perron Test: A Comparative Analysis 58 with KB and AR estimators of spectral density at nonzero lag length/ lag truncation. Our result shows that the test has very low effective power in a small sample size with both estimators. Figures 3 and 4 depict the size distortion and the effective power of the test with a positive value of moving average coefficient. Figure 5.5 The Size Distortion and Effective Power Ng-Perron Test with AR and KB Estimator when MA = 0.2 for DGP-I Figure 5.6 The Size Distortion and Effective Power Ng-Perron Test with AR and KB Estimator when MA = 0.6 for DGP-I According to the figures above, the distortion in size and effective power of the test increases with lag length when we use an AR estimator of spectral density; on the other hand, when using a KB estimator, the effective power of the test improves with large lag truncation without any distortion in the size of the test. Therefore it could be deducted that in the case of a positive moving average, the KB estimator outperforms the AR estimator. The behavior of the effective power and distortion remains similar, for experiments with different values of MA and autoregressive parameters. A different picture emerges when we have a negative moving average in the data generating process. Lag length selection has significant consequences on the performance of the test with an AR estimator. As evident in Figures 5 and 6, we observed that for a weaker negative -5 0 5 10 15 20 25 30 0 1 2 3 4 5 Percentage Lag Length/ Lag Truncation Size Distortion at sample size 40 and 250 AR(40) KB(40) AR(250) KB(250) 0 10 20 30 40 50 60 70 80 90 0 1 2 3 4 5 Percentage Lag Length/ Lag Truncation Effective Power for Rho=0.95 at sample size 40 and 250 AR(250) KB(250) AR(40) KB(40) -5 0 5 10 15 20 25 30 0 1 2 3 4 5 Percentage Lag Length/ Lag Truncation Size Distortion at sample size 40 and 250 AR(40) KB(40) AR(250) KB(250) 0 10 20 30 40 50 60 70 80 90 0 1 2 3 4 5 Percentage Lag Length/ Lag Truncation Effective Power for Rho=0.95 at sample size 40 and 250 AR(250) KB(250) AR(40) KB(40)