International Journal of Statistics and Probability; Vol. 7, No. 5; September 2018 ISSN 1927-7032 E-ISSN 1927-7040 Published by Canadian Center of Science and Education 19 Optimized Dickey-Fuller Test Refines Sign and Boundary Problems Compare to Traditional Dickey-Fuller Test Masudul Islam 1,4 , Afroza Akhtar 2 , Sirajum Munira 3 , Md. Salauddin Khan 4 & Md Monzur Murshed 5 1 Department of Mathematical Science, Ball State University, USA. 2 Beximco Pharmaceuticals Limited, Bangladesh. 3 Department of Statistics, Jahangirnagar University, Savar, Dhaka. 4 Statistics Discipline, Khulna University, Khulna, Bangladesh. 5 Department of Mathematical Science, Ball State University, USA. Correspondence: Masudul Islam, Department of Mathematical Science, Ball State University, USA. Tel: 1-765-400-9873. E-mail: mislam4@bsu.edu Received: May 21, 2018 Accepted: June 8, 2018 Online Published: August 3, 2018 doi:10.5539/ijsp.v7n5p19 URL: https://doi.org/10.5539/ijsp.v7n5p19 Abstract Impede nonstationarity is vigorous to study performance of time series data and removes long-term components to expose any regular short-term regularity. So, we find miscellaneous unit root tests for instance Dickey-Fuller test, Augmented Dickey-Fuller plus DF-GLS Tests and identify that almost all unit root tests with the estimated model suffer from sign and boundary problems of the parameters to smooth the progress of the non-stationarity problem. In this paper, we usage Dickey-Fuller test and impose some limits on the parameter. Our proposed optimized DF test based on error sum of square (ESS). Monto Carlo simulation method is used to generate simulated critical values for different sample size. Our proposed optimized DF test gives better result than the ordinary DF test with effectiveness, uniformity and power properties. Also, optimized DF improves the sign and boundary problems through imposing some limit on error sum of squares and capture more nonstationarity of time related data. Keywords: Optimized Dickey-Fuller test, Non-stationarity, ESS, Sign and boundary problems 1. Introduction Socio-economic, statistical, time series, econometrics or econometric era are pedestal on a few exact postulations. Infringement of theories greatly influences the guess of the parameter over and above test of hypothesis (Akter, 2014). Nonstationary test is necessary for analyzing the activities of advance time series research. Usually non-stationarity can be tested by different unit root tests for example Dickey-Fuller (DF) test, Augmented Dickey-Fuller (ADF) test, DF-GLS Tests, Phillips-Perron (PP) test and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test discussed by Dickey and Fuller,1979; Kwiatkowski et al.,1992; Kodde and Palm, 1986). But all the unit root tests as well as the estimated model suffer from sign and boundary tribulations of the parameters (Akter and Majumder, 2013). So, appropriate testing procedure plays key role at the preliminary arena of any inquiries. In keeping with the assumption of the Dickey-Fuller test, || < 1 or −2 <  < 0 of the time series models, such as   =  −1 +  . Any estimated value of  < −2 or >0 may fallout in invalid model for making decision regarding nonstationarity (Naznin et al., 2014). To triumph over this condition, it is required to impose appropriate limits on the parameters, which is larger than zero. Very diminutive quantity of literatures is on this concern such as Majumder and King (1999) proposed one sided tests. Basak et al. (2005) and Rois et al. (2008) worked on distance based approach. Aktar and Majumder (2013) developed one sided DF testing procedure. Naznin et al (2014) showed the sign and boundary problems and solution by Augmented Dickey-Fuller (ADF) test. Hence the usual DF test for testing unit root is not always fit and we need to enlarge constrained parameter estimation on restricted DF test. So, we are provoked to expand a suitable testing technique. The principle of this paper is, firstly test the stationarity of some observed time series. Secondly, propose the testing approach due to arising some unit root problems. Finally, compare the proposed test with the usual test.