Research Of Unit Root Test With Nonlinear Trend

Unit root test is very important and widely used in both theoretical analysis of econometrics and empirical research of various branches of applied economics. It is the base and precondition for analyzing spurious regression, testing cointegration relation and building ARMA model. Meanwhile, many economic theories and assumptions can deduce that the researched objects are stationary or of unit root processes, so they can be verified by unit root test directly.For example, assumptions of efficient capital market deem that the price of capital is random walk process, it means the price series of capital is of unit root process. Purchasing parity theory deems that exchange rate trends to purchasing parity in a long period, it entails the real exchange rate being stationary. Inter-temporal budget constraint theory entails government fiscal surplus being of stationary process. So the theoretical assumptions can be tested and verified by the unit root test based on the real historical data.As to the real economic data, time series usually includes a deterministic trend which increases with time going, therefore unit root test with trend is easy to encounter in real application. And the trend is usually assumed to be linear in conventional handling, although various non-linear trends may be of reality.In view of the importance and widely using, literate references concerning unit root test are affluent. Unfortunately a wealth of literate references tend to have incomplete or even wrong applications, leading to unreliable as far as to completely false conclusions. Unit root empirical test invariably follows the steps as data acquiring, selection of test approaches, statistic computing as well as drawing deductive conclusions. Errors may occur at any step in the process. The frequently committed mistakes result from small sample size, leading to degraded testing power without convincing conclusions. The most serious problem is of wrong specification, e.g. the trend is taken for linear one by default without testing in advance, but in fact it may be nonlinear. The wrong specification usually results in completely wrong test conclusions.ADF or PP unit root test is the prevalently used method to test unit root in theoretical and empirical analysis with the premise of linear trend assumption. The standard ADF or PP Unit root test methods offered by most conventional econometric softwares are invariably based on the assumption of linear trend. No thought is given to check if there is any trend or if the trend is genuine linear or not, as well no tests are conducted. Handling like this imposes great risk because wrong specification may cause fatal mistake. The research indicates that ADF or PP with the assumption of linear trend will draw dubious conclusions as far as possibly completely false conclusions to test those with non-linear trends.Is the real economic data definitely of linear trend process? In most cases it is not sure. The unit root process with linear trend is of random walk process with fixed drifting velocity. In the event that the drifting velocity changes, the unit root process will become to include nonlinear trend. Many empirical literatures refer to the test results of ADF or PP with the premise of linear trend, the reliability of their conclusions may have big doubts.One key purpose of this paper targets the unit root tests of non-linear trend series (regarding linear trend as the special case), i.e., a unit root test algorithm is worked out to conduct unit root test of any trend series, no matter it has trend or not or the trend is linear or non-linear. It can achieve effective testing result with same methods and steps.Because sample data has great impact on the unit root testing power to both linear and nonlinear trend, this paper addresses the small sample size in data collection and measurement errors of sample data, which will influence the testing result.The main contents of this paper are as below.At first this paper summarizes the conventional mistakes about unit root test found in many literate references and gives out detailed analysis of various errors and the causing reasons. The mistakes range from calculation errors caused by formula deduction or program design, the pitfalls of small sample size caused by difficulty to collect data, the logic mistakes of wrong generalization based on special cases, test failure arising from neglect of test power, to regression specification errors which have the profound influence on the test results, etc. This paper addresses the general rules of selection of regression functions in unit root test by way of Mont Carlo simulation approaches. The basic conclusions drawn from the study show that the best test result will be achieved if the test regression formula matches the data generation process. If there are less parameters in the test regression formula than those produced in the data generation process, i.e. the former one is unable to encompass the latter one, even with large sample size, the test power is possibly lowered to 0, that means test failure. Otherwise if there are more parameters in the test regression formula than those produced in the data generation process, i.e. the former one is able to encompass the latter one, it usually results in lowering of the test power. However the lowering is not very awful, it will be improved with the increase of sample size. Therefore it can achieve acceptable test results in case of large sample size.In case of small sample sizes(less than 100) and greatρ(>0.9), no matter whether the residuals are correlated, all sorts of data generation processes tested by various approaches may have low test power. Here the main problem is of small sample size, and the test power will be further degraded if residual series is correlated. In case of medium sample size ([100,500]), careful distinguishing of different data generation process has merits to result in improved test power, it is necessary to make a judge whether the DGP includes interception, time trend and whether the unit root process includes drift. However from the angle of data test, the distinguishing is dramatically difficult because no reliable result can be achieved other than in cases of large sample size test. But the distinguishing makes less sense in case of large size sample, good test results can be achieved directly by way of regression with enough parameters.When the residual series is correlated, the test power is usually lowered. Thinking of correlation of residuals, if the lagged difference parts are added in the regression formula, the test power can be usually improved. In case of medium sample size, the improvement is prominent when the lagged difference parts match the residual correlated structure. Although various considerations concerning correlation may cause differences in test power, it may diminish with the enlarged size of samples. Concerns of correlation of residuals may be ignored in case of large sample size.Generally much attention is paid to test size in unit root empirical test analysis, but less attention is drawn to test power's influence. No reliable test result can be achieved in case of small sample size. As the most fundamental DF unit root test approach is concerned, this paper works out the estimating formula for the unit root test power, researches the influencing factors of the test power. On base of them, this paper deduces the estimating formula with the requirement of minimum length of samples in unit root test. The fitted formula for estimating smallest length is deduced combining the simulation data with curve fit, which offers theoretical and practical basis of sample choice in empirical analysis of unit root test. At the same time, because samples are always finite in empirical analysis, it's impossible to make a judge whetherρequals to 1 or not strictly, so we have to treat it as unit root process whenρis greater than some value, and it's better to offer the test power as reference.Sample selection is the base of unit root test. Other than the sample size, the influence of measurement errors, which exist almost forever, should be addressed in unit root test, too. This paper studies the limiting distribution of two statistics, T (ρ? ? 1) andτ, of time series with additive stationary measurement error in unit root test, under two kinds of regression situations with or without interception. Theoretical analysis and Monte Carlo tests both show that measurement errors usually make the statistics left biased, resulting in distorted size but improved power, only when variance of measurement error is relatively small, its influence to unit root test can be ignored. The degree of left bias is decided by the relative size of variance and first order covariance, otherwise it has no relation with the mean and probability distribution of measurement error. Left bias becomes more serious when variance of measurement error increases, but lessens and is counteracted by positive first order covariance. However negative first order covariance may intensify left bias. Left bias is small when first order covariance is close to its variance.This paper then turns to research the regression specification of unit root test, focusing on the situations with non-linear trend or unidentified trend.ADF or PP unit root test is the widely used method for unit root test. In theoretical analysis or conventional econometric software, the two test methods are usually applied to deal with time series with linear trend. So in case of non-trend or linear trend stationary process with sufficient samples, ADF or PP unit root test can give right test results. But to time series with nonlinear trend, Monte Carlo test shows that they tend to take a stationary series as a unit root process, resulting in false test. Only when the disturbance is relatively strong referring to the trend and then nonlinear trend is immersed in noises, various non-linear trends may be treated as quasi-linear or as non-trending, ADF or PP test method can give right result.Therefore it is imperative to test whether or not the trend is really linear before using ADF or PP test which usually assumes implicitly that the series has linear trend, as a result false conclusions are easily drawn to test series with non-linear trend by using them. However it is ignored in most literatures concerning positive analysis. This paper intensively researches the test of deterministic trend in unit root test, proposes test method to determine whether the time series has trend or not. If with trend, the regression coefficient and equivalent mean test methods are proposed to test linear or non-linear trend. Meanwhile this paper discusses the impact on the test result for serial correlations and suggests generalized difference method or sampling child-series method to remove the impact of residual correlation.This paper goes in deep to research the VR unit root test method based on the ratio of long run variance and short run variance, deduces the limiting distribution of statistic under situations of unit root process and stationary process from time domain, offers the selection of truncation length, the critical value of test by simulation, the test size and power when residual is correlated or not. VR test method has many merits, such as the rapid speed for VR statistic to converge to limiting distribution, the strong ability to anti-jamming, no need to amend VR statistic because of residual correlation, non-sensitivity to regression formula, etc. But VR has also a severe drawback resulting in test size being distorted when residual series has negative correlation. Of course PP test has the same flaw.At last, this paper respectively propose three unit root test ways which can test any deterministic trend using the same way and following the same steps. Right conclusions can be drawn without bothering to test whether the time series has trend, linear or nonlinear trend. I.e., the three methods are designed to estimate any deterministic trend. After estimation of trend, it's very easy to test whether the residual series has a unit root by using non-trend test methods such as RMA, VR, etc.The first proposed way to estimate deterministic trend is orthogonal polynomial approximation. This paper introduces how to generate normal orthogonal polynomial, researches the method and properties of approximation of time polynomial and non-polynomial of trend by using orthogonal polynomial. It deduces the limiting distribution of unit root test statistic based on orthogonal polynomial approximation and proposes how to decide the highest order of orthogonal polynomial. By simulation test, the paper offers the test critical values of different highest order, the test size and power when the time series has non- trend, linear or non-linear trend and residual is correlated or non-correlated. The results show that the test size has not any distortion, the test power is ideal in case of various trends.The following is the second proposed way to estimate deterministic trend. The test approach separates the deterministic trend and random disturbance parts of pending test time series with singular value decomposition and does unit root test for residuals with recursive mean adjustment. Also by simulation test, the paper offers the test critical values of different sizes, the test size and power when the time series has non- trend, linear or non-linear trend and residual is correlated or non-correlated. The results show that the test size has not any distortion and the test power is good in case of various trends. Monte Carlo test shows that it has effective power to linear or nonlinear trend, even structure changing process and it has no much distortion even when the residual has severe negative correlation.The third proposed way is to estimate deterministic trend with local polynomial regression, then make unit root test to residual. The method does not care the form and specification of trend. The paper introduces the properties of local polynomial regression, deduces the limiting distribution of unit root test based on VR statistic. By simulation test, the paper researches how to select the width of window and offers the test size and power when the residual is correlated or non-correlated. The results show that the test approach is effective. Of course, strong series correlation of residual will affect the test size and power based on VR test. Positive correlation will lower test size and power, and negative correlation will increase the size and power. But this affection will diminish when the samples enlarge. The increase of width of window will lessen the size distortion of VR test. The size distortion can be improved by using RMA or other methods applied in non-trend situations.Finally, using all kinds of nonlinear trend unit root test methods proposed by this paper, the purchasing power parity theory, random walk theory of securities markets and intertemporal government budget constraint theory are tested by empirical data. The test results of all approaches turn out to be the same. The positive tests don't support PPP theory assumption, but support random walk assumption and intertemporal government budget constraint theory. Meanwhile the tests show that a plenty of economic time series have nonlinear trends. It means conventional unit root rest methods can't be used to get right conclusions.The main initiatives of this paper include how to estimate the least sample length and analyzing the impacts on the test consequences resulting from the sample measurement errors in unit root test, testing of linear or nonlinear property of deterministic trend in unit root test, proposing three unit root testing methods that can test time series with any deterministic trends that may be linear,nonlinear or non-trend, using uniform approaches and steps without any need for prior assumption, judgement and test.