Testing for structural change in cointegrated regression models: some comparisons and generalizations

This paper compares and generalizes some testing procedures for structural change in the context of cointegrated regression models. The Lagrange Multiplier (LM) tests proposod by Hansen (1992) are generalized to testing for partial structural change. An exponential average LM test is also suggested following the idea of Andrews and Ploberger (1992). In particular, an optimal test for cointegration is developed. We also propose a new cointegration test which is robust to a possible one-time discrete jump in the intercept. We tabulate the asymptotic critical values for the above tests and conduct a small Monte Carlo simulation to investigate their finite sample performance.     

Bootstrap LR tests of stationarity,common trends and cointegration

This paper considers the likelihood ratio (LR) tests of stationarity, common trends and cointegration for multivariate time series. As the distribution of these tests is not known, a bootstrap version is proposed via a state- space representation. The bootstrap samples are obtained from the Kalman filter innovations under the null hypothesis. Monte Carlo simulations for the Gaussian univariate random walk plus noise model show that the bootstrap LR test achieves higher power for medium-sized deviations from the null hypothesis than a locally optimal and one-sided Lagrange Multiplier (LM) test that has a known asymptotic distribution. The power gains of the bootstrap LR test are significantly larger for testing the hypothesis of common trends and cointegration in multivariate time series, as the alternative asymptotic procedure – obtained as an extension of the LM test of stationarity – does not possess properties of optimality. Finally, it is shown that the (pseudo-)LR tests maintain good size and power properties also for the non-Gaussian series. An empirical illustration is provided.     

Testing for the Null Hypothesis of Cointegration with a Structural Break

In this paper we propose residual-based tests for the null hypothesis of cointegration with a structural break against the alternative of no cointegration. The Lagrange Multiplier (LM) test is proposed and its limiting distribution is obtained for the case in which the timing of a structural break is known. Then the test statistic is extended to deal with a structural break of unknown timing. The test statistic, a plug-in version of the test statistic for known timing, replaces the true break point by the estimated one. We show the limiting properties of the test statistic under the null as well as the alternative. Critical values are calculated for the tests by simulation methods. Finite-sample simulations show that the empirical size of the test is close to the nominal one unless the regression error is very persistent and that the test rejects the null when no cointegrating relationship with a structural break is present. We provide empirical examples based on the present-value model, the term structure model, and the money-output relationship model.     

Testing for the Null Hypothesis of Cointegration with a Structural Break

In this paper we propose residual-based tests for the null hypothesis of cointegration with a structural break against the alternative of no cointegration. The Lagrange Multiplier (LM) test is proposed and its limiting distribution is obtained for the case in which the timing of a structural break is known. Then the test statistic is extended to deal with a structural break of unknown timing. The test statistic, a plug-in version of the test statistic for known timing, replaces the true break point by the estimated one. We show the limiting properties of the test statistic under the null as well as the alternative. Critical values are calculated for the tests by simulation methods. Finite-sample simulations show that the empirical size of the test is close to the nominal one unless the regression error is very persistent and that the test rejects the null when no cointegrating relationship with a structural break is present. We provide empirical examples based on the present-value model, the term structure model, and the money-output relationship model.     

Tests for Parameter Instability in Regressions with 1(1) Processes

This article derives the large-sample distributions of Lagrange multiplier (LM) tests for parameter instability against several alternatives of interest in the context of cointegrated regression models. The fully modified estimator of Phillips and Hansen is extended to cover general models with stochastic and deterministic trends. The test statistics considered include the SupF test of Quandt, as well as the LM tests of Nyblom and of Nabeya and Tanaka. It is found that the asymptotic distributions depend on the nature of the regressor processes—that is, if the regressors are stochastic or deterministic trends. The distributions are noticeably different from the distributions when the data are weakly dependent. It is also found that the lack of cointegration is a special case of the alternative hypothesis considered (an unstable intercept), so the tests proposed here may also be viewed as a test of the null of cointegration against the alternative of no cointegration. The tests are applied to three data sets—an aggregate consumption function, a present value model of stock prices and dividends, and the term structure of interest rates.     

Reconsidering LM unit root testing

Non-rejection of a unit root hypothesis by usual Dickey & Fuller (1979) (DF, hereafter) or Phillips & Perron (1988) (hereafter PP) tests should not be taken as strong evidence in favour of unit root presence. There are less popular, but more powerful, unit root tests that should be employed instead of DF-PP tests. A prime example of an alternative test is the LM unit root test developed by Schmidt & Phillips (1992) (hereafter SP) and Schmidt & Lee (1991) (hereafter SL). LM unit root tests are easy to calculate and invariant (similar); they employ optimal detrending and are more powerful than usual DF-PP tests. Asymptotic theory and finite sample critical values (with inaccuracies that we correct in this paper) are available for SP-SL tests. However, the usefulness of LM tests is not fully understood, due to ambiguity over test type recommendation, as well as potentially inefficient derivation of the test that might confuse applied researchers. In this paper, we reconsider LM unit root testing in a model with linear trend. We derive asymptotic distribution theory (in a new fashion), as well as accurate appropriate critical values. We undertake Monte Carlo investigation of finite sample properties of SP-SL LM tests, along with applications to the Nelson & Plosser (1982) time series and real quarterly UK GDP.     

Embodied Technological Change and Tests of the Internal-Adjustment-Cost Hypothesis

Standard unit-root and cointegration tests are sensitive to atypical events such as outliers and structural breaks. In this article, we use outlier-robust estimation techniques to examine the impact of these events on cointegration analysis. Our outlier-robust cointegration test provides a new diagnostic tool for signaling when standard cointegration results might be driven by a few aberrant observations. A main feature of our approach is that the proposed robust estimator can be used to compute weights for all observations, which in turn can be used to identify the approximate dates of atypical events. We evaluate our method using simulated data and a Monte Carlo experiment. We also present an empirical example showing the usefulness of the proposed analysis.     

储蓄、投资与中国资本流动——基于面板协整分析

在对Feldstein—Horioka之谜及其相关研究的基础上,利用中国的省级数据,通过建立、估计和检验面板协整模型,分析各区域之间的资本流动关系。研究结果表明：中国东部地区的资本流动性最强,中部地区的资本流动性次之,西部地区资本流动性最差;东部地区各省份以资本净流人为主,西部地区各省份以资本净流出为主。因此,要提高资本的自由流动程度,尤其要引导资本向中西部地区流动,加速中国经济的一体化进程,缩小不同区域经济发展的差距。     

Quasi-likelihood ratio tests for cointegration,cobreaking, and cotrending

We consider a set of variables with two types of nonstationary features, stochastic trends and broken linear trends. We develop tests that can determine whether there is a linear combination of these variables under which the nonstationary features can be canceled out. The first test can determine whether stochastic trends can be eliminated and thus whether cointegration holds, regardless of whether structural breaks in linear trends are eliminated. The second test can determine whether both stochastic trends and breaks in linear trends are simultaneously removed and thus whether cointegration and cobreaking simultaneously hold. The third test can determine whether not only breaks in linear trends but also linear trends themselves are eliminated along with stochastic trends and thus whether both cointegration and cotrending hold.     

A residual-based test of the null of cointegration in panel data

This paper proposes a residual-based Lagrange Multiplier (LM) test for the null of cointegration in panel data. The test is analogous to the locally best unbiased invariant (LBUI) for a moving average (MA) unit root. The asymptotic distribution of the test is derived under the null. Monte Carlo simulations are performed to study the size and power properties of the proposed test.

overall, the empirical sizes of the LM-FM and LM-DOLs are close to the true size even in small samples. The power is quite good for the panels where T ≥ 50, and decent with panels for fewer observation in T. In our fixed sample of N = 50 and T = 50, the presence of a moving average and correlation between the LM-DOLS test seems to be better at correcting these effects, although in some cases the LM-FM test is more powerful.

Although much of the non-stationary time series econometrics has been criticized for having more to do with the specific properties of the data set rather than underlying economic models, the recent development of the cointegration literature has allowed for a concrete bridge between economic long run theory and time series methods. Our test now allows for the testing of the null of cointegration in a panel setting and should be of considerable interest to economists in a wide variety of fields.     

Testing purchasing power parity: some evidence of the effects of transaction costs

Empirical tests of purchasing power parity often recognize the problems created by simultaneous equations, but seldom recognize the effects of measurement error or transaction costs. Presumably because most researchers believe that they are unimportant. We present evidence that shows that measurement error and transaction costs and create serious econometric problems for testing purchasing power parity. One effect of these problems is that conventional tests of purchasing power parity can accept PPP when predictive errors are relatively large and reject it when predictive errors are relatively small. Another effect is to bias test of cointegration toward accepting the null of no cointegration between exchange rates and relative price indexes. We also construct a simple model of the determination of exchange rates that shows how transaction costs lead to regression switiching.     

A joint test of fractional cyclic integration and a linear time trend

We propose in this article a joint test for testing simultaneously a deterministic trend component and the degree of integration of the cyclical component in a given time series. The test is directly derived from Robinson's (1994) procedure, which is based on the Lagrange Multiplier (LM) principle. Thus, it has standard null and local asymptotic distributions. However, finite-sample critical values of the tests are evaluated and, an empirical application using historical annual data, is also carried out at the end of the article.     

Testing linearity against nonlinear moving average models

Lagrange multiplier (LM) test statistics are derived for testing a linear moving average model against an asymmetric moving average model and an LM type test against an additive smooth transition moving average model. The latter model is introduced in the paper. The small sample performance of the proposed tests are evaluated in a Monte Carlo study and compared to Wald and likelihood ratio statistics. The size properties of the Lagrange multiplier test are better than those of other tests.     

Testing for cointegration at any frequency using spectral methods

Summary In this paper a new simple test for cointegration at any frequency is presented. This method can thus be applied to test for cointegration both at the zero and at the seasonal frequencies. It requires the estimation of the coherency spectrum of weakly stationary processes, therefore only standard spectral theory is involved. The testing procedure is similar to the one suggested by Phillips and Ouliaris (1988) and recently generalized by Joyeux (1992) to frequencies different from zero, but it does not suffer of some problems connected with the use of principal components methods in the frequency domain. Invited paper at the Conference held in Bologna, Italy, 27–28 May 1993, on ?Statistical Tests: Methodology and Econometric Applications?.     

Testing for multivariate heteroscedasticity

In this article, we propose a testing technique for multivariate heteroscedasticity, which is expressed as a test of linear restrictions in a multivariate regression model. Four test statistics with known asymptotical null distributions are suggested, namely the Wald, Lagrange multiplier (LM), likelihood ratio (LR) and the multivariate Rao F-test. The critical values for the statistics are determined by their asymptotic null distributions, but bootstrapped critical values are also used. The size, power and robustness of the tests are examined in a Monte Carlo experiment. Our main finding is that all the tests limit their nominal sizes asymptotically, but some of them have superior small sample properties. These are the F, LM and bootstrapped versions of Wald and LR tests.     

On the usability of the fluctuation test statistic to identify multiple cointegration break points

The fluctuation test suggested by Hansen and Johansen [Some tests for parameter constancy in cointegrated VAR models, Econometrics J. 2 (1999), pp. 306–333] intends to distinguish between the presence of zero and one break in cointegration relations. In this article, we provide evidence by Monte Carlo simulations that it also serves as a graphical device to detect even multiple break locations. It suffices to consider a simplified and easy-to-implement version of the original fluctuation test. Its break detection performance depends on the sign of change in cointegration parameters and the break height. The sign issue can be approached successfully by a backward application of the test statistic. If breaks are observable, the break locations are detected at the true location on average. We apply the graphical procedure to assess the cointegration of bond yields of Spain, Italy and Portugal with German yields for the period 1995–2013 which is surprisingly supported by the trace test. However, the recursive cointegration approach shows that a stable relationship with German yields is only present for sub-periods between the introduction of the Euro and the global financial crisis which is in line with expectations. The statistical robustness of these results is supported by a forward and backward application of the cointegration breakdown test by Andrews and Kim [Tests for cointegration breakdown over a short time period, J. Bus. Econom. Stat. 24 (2006), pp. 379–394].     

A monte carlo study of the wald,lm, and lr tests in a heteroscedastic linear model

In this paper, we examine by Monte Carlo experiments the small sample properties of the W (Wald), LM (Lagrange Multiplier) and LR (Likelihood Ratio) tests for equality between sets of coefficients in two linear regressions under heteroscedasticity. The small sample properties of the size-corrected W, LM and LR tests proposed by Rothenberg (1984) are also examined and it is shown that the performances of the size-corrected W and LM tests are very good. Further, we examine the two-stage test which consists of a test for homoscedasticity followed by the Chow (1960) test if homoscedasticity is indicated or one of the W, LM or LR tests if heteroscedasticity should be assumed. It is shown that the pretest does not reduce much the bias in the size when the sizecorrected citical values are used in the W, LM and LR tests.     

Applying estimated score tests in econometrics

The estimated score test is a hypothesis testing procedure that can improve on the standard score, or Lagrange multiplier (LM) test. The score for a parameter of interest will usually contain nuisance parameters. Essentially, the test replaces nuisance parameters by estimates and then takes the difference between this estimated score and its expectation as the critical region for a test. If the expectation is zero the test coincides with the standard score, or LM, test, but if the expectation is non-zero the small sample properties of the tests differ. In some cases even asymptotic properties differ. This paper examines the scope for the application of estimated score tests in econometrics and illustrates with examples. Comparisons with the standard tests emphasize differences between the tests in terms of the true, as distinct from nominal, sizes of tests.     

The Comparison of Performances of Widely Used Cointegration Tests

The aim of this study is to compare performances of commonly cointegration tests used in literature in terms of their empirical power and type I error probabilty for various sample sizes. As a result of the study, it has been found that some tests are not appropriate in testing cointegration in terms of empirical power and type I error probability. As a result of simulation study, λ_max test for any values of ρ and sample sizes have been found most appropriate test in conclusion.