TESTING FOR CO-INTEGRATION

Transcription

1 Bo Sjö TESTING FOR CO-INTEGRATION To be used in combination with Sjö (2008) Testing for Unit Roots and Cointegration A Guide. Instructions: Use the Johansen method to test for Purchasing Power Parity in Mozambique Think about how you can convince the reader that your decision is OK, or at how you can invite the reader to a meaningful discussion about your conclusions. Hand in Dec 22, 2010 at 17:00. Lab on Dec 20 will deal with further testing of the cointegration vector, testing for Granger causality and impulse responses (a form of dynamic simulation). Lab on Jan 5 will deal with ARCH/GRACH models. 1. Purchasing Power Parity In this lab you will learn to test for cointegration. The problem is to test if purchasing power parity (PPP) holds for Mozambique. PPP is a basic relation, which states that the domestic price level in an open economy if not determined, so in a parity condition with prices in the rest of world. Thus, domestic price level (P d ) is given by the foreign price (P f ) multiplied with the exchange rate (E), P d = P f E, Where the exchange rate E is defined as number of domestic currency units per foreign currency unit. Alternatively, the relation can be rewritten as the real exchange rate (q), q = [P f E] / P d This relation will hold exactly if all goods are equal, there are no trade barriers etc. There is a discussion about the proper way to define the prices used in the parity conditions; CPI, Whole Sale Price Index, unit labour cost, etc. Tests of PPP have shown that the relationship is at most a long-run relationship, years. Since price levels and exchange rates are typically integrated variables, it is natural to test if the series follows the same stochastic trend in the long-run. The most typical test is to make a bivariate test, and use CPI or WSPI. In this lab the data series are, 1

2 The variables are Lp: ln CPI for Mozambique Le: ln exchange rate (metical/rand) Lpsa: ln South Africa industrial production price. The economic relation (PPP) to be tested is Lp = Le + Lpsa, In linear econometric model form, after adding stochastic deviations, Lp t = β 0 + β 1 Le t + β 2 lpsa t + e t, If PPP holds in it most restrictive form β 1 = β 2 = 1.0, and the residual term in e t is stationary. We can allow β 0 to capture given constant differences over time. Since price indexes and exchange rates typically are integrated variables of order one, it is not possible to test these restrictions, or PPP, using ordinary t- or F-tests, since under the null of integrated variables, the estimated parameters follow non-standard distributions. Our questions are therefore. - What is the order of integration of each variable? - Does the order of integration match among so that PPP forms a meaningful economic relationship for this data? - Is the linear combination of variables formed by (Lp t, Le t, Lpsa t ) forming a stationary relationship (=Do the integrated variables cointegrate so that they form a linear stationary relationship)? - And, if they are cointegrating, is it then also possible to assume (impose) that β 1 = β 2 = 1.0? 1 The question is if these variables form a stationary relation (a stationary real rate), and if the parameters are those suggested by theory? We can also ask which variables are predicted by the vector, this will indicate if the (perhaps) stationary vector is driving the exchange rate, domestic prices, or both. The significance and sign of the alpha parameters, the error correction parameters will tell you this) Data is kept in the EXCEL file: lcoint.xls All data are on a monthly basis. You might want to confirm that the series are I(1), use graphs and unit root tests. In this lab we don t bother about seasonal effects, since they are not clearly visible, and the sample period is quite short, 10 years implies only 10 observations for each seasonal dummy, which is quite small. Build a VAR and find the appropriate lag length. Test the model for misspecification. (There might be number of problems, but no autocorrelation is top priority here) 1 Which implies a vector of cointegration parameters β = [1, -1, -1] and x t = [Lp, Le, Lpsa] 2

3 Use Johansen s method to test for cointegration. Are there any significant stationary (cointegrating) vectors? Johansen's method will get you a number of eigenvectors. The number of significant eigenvalues translates into a number of eigenvectors and Beta variables. The eigenvectors, representing cointegrating relations must be identified in some way. The firs step is to normalise the eigenvector arond some variable. Typically econometric programs will do this automatically in some way around the first coefficient (the first variable in the model). This must be considered when setting up the system, but might not be what the data is telling you. Hence it might be good to look at the raw eigenvectors prior to any normalisation. Eviews will normalise the vector on the first variable in the system. (Can this be done in eviews: Look at estimated Beta-vectors. Can you identify a PPP/real exchange rate vector in any vector?) Estimate again but with rank =1 imposed, (Pick Cointegrated VAR, after Estimation of VAR under Model Settings). You will get a Beta vector and an alpha vector (if r=1). If we have a stationary relation, the estimated alpha coefficient will have an asymptotic normal distribution and ordinary t-test are possible. Look at the alpha values, and their significance (estimate /stand error). What are the cointegrating vectors representing? Which of the variables are predicted, in this reduced system? Notice that once you change anything here the results will change. (As an example, identify a residual outlier, in an equation with problems in the diagnostic tests, put in a dummy and redo the cointegration test) 2. Introduction and Objectives to the Lab. Most economic time series are integrated or near integrated. Because of this inference regarding parameter significance cannot be based on standard distribution like normal, t-, F- or chi-square distributions. The way around this problem is to test for integration and cointegration in order to transform the model into stationary relations. Objectives: This lab will introduce you to testing for cointegration using Johansen s VAR. Let {x t } be a vector process of x t ' = [x 1,t, x 2,t ], where both elements are integrated of order one, {x t } I(1). Our interest is linear equations of the type. x 1,t = a + bx 2,t + u t or u t = x 1,t - a - bx 2,t Cointegration implies that there are parameters such that a linear combination of the integrated non-stationary variables form one (or more) stationary relation(s). A linear combination of integrated variables can integrated of the same order as the individual elements or is cointegrating having a lower order of integration. Furthermore, regressing two or more integrated variables against each other leads to a problem of spurious regression. 2 In the regression x 1,t = a + bx 2,t + u t the variances of a and b follows unknown distributions under the null that the variables are integrated. To test if b is significant, indicating a relation between the variables is to test for cointegration. 2 Students are supposed to know the "spurious regression problem" and explain its causes. 3

4 3. Testing for Cointegration 3.1 Johansen's method. Johansen's method is a multivariate method based on a VAR representation of the stochastic process. Once the VAR has been formulated, you can determine the number of significant eigenvalues (0 number of cointegrating vectors) in the system. However, two things are important. 1) Find the optimal lagt structure. 2) Make sure the residuals are normally distributed white noise. The method requires normal distribution in the residuals, which often necessitates the use of some dummies. Eviews does not fully understand this, and is thus not giving the support diretly needed to formulate the VAR. The test can be described as a multivariate form of the ADF-test for unit roots. It is advisable to start from a general model, with many, lags and later reduce the VAR by eliminating insignificant lags downwards, as far as possible without destroying the assumption of white noise variables. 3 The estimated VAR is (1) xt = A( L) xt 1 + ΨDt + µ + ε t, where D t represents a vector of dummy variables, µ is a vector of constant and ε is a residual vector. The VAR is constructed by choosing a common lag length across the variables and equations such that the it becomes impossible to reject the null that the residuals are normally white noise variables. (This can be tricky in real life, because by adding a large number of lags there is no degrees of freedom left to test the null. We would like to have as few lags as possible, say 2 lags on each variable, plus perhaps a few dummies. A large part of the literature discusses how to choose the optimal lag length in a VAR.) The estimated VAR can be rewritten as, (2) x t = l i= 1 Γ x i t i + Πxt 1 + ΨDt + µ + εt, where Πx t = αβ x t-1 represent the stationary cointegrating relations, where the Beta parameters are the cointegrating parameters that forms linear stationary relations with the non-stationary data series in y t. Please notice the similarity with Equation (2) and the Dickey-Fuller regression. If all x:s are I(1), they must be stationary in first differences. It follows that the only way for the Π-matrix not be filled with only zeros is that the variables cointegrate and form stationary I(0) relations. We can test if the matrix contains rows which are different from zeros by testing the rank of the matrix. If 3 If seasonal effects are in the data, and there is sufficient number of observations, add (centered) seasonals to the model. To get the best results estimate only complete years Jan-Dec. 4

5 we find a reduced rank, it follows that there must be stationary I(0) combinations among the variables such that there are non-zeros in the matrix. 4 We test carefully all aspects of white noise residuals. If there is autocorrelation add more lags. However, in most cases 2 lags will do fine, sometimes in combination with some dummy variables. You can also inspect the estimated residuals in order to identify huge outliers that can be eliminated with impulse dummy variables. Create Impulse dummies for a specific observation. As soon as you include step dummies, the critical values must be re-simulated. (They will increase in general) You should also try to formulate a model with white noise errors, in all respects. This is however quite difficult. In this lab, we will be satisfied with no autocorrelation at the 5% risk level. 3 Perform the cointegration test Depending on the number of equations in the VAR you get a number of estimated eigenvalues. Under the null of no cointegration these eigenvalues have a non standard distribution. 5 An n- dimensional system gives n estimated eigenvalues. The program keeps track of the relevant critical values for your model, and gives you probability values. With five variables in the VAR you get five eigenvalues to test. The test is done in a specific order, from the largest to the smallest. You test for significance until you can longer reject the null, at that point you have to stop. After the point where you cannot reject the null hypothesis, the following test statistics are no longer valid for the given critical values. (The Pantula Principle) The first null is that there are no stationary relations in the data. If the probability value is not below say the eigenvalue is not significant. No significance means that the null of no stationary relations (=no cointegration) is not reject. The test is over, and we conclude that there is no cointegration. In case of rejection of the null of no stationary relations, the conclusion is so far, that there is at least one cointegrating vector (r=1). We proceed to test eigenvalue number 2. The null is now only one stationary relation (cointegrating vector). The alternative is that there are more than one cointegrating vectors. In the case of no rejection of the null the test is over. We conclude that there is one cointegrating vector (r=1). If the null is rejected, the conclusion is at least two cointegrating vectors (r=2), and we might proceed to test the next eigenvalue, etc. The program gives you various outputs, other than the eigenvalue test, but this can be neglected at this stage. Once the number of cointegrating vectors has been determined, we can return to the model formulation, reestimate the model under the restriction that there are, say r cointegrating vectors. We will then estimate a VECM with only stationary variables, and we will learn about αβ x t-1. At 4 You should be able to set up the equation, explain what the variables stand for, how to build the VAR, and what you are testing for! 5 Johansen (1995) and Sjö (2008) give the Tables of critical values for the different types of model. 5

6 this point we only know that there are stationary relations among the data series, we don t know what they represent yet. What can go wrong in the test? We need normally distributed white noise residuals for the test. The test is an asymptotic test, which can be sensitive how we formulate the model in a limited sample. The test assumes that there are no structural breaks in the data. If we put a stationary variable, of course, the test will give at least one stationary cointegrating vector. This vector will consist of the stationary variable. The judgment of the modeler plays an important role. 5. Estimate and test the cointegrating vectors in Eviews. In Eviews after loading the data create a workfile, a new objective, select a (a VAR). Put in the names of the variables in the VAR system. The order of the variables is important for understanding the output. Define the number of lags, say 4. The programme will start with 2 lags only. Test the residuals. Under View pick residual test. If the residuals log ok, in this case no autocorrelation. It is with this data difficult to get normal distribution in the residuals. Next under view go to cointegration test, if the model is ok. The trace test is presented at the top of the result. The output gives you the matrixes of adjustment parameters (α) and the beta parameters that form the stationary steady state relations among the variables. In matrix form. Π x t-1 = αβ' x t-1. The estimated Beta vectors you see in the output, are like the e t in the two-step model. All variables (and there parameters) have been shifted to the right hand side in the VECM model. x t =. + α [β 1 Lp t + β 2 Le t + β 3 lpsa tt ] It is up to you to decide which variable you would like to put as the left hand variable in the steady state formed by the cointegrated vector(s). There is also a constant term in the vector (β 0 ), if you estimated a VAR with an UNRESTRICTED constant term, but it is included in a constant term of each equation. The parameters in the beta matrix represent a space of values of the parameters. Our job is to economically identify the relationships. The beta vectors, in combination with the variables, can be understood as the residual term in the two-step procedure. The first thing is to normalize the vector, decide which variable is being determined. Eviews does a (preliminary) normalization of all Beta vectors using the diagonal of your input matrix. The program simply assumes that you start with your most important variable, which you will normalize around, just like in the two-step procedure. Of course, this might turn out to wrong. Once the cointegrating rank is imposed on the model, the estimated alpha parameters have a standard distribution. Their significance can be tested using ordinary t-statistics. At this stage you 6

7 have a statistical representation of your data in the form of an VECM, and you have a general economic interpretation of your estimated long run (if it exists). 5.1 Graph the cointegrating vectors The graphics menu can graph all cointegrating vectors. Make graphs of the estimated vector. Under Test, Graphic Analysis indicate Cointegrating relations and (1) pick Use (Y: Z) and next use (Y_1:Z) with lagged DY and U removed. You will get two graphs of the estimated vector(s). The first is β x t the second is β x t-1 y t-i, D t,. It is the second equation/graph which is stationary in the estimated model. Conditional on the lag length and deterministic variables (including trends). 5.2 Test the cointegrating parameters Given the estimate VECM model, which we now have. We can impose restrictions on alpha and beta, and ask if the restrictions are consistent with the data and the model. These tests will have a chi-square distribution under the null. In particular we might be interested in testing for specific assumptions about the Beta values, if they are unity, minus unity or zero. In this case we would like to test the parameter vector [1, -1, -1]. Impose the restricted vector, study the test statistic. The Null is that the imposed vector is included in the space spanned by the Betas, the alternative that it is not implying non-stationarity of the vector. You can use this test to test if individual variables are I(0) or I(1). The null is I(0) in this case. You can also test quite complex restrictions, but this will require some more skills beyond this lab. Warning! (1) The degrees of freedom in complex tests are hard to calculate for a computer. You need to learn how to do this by hand (takes some practice). (2) You are looking at a reduced form model. All you say and test for the Betas hold irrespectable of the structural model you might formulate from this reduced system. Questions regarding alphas on the other hand depend on how you condition the model in the future. Thus alphas can change value and significance if the model is respecified from this point, i.e. you introduce other (stationary) explanatory variables or formulate a structural model. 6. What to do in this lab Use the Johansen method to test for cointegration (= test of long-run PPP, or a long-run stationary real exchange rate.) Think about how you can convince the reader that your decision is OK, or how you can invite the reader to a meaningful discussion about your conclusions. Test if PPP holds in this sample. Which variables are predicted by the cointegrating vector(s)? Does PPP hold for Mozambique? What is your economic interpretation of the estimated cointegrating vectors? 7

8 The continuation for self studies. You can test if the variables are stationary I(0) or non-stationary I(1). You can test for specific values of beta parameters. You test which variables, all or only some should be included in the vector to achieve stationarity. You can test the significance of the alpha variables to see which variables are actually driven, or at least predicted by the vectors. You can also test for I(2) relations conditional on the assumption that the model is a correctly specified I(1) system. (Quite difficult for the beginner) Finally, you can specify a system of error correction equations. All tests have a standard distribution, once the number of cointegrating vectors (cointegrating rank) has been imposed on the model. The alpha and the beta matrices can be tested. The easies test to start with is the general restriction test. Be aware of the degrees of freedom problem though. First run Engle and Granger s two step procedure, as a quick but dirty method for testing cointegration. The Two-step procedure Engle and Granger developed a simple, but not so good test in 1987 called the two-step procedure. The test is done by first running the co-integrating regression using OLS, and then test if the residual in the estimated equation comes out being is stationary. If it is stationary, it indicates that there is a stationary cointegrating relationship. Run the following model by OLS, from 1992:01: Lp t = β 0 + β 1 Le t + β 2 lpsa t + e t, Inspect the estimated coefficients, do they make sense? Take out and save the residual, from the Test menu. Do an ADF-test on the residuals with say 6 lags in the augmentation. Find the correct critical value for this test. (Use my guide) YOU CANNOT USE THE DICKEY- FULLER DISTRIBUTION HERE. THE ADF CRITICAL VALUES DO APPLY HERE, BECAUSE THE RESIDUAL IS A DERIVED VARIABLE WITH LESS DEGRESS OF FREEDOM THAN A SINGLE SERIES. There is problem with the data. There is a clear structural break that causes problems for the test. One way of checking if the structural break matters is to re-estimate the model and include a step dummy for the period up to 1996:06, when the inflation rate goes down. The results of this model can be compared with the previous model. BUT, the distribution of the critical values needs to simulated for this particular model. The old ones do apply when I introduce a step dummy like this into the model. Here I have to use my judgment, look at the results and judge if the test statistic increases with some margin, and then form an opinion. And, if I think the process is stationary with reasonable parameter values I can use the error term as a error correction term, in an error correction model. However, this is not a regular test though. 8

9 Relevant critical values are found in Engle and Granger (1987), Banerjee et.al (1992) and Sjöö (2004). In the example given here the number of variables in the model is n=2. Critical values are simulated for ADF(1) and ADF(4) models. If the ADF test, in the second step, is significant the null of e t being I(1) is rejected. The conclusion, if rejection is that we cannot reject that the variables in the model are cointegrating. Critical values have been estimated for equations with up to five variables (n=5). 6 If the estimated e t is regarded as stationary it can be used as an error correction expression in a single equation error correction model. Engle and Granger, 1987, and Johansen (1987) proof the "representation theorem stating that cointegration implies an error correction representation. Form an ADF equation with the estimated residual. Again, notice that the Critical values for the test of integration are not valid in a cointegration test. There are two big problems here (1) the test is asymptotically consistent. Hence, we use it as a guide for our modeling rather than as a test of cointegration. (2) If there are more than two variables, the test is not telling us which variables are actually cointegrating. We only know that the dependent variable is cointegrating with at least one of the right hand side variables. 6 For n=1 the test becomes an ordinary ADF test of a single integrated variable. 9