TESTING FOR CO-INTEGRATION PcGive and EViews 1

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1 Bo Sjö Lab 3 TESTING FOR CO-INTEGRATION PcGive and EViews To be used in combination with Sjö (203) Testing for Unit Roots and Cointegration A Guide and the special instructions below for EViews 2. Hand in by January 7, 7:00 Instructions: Use the Johansen method to test for Purchasing Power Parity (PPP) in Mozambique. Test against South Africa whole sale prices and US CPI. Start the model and testing from 997:04, perhaps some dummies are needed but perhaps not, you simply have to try. 3 Whether or not PPP holds depends on how you ask the question: i) Does the PPP relation form a stationary vector?, ii) does the PPP relation form a stationary vector with unit elasticities?, iii) is the real exchange rate stationary? The latter is basically the same as question ii). Present the results in such a way that you convince the reader that you tests and final conclusions are accurate. You should present relevant material and just not copy and past indiscriminately from the program. (See also questions at the end of this paper). Introduction: Purchasing Power Parity In this exercise 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 (S), P d = P f S, Where the exchange rate S 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 S] / 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 See Appendix for more detailed instructions. 2 PcGive is the better program of the two (compared with EViews), alternatively you can use CATS in RATS, or gretl. 3 After testing for the shorter period, you can try the longer period.

2 Sale Price Index, unit labour cost, etc. Tests of PPP have shown that the relationship is at most a long-run relationship, 0-20 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 use CPI or WSPI. In this exercise the data series are, The Data is in lcoint22.xls The variables in logs are Lp: ln CPI for Mozambique Lssa: ln exchange rate (metical/rand) Lpsa: ln South Africa industrial production price Lsus: ln exchange rate metical/usd Lpus: ln US CPI The economic relations (PPP) to be tested are Lp t = Lssa t + Lpsa t, and Lp t = Lsus t + lpsus t. In linear econometric model form, after adding stochastic deviations, Lp t = β 0 + β Ls t + β 2 Lpsa t + e t, PPP holds if this is a stationary form and in it most restrictive form then β = β 2 =.0. However, the data series are I(), therefore normal inference is not possible, and nothing says that PPP must hold in the short run. If prices and markets adjust slowly we might expect a dynamic relationship, which is not the way in which the equation above is set up. If the data is non-stationary, the first and most basic test is whether the three variables form a stationary relation (vector) with parameters that can be understood as a PPP relation. The latter means we that we have to look at the signs and sizes of the estimated parameters. Thus, if PPP holds as long-run relationship we expect that Lp, Ls and Lpsa form a long-run stationary cointegrating vector so that they together form a I(0) stationary relation. (Or in other words that they share a common stochastic trend). Our questions are therefore: - What is the order of integration of each variable? - Does the order of integration match among the variables such that a PPP relation is possible, and therefore make sense to test for? - Is the linear combination of variables formed by (Lp t, Ls t, Lpsa t ) forming a stationary relationship (=Do the integrated variables cointegrate so that they form a linear stationary relationship)? - If they are cointegrating, is it then also possible to assume (impose) that β = β 2 =.0, alternatively can we assume that β =, or β 2 =, or simply test if β = β 2? 4 Additionally, we can also ask which variables are predicted by the cointegrating vector(s), this will indicate if the (perhaps) stationary vector(s) in the data are driving the exchange rate, domestic prices, or both. The significance and sign of the alpha parameters, the error correction parameters, will tell you. 4 Which implies a vector of cointegration parameters β = [, -, -] and x t = [Lp, Ls, Lpsa] 2

3 Data is kept in the EXCEL file: lcoint22.xls All data are on a monthly basis. You might want to confirm that the series are I(), use graphs and unit root tests. In this exercise we don t bother about seasonal effects, since they are not clearly visible, and the sample period is quite short, 0 years implies only 0 observations for each seasonal dummy, which is quite small. 2. Testing for Cointegration 2. 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 (number of cointegrating vectors) in the system. However, two things are important. ) Find the optimal lag structure, and 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. The test can be described as a multivariate form of the univariate 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. 5 Of course, you must test if the assumption of white noise normally distributed residual holds as you reduce the model. The estimated VAR is k () xt = π i xt i + ΨDt + µ + ε t, i= where D t represents a vector of dummy variables, µ is a vector of constant and ε τ is a residual vector. The VAR is constructed by choosing a common lag length (k) across the variables and equations such that 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 an Vector Error Correction Model (VECM) as, 5 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. 3

4 k (2) x t = i= Γ x i t i + Πxt + ΨDt + µ + ε t, where Πx t = αβ x t- 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. If there is cointegration, the matrix Π will have reduced rank, the number of unique independent rows in the matrix will be less than the number of variables in the system. The VECM has a shorter lag length than the corresponding VAR. This is why the lag structure of the x t i : s is k- instead of k. (This is something EViews users must monitor and adjust this manually! PcGive and most other programs handle this automatically. Notice the similarity with Equation (2) and the Dickey-Fuller regression. If all x:s are I(), 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 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. 6 In a similar way to the ADF-test, the VAR can be set up with and without constant and trend terms. At the moment, we ignore this and use the standard test with unrestricted constant terms in the VAR. We must test carefully all aspects of white noise residuals. Rule of thumb: if there is autocorrelation add more lags! However, in most cases 2-4 lags will do fine, often in combination with dummy variables for outliers. However, in this exercise you cannot solve any problems with dummies for outliers. 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. In this exercise, we will be satisfied with no autocorrelation at the 5% risk level. We can say that we formulate the least bad model we can given this data. 3. Detailed Introduction to and Objectives of the Exercise 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 exercise will introduce you to testing for cointegration using Johansen s VAR. Let {x t } be a vector process of x t ' = [x,t, x 2,t ], where both elements are integrated of order one, {x t } I(). Our interest is linear equations of the type. x,t = a + bx 2,t + u t or u t = x,t - a - bx 2,t 6 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! 4

5 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. 7 In the regression x,t = a + bx 2,t + u t the variances of a and b follows unknown distributions under the null that the variables are integrated. In this regression with integrated variables it is not possible to test for significance of the b parameter using ordinary test statistics. The only way to proceed is to test for cointegrating, that do the variables share the same stochastic trend or not. If they do they will be correlated and for a linear steady state in the long run, and b can the said to be significant. Build a VAR and find the appropriate lag length and normally distributed white noise residuals. Test the model for misspecification. There might be number of problems, but no autocorrelation is top priority here. (THE DATA IN THIS EXERCISE WILL NOT LEAD TO INSIGNIFICANT TESTS FOR RESIDUAL NORMALITY) 8. Use Johansen s method to test for cointegration. Are there any significant stationary (cointegrating) vectors? Johansen's method will get you a number of significant eigenvectors in the Π matrix. The number of significant eigenvalues translates into significant eigenvectors and Beta variables. The eigenvectors, representing cointegrating relations must be identified in some way, to find Beta parameters. The first step is to normalise the eigenvector around some variable. Typically econometric programs will do this automatically in some way around the first coefficient (the first variable in the model). EViews and PcGive will normalise the vector on the first variable in the system, but will also present the raw vector without normalization. One the number of cointegrating vectors (r) in the VAR has been found. That value of r can be imposed on the model and the model rewritten in VECM form. In the VECM form In the VECM form all variables are stationary and the analysis usually continuous with testing the vectors. Estimate again but with rank = imposed. You are now looking at, from Equation (2) above, Πx αβ, t = ' xt which is now consisting of stationary relations (steady state, or cointegrating vectors) β. In the program, when you see the beta vector, you see all parameters on the right hand side of the equation. There you must ask which parameter should be given the value.0, and appear on the left hand side in a long run steady state, and which should be on the right hand side of the steady x t 7 Students are supposed to know the "spurious regression problem" and explain its causes. 8 The usual case is that by imposing a few dummies for outliers one can have few lags in the VAR, no autocorrelation and a normal distribution in the residuals. 5

6 state and thus change signs according to what they shoe in the cointegrating vector. (Look at the program out put and you will set how it works.) You will get a beta vector and an alpha vector (if r = ). 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 (estimated /stand error). The alpha parameters are associated with stationary variables (relations) so they can be tested with ordinary t-statistics. The question is, what are the cointegrating vectors representing? On this exercise/data is the vector representing a PPP relation? Does the signs and the size of the parameters make sense? Which of the variables are predicted (by the alpha parameters), in this reduced system? The solution of this model is sensitive to the choice of sample, and model specification. 4 Perform the cointegration test and the Pantula principle again 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. 9 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. In the Johansen test you only consider the so-called Trace-test, the Lambda-max (etc) is no longer used, see Johansen 996. With five variables in the VAR you get five estimated eigenvalues to test. The test must be done in a specific order, from the largest eigenvalue 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 test statistics that follows are no longer valid for the given critical values. (The Pantula Principle!) The first null hypothesis to be tested is that there are no stationary relations in the data. If the probability value is not below say 0.00 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 = ). We proceed to test the significance of eigenvalue number r = 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 = ). 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, re-estimate 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-. At 9 Johansen (995) and Sjö (20,... A Guide) give the Tables of critical values for the different types of models. 6

7 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 latter is absolutely necessary since the test relies on a full information maximium likelihood (FIML) estimation method. The test is an asymptotic test, which is typically quite sensitive to the sample and 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. The test outcome of the test is sensitive to the choice of constant and trends. Johansen offer five different model from no trends or constant to constants and trends in the cointegrating vectors. The first model is almost never used in practice. These models should also be tested in the correct sequence, from no trends and constant an upwards. (The Pantula principle). If the test shows no cointegration, or no meaningful results, additional variables and relations might be missing. In the end, it is about making judgments. Those who read Johansen and Juselius will see how they actively look for vectors that have an economic meaning interpretation. 5. Estimate and test the cointegrating vectors in EViews + Read the additional guide I have for EViews, and PcGIVE. 5. 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. 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- = αβ' x t-. 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 Lp t + β 2 Ls t + β 3 lpsa tt ] + e t 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. 7

8 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 offers a 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 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). 5.2 PcGive In PcGive, under Model, chose Multiple Equation Dynamic Modeling. You are then asked to set up a VAR, in levels. Indicate the variables and the lag length. All variables should have the same lag length. (See the box below in the window to the right) It is advisable to start from a general model, with many, lags and reduce the VAR by eliminating insignificant lags downwards, as far as possible without destroying the assumption of white noise variables. 0 Once you have found a VAR, white noise residual normal distribution, few lags according to information criteria. To you help, graph residual, ask for print-out of large residuals (Under further output). Create dummies for large outliers. Add seasonal if necessary. Analyse the lag structure under Test, Dynamic Analysis, and Lag structure analysis. Here you get significance of each lag, and AIC and SC information criteria, which will help in the specification of the model. Test for cointegration once you find a good VAR, Coint test is found under Test and Dynamic Analysis, I() cointegration test. More on PcGive k (2) x t = i= Γ x i t i + Πxt + ΨDt + µ + ε t, Lokk at equation 2 again and notice the similarity with Equation (2) and the Dickey-Fuller regression. If all x:s are I(), 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 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. 0 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. 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! 8

9 In your first model, all deterministic variables, like the constant, seasonals etc. should be marked as "unrestricted". Variables in the model can be given different statues, see menu to the left. Endogenous, repressor (=exogenous), unrestricted/restricted. Setting a deterministic variable to unrestricted means that it will be included in both the cointegrating relation(s) and in the Error correction model. If it is restricted the constant will only appear in the cointegrating relation in the ECM, which is a bit restrictive. Let the program choose the estimation method for you. Check the test statistic (test summary) of the equations, at the end of the output. Under Model, under Options, you can set the program to give you additional out put. Make sure that summary statistics are indicated. In the Johansen test we are very concerned with the properties of our estimated model. We test more 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. Look under Test and Graphic Analysis. Use the Calculator to 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 exercise, we will be satisfied with no autocorrelation at the 5% risk level. 5.3 Graph the cointegrating vectors The cointegrating vector can be graphed. There are two graphs. ) The raw vector as it is created by the beta vector multiplied with the x:s. 2) The raw vector conditional other variables in the VECM model. The latter is what the cointegrating vector look like in the model. EViews: To be written. PcGive: From the Graphics menu all cointegrating vectors can be graphed. Make graphs of the estimated vector. Under Test, Graphic Analysis indicate Cointegrating relations and () pick Use (Y: Z) and next use (Y_: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- 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). Saving the cointegrated vector as a variable. What you see in the graph(s) is the result of the cointegrating vector. You can calculate the vector and manually create an error correction term, or ask the program to do it. PcGive: Go to test menu, look under save residuals etc. The algebra for calculating the vector can also be found and pasted from, under Model, and Batch or Ox Batch Code. 9

10 EViews: If call the VAR model say myvar, and you tested for and put in the number of cointegrating vectors (r) in the VAR, then you can ask EViews to calculate the cointegrating vectors and put them in the Workfile as variables. The command is myvar.makecoint gcoint The cointegrated vectors will then appear in the Workfile data under the name gcoint. The latter name can any name. 5.4 Test the cointegrating parameters Given the estimated VECM, we can impose restrictions on alpha and beta parameters, and ask if the restrictions are consistent with a stationary vector(s). 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 [, -, -]. Impose the restricted vector, study the test statistic. The Null is that the imposed vector can be assumed to be included in the stationary space spanned by the Betas, the alternative is the restricted vector is not stationary which implies a rejection. And, in the latter case you reject that the restricted vector(s) is/are forming stationary relations. In the testing, you must impose the value of unity on one parameter, to normalize the vector around that variable, this is not counted as a restriction. Think about the left hand variable in a single equation steady state. After normalization you can impose restrictions to test on the other parameters. For one vector the program will be accurate in calculating the degrees of freedom of the test. You can use this test to test if individual variables are I(0) or I() as well. The null is I(0) in this case. This is done by restricting all other parameters, except the test variable, to zero in the vector(s). PcGive: Go to Model, re-estimate the VAR, impose the rank (=number of cointegrating vectors), Choose to estimate Cointegrated VAR, and then chose General restrictions. This will bring up a window where you can restrict your estimated alpha and beta parameters. Start by normalizing around one parameter in each cointegrated vector. This normalization was done automatically before. This normalization is not a restriction, and will not affect the of degrees of freedom for the following test. Use &x =.0, etc to restrict parameters. You can test quite complex restrictions, especially if you have two vectors, but this will require some more skills beyond this exercise. Warning! () 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 independently 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. 0

11 6. Instructions Specific Questions Use ADF-test to find out the order of integration of the variables. 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 for Mozambique in this sample. (Define what you mean with PPP, parameters equal to unity or simply a stationary relation?) Which variables are predicted by the cointegrating vector(s)? Does PPP hold for Mozambique? What is your economic interpretation of the estimated cointegrating vectors? Present the results in a way that is meaningful and convincing to the reader. The continuation for self studies 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. Finally, you can model a single equation error correction equation for inflation. Appendix: Testing for Cointegration in EViews - Instructions Read in the data Open Workfile Under Object chose New Object Next chose VAR In the window VAR Specification, under VAR Type, chose Unrestricted VAR In Endogenous Variables write in the variable names In this case: Lp Ls Lpsa Specify the number of lags in VAR in Lag Intervals for Endogenous. For 4 lags in the VAR you write 4 (WARNING NOT LOGICAL COOMAND STRUCTURE) Normally, you will try different lag structures here. Now, Estimate the model Test the model: Under View you will find Residual tests, since the residuals should be normally distributed white noise you have to test for that. In this case it is more serious than under ARIMA modeling. Test for autocorrelation, (Q and LM test, Normality (JB), ) If you use other programs developed in cooperation with Johansen, like CATS in RATS and PcGive, you will see more test options. Typically, there will be ARCH test and RESET test. In

12 addition LM, JB and ARCH will also be shown as vector tests as well, meaning that the residuals are pooled across the equations to represent the whole system. With this data you can only opt for no autocorrelation. The problems with non-normal residuals cannot be solved in this data. EViews has advantage for the VAR, which is the test for lag length in the VAR. Under View, chose Lag Structure and Lag Length Criteria. In the latter window you get various information criteria. However, these criteria are somewhat useful, but they are only optimal for those who want to forecast. In a cointegration test you are looking for a correctly specified model with normally distributed error terms with no autocorrelation with as few lags as possible, which is needed for the test. Typically, impulse dummies are needed to improve the model (Not with this data), which is also the reason why you should inspect the residuals for outliers. Once you find a good VAR representation for the data. You can test for cointegration. Under View chose Cointegration Test, use the standard option for testing, Model 3) with unrestricted constant. The other options are only suitable if Model 3) does not work (no cointegration. Model 2) or Model 4) might be alternative options. Notice: You do not bother about the Eigen value(lmax) max test. You only look at the Trace-test. Both EViews and Enders need to be updated on this subject. The so-called Eigenvalue max-test is not asymptotically correct, as discussed in Johansen (996). Never bother about the standard errors of estimated Beta vector variables. Once you decided on the number of cointegrating vectors you can look a study what the VECM looks like in terms of alpha and beta parameters. Next, the estimated vector can be tested. By imposing restrictions on the vector the test becomes a question if the vector is still stationary after the restrictions have been imposed. If the vector is still stationary the restrictions are valid. Under View, Cointegration Test and VEC Restrictions is it possible, but this is not advisable. EViews has too many peculiarities when it comes to cointegration models to use this option. Therefore: To impose restrictions, go to Estimate, under Basics indicate Vector Error Correction. NEXT AND THIS IS IMPORTANT REDUCE THE LAG LENGTH BY - IN THE LAG LENGTH FOR ENDOGENOUS. AS you go from the VAR to the VECM you drop one lag in the model. (all other programs do this automatically, but not EViews). Under Cointegrating impose the number of significant vectors. You can now estimate the VECM. Next back to Estimation and chose VEC Restrictions. (What EViews call VEC is called VECM in all other programs and textbooks.) B(r,) indicate the cointegrating vector number r, and parameter in that vector. Thus to normalize the vector around the first variable, in the first vector around unity B(,) = To set the second parameter in the first vector to say -0.55, write B(,2) = etc. To restrict an alpha value write, A(k,r) where k refers to the number of the alpha in cointegrating vector number r. 2

13 Remember that you are entitled to impose the value of.0, to normalize the vector and this is not counted as a restriction. With more than one vector things are more complex when it comes to imposing restrictions. You must normalize each vector but also identify the vectors, by imposing valid restrictions that separates the vector from each other. This can be done intuitively to some extent, but in the end one must calculate the correct degrees of freedom, which is a bit more complex exercise. 3