Chapter 14: Cointegration

Introduction

Test for a Cointegration Relation in the Bivariate Case

Cointegration Test Using PROC VARMAX for Two Price Series

Cointegration Tests in a Five-Dimensional Series

Initial Estimates for the β Values

A Model with Rank 2

Use of the RESTRICT Statement to Determine the Form of the Model

Stock-Watson Test for Common Trends for Five Series

A Rank 4 Model for Five Series Specified with Restrictions

An Alternative Form of the Restrictions

Estimation of the Model Parameters by a RESTRICT Statement

Estimation with Restrictions on Both the α and β Parameters

Conclusion

Introduction

In this chapter, you will learn how to test for the existence of error correction relations among multivariate time series. In statistical terms, such relations are denoted cointegrated relations. The setup for such problems is discussed in the situation involving more than just two series as in the example in Chapter 13, which involved with only a bivariate series. The underlying notation that explains this naming is presented in the next section.

The existence of error correction relations is often seen as underlying scientific “laws” for the series in question. The idea is fundamental in economic science where long-run relations are considered as equilibria, such as a market equilibrium, which represent something of global and everlasting importance. In other sciences, a long-run relation could be an empirical established relation between the temperature measured on the American East Coast and in Europe. Or it could be the relations among prices of some commodities that in some way move in parallel.

The question is whether such relations can be established by the data. A theoretically postulated long-run relation in some scientific fields might be supported by testing whether the relation is accepted for a data set. In this way, cointegration testing is a tool to test scientific hypotheses.

If a relation is found, it can lead to new thoughts about the structure of the underlying mechanisms. In this way, cointegration can be seen as an empirical driver for scientific development. In more practical situations, a long-run relation among the sales of various goods on a market can improve business decisions. The search for cointegrated relations is in this sense a search for fundamentals of observed time series with important implications.

The concept of cointegration in this chapter is like the theory of error correction models and is mainly demonstrated by examples. The underlying theory is outlined in the next section, but without many mathematical details. The main part of this chapter is devoted to applications of the VARMAX procedure to cointegration testing in practice.

One problem is that the test statistics are mathematically involved and rely on advanced matrix algebra. Moreover, the distributions of the test statistics are formulated as stochastic integrals, so they are not standard distributions as, for instance, chi-square distributions. The theory is, of course, important, and the reader is referred to the rich literature. See, for instance, Juselius (2006) or Johansen (1995).

Test for a Cointegration Relation in the Bivariate Case

For a simple example, consider two time series x_t and y_t. In Chapter 13, the error correction model was estimated by the assumption that a linear relation was present as a long-run relationship between the two series. This stability should be seen in contrast to the original series x_t and y_t which individually could include trends or be drifting, having changing means.

In more precise mathematical terms, stability is expressed by saying that the series

β₁y_t + β₂x_t + β₀

is stationary. Here, the coefficients are numbered and ordered in order to cope with the mathematics. Moreover, a coefficient β₁ is written to the y_t series.

The stationarity of this linear combination of the two series is in contrast to the original series x_t and y_t not being stationary but instead being stationary only after a differencing. See Chapter 4. The original series x_t and y_t are assumed to be integrated of 1. The linear combination of the two series and perhaps a constant or linear term are denoted a cointegration relation if the error series in the linear relation has no unit root.

In the bivariate situation, it is obvious that only a single cointegration relation is possible. The other possibilities are none or two. No cointegration relations means that all linear combinations between the series are integrated of order 1, which is often the case if no stable relation exists. The existence of two independent, stationary, linear relations between the series implies that both series are stationary, which is in contrast to the assumption that they both are integrated of order 1.

The number of cointegration relations is usually called the cointegration rank and is denoted r. For the bivariate case, the only interesting situation is that this rank could be r = 1.

In order to test the existence of a cointegration relation, first the assumption of a unit root has to be tested—that both series should be differenced in order to be stationary. This test situation was the subject of Chapter 5. It is easily performed by PROC VARMAX using the table with the Dickey-Fuller test statistics. If this test accepts that both series are integrated of order 1, then the interesting question is whether one or no cointegration relation exists.

PROC VARMAX performs the Johansen trace test for this hypothesis. The test is based on the trace of a particular matrix, and the actual formula is rather involved. Moreover, the distribution of the test statistic is nonstandard because it is expressed as integrals of Wiener processes. The reader is referred to the literature for details (for example, Johansen, 1995, or Juselius, 2006). However, the test is easily seen from a user’s perspective because PROC VARMAX simply presents the p-values for the test statistics.

Cointegration Test Using PROC VARMAX for Two Price Series

In this section, the existence of a cointegration relation between the price series for potatoes in Ohio and Pennsylvania is tested by PROC VARMAX. In Chapter 13, such a cointegration relation was simply assumed because it is intuitively evident that the price of potatoes in two neighboring American states should be related. Moreover, the plots, Figure 13.1 and Figure 13.2, clearly documented the relation. So the test in this section is in some way superfluous. Anyway, the test is useful as an example before more complicated examples emerge in later sections of this chapter.

Program 14.1 gives the simple code to perform the test. The COINTEST=(JOHANSEN) option to the MODEL statement is enough. Remember the parentheses! In Program 14.1, an AR(4) model is applied as in Chapter 13. But, otherwise, all unnecessary options are omitted.

Program 14.1: Testing for a Cointegration Relation by the Johansen Test

The test results (Output 14.1) indicate rejection of the hypothesis of no cointegration relations, p = .003, so it is concluded that at least one relation is present. The hypothesis of one cointegration relation is accepted, p = .74, toward the alternative hypothesis that more relations exist. The conclusion is that the analysis using the error correction model in Chapter 13 was valid.

Output 14.1: The Johansen Cointegration Trace Test for a Bivariate Series

The p-values presented in Output 14.1 are only supported by SAS/ETS 13.2 and newer. Up to version 13.1 of SAS/ETS, the value of the test statistic was presented together with a critical value for a test on a 5% level. The test level could be changed to 1% or 10% by an option. The user then had to conclude the test by comparing the test values to the critical values.

Cointegration Tests in a Five-Dimensional Series

In this section, a five-dimensional series of potato prices for five American states is studied in order to find cointegrating relations. The data set POTATOES_YEAR includes price series for all American states where 148 observations are available. For all these states, yearly observations from 1866 to 2013 are published. Many of these states are geographically in the Northeastern United States, and it is expected that the prices are similar as trading costs are smaller than for states having longer distances.

In Program 14.2, the price series for five states in the Northeast are tested for cointegration by PROC VARMAX.

Program 14.2: Cointegration Test for a Five-Dimensional Series

In Program 14.2, normalization as in Chapter 13 by setting one of the β coefficients to equal 1 is left out. A normalization seems unnatural because it is possible that some of the long-run relations include only a few states and not all five states. So it is not possible to guess which variable to use in the NORMALIZE option.

The Dickey-Fuller tests for unit roots again support that each of the five series are all integrated of order 1, which means that cointegration is a relevant issue for this five-dimensional time series. Output 14.2 presents the cointegration test output generated by Program 14.2.

Output 14.2: The Johansen Trace Test

The hypothesis of Rank 1 is rejected (p = .01), while the hypothesis of Rank 2 is accepted having p = .21. These tests are in both situations toward the alternative of a larger cointegration rank. The conclusion from Output 14.2 is that the cointegration rank is at least 2.

Another version of the Johansen cointegration test exists, the maximum eigenvalue test. This test applies a test statistic, which is the maximal eigenvalue of a relevant matrix instead of the trace of the matrix. The trace is actually the sum of all eigenvalues, so these two versions of the test are in some aspects closely related. The maximal eigenvalue test statistic is calculated by a further option to the JOHANSEN option in Program 14.2:

COINTTEST=(JOHANSEN=(TYPE=MAX))

The output is presented in Output 14.3. The conclusion that a cointegration rank r = 2 suffices is similar to the conclusion using the trace test in Output 14.2. It seems that there is no need to consider both test versions because the information using only the default trace test is sufficient.

Output 14.3: The Johansen Maximum Eigenvalue Test

The actual specifications of the long-run relations are more difficult when two relations are present than when only one relation is present as in the bivariate case considered in Chapter 13. The problem is that the parameterization is uniquely defined only up to rotations. In this context a rotation means that if, for instance, two long-run relations are present, then any linear combination of these two relations is also present.

The log-likelihood value is also printed to output for SAS/ETS 14.1 of August 2015 and onwards. The value is printed under the heading “Multivariate Diagnostics” in the Results window. This log-likelihood value could be used to test for various modifications of the model that are not tested in the default procedure output. For the model in this section the log-likelihood value is 965.68. This value of the log-likelihood is for the most general model and the values for the various restrictions to this model in the following sections are of course smaller.

Initial Estimates for the β Values

PROC VARMAX prints initial estimates for the long run β parameters as a part of the cointegration test output, see Output 14.4, which is generated by Program 14.2. In Program 14.2, no normalization is used, so the user has to look for numerical large β values, which means that β values close to zero could possibly be neglected. This inspection is not helped by standard deviations, so the formulation of hypothetical long-run relations is in some sense pure guesswork. However, the analyst has to remember the data context. For this particular data example, it is relevant to search only for states for which potato prices are identical in the long run.

Output 14.4: Initial Estimates for the β Values

The second column gives the clearest picture. The coefficients for Delaware and Maryland are both close to 16 but with different signs. But all other values are numerically less than 1. This corresponds to a long-run relation of the form

−16x_1t + 16x_2t ≈ 0

between the two series. This is, of course, the same as an identity between the Delaware and Maryland price series when the relation is normalized.

The first column seems to be more involved. One possibility is to regard the numerical small values for Delaware and Maryland as 0. In this case, the coefficients to Ohio and Virginia add up to the coefficient to Pennsylvania with the opposite sign. This leads to a relation like

8x_3t + 8x_4t − 16x_5t ≈ 0

between the three series if the coefficients to Ohio and Virginia are considered equal.

When normalized, this relation states that the Pennsylvania price equals the average of the Ohio and the Virginia price

$$x_{5t}\approx \frac{1}{2}{x}_{3t}\text{+}\frac{1}{2}{x}_{4t}$$

Alternatively, it could be a weighted average if the coefficients to Ohio and Pennsylvania are considered unequal.

A Model with Rank 2

In Program 14.3, a model with two cointegration relations is fitted. Moreover, the hypothesis that the two relations have the form as specified above is tested by the COINTEG statement. The H matrix is in this situation a 5 ×2 matrix. The number of rows in the H matrix is the dimension of the time series, which is 5 in this example. The number of columns in the H matrix is the number, here 2, of long-run equations.

Program 14.3: Estimation of Two Long-Run Relations for Five Series

The log-likelihood value is reduced from 965.68 in the previous general model to 953.79 for the present model. This reduction in log-likelihood is due to the reduction from rank r = 5 to rank r = 2. The hypothesis stated by the H matrix in the COINTEG statement is not included in this log-likelihood value. As the actual number of restrictions imposed by the reduction of the cointegration rank from 5 to 2 is unclear, a direct test is not possible. Moreover, the distribution of a likelihood ratio test is possibly not standard when restricting β parameters.

The H matrix is printed (Output 14.5) to control the specification in the code.

Output 14.5: The H Matrix Used in Program 14.3

The test accepts the hypothesis that the two relations have the form as specified by the H matrix in Program 14.3; see Output 14.6. The test gives p = .18.

Output 14.6: Test Results for Hypothesis on the β Parameters

The cointegration part of the model is in this five-dimensional example written by matrices as

$$\Delta y_t=\left(\begin{array}{c}\Delta y_{t1}\\ \Delta y_{t2}\\ \Delta y_{t3}\\ \Delta y_{t4}\\ \Delta y_{t5}\end{array}\right)=\alpha {\beta }^T\left(\begin{array}{c}\Delta y_{t1}\\ \Delta y_{t2}\\ \Delta y_{t3}\\ \Delta y_{t4}\\ \Delta y_{t5}\end{array}\right)+\mathrm{...}$$

$$\alpha =\left(\begin{array}{cc}{\alpha }_{11}& {\alpha }_{12}\\ {\alpha }_{21}& {\alpha }_{22}\\ {\alpha }_{31}& {\alpha }_{32}\\ {\alpha }_{41}& {\alpha }_{42}\\ {\alpha }_{51}& {\alpha }_{52}\end{array}\right),\beta =\left(\begin{array}{cc}{\beta }_{11}& {\beta }_{12}\\ {\beta }_{21}& {\beta }_{22}\\ {\beta }_{31}& {\beta }_{32}\\ {\beta }_{41}& {\beta }_{42}\\ {\beta }_{51}& {\beta }_{52}\end{array}\right)$$

This parameterization by the matrices α and β is not unique. This is easily seen as αβ^T = αMM^-1β^T for every regular 2 ×2 matrix M.

A hypothesis stated for the β parameters has the form of a parameterization of the β matrix, using a smaller number of parameters. In the present situation, the hypothesis is that the β matrix is of the form β = Hφ as

$$\beta =\left(\begin{array}{cc}{\beta }_{11}& {\beta }_{12}\\ {\beta }_{21}& {\beta }_{22}\\ {\beta }_{31}& {\beta }_{32}\\ {\beta }_{41}& {\beta }_{42}\\ {\beta }_{51}& {\beta }_{52}\end{array}\right)=H\phi =\left(\begin{array}{cc}1& 0\\ -1& 0\\ 0& 0.5\\ 0& 0.5\\ 0& -1\end{array}\right)\left(\begin{array}{cc}{\phi }_{11}& {\phi }_{12}\\ {\phi }_{21}& {\phi }_{22}\end{array}\right)=\left(\begin{array}{cc}{\phi }_{11}& {\phi }_{12}\\ -{\phi }_{11}& -{\phi }_{12}\\ 0.5{\phi }_{21}& 0.5{\phi }_{22}\\ 0.5{\phi }_{21}& 0.5{\phi }_{22}\\ -{\phi }_{21}& -{\phi }_{22}\end{array}\right)$$

The estimated β parameters reflect this parameterization using φ's. The two individual long-run relations are not presented because the zeros in the H matrix are absent in the β matrix; see Output 14.7. But the presented parameter values expressed by the φ's are just linear combinations of the β parameters in the two specified long-run relations.

Output 14.7: Estimated β Parameters Subject to the Restrictions

The α parameters are of minor interest in this situation. The printed α estimates are the parameter values when the printed β estimates are used. If a linear combination of the printed β values is used instead—for instance, with the series as imposed by the matrix H, the α parameters change by the inverse transformation. Such linear combinations are obtained by a rotation of the α and β matrices.

The factorization αβ^T is not unique as αβ^T = αMM⁻¹β^T for every regular matrix M. In the situation of the hypothesis β = Hφ, we have αβ^T = αMM⁻¹β^T  = αMM⁻¹φ^TH^T  saying that the estimated α and φ matrices are not unique. In the estimation algorithm, a particular identification by an M matrix is applied, and this does not necessarily lead to interpretable output. This is also the reason for the nonzero entries in the estimated β matrix.

Use of the RESTRICT Statement to Determine the Form of the Model

The RESTRICT statement was applied in Chapter 13 to formulate a model with specific constraints on the α and β matrices. In this section, the RESTRICT statement is used to find a model, which is similar to what the user wants and is not just a rotated version of the β matrix.

In the previous section, a specific form of the β matrix was formulated by an H matrix in Program 14.3. The code in Program 14.3 produced a useful test statistic which accepted the hypothesis. See Output 14.6. But the printed β parameters did not follow the form that was specified by the H matrix. It is possible to find β parameters that meet the restrictions as formulated by the H matrix by the RESTRICT statement.

Program 14.4 includes a RESTRICT statement for the β parameters. The number of restrictions is very large. The last line in the RESTRICT statement specifies the five zeros in the β matrix.

The two stable relations are specified in the first line.

First, the relation

x_1t + x_2t ≈ 0

is stated. This relation is equivalent to the restriction on the β’s in

β₁₁ + β₂₁ =  0

for the Delaware and Maryland price series.

Second, the relation

x_3t +x_4t + 2x_5t ≈ 0

is stated. This relation is equivalent to

β₃₂ + β₄₂ + 2β₅₂ = 0

for the Ohio, Virginia, and Pennsylvania price series.

The second line is a normalization setting where two β’s equal 1. This makes the printed estimates easier to understand, but the normalization does not in any way change the overall fit of the model.

Program 14.4: A RESTRICT Statement for the β Parameters

The estimated β matrix now has the form we wanted. See Output 14.7.

Output 14.8: Estimated α and β Parameters

The first relation between the potato prices in Delaware and Maryland seems well established with large α parameters. But the second relation is more dubious. The effect seems to be large for the three involved data series and largest for the Virginia price. This means that the Virginia price series gives the largest contribution for establishing the long-run relation if the relation is away from equilibrium. In the section “A Rank 4 Model for Five Series Specified with Restrictions,” a more elaborate model is estimated.

Stock-Watson Test for Common Trends for Five Series

All five series for potato prices in different states in the US are plotted in Figure 14.1. The five series move in parallel mainly because they all have an upward trend that is of the same form for all series. This is, of course, intuitively obvious. But the plot of the five series can also reveal that some of the series occasionally drift away from the other for periods of some years. This means that even if the basic shape of all five series is the same, some departures exist.

Figure 14.1: Plot of All Five Series for Potato Prices

PROC VARMAX includes the Stock-Watson test for common trends. The idea is that many series—say five as in this example—vary around a smaller number of trends. In this example, a single trend could be the correct number of trends as judged from Figure 14.1. The test is performed by the SW option to the COINTEST in the MODEL statement as in Program 14.5.

Program 14.5: Testing for Common Trends and Estimating Four Restrictions

The table with the test results for the Stock-Watson tests for common trends is shown in Output 14.9. This table shows that Rank 3 supports the alternative hypothesis that fewer trends are present. This means that one common trend for the five series is not enough, but rather that three trends are necessary. The hypothesis that even four trends are present is rejected for the alternative hypothesis that three trends suffice. But Rank 4 is accepted toward the alternative of Rank 2.

The reason for these rather messy test conclusions is probably that the test has as few restrictions as possible (a small value for the rank, r) as the null hypothesis. The test-power for rejecting the hypothesis is too small for rejection that the rank is only 2.

Output 14.9: Stock-Watson Test Results

A Rank 4 Model for Five Series Specified with Restrictions

If the intuitive idea that the five potato price series move in parallel is correct, there ought to be four stable restrictions among the five series. One possibility is to suppose that the four stable relations are stated by saying that all differences between two series among the five series are stationary. This means, for instance, that the following four differences are stationary.

x_1t − x_2t,  x_2t − x_3t,  x_3t − x_4t,  x_4t − x_5t

A β matrix for this assumption in a Rank 4 model could be specified by a RESTRICT matrix as in Program 14.6. This takes a large number of individual restrictions. First, the four stable relations are specified. Then the normalizations letting four coefficients equal 1 are listed. And at last, all the zero entries in the β matrix are specified as in a total of 12 identities.

Program 14.6: Identification of the β Matrix by Many Restrictions

The estimated α and β matrices are presented in Output 14.10. The β matrix is seen to be of the right form. But the α parameters are messy. The only substantial parameter values are for the Delaware versus Maryland stable relation, saying that the prices in these two states move in parallel.

Output 14.10: Estimated α and β Parameters in the Rank 4 Model

An Alternative Form of the Restrictions

In the previous subsection, 4 pairs were selected randomly out of the five series. Then the cointegrating relations were expressed using the differences between these pairs of series. In this subsection, an alternative idea is followed.

If the price for potatoes in one state for some reason is out of balance compared to the price in the other four states, the price has to move toward the average of the prices in the other four states. For State No. 1 (in this setup it is Delaware), this could be formulated as follows:

β₁y_t1 + β₁₂y_t2 + β₁₃y_t3 + β₁₄y_t4 + β₁₅y_t5 ≈ 0

Normalizing by letting β₁ = 1, this relation states that

y_t1 ≈  − β₁₂y_t2 − β₁₃y_t3 − β₁₄y_t4 − β₁₅y_t5

For simplicity, the weighted average on the right side of this relation could be taken as a simple average by taking all coefficients equal to .25:

y_t1 ≈  .25 y_t2 + .25 y_t3 + .25 y_t4 + .25 y_t5

The hypothesis is that such a relation exists for the first four series. The similar relation for the fifth series is established by summing the first four series.

Estimation of the Model Parameters by a RESTRICT Statement

Program 14.7 gives the code for estimation of  the β matrix exactly in the form specified by the four stable relations by a long RESTRICT statement.

Program 14.7: Estimating the Model Parameters by a RESTRICT Statement

In Program 14.7, all restrictions on the β matrix are stated in the RESTRICT statement even if some of them are implied by the remaining restrictions. In the log window, this is noted by the remark “NOTE: Deleted redundant linear constraints”.

The output (Output 14.11) is promising, as the coefficient to the highlighted variable. E. g., the value  α₁₁ = −.39 to the first variable, that is the Delaware price, in the first line of the α matrix, is numerically larger than the other coefficients in the same row and column. The negative sign of α₁₁ is expected as discussed later.

Output 14.11: Estimated β and α Matrices by Program 14.7

This formulation of the cointegration model is written as four equations in which only two decimal places are used.

For Delaware there are three similar terms with coefficients close to 0 + AR terms:

y_t,1 ≈  y_t−1,1 − .39( y_t−1,1 − .25y_t−1,2 −.25y_t−1,3 −.25y_t−1,4 −.25y_t−1,5 ) +

.10( y_t−1,2 − .25y_t−1,1 −.25y_t−1,3 −.25y_t−1,4 −.25y_t−1,5 ) +

The first term says that if the Delaware potato price at time t − 1 is larger than the average of the potato prices in the other four states, the price in Delaware will lower from time t − 1 to t by .39 times the difference between the Delaware price and the average price. This tendency is very intuitive.

The other negative α's, α₂₂, α₃₃, and α₄₄ also indicate that if the price in a particular state is larger than the average of the price in the other four states at time t – 1, then the price will lower from time t − 1 to t.

The next term in the Delaware equation says that if the Maryland price is higher than the average price in the other four states (including Delaware), then the Delaware price will tend to increase by a factor α₁₂= .10 times the imbalance. This is also intuitive, but of negligible magnitude. One large effect of this kind is found for the coefficient α₂₁ = 0.21, which is the coefficient to the effect of an imbalance of the Delaware price to the Maryland price. This coefficient says that an imbalance for the Delaware price (the first series) at time t - 1 has an effect for the Maryland price at time t.

In total, the estimated α and β parameters in Output 14.11 give four such equations. After some rearranging and normalizations, the sum of these equations is a similar equation for the fifth series, the Pennsylvania price series.

Estimation with Restrictions on Both the α and β Parameters

The reported standard deviation for the estimated α parameters are, however, very large. All the α's—even  the negative coefficients α₁₁ = .−39 that were discussed above—are insignificant. The output table presenting these standard deviations is too long to be presented here. The insignificance could be due to the huge number of parameters. Program 14.8 gives the code for restricting α_ij = 0 for all i ≠ j in order to reduce the number of free model parameters. All these α_ij parameters could be 0 according to intuition.

Program 14.8: Estimation with Restrictions on Both the α and β Parameters

The table of estimated α and β parameters including their significance is also too large to be printed here. The estimated α₁₁, α₂₂, α₃₃, and α₄₄ are all significantly negative as they should be according to intuition. The values are all close to −.4.

The table of restrictions is also very long because total 35 restrictions are applied on the model. Output 14.12 presents only the α parameter part of this table. No test statistics are printed for the restrictions on the β parameters.

Output 14.12: Tests for Restrictions on the α Parameters

Some of the restrictions on the α parameters are rejected by the tests in Output 14.12. This means that the model could be improved by estimating these parameters. But as a total of 16 restrictions on the α parameters are tested, it is quite possible that the significance of a few α parameters could be spurious. Only the parameter α₂₁ is strongly significant. The sign of the reported test statistic for α₂₁ is positive, which corresponds to the positive value α₂₁ = .21 in Output 14.11. This sign of the coefficient is intuitive as explained in the discussion below Output 14.11.

Conclusion

This chapter demonstrates how to test for cointegration relations in two practical examples. First, there is the conclusion from Chapter 13 that the potato prices in Ohio and Pennsylvania move in parallel in the sense that the hypothesis that their difference forms a stationary series is accepted.

In the second example, the dimension is increased so that potato prices in five states in the United States are studied. For this five-dimensional series, various specifications of the models are tested and estimated, leading to the conclusion that the five series move in parallel.