Summary

1. Unrestricted VAR models suffer from the problem of overparametriza-tion. The Bayesian version (BVAR) has been found to give better results and has a good forecasting record.

earlier) has only one unit root, the bivariate and multivariate tests will give the same results.

Another issue that arises in estimating the cointegrating regressions is the choice of the dependent variable. In the two-variable case, whether we regress on x or X on (and take the reciprocal of the regression coefficient) does not make any difference asymptotically, but it does in small samples. The issue is the same in the case of more than two variables. However, this problem does not arise if one is using the maximum likelihood (ML) method, as in the Johansen procedure (which we have only referred to, not discussed). The problem is similar to that of 2SLS versus LIML discussed in Chapter 9. The 2SLS estimates, which depend on the regression method are not, in general, invariant to the normalization adopted, whereas the LIML estimates, which depend on the ML method, are invariant.

While estimating cointegrating regressions, many of the problems that we often discuss in the case of the usual regression and simultaneous equations models (e.g., omitted variables, parameter instability due to structural change, outliers, multicoUinearity, heteroskedasticity, etc.) are often ignored, and the attention is concentrated on testing for cointegration, as if that were the ultimate objective of all analysis. Even if one is doing an analysis with 1(1) variables, many of these problems should not be ignored, and they do affect tests for cointegration as well.

One final issue is that of the long-run equilibrium economic relationships that the cointegrating regressions are supposed to capture. The earlier literature on partial adjustment models (discussed in Sections 10.6 and 10.7) was concerned with the estimation of long-run equilibrium relationships as well as the time lags involved in achieving equilibrium. In discussions of cointegration, the long-run relationships are estimated through static regressions and not much is said regarding the time lags required to achieve the equilibrium unless an ECM is also estimated. As argued earlier, a procedure of estimating both the long-run parameters and short-run dynamics jointly would be a better one and would also be in the spirit of the earlier discussions on the estimation of dynamic models. Also, given that the evidence in favor of unit root processes in most economic time series has been found to be fragile, preoccupation with cointegration as the sole vehicle for studying dynamic economic relationships is unwarranted. Estimation of standard ECM and VAR models with attendant diagnostics might lead to less fragile inference. The ECMs can be used to merge short-run and long-run forecasts in a consistent fashion.

Exercises

1. Explain whether the foUowing statements are true (T), false (F), or uncertain (U). If a statement is not true in general but is true under some conditions, state the conditions.

(a) An unrestricted VAR model gives bad forecasts because of the large number of lagged variables in each equation.

(b) The overparametrization problem in the VAR model can be solved by using Almon or Koyck lags for the lagged coefficients.

(c) The Bayesian VAR (BVAR) model introduces very unnatural restrictions on the parameters in the model. Hence it cannot be expected to give good forecasts.

(d) Unit root tests are all biased toward acceptance of the unit root null hypothesis.

(e) It is better to have the null hypothesis of stationarity with the alternative as a unit root rather than the other way around.

(f) The Dickey-Fuller tests are not very useful for testing unit roots. One should use the Phillips tests.

2. There have been many tests suggested for unit roots. In most of these tests, the null hypothesis is that there is a unit root, and it is rejected only when there is strong evidence against it. Using these tests, most economic time series have been found to have unit roots. However, some tests have been devised that use stationarity as the null and unit root as the alternative hypotheses. Using these tests, most economic time series have been found not to have a unit root. Some other limitations of unit root tests have also been discussed. The evidence of unit roots in economic time series appears to be fragile.

3. The theory of cointegration tries to study the interrelationships between long-run movements in economic time series. Most economic theories are about long-run behavior, and thus much important information relevant for testing these theories is lost if the time series is detrended or differenced, as in the Box-Jenkins approach, before any analysis is done.

4. Cointegration impUes the existence of an error correction model (ECM). It also implies some restrictions on the VAR model.

5. Tests for cointegration specify the nuU hypothesis as no cointegration. This is because the unit root tests have the null hypothesis of unit root. Some problems with this have been discussed.

6. Cointegration theory has been used to test the rational expectations hypothesis (REH) and the market efficiency hypothesis (MEH). The former rests on rejecting the hypothesis of no cointegration and the latter on acceptance of this hypothesis. The results of these tests are sensitive to whether we consider bivariate or multivariate relationships. For instance, x and may not be cointegrated, but X, y, and z may be cointegrated.

(g) The Box-Jenkins method of visual inspection of the correlogram in levels and first differences gives the same results as the Dickey-FuUer unit root tests.

(h) If x, and y, both have unit roots, the coefficient of , in the equation for X, and the coefficient of y,, in the equation for y, will both be close to 1 when we estimate a VAR model for these two variables.

(i) Unit root tests all have low power because the alternative is not well specified.

Cj) Unit root tests applied to macroeconomic time series have low

power because of changes in the structure of the economy, (k) When considering simultaneous equations models in 1(1) variables, we do not have to worry about simultaneity bias. We can estimate the equations by OLS and get consistent estimates of the parameters.

(1) If X, and y, are both 1(1) variables, since estimates of the regression coefficient are superconsistent, it really does not matter whether we regress x, on y, or y, on x,. (m) Cointegration implies Granger causality.

(n) Suppose that we have the following pth-order representation of a vector x, of random variables:

x, = A,x,„, + A2X, 2 + • • • + ApX,„p + e,

The rank of A gives the number of cointegrating relationships, (o) The representation in part (n) can be written as

, = , , , + • • • + „, , , + , + .

The rank of Bp gives the number of cointegrating relationships.

2. Explain how to construct hypotheses to test the following economic theories by cointegration tests.

(a) Market efficiency hypothesis.

(b) Purchasing power parity theory.

(c) Rational expectations hypothesis.

3. Consider the following hypotheses:

: a = 1 in , = + ay,, + , ((x 5 0)

,: a < 1 in Z, = (x + aZ, , + u, where Z = y, - , -

Show that the OLS estimator a of a in y, = + ay, , + , tends to 1 under ,.

Exercises 4 to 10 are similar except that they use different data sets. Students should select the data set and the appropriate question.

4. Using the data in Table 13.4, estimate a VAR model for C, and F, with one lag and two lags.

(a) Is the model with two lags better than the model with one lag? Use the AIC and BIC criteria (see Section 13.5). Also check for residual autocorrelations.