Incorrect! If non-stationary data are used in levels form, a phenomenon known as "spurious regression" may result. This will not cause the coefficients to be estimated wrongly (they will still be unbiased, so d will not be a consequence). But a problem will arise with the measured strength of the relationship and with the standard error estimates. If x and y are completely independent non-stationary series, a regression of y on x could lead to a spuriously high R-squared (the R-squared should be zero if x and y are unrelated), and the actual distributions of the test statistics will be much flatter (will have fatter tails) than the assumed distributions (the t- or F-distributions). This is likely to result in invalid inferences, and in particular, a higher probability of type I error than the significance level employed.

Incorrect! A shock is just another word for the disturbance. If a process is a stationary autoregressive one (e.g. the equation in 2, but replacing the parameter "1.5" with some value between -1 and +1), then when the series is expressed as an infinite sum of past shocks, the series will be converging. That is, shocks further in the past will have less impact on the current value of y than more recent shocks. Thus the effect of a given shock will eventually die away to zero as time passes for a stationary autoregressive process.

Incorrect! This model is clearly a deterministic trend model, since it includes the "lambda t" deterministic trend term. Such a process will have a straight line underlying trend that will slope upwards (downwards) if lambda is positive (negative). It is clearly not a stationary process since it has a trend (unless lambda = 0!). Neither is it a unit root process or a random walk with drift since there is no lagged term in y on the right hand side.

Incorrect! If a series is integrated of order 2, this implies by definition that it contains exactly two unit roots, and that differences would have to be taken twice to obtain a stationary process. It is also the case, however, that if the series is differenced more than twice (e.g. 3 or 4 times), the resulting series will still be stationary. This is termed "overdifferencing", and may lead to some undesirable properties in the resulting series, but it will still be stationary. d is not a plausible model for the series since no value of phi could be used in this equation to generate an I(2) process. Even setting phi = 1 would only create a process containing one unit root. The data generating process for an I(2) variable would have to be more complex, including a two-period lag and a one-period lag of y on the right hand side.

Incorrect! The Dickey-Fuller test will only be "well-behaved" - i.e. appropriately sized with reasonably large power - if the residuals from the test regression are "white noise". This means that they must be completely random, and in particular, that they must be free from autocorrelation. Augmenting the test regression involves adding lagged values of the dependent variable (lags of the changes in y). Thus, the purpose of augmenting the test regression is to soak up any autocorrelation that may be present in the dependent variable so that the residuals are free from autocorrelation. Note that the coefficient values on the lags of the change in y are of no interest whatsoever, we simply allow for them in the same way that lags may be used in any regression model to soak up the dynamic structure in y.

Incorrect! The reason that you do not need statistical tables to answer this question is that the test statistic is so far away from the rejection region that the conclusion is indisputable! Recall that the null hypothesis of a unit root in y would be rejected if the test statistic is more negative than the critical value. The critical values are typically about -3 for a 5% significance level, so clearly the test statistic will be in the non-rejection region since it isn't even negative! Thus the null hypothesis would be rejected, and we would say that y contains a unit root. In fact, the null hypothesis would still have not been rejected here if y had contained more than one unit root, so therefore strictly the appropriate conclusion is that y contains at least one unit root. In order to test whether y contains one unit root or more than one, we would have to "move up a gear". This would involve regressing the second difference of y on the lag of the first difference, as in question 11.

Incorrect! Again, the test statistic is sufficiently extreme that the appropriate conclusion is obvious without statistical tables. The critical values for Dickey-Fuller type regressions are typically of the order of -3 for a 5% significance level, so clearly in this case the result is a rejection of the null hypothesis. But what was the null hypothesis? The null is that y has at least 2 unit roots. Since it is stated in the question that this test in question 10 is thought necessary and is being conducted after a test for one unit root as in question 9, the alternative hypothesis would be that there is one unit root. Thus the conclusion would be that the series y contains exactly one unit root, i.e. that it is I(1). If a test of whether the series y is I(1) had not been conducted previously, it is possible that y could be I(0), i.e. that it contains no unit roots and is therefore stationary.

Incorrect! If the 2 variables x and y are individually I(1), clearly they cannot be stationary so that (i) is wrong. Although in general there can be more than one independent linear combination of I(1) variables that are stationary, the maximum number of linearly independent cointegrating combinations is given by "n -1" where n is the total number of variables in the regression equation (y and all of the x's). Since there are only two variables here (y and x), there can only be at most one independent linear combination of the variables that is stationary. The cointegrating equation describes the long run behaviour of the series. This can be seen by calculating the long run static equilibrium solution to the error correction model (ECM), which will result in everything except the cointegrating terms disappearing, so that the latter must describe the long-run relationship. The first-differenced terms will describe the short-run relationships between the series. So (iii) is wrong. Finally, by definition, for there to be cointegration between y and x, the residuals from the cointegrating regression must be stationary. So (ii) and (iv) are correct.

Incorrect! The null hypothesis for the Engle-Granger test is that the residuals from this regression are non-stationary. That being the case, a stationary linear combination of the variables has not been found so that under the null hypothesis, the variables are not cointegrated. For the variables to be cointegrated, we require that the null hypothesis from the Engle-Granger regression to be rejected.

Incorrect! The gamma terms in the cointegrating part of the equation do indeed describe the long-run relationship between y and x, as described above, since it is only these terms that will be left when the long run static equilibrium solution to the model is calculated. Clearly, then, they cannot also describe the short-term relationship between y and x. Hypothesis tests cannot validly be conducted on the gamma terms since these terms will have been estimated using the cointegrating regression. Even if the residuals from this regression are stationary (which we can assume they are since the term is present in an ECM), they are likely to have strong positive autocorrelation since the cointegrating regression is a completely static one. As a result, it is not valid to conduct hypothesis tests on the gamma terms of the cointegrating relationship. It is, however, perfectly valid to conduct hypothesis tests on the betas since all of the elements attached to a beta in the ECM regression given in the question are stationary.

Incorrect! Both the ADF and the PP approaches are based on estimation of a single unit root testing equation, and they are therefore not systems methods where several equations are estimated at the same time. Both the ADF and PP tests have a unit root under the null hypothesis, and are thus unit root tests rather than stationarity tests. Thus, the ADF and PP tests are very similar - in fact, the only difference is that the PP test incorporates an automatic correction for autocorrelated residuals in the test regression. Therefore PP tests are also likely to have low power to reject the unit root null hypothesis in small samples.

Incorrect! Whilst all of (a) to (d) are legitimate criticisms of the Dickey-Fuller/Engle-Granger approach to testing for unit roots and modelling cointegrated systems, only (d) is actually addressed by Engle-Yoo. Specifically, EY have all steps in common with EY for the formulation of an error correction model, but then EY adds a third step in the procedure which provides an updated estimate of the cointegrating relationship and the standard errors within this relationship. This then allows us to validly conduct hypothesis tests concerning the actual cointegrating relationship.

Incorrect! This matrix is of reduced rank (that is, it is not of full ranks) since it has one zero eigenvalue and its rank is 1 (it would have to be of rank 2 to be described as being of full rank). Clearly, it is not a zero matrix, which would mean that all its elements would be zero. A matrix with non-zero determinant would have to have to be of full rank - that is, it would have to have all of its eigenvalues non-zero. The determinant of a matrix is the product of its eigenvalues, and so this matrix has zero determinant. A matrix that is of reduced rank is also known as a singular matrix, and therefore c is correct.