This paper is forthcoming in the Journal of Monetary Economics. I am grateful to an anonymous referee for very helpful comments on earlier drafts and to Morgan Stanley and Co., Kenneth Kasa, and David Cutler for providing some of the data used in this study. Thanks also to Peter Cornelius, Graham Elliott, Albert Jaeger, Mark Klock, Karen Parker and Heidi Willmann Richards for comments, and Shahbaz Khan for computing advice. The views expressed in this paper are those of the author, and do not necessarily reflect those of the International Monetary Fund.
To be specific, there is an upward bias to the test statistics as the number of degrees of freedom approaches zero, because the canonical correlations between the Xt and AXt (corrected for the lagged differences) will approach unity even if the series are not cointegrated (in which case they should approach zero); see also Hall (1991).
Cheung and Lai (1993) suggest equations for small-sample critical values that are approximately given by CR∞*[0.1 + 0. 9*T/(T-nj)], implying that the Reinsel-Ahn scaling factor overcorrects slightly. This conclusion is based on their exclusion of cases of very few degrees of freedom. As is shown in Section 3b, the Reinsel-Ahn correction may actually undercorrect in such cases.
A simple example illustrates how cointegration of stock prices (without dividends) might be theoretically feasible in an efficient market. Assume that managers believed there is an optimal and changing trading price range for their stock where the stock will attract the greatest investor interest. Managers might use dividend policy to keep the stock price within this range, paying very high dividends when earnings are high, and omitting the dividend if the price fell below the desired range. The prices of unrelated stocks might thus appear to be cointegrated around this range, though it is most unlikely that their total returns indices would also be cointegrated. Of course, dividends do not show the level of volatility that would be required in this example (or a related example in Dwyer and Wallace, 1992), so cointegration of stock prices may not be especially likely in practice.
The following analysis is implicitly carried out in a common currency. For assets denominated in different currencies, the unexpected return could be disaggregated into its exchange-rate and domestic-currency components; this would simply add a third cumulated error term to equation 6.
A cursory search located nearly 20 recent journal articles testing for cointegration between national market indices, with a substantial majority claiming to find some evidence for cointegration. Three caveats might be noted. First, many papers conduct a large number of tests, and the highlighted rejections of the null are often only slightly more frequent than implied by the size of the tests, which is often set higher than five percent; this suggests a publication bias towards papers that find, rather than fail to find, cointegration. Second, rejections are more common in multi-country tests using the Johansen methodology without a small-sample correction. Third, an economic interpretation of the estimated stationary vectors is rarely provided. How, for example, should one interpret the following near-stationary vector in the five-country dataset studied in this section: (1.20*United States + 2.00*United Kingdom - 0.21*Canada -0.65*Germany - 0.98*Japan)?
The Akaike and Schwarz-Bayes information criteria both suggest only one lag in the VAR. Using the Sims likelihood-ratio test with a five percent significance level, one cannot reject successive reductions from j-10 until the restriction of j-3 is rejected against j-4, though this rejection may be spurious; the restriction of j-1 is not rejected against j«4 at conventional significance levels.
By simply reshuffling the innovations here and in the remainder of the paper, all simulations impose zero coefficients on all but the first lag in the VAR. This is based on the absence of ARCH effects in these quarterly data, the weak evidence for a higher-order VAR (see footnote 7), and the fact that the data are financial market data which should be close to random walks under an efficient markets null. Alternative simulations suggest that estimates of the small-sample bias are little affected by this assumption. Furthermore, Monte Carlo (with normal innovations) and bootstrapped simulations (sampling with replacement) yield very similar results to the results shown.
The Reinsel-Ahn scaling factor is equal to 8.14 (i.e. 57/(57-5*10)); the simulations suggest scaling factors for the null of r«0 of 8.8 (trace statistic) and 10.6 (maximal eigenvalue). If a further lag had been added to the VARs, there would have been 56 explanatory variables and 56 observations, rendering tests of cointegration impossible.
Data for two series calculated by Morgan Stanley since 1969--Belgium and Singapore/Malaysia–were not available and were not included in the study.
Similar results (available upon request) are obtained for both simple (not excess) return indices and hedged domestic-currency return indices, where the latter were proxied by assuming the currency risk on a foreign equity position is offset by a short position in a foreign short-term interest-bearing security.
An alternative approach is to test for pairwise cointegrating relationships between the 16 countries in the sample. Based on the Engle-Granger test, the 120 country pairs yield only 6 rejections of the null of no cointegration at the 5 percent level, and 18 rejections at the 10 percent level.
Tests using data ending in December 1986 suggest, however, that these results are not driven by the rise and fall of Japanese stock prices in the late 1980s and early 1990s.
Monte Carlo simulations show that when innovations for all countries have a similar mean, winner-losers tests have a zero expected return, so that there is no bias corresponding to the autocorrelation bias in regression tests. Simulations using bootstrapped series yield slightly wider confidence intervals than the randomized series but yield similar conclusions on statistical significance.
Similar results indicating winner-loser reversals are also obtained using data for a group of emerging markets (see Richards, 1996b).
The implied autocorrelation coefficients for the five horizons are: +.07 (1 year),-.16 (2 years), -.25 (3 years), -.31 (4 years), and -.18 (5 years).
Indeed, Loughran and Newbold (1995) conclude from an empirical study that “It is perhaps overly cynical to suggest that applying cointegration tests to financial markets data is a superb mechanism for generating random numbers. Nevertheless, the inconsistencies in results reported in applied studies … suggest extreme caution in inferring structural conclusions from the test statistics.”