to give misleading results. The possible presence of cointegration must be taken into account when choosing a technique to test hypotheses concerning the relationship between two variables having unit roots (i.e. integrated of at least order one). The usual procedure for testing hypotheses concerning the relationship between non-stationary variables was to run ordinary least squares (OLS) regressions on data which had been differenced. This method is biased if the non-stationary variables are cointegrated. For example, regressing the consumption series for any country (e.g. Fiji) against the GNP for a randomly selected dissimilar country (e.g. Afghanistan) might give a high R-squared relationship (suggesting high explanatory power on Fiji's consumption from Afghanistan's GNP). This is called spurious regression: two integrated series which are not directly causally related may nonetheless show a significant correlation; this phenomenon is called spurious correlation. Tests The six main methods for testing for cointegration are: Engle–Granger two-step method If and are non-stationary and order of integration d=1, then a linear combination of them must be stationary for some value of and . In other words: where is stationary. If we knew , we could just test it for stationarity with something like a Dickey–Fuller test, Phillips–Perron test and be done. But because we don't know , we must estimate this first, generally by using ordinary least squares (by regressing on and an intercept) and then run our stationarity test on the estimated series, often denoted . A second regression is then run on the first differenced variables from the first regression, and the lagged residuals is included as a regressor. Johansen test The Johansen test is a test for cointegration that allows for more than one cointegrating relationship, unlike the Engle–Granger method, but this test is subject to asymptotic properties, i.e. large samples. If the sample size is too small then the results will not be reliable and one should use Auto Regressive Distributed Lags (ARDL). Phillips–Ouliaris cointegration test Peter C. B. Phillips and Sam Ouliaris (1990) show that residual-based unit root tests applied to the estimated cointegrating residuals do not have the usual Dickey–Fuller distributions under the null hypothesis of no-cointegration. Because of the spurious regression phenomenon under the null hypothesis, the distribution of these tests have asymptotic distributions that depend on (1) the number of deterministic trend terms and (2) the number of variables with which co-integration is being tested. These distributions are known as Phillips–Ouliaris