Stephen M. Miller TABLE 1. Tests for Stationarity DF Test ADF Test Levels lst-Differences Levels 1st-Differences variable(x) 6.75 3.461 In mia l.345 7.132* 0.115 3.143* In M2 2.182 5.136* 0.430 y 0.964 In P 0.517 3.974* l.652 6.875* 2.000 5.215* A. NOTES: The augmented Dickey- Fuller(ADF)test is based on the following Dx,=a+σx-1+∑ΦDx1-+e1 here D is the first-difference operator, and e, is a stationary lull hypothesis is that r, is a ne negative. The Dickey- Fuller(DF) test deletes the sur quation. The sample period runs from 1959: i to 1987: io *significant at the 5% level *asignificant at the 10% level grated. They examine four measures of the money stock--Ml, M2 M3, and liquid assets (L)from 1959: i to 1981: ii. They conclude that the money stock and nominal gross national product(GNP)are not cointegrated with the possible exception of M2, which passes the adf test when the natural logarithm of m2 is regressed on the natural logarithm of GNP. In other words, the velocity of circula tion is generally non-stationary. The absence of cointegration between the money stock and nominal gnP does not rule out cointegration in some higher order system, including the money stock and nominal GNP. The omission of relevant variables may lead to the non-cointegration finding. The demand for money literature provides the avenue for a logical ex tension,since velocity potentially depends on nominal income and the interest rate IGould and Nelson(1974), Gould, Miller, Nelson, and Upton(1978), and Ahk (1984)also find non 574Stephen M. Miller TABLE 1. Tests for Stutionarity DF Test Levels lst-Differences ADF Test Levels lst-Differences Variable (x,) In Ml 6.758 In MlA 1.345 In M2 2.182 ln Y -1.425 In P 3.496 In r, -1.871 ln rd - 1.652 -6.096* 3.461 -3.193* -7.132* 0.115 -3.143* -5.136* 0.430 -3.662* -8.215* -0.964 -4.4a5* -3.856* -0.517 -1.774 -7.813* - 1.935 -3.974* -6.875* -2.ooo -5.215* NOTES: The augmented Dickey-Fuller (ADF) test is based on the following regression: 4 Dr, = Q. + mm, + c @,Dx,-, + e, , where D is the first-difference operator, and e, is a stationary random error. The null hypothesis is that x, is a non-stationary series, and it is rejected when o is significantly negative. The Dickey-Fuller (DF) test deletes the summation from the equation. The sample period runs from 1959:i to 1987:iu. *signi&ant at the 5% level. **significant at the 10% level. grated. They examine four measures of the money stock-Ml, M2, M3, and liquid assets @J-from 1959:i to 1981:ii. They conclude that the money stock and nominal gross national product (GNP) are not cointegrated with the possible exception of M2, which passes the ADF test when the natural logarithm of M2 is regressed on the natural logarithm of GNP. In other words, the velocity of circula￾tion is generally non-stationary.” The absence of cointegration between the money stock and nominal GNP does not rule out cointegration in some higher order system, including the money stock and nominal GNP. The omission of relevant variables may lead to the non-cointegration finding. The demand for money literature provides the avenue for a logical ex￾tension, since velocity potentially depends on nominal income and the interest rate. “Gould and Nelson (1974), Gould, Miller, Nelson, and Upton (1978), and Ahk￾ing (1984) also find velocity to be generally non-stationary. 574