Disequilibrium macroeconomics plicit price deflator(P) Cointegration and crror-corrcction modeling proceeds as fol- lows. First, determine the orders of integration for each of the va econd, estimate with ordinary least squares, using variables with the same order of integration. Third, test for stationary residuals of the cointegration equations. And finally, construct the error-correction models Table 1 reports Dickey-Fuller(DF)and adjusted Dickey-Fuller (ADF) tests for stationarity of the natural logarithm of each variable (Fuller 1976 and Dickey and Fuller 1979 For levels of the series none reject the null hypothesis of non-stationarity at either the 5% or 10% levels. After first differencing, each series, save one, rejects of th of the implicit price deflator(that is, In P), the exception, rejects non-stationarity by the df test, but not by the adf test Further analysis of the implicit price deflator suggests that first differencing probably induces stationarity. The DF and ADF tests on second differences indicate over differencing, since the coeff cients of the lagged level (that is, the first difference of the series) are significantly less than minus one. Also, examination of autocor relation and partial autocorrelation charts reveals that first differ encing leaves a highly autocorrelated series with a slowly declining autocorrelation function and significant partial autocorrelation spikes of 0. 76 and 0. 34 at lags one and two, but that second differencing oduces only one significant spike pike in the autocorrelation and par- tial autocorrelation functions at lag one of0. 43, suggesting pos sible over differencing In sum, the evidence suggests that each series is stationary in first differences Cointegration equation Engle and Granger(1987) consider whether alternative mon- etary aggregates and nominal gross national product are cointe- These choices are not exhaustive, but do reflect frequently examined variables Since I find a set of co-integrated variables, I search no further. Ml and M2 are the current Federal Reserve definitions and MIA subtracts other checkable de- posits from M1. Renewed interest surrounds MIA( Darby, mascaro 1989). Hamburger(1987) supports the use of the dividend-price ratioDisequilibrium Macroeconomics that is decomposed into real gross national product (y) and the im￾plicit price deflator (P).‘” Cointegration and error-correction modeling proceeds as fol￾lows. First, determine the orders of integration for each of the vari￾ables under consideration. Second, estimate cointegration equations with ordinary least squares, using variables with the same order of integration. Third, test for stationary residuals of the cointegration equations. And finally, construct the error-correction models. Testing for Stationary Series Table 1 reports Dickey-Fuller (DF) and adjusted Dickey-Fuller (ADF) tests for stationarity of the natural logarithm of each variable (Fuller 1976 and Dickey and Fuller 1979). For levels of the series, none reject the null hypothesis of non-stationarity at either the 5% or 10% levels. After first differencing, each series, save one, rejects non-stationarity at the 5% level. The test of the natural logarithm of the implicit price deflator (that is, In P), the exception, rejects non-stationarity by the DF test, but not by the ADF test. Further analysis of the implicit price deflator suggests that first differencing probably induces stationarity. The DF and ADF tests on second differences indicate over differencing, since the coeffi￾cients of the lagged level (that is, the first difference of the series) are significantly less than minus one. Also, examination of autocor￾relation and partial autocorrelation charts reveals that first differ￾encing leaves a highly autocorrelated series with a slowly declining autocorrelation function and significant partial autocorrelation spikes of 0.76 and 0.34 at lags one and two, but that second differencing produces only one significant spike in the autocorrelation and par￾tial autocorrelation functions at lag one of -0.43, suggesting pos￾sible over dilferencing. In sum, the evidence suggests that each series is stationary in first differences. Cointegration Equations Engle and Granger (1987) consider whether alternative mon￾etary aggregates and nominal gross national product are cointe- “These choices are not exhaustive, but do reflect frequently examined variables. Since I find a set of co-integrated variables, I search no further. Ml and M2 are the current Federal Reserve definitions, and MlA subtracts other checkable de￾posits from Ml. Renewed interest surrounds MlA (Darby, Mascara, and Marlow 1989). Hamburger (1987) supports the use of the dividend-price ratio. 573