stationary

The endogeneity of the money supply, however, makes equation (5) econometrically unidentified, as noted by King and Watson (1992). In fact, they show that one additional restriction is required in order to identify the model and test long-run neutrality restrictions. In the literature various identifying restrictions can be found. One approach is to assume that the model is recursive, so that either gamma^sub ym^ = 0 [see, for example, Geweke (1986), Stock and Watson (1988), Rotemberg, Driscoll, and Poterba (1995), and Fisher and Seater (1993)] or lambda^sub my^ = 0 [see Geweke (1986) who also presents results under this restriction]. Alternatively, long-run neutrality of money with gamma^sub ym^ = 0 may be assumed [as in Gali (1992), King, Plosser, Stock, and Watson (1991), and Shapiro and Watson (1988)] or money exogeneity with gamma^sub my^ = 0 (which holds, for example, if lambda^sub my^ = aalpha^sup 1^^sub my^ = alpha^sup 2^^sub my^ = . . = alpha^sup p^^sub my^ = 0).

In this paper, we follow King and Watson's (1992) more eclectic approach and instead of focusing on a single identifying restriction, we report results for a wide range of identifying restrictions. In particular, we iterate each of lambda^sub my^, lambda^sub ym^, lambda^sub my^ and ym within a reasonable range, each time obtaining estimates of the remaining three parameters and their standard errors. This Paloma Picasso Loving Heart pendant strategy is clearly more informative in terms of the robustness of inference about long-run neutrality to specific assumptions about lambda^sub ym^, lambda^sub my^, or lambda^sub my^ The model is estimated by simultaneous equation methods, as described in King and Watson (1992).

INTEGRATION AND COINTEGRATION PROPERTIES OF THE   DATA

The Data

The data analyzed in this and the following section come from Backus and Kehoe (1992) and consist of over a hundred years of annual observations on real GNP/ GDP and money for Australia, Canada, Denmark, Germany, Italy, Japan, Norway, Sweden, the United Kingdom, and the United States. Due to missing observations, the years 1915 to 1920 were removed for Denmark, the years 1914 to 1924 and 1939 to 1949 for Germany, the years 1941 to 1951 for Japan, and the years 1940 to 1945 for Norway; see Backus and Kehoe (1992) for more details.

B. Univariate Tests for Unit Roots

As it was argued in the introduction, meaningful neutrality tests can only be conducted if both nominal and real variables satisfy certain nonstationarity conditions. In particular, neutrality tests require that both nominal and real variables are at least integrated of order one and of the same order of integration, and superneutrality tests are possible if the order of integration of the nominal variables is equal to one plus the order of integration of the real variables. Hence, the first step in conducting neutrality and superneutrality tests is to test for stochastic trends (unit roots) in the autoregressive representation of each individual time series. In doing so, in what follows we use four alternative unit root testing procedures to deal with anomalies that arise when the data are not very informative about whether or not there is a unit root.

In the first three columns of panel A of Tables 1 and 2 we report p-values for the augmented weighted symmetric (WS) unit root test (see Pantula, Gonzalez-Farias, and Fuller, 1994), the augmented Dickey-Fuller (ADF) test (see Dickey and Fuller 1981), and the nonparametric, Z(t^sub alpha^), test of Phillips (1987) and Phillips and Perron (1988). These p-values (calculated using TSP 4.3) are based on the response surface estimates given by Mackinnon (1994). As discussed in Pantula et al. (1994), the WS test dominates the ADF test in terms of power. Also the Z(t^sub alpha^) test is robust to a wide variety of serial correlation and time-dependent heteroskedasticity. For the WS and ADF tests, the optimal lag length was taken to be the order selected by the Akaike information criterion (AIC) plus 2; see Pantula et al. (1994) for details regarding the advantages of this rule for choosing the number of augmenting lags. The Z(t^sub alpha^) test is done with the same Dickey-Fuller regression variables, using no augmenting lags. Based on the p-values for the WS, ADF, and Z(t^sub alpha^) test statistics reported in panel A of Tables 1 and 2, the null hypothesis of a unit root in log levels cannot be rejected. This is consistent with the Nelson and Plosser (1982) argument that most macroeconomic time series have a stochastic trend.

It is important to note that in the tests that we have discussed so far the unit root is the null hypothesis to be tested and that the way in which classical hypothesis testing is carried out ensures that the null hypothesis is accepted unless Paloma Picasso Loving Heart ring is strong evidence against it. In fact, Kwiatkowski et al. (1992) argue that such unit root tests fail to reject a unit root because they have low power against relevant alternatives and they propose tests (known as the KPSS tests) of the hypothesis of stationarity against the alternative of a unit root. They argue that such tests should complement unit root tests and that by testing both the unit root hypothesis and the stationarity hypothesis, one can distinguish series that appear to be stationary, series that appear to be integrated, and series that are not very informative about whether or not they are stationary or have a unit root.

KPSS tests for level and trend stationarity are also presented in panel A of Tables 1 and 2 under the KPSS columns. As can be seen, the t-statistic eta^sub mu^, that tests the null hypothesis of level stationarity is large relative to the 5 percent critical value of .463 given by Kwiatkowski et al. (1992). Also, the t-statistic eta^sub tau^ that tests the null hypothesis of trend stationarity exceeds the 5 Paloma's Crown of Hearts pendant critical value of .146 (also given by Kwiatkowski et al. 1992), except for Denmark's output series and the money series for Canada, Denmark, Sweden, and the United States. Although the evidence in favor of the trend stationarity hypothesis is significant for the money series for Norway and the United States, combining the results of our tests of the stationarity hypothesis with the results of our tests of the unit root hypothesis, we conclude that all the (logged) output and money series have at least one unit root.