Abstract
 Top of page
 Abstract
 1. INTRODUCTION
 2. MODELLING CHANGING GRANGER CAUSALITY
 3. A MODEL OF MARKOV SWITCHING GRANGER CAUSALITY
 4. MONEY–OUTPUT CAUSALITY
 5. SUMMARY
 Acknowledgements
 REFERENCES
 Supporting Information
The causal link between monetary variables and output is one of the most studied issues in macroeconomics. One puzzle from this literature is that the results of causality tests appear to be sensitive with respect to the sample period that one considers. As a way of overcoming this difficulty, we propose a method for analysing Granger causality which is based on a vector autoregressive model with timevarying parameters. We model parameter timevariation so as to reflect changes in Granger causality, and assume that these changes are stochastic and governed by an unobservable Markov chain. When applied to US data, our methodology allows us to reconcile previous puzzling differences in the outcome of conventional tests for money–output causality. Copyright © 2005 John Wiley & Sons, Ltd.
1. INTRODUCTION
 Top of page
 Abstract
 1. INTRODUCTION
 2. MODELLING CHANGING GRANGER CAUSALITY
 3. A MODEL OF MARKOV SWITCHING GRANGER CAUSALITY
 4. MONEY–OUTPUT CAUSALITY
 5. SUMMARY
 Acknowledgements
 REFERENCES
 Supporting Information
The relationship between money and output has attracted a phenomenal amount of interest over the years in both empirical and theoretical work. An important strand of the literature has examined the ‘causal’ links between money and output, testing for Granger (or Sims) causality from money to output.1 The interest in this issue is hardly surprising given the vital role typically ascribed to monetary policy over the business cycle. The conclusions drawn from these exercises, however, have changed a number of times over the last three decades as researchers have modified their empirical models in the light of developments in measurement, economic theory and econometric techniques.
Our interest in this issue is partly due to the observation that empirical evidence on money–output causality appears to be ‘unstable’. The particular type of instability that we focus upon is sample dependence (or time dependence) of the results from causality tests.2 An example of this type of instability is the contrast between the empirical results of Eichenbaum and Singleton (1986) and Stock and Watson (1989) on the one hand and those of Friedman and Kuttner (1992, 1993) on the other. Eichenbaum and Singleton (1986) and Stock and Watson (1989) find that the causal role for money is much weaker in a sample that excludes data from the 1980s than in data sets that include the 1980s. In stark contrast, Friedman and Kuttner (1992, p.472) state that: ‘Including data from the 1980's sharply weakens the postwar timeseries evidence indicating significant relationships between money (however defined) and nominal output or between money and either real output or prices separately. Focusing on data from 1970 onward destroys this evidence altogether’.3 The sensitivity of results from causality tests with respect to the sample period under consideration was further examined by Thoma (1994) and Swanson (1998), who used recursive and rolling window techniques to analyse Granger causality between money and output. This type of evidence indicates that causal relationships may change over time and/or that such links between money and output are very sensitive to the exact sample period under consideration.
In this paper, we propose a method for analysing causality patterns in environments where there may be good reasons to suspect that causality relationships have changed over the sample period of interest. The main motivation for our approach is that, in addition to the implications for economic theory and policy analysis, nonconstancy of causality patterns also poses significant econometric problems in the application of standard tests for Granger causality. Although there may be many reasons for time dependence of causality patterns (changes in operating procedures and other aspects of monetary policy, large shocks to the economy, etc.), unless researchers have strong a priori indications of when and why the links between the variables of interest change, accounting for such changes may be difficult. Furthermore, it is by now well recognized (see, e.g., Cooley and LeRoy, 1985; Cooley and Dwyer, 1998; Hamilton, 1994, chap. 11) that there is no direct relationship between causality in an economic sense—one variable being responsible for changes in another variable—and causality in an econometric sense—one variable helping to predict another. In other words, information that is useful in terms of understanding changes in causality in an economic sense may not be useful, or sufficient, for guiding one's choice of dates at which Granger causality changes.
To overcome these difficulties, we propose a framework of analysis based on vector autoregressive (VAR) models with timevarying parameters. Crucially, however, parameter timevariation is modelled so as to reflect directly changes in causality between the variables of interest. The starting point of our analysis is the observation that the number and location of points at which causal links change are typically unknown a priori. For this reason, we treat changes in causality as random events and allow the data to select the change points. More specifically, we model changes in causality as being governed by a hidden Markov chain with stationary but unknown transition probabilities. Such a formulation is flexible enough to allow for a variety of stochastic changes in causality, ranging from onetime permanent changes to frequent shortlived changes. What is more, even though these changes are unobservable, inferences about them can be made using likelihoodbased methods. Given the welldocumented success of Markov switching models in describing the behaviour of economic time series subject to changes in regime—see, for example, Kim and Nelson (1999) and the references therein—there are good reasons to expect such a characterization of the data to yield a good representation of the changes in causality that are often observed in practice.
The new methodology is used to reexamine the empirical evidence on the predictive content of money and interest rates for postwar US real output using quarterly data for the period 1959 : 1 to 2001 : 2. We find that the causality patterns have changed over the sample for each of these three variables under consideration (namely, M1, M2 and the Federal Funds rate). We identify two main periods during which M1 growth has had predictive power for output growth. The first is a sevenyear period that spans the interval from 1976 to mid1982, while the second is a brief period around the early 1990s recession. M2 growth is found to have Grangercaused output in an interval from 1970 to 1983 and in a short spell at the beginning of our sample. It is noteworthy that our results reveal money to have lost most of its predictive power at the end of the Volcker disinflation period, which explains the difference between the results of Stock and Watson (1989) and Friedman and Kuttner (1992). We also find that there is a marked tendency for the moneystock growth rates to gain predictive power during, or immediately before, recessions. This may indicate that monetary policy is used more actively during recessions, or just before recessions, in order to prevent a deterioration in the economy, while monetary policy might be more accommodating during expansions. The results for the Federal Funds rate are somewhat different. We find that the Federal Funds rate has significant predictive power for output growth in the post1985 sample and in the pre1970 sample. In the intermediary period, from 1970 to 1985, we find more frequent changes in the relationship between interest rates and output growth. Again, the post1985 results appear natural given the importance of the Federal Funds rate for the conduct and stance of US monetary policy.
The plan of the paper is as follows. In the next section, we briefly discuss alternative approaches to modelling changing causality. In Section 3, we propose a multivariate autoregressive model with Markov switching causality and discuss its interpretation and implications. Our empirical results are presented in Section 4. Finally, Section 5 summarizes and concludes.
2. MODELLING CHANGING GRANGER CAUSALITY
 Top of page
 Abstract
 1. INTRODUCTION
 2. MODELLING CHANGING GRANGER CAUSALITY
 3. A MODEL OF MARKOV SWITCHING GRANGER CAUSALITY
 4. MONEY–OUTPUT CAUSALITY
 5. SUMMARY
 Acknowledgements
 REFERENCES
 Supporting Information
As mentioned before, numerous researchers have noted that results from Granger causality tests tend to be sensitive with respect to changes in the sample period. The most common approach to dealing with such instabilities is to test for causality after splitting the sample under consideration into subsamples. This method requires the researcher to specify the dates at which causal relationships are supposed to have changed. In practice, this approach may not be straightforward. Although there may be cases in which researchers have reliable a priori information about dates at which the economy has been subject to a structural change (e.g., changes in operating procedures), such information may not be directly related to Granger causality (i.e., changes in the predictive power of one variable for another). Furthermore, such information may often be incomplete and not be precise enough to allow one to choose the break dates accurately. Hence, break points will often have to be estimated in practice rather than be prespecified. However, in many cases, there is no reason to assume that only one break takes place, especially when the sample covers a long time period. In the absence of prior information about the location of breaks, a main source of difficulty is that the number of possible locations of break points quickly becomes very large as the sample size increases (in a sample with T observations and b break points, there are possible locations for the breaks). Procedures such as those proposed by Bai and Perron (1998) and Bai (1999) offer a way to test for the presence of multiple (deterministic) changes in the parameters of the model of interest and estimate their number and locations. Such changes may not, however, be directly related to changes in Granger causality (as the relevant coefficients do not necessarily take a zero value before or after a structural break).
An alternative approach, adopted, for example, by Thoma (1994) and Swanson (1998), involves a rollingwindow technique. This consists of analysing only a fixedlength window of observations from the sample and then investigating the stability of the results as the window is rolled through the sample. Although this approach offers a sensible way of identifying changes in causality, a potential difficulty is that results may be sensitive with respect to the size of the window used—a very narrow window makes the results sensitive to outliers and sampling errors, while the use of too wide a window makes it unlikely that shortlived changes will be detected—and there are no firm statistical guidelines available for choosing the window size (see Swanson, 1998 for a discussion). Furthermore, although this technique might enable one to check the timeinvariance of the results, it does not provide a statistically sound way of identifying the exact dates at which changes have taken place, let alone test for the statistical significance of such changes.
Our starting point in this paper is to substitute the notion of permanent causal relations with a notion of ‘temporary’ Granger causality, that is causality which holds during some periods but not in others. Our strategy then consists of identifying the periods during which a variable Grangercauses another variable. Our methodology is based on a VAR model with timevarying parameters where, given our objectives, parameter timevariation directly reflects changes in causality. By treating changes in causality as random events governed by an exogenous Markov process, inferences about these changes can be made on the basis of the estimated probability that each observation in the sample comes from a particular causality regime.
It should be noted that VAR models with Markov switching have been used before by many researchers, primarily to examine how information about relevant economic variables affects transitions between alternatives states of the economy; see, inter alia, Sola and Driffill (1994), Krolzig (1997, 2000) and Warne (1999). The models used in these papers are variants of a VAR model where the mean (or intercept), error variance and/or autoregressive coefficients are subject to Markov switching. However, simple VAR models with switching intercepts and/or error variances are not wellsuited to studying regimedependent causality. Furthermore, even models with switching coefficients are not ideal for the problem in hand for at least three reasons. First, the separation of regimes in such models need not be related to changes in causality and hence a regime may include observations from states of nature associated with very different causal patterns. Second, the endogenous variables in a VAR model may be subject to changes that occur at very different points in the sample, and hence an adequate description of these changes may require the use of a model with a large number of regimes. Finally, a general Markov switching VAR model is not necessarily consistent with our notion of temporary causality, i.e., causality which holds during some periods but not in others, since the coefficients that parameterize Granger causality are not restricted to be zero in one or more of the states of nature.4
The approach advocated here is based on a switching VAR model in which the regimes represent alternative causal states of nature. In such a model, there are always four regimes (if the model is bivariate), which are associated with the four possible causal relationships between the variables. Let us now explain our methodology in some detail.
3. A MODEL OF MARKOV SWITCHING GRANGER CAUSALITY
 Top of page
 Abstract
 1. INTRODUCTION
 2. MODELLING CHANGING GRANGER CAUSALITY
 3. A MODEL OF MARKOV SWITCHING GRANGER CAUSALITY
 4. MONEY–OUTPUT CAUSALITY
 5. SUMMARY
 Acknowledgements
 REFERENCES
 Supporting Information
Consider the problem of analysing the Granger causality links between the components of the bivariate time series {X′_{t} = [X_{1, t} : X_{2, t}]} conditionally on the scalar time series {Z_{t}}.5 Our analysis is based on the following Markov switching VAR model:
 (1)
Here, S_{1, t} and S_{2, t} are latent random variables that reflect the ‘state’ (or ‘regime’) of the system at date t and take their values in the set {0, 1}, and {ε′_{t} = [ε_{1, t} : ε_{2, t}]} is a whitenoise process, independent of {S_{1, t}} and {S_{2, t}}, with mean zero and covariance matrix which depends on S_{1, t} and S_{2, t} in a way to be specified below.
The specification in (1) allows for four alternative states of nature, which may be conveniently indexed by using the following stateindicator variable:
The covariance matrix of the disturbances in (1) is then specified as
Before describing the probabilistic properties of the state variables S_{1, t} and S_{2, t}, it is useful to make clear how these variables affect the parameters of the VAR. Observe that the model in (1) implies that
It is evident, therefore, that the regime indicators S_{1, t} and S_{2, t} (and hence S_{t}) determine the causal links in the model. In particular, S_{1, t} determines whether X_{2, t} is Grangercausal for X_{1, t} while S_{2, t} dictates whether X_{1, t} is Grangercausal for X_{2, t}. Given that at least one of the parameters is nonzero, X_{2, t} Grangercauses X_{1, t} when S_{1, t} = 1 (S_{t} = 1 or S_{t} = 3) and is Granger noncausal for X_{1, t} when S_{1, t} = 0 (S_{t} = 2 or S_{t} = 4). Likewise, X_{1, t} Grangercauses X_{2, t} when S_{2, t} = 1 (S_{t} = 1 or S_{t} = 2), provided that not all of the parameters are zero, and is Granger noncausal when S_{2, t} = 0 (S_{t} = 3 or S_{t} = 4). Note that this parameterization also encompasses the case where one or both variables are Granger noncausal for the whole sample period ( and/or ).
The specification of the model is completed by assuming that nature selects the state of the system at date t with a probability which depends only on the state prevailing at date t − 1. More specifically, we assume that the random sequences {S_{1, t}} and {S_{2, t}} are timehomogeneous, firstorder Markov chains with transition probabilities
Moreover, it is assumed that {S_{1, t}} and {S_{2, t}} are independent.6 Hence, if P denotes the stochastic matrix whose (i, j) element is the probability ℙ(S_{t+1} = iS_{t} = j), for i, j = 1, …, 4, the random sequence {S_{t}} is a timehomogeneous, firstorder Markov chain with transition probability matrix
Causality analysis based on the Markov switching model in (1) is attractive for a number of reasons. First, the states of nature are defined directly in terms of the causal relationships between the two variables; this is a natural way of classifying regimes when the focus of the analysis is changes in causality. Second, it allows for the possibility of arbitrarily many changes in causality at unknown locations in the sample. What is more, these changes are parameterized in a parsimonious way, driven as they are by a simple, timehomogeneous Markov chain. Finally, our approach allows one to make probabilistic inferences about the dates at which changes in causality occurred during the sample period. More specifically, although the state variables governing the manner in which regime shifts occur are unobservable, the likelihood of each of the possible regimes being operable at each date in the sample period can be inferred on the basis of the estimated conditional probabilities , where W′_{t} = [X′_{t} : Z_{t−1}] and is an estimator of the unknown parameters in (1). Thus, we can evaluate the extent to which changes in causality have actually occurred and identify the locations of such changes in the sample.
4. MONEY–OUTPUT CAUSALITY
 Top of page
 Abstract
 1. INTRODUCTION
 2. MODELLING CHANGING GRANGER CAUSALITY
 3. A MODEL OF MARKOV SWITCHING GRANGER CAUSALITY
 4. MONEY–OUTPUT CAUSALITY
 5. SUMMARY
 Acknowledgements
 REFERENCES
 Supporting Information
In this section, we use the proposed model for Markov switching Granger causality to address the question of whether financial variables have predictive power for output. Our analysis is based on quarterly US data for the period 1959 : 1 to 2001 : 2. We measure output by real GDP and examine three financial variables, namely, M1, M2 and the Federal Funds rate (r). We logarithmically transform real GDP (y), M1 (m_{1}) and M2 (m_{2}), and use the growth rate of the consumer price index as a measure of inflation (π). Except for the interest rates, all data are seasonally adjusted.
4.1. Trending Properties of the Data
It is well documented that causality inferences are sensitive with respect to the presence of trends in the data, and correct modelling of the longrun characteristics of the data is hence very important (cf. Christiano and Ljungqvist, 1988; Stock and Watson, 1989). With this in mind, we report in Table I the ordinary leastsquares (OLS) estimate of the sum of autoregressive coefficients (ρ) in a univariate autoregression for each of the series of interest and its first difference.7 We also report 90% equaltailed confidence intervals for ρ obtained by Hansen's (1999) gridt bootstrap method, using 999 bootstrap replications at each of 200 gridpoints. In addition, we give 90% symmetric confidence intervals for ρ obtained by the subsampling method of Romano and Wolf (2001) applied to the studentized OLS estimator of ρ.8, 9
Table I. Trending properties of the dataSeries  q   SE  90% Confidence interval for ρ  tRatio 

Grid bootstrap  Subsampling   


y  2  0.948  0.018  (0.934, 0.993)  (0.888, 1.008)  2.778  2.904 
m_{1}  5  0.988  0.007  (0.985, 1.007)  (0.966, 1.009)  1.629  1.847 
m_{2}  5  0.995  0.004  (0.993, 1.006)  (0.995, 1.011)  1.155  1.574 
r  5  0.901  0.034  (0.875, 0.984)  (0.806, 0.996)  0.014  2.291 
Δy  1  0.376  0.093  (0.258, 0.562)  (0.206, 0.546)  −0.759  3.762 
Δm_{1}  4  0.793  0.069  (0.739, 0.959)  (0.642, 0.944)  −0.903  2.284 
Δm_{2}  4  0.766  0.071  (0.701, 0.930)  (0.597, 0.935)  −1.426  2.961 
π  4  0.923  0.036  (0.905, 1.017)  (0.831, 1.014)  −1.025  1.950 
At least one of the confidence intervals for ρ contains unity in the case of y, m_{1}, m_{2} and π but not in the case of r, Δy, Δm_{1} or Δm_{2}. This suggests that y, m_{1} and m_{2} are integrated of order one, while the interest rate and the differenced series are integrated of order zero. We will, therefore, firstdifference y, m_{1} and m_{2} in subsequent analysis. We will not difference π since the unit autoregressive root suggested by the results in Table I is most likely an artefact due to a substantial shift in the level of inflation in the early 1970s.10
Granger causality results are also sensitive with respect to the specification of deterministic trends, as Stock and Watson (1989), for example, demonstrated. We investigate this issue in the last two columns of Table I. Unlike Stock and Watson (1989) and Swanson (1998), who report evidence of significant deterministic trends in some of their measures of money growth, we find a significant drift in Δy, Δm_{1}, Δm_{2} and π but no statistically significant linear trend (at significance levels lower than 16%) over our sample period.11 Hence, no specifications with deterministic trends will be considered in subsequent analysis.
4.2. Results from Linear VAR Models
As a starting point for the analysis, we consider conventional fullsample tests for Granger noncausality from m_{1}, m_{2} or r to y, based on linear VAR models of the form
 (2)
where X′_{t} = [Δy_{t} : Δm_{1, t}], X′_{t} = [Δy_{t} : Δm_{2, t}] or X′_{t} = [Δy_{t} : r_{t}], and ξ_{t} is a vector of disturbances.12 Following the practice of some authors (e.g., Friedman and Kuttner, 1992), lagged inflation is included as a conditioning variable when testing for Granger causality between the financial variables and output. We set n_{1} = n_{2} = 2 for X′_{t} = [Δy_{t} : Δm_{1, t}] or X′_{t} = [Δy_{t} : Δm_{2, t}] and n_{1} = n_{2} = 3 for X′_{t} = [Δy_{t} : r_{t}].13
The pvalue for a conventional noncausality Ftest (from each of the nominal variables to real output) is 0.1484, 0.0002 and 0 for Δm_{1}, Δm_{2} and r, respectively. Thus, in the full sample there is evidence in favour of Δm_{2} and the Federal Funds rate having significant predictive content for output growth, while Δm_{1} does not appear to have any such predictive power. This difference between the predictive content of M1 and M2 growth is in line with previous findings in the literature (see, e.g., Friedman and Kuttner, 1992).
To examine the temporal stability of causal relationships, we now test the constancy of the parameters of the VAR models on which the causality tests are based using the procedures discussed in Hansen (1990), Andrews (1993) and Andrews and Ploberger (1994). These tests are based on functionals of the sequence of Lagrange multiplier statistics which test the null hypothesis of parameter stability against the alternative of a onetime change at each possible break point in the sample. The test statistics are defined as
where, for 0 < ω < 1, LM(ω) stands for the Lagrange multiplier statistic for testing the null hypothesis of constant parameters against the alternative of a change at date t = [ωT] (square brackets designating the integer part). Following Andrews (1993), the tests are implemented with ω_{1} = 1 − ω_{2} = 0.15. We also consider Nyblom's (1989) test for randomly timevarying parameters (denoted NYB), for which the alternative hypothesis is that the coefficients follow a random walk.
These tests are used to test the constancy of the coefficients in the VAR models defined in (2). In Table II, we report the outcome of the tests for the output equation and the associated pvalues. These pvalues are obtained from a bootstrap approximation to the null distribution of the test statistics, which was constructed by means of Monte Carlo simulation using 999 artificial samples generated from a VAR model with constant parameters. The sequential Lagrange multiplier tests suggest that there is significant evidence of parameter nonconstancy in the output equation of the output–M1 and output–M2 models, although this evidence is not very strong in the latter model. No significant evidence of nonconstancy is found in the model with the Federal Funds rate. It should be borne in mind, however, that these tests for parameter instability are not designed to be optimal against Markov changes in parameters and may not be particularly powerful in the presence of Markov regime switching of the type discussed in Section 2 (cf. Carrasco, 2002).
Table II. Stability tests for output equationTest  Δm_{1}  Δm_{2}  r 


MLM  13.22  11.81  13.26 
(0.038)  (0.098)  (0.385) 
ELM  10.28  7.129  8.657 
(0.004)  (0.104)  (0.268) 
SLM  27.21  19.20  23.41 
(0.004)  (0.107)  (0.193) 
NYB  1.428  1.227  1.119 
(0.162)  (0.322)  (0.879) 
To investigate further the potential changes in causality over the sample period, we consider next Ftests for Granger noncausality computed from rolling subsamples of a fixed size (cf. Thoma, 1994; Swanson, 1998). More specifically, the pvalues of Fstatistics testing the lack of Granger causality from Δm_{1}, Δm_{2} or r to Δy are computed from the VAR models defined in (2) fitted to rolling windows of 24 observations (six years of data). To avoid the potential inaccuracies of conventional asymptotic inference procedures in small samples (only 24 observations in our case), the pvalues were obtained from a bootstrap approximation to the sampling distribution of the test statistics. The bootstrap approximation was constructed, as before, by means of Monte Carlo simulation, using 999 artificial samples generated from a VAR model which satisfies the null hypothesis under test and the disturbances of which are obtained from the leastsquares VAR residuals by equiprobable sampling with replacement.
The bootstrap pvalues of the rolling test statistics are shown in Figure 1 (the horizontal axes show the final observation in each of the sixyear rolling windows). In line with the fullsample results, the plots indicate that, at least during certain parts of the sample, the monetary variables have predictive power for output. However, the plots also suggest that there have been important changes in causal links over the sample period. We find that there is only a brief period during which M1 appears to have had predictive content for output. This period consists of a fouryear period from 1983 to 1987. During the rest of the sample, the results do not indicate a significant causal role for M1. The results for M2 are somewhat similar, but for this variable the evidence suggests that M2 had predictive power for output mainly in the period 1973–1978. For both moneystock measures, however, we also observe that the pvalues change substantially over the sample, although the causality from money to output cannot be rejected at the 10% level during most of the sample. Finally, the rollingsample results indicate that the Federal Funds rate was Grangercausal during a long period from 1970 to around 1987.
4.3. Results from Markov Switching Models
Let us now consider the results from estimating Markov switching VAR models of the type specified in (1) with X′_{t} = [Δy_{t} : Δm_{1, t}], X′_{t} = [Δy_{t} : Δm_{2, t}] or X′_{t} = [Δy_{t} : r_{t}] and Z_{t} = π_{t}. We set h_{1} = h_{2} = 2 in all three cases, a choice of lag length which is supported by the data since the resulting models have standardized residuals which exhibit no significant autocorrelation (in either their levels or their squares) according to the portmanteau test of Ljung and Box (1978).14
The parameters of Markov switching models are estimated by maximum likelihood (ML), assuming that the conditional distribution of X_{t} given {W_{t−1}, …, W_{1}, S_{t}, S_{t−1}, …, S_{0}} is normal. The likelihood function is evaluated by means of an iterative filtering algorithm similar to the one discussed in Hamilton (1994, chap. 22), and the ML estimates are found by a quasiNewton optimization algorithm which uses the Broyden–Fletcher–Goldfarb–Shano secant update to the hessian. The estimates of all 40 parameters of each switching VAR model and the associated asymptotic standard errors—computed from the inverse of the empirical Hessian—are reported in Tables III–V. (The output equation is always the first equation of the VAR.)
Table III. Estimates of parameters of the model for output and money (M1)  Estimate  Std. error   Estimate  Std. error 

 0.027  0.095   −0.049  0.068 
 0.052  0.105   0.273  0.109 
 0.088  0.089   −0.084  0.084 
 0.097  0.117   −0.112  0.105 
 0.418  0.140   0.737  0.086 
 0.331  0.139   0.008  0.092 
 0.044  0.327   0.636  0.239 
 0.013  0.720   −0.257  0.202 
 −0.892  0.357   0.174  0.209 
 0.177  0.524   0.285  0.223 
µ_{10}  0.650  0.189  µ_{20}  0.325  0.133 
µ_{11}  0.461  0.366  µ_{21}  0.256  0.135 
σ_{11, 1}  0.979  0.373  σ_{22, 1}  1.269  0.856 
σ_{11, 2}  0.608  0.125  σ_{22, 2}  0.362  0.072 
σ_{11, 3}  0.964  0.328  σ_{22, 3}  0.240  0.037 
σ_{11, 4}  0.191  0.036  σ_{22, 4}  0.402  0.262 
σ_{12, 1}  1.653  0.402  σ_{12, 3}  0.079  0.091 
σ_{12, 2}  0.237  0.077  σ_{12, 4}  0.876  0.081 
 0.975  0.019   0.931  0.043 
 0.977  0.018   0.979  0.014 
loglikelihood  −48.646     
Table IV. Estimates of parameters of the model for output and money (M2)  Estimate  Std. error   Estimate  Std. error 

 0.047  0.101   0.562  0.131 
 0.314  0.110   0.834  0.091 
 −0.019  0.093   0.076  1.137 
 0.244  0.114   −0.167  0.087 
 0.146  0.191   −0.027  0.074 
 0.390  0.162   −0.056  0.071 
 −0.138  0.316   −0.294  0.222 
 0.005  0.244   −0.382  0.186 
 −0.394  0.314   0.373  0.225 
 −0.205  0.272   0.089  0.200 
µ_{10}  0.479  0.182  µ_{20}  0.625  0.142 
µ_{11}  0.316  0.359  µ_{21}  0.749  0.155 
σ_{11, 1}  1.124  0.223  σ_{22, 1}  0.610  0.090 
σ_{11, 2}  0.423  0.116  σ_{22, 2}  0.207  0.038 
σ_{11, 3}  1.807  1.875  σ_{22, 3}  2.620  4.765 
σ_{11, 4}  0.234  0.039  σ_{22, 4}  0.238  0.063 
σ_{12, 1}  0.331  0.104  σ_{12, 3}  3.869  2.291 
σ_{12, 2}  0.103  0.053  σ_{12, 4}  0.248  0.039 
 0.993  0.008   0.973  0.021 
 0.992  0.010   0.983  0.012 
loglikelihood  0.816     
Table V. Estimates of parameters of the model for output and interest rate  Estimate  Std. error   Estimate  Std. error 

 0.218  0.088   0.179  0.043 
 0.053  0.094   1.189  0.067 
 0.117  0.096   0.056  0.044 
 0.180  0.072   −0.259  0.059 
 0.121  0.055   0.699  0.175 
 −0.238  0.061   0.067  0.195 
 −0.173  0.380   0.270  0.135 
 0.248  0.230   −0.781  1.350 
 0.749  0.362   0.174  0.209 
 −0.594  0.240   1.903  1.380 
µ_{10}  0.480  0.299  µ_{20}  −0.121  0.154 
µ_{11}  0.887  0.145  µ_{21}  −0.137  0.108 
σ_{11, 1}  0.735  0.137  σ_{22, 1}  0.204  0.017 
σ_{11, 2}  0.351  0.225  σ_{22, 2}  0.972  2.555 
σ_{11, 3}  0.172  0.048  σ_{22, 3}  0.291  0.166 
σ_{11, 4}  1.704  0.734  σ_{22, 4}  3.444  5.296 
σ_{12, 1}  0.074  0.039  σ_{12, 3}  0.546  0.061 
σ_{12, 2}  4.491  0.420  σ_{12, 4}  11.540  1.713 
 0.834  0.061   0.973  0.016 
 0.810  0.070   0.871  0.070 
loglikelihood  −47.106     
In the case of M1, the results in Table III how that both and are significantly different from zero so M1 growth Grangercauses real GDP growth when S_{t} = 1 or S_{t} = 3. From Figure 2, we see that the strongest evidence in favour of money–output causality is obtained for the period after the first oilprice crisis, namely from 1977 up to the end of the Volcker disinflation. We also find evidence of money–output causality in a short period around the early 1990s recession, and less significantly around the early 1960s recession.
These results are interesting for many reasons. Firstly, while fullsample evidence does not support the hypothesis that M1 is Grangercausal for output, the results based on the switching model clearly show that M1 growth did have predictive power for output during various subperiods in the sample. Secondly, we find, as expected, that the predictive power of M1 is highest around the Volcker period. Furthermore, the results indicate that monetary policies directed towards instruments that affect highpowered money may have been used actively during the two recessions in the early 1980s and the recession in the early 1990s, and that these policies have had some effect on the predictive power of money for output growth. For the early 1980s episodes, this result seems natural given the money growth policies adopted under Paul Volcker's disinflation and the subsequent recessions. Finally, the results also make it clear why Friedman and Kuttner (1992) found that the evidence on money–output causality is much weaker if data for the 1980s are considered, while Stock and Watson (1989) documented the opposite result. We find that the 1980s are divided into two distinct periods, with evidence in favour of causality from money to output found in the first half and evidence of noncausality found in the second. Thus, when using conventional fullsample tests of Granger causality, it is crucial to the results whether one includes in the sample data from only the first half of the 1980s, as in Stock and Watson (1989), or data from the entire 1980s, as in Friedman and Kuttner (1992).
The results for M2, shown in Table IV, are in many ways similar to those for M1. We find that is significantly different from zero so that M2 growth causes real GDP growth in the states represented by S_{t} = 1 and S_{t} = 3. The plot of the estimated probability that M2 growth is Grangercausal for output, shown in Figure 3, is similar to that for M1 growth with the following exceptions. First, the period where M2 growth causes output growth is longer than the period identified for M1 growth. For M2, money causes output during most of the period 1970–1983, apart from a very short period that coincides with the first oilprice crisis. Second, we no longer find causality around the early 1990s recession. These differences may occur for a number of reasons but, given the difference in the content of M1 and M2, it is not surprising that these money stock measures may behave differently. We note again that the fullsample results are somewhat misleading in so far as they would lead one to conclude that M2 growth has predictive power over output when we find that the predictive content of M2 has in fact vanished since the end of the Volcker disinflation.
It is interesting to note that our results suggest that moneystock growth is more likely to have predictive power for output growth during recessions than during expansions. In particular, M1 growth has predictive power for output growth during four of the six recessions in the sample and very little predictive power during any expansionary periods. M2 growth has predictive power for output growth during all recessions apart from the early 1990s recession, while during expansions M2 growth has predictive power only during parts of the 1970s. Thus, there seems to be asymmetry in the causal links between money and output related to the state of the business cycle. A possible explanation for this result is that monetary policy might assume a more active role during—or just before—recessions in order to prevent further deteriorations in the economy, while more accommodating policies are applied during expansions.
The results for the Federal Funds rate are shown in Table V nd the estimated probabilities that the interest rate is Grangercausal for output are plotted in Figure 4. Evidently, there is a clear division of the sample period into periods of causality from interest rates to output. The interest rate Grangercauses output in the period 1960–1969 and then again from 1983 onwards. In the intermediate period, we find that the probability that the interest rate causes output drops in each of the recessions in the late 1960s, the oilprice crises in the 1970s and the early 1980s recession (when the Fed targeted money growth and the Federal Funds rate soared). Furthermore, the interest rate appears to regain predictive power for output in each of the intermediate periods between these recessions. It is worth noting that the Federal Funds rate maintains its predictive power in the 1990s when the money stocks lose their predictive ability. This may reflect the fact that the Federal Funds rate is endogenous and hence has predictive power for output growth—to the extent that the demand and supply of Federal Funds have some predictive power for output (over and above the predictive power of past output growth itself). Notwithstanding the lack of a direct link between predictive ability and causation in an economic sense, this evidence may offer an explanation as to why the Federal Funds rate seems to work better than money stocks as a basis for evaluating the stance and effects of US monetary policy.17
4.4. Some Simulation Results
One issue that we have not addressed so far is the effectiveness of our method as a means of detecting changes in causality. In order to get some insight into the ‘precision’ of the method in identifying the different regimes in cases where there are periods of causality and noncausality, we carry out a few Monte Carlo experiments which are based on the empirical results reported in Tables III–V.
To be more precise, the basis of our calculations is 500 independent samples of size T = 170 from a bivariate Markov switching model like (1) with Gaussian errors and h_{1} = h_{2} = 2. To ensure the empirical relevance of the simulations, the coefficients and error covariance matrix of the model are chosen to be the ML estimates reported in Tables III–V (referred to in the sequel as models 2, 3 and 4, respectively). In all cases, observed historical values of inflation are used as {Z_{t}}. The transition probabilities of {S_{1, t}} and {S_{2, t}} are chosen as: (i) ; or (ii) . The difference between the two sets of transition probabilities is that the expected duration of the regime in which X_{2, t} is Grangercausal for X_{1, t} is much higher in case (ii) (50 time periods, compared with an expected duration of 10 periods for case (i)).
For each artificial sample, ML estimates of the parameters of model (1) (with h_{1} = h_{2} = 2) are obtained and the probabilities , are computed using these estimates. To assess how accurately the four regimes can be classified by using the filtered probabilities, we then compute the value of the criterion
where ��(A) denotes the indicator of the event A. Note that 0 ≤ C_{ℓ} ≤ 1 and low values of C_{ℓ} imply that the method works well while high values imply inaccurate classification of regimes.
Table VI reports the average value of C_{ℓ} across the 500 Monte Carlo replications. For all design points, the average value of the regimeclassification indicator is fairly small, suggesting that our method is quite accurate in identifying the points in the sample at which changes in causality have occurred. Combined with the other attractive features of our approach discussed earlier, these results reinforce the case for using Markov switching models of the form (1) to analyse causality patterns which are suspected to be unstable over the sample period.
Table VI. Monte Carlo mean of C_{ℓ}  ,  ,  C_{1}  C_{2}  C_{3}  C_{4} 

Model 2  0.90  0.96  0.14  0.07  0.07  0.11 
0.98  0.96  0.08  0.03  0.05  0.10 
Model 3  0.90  0.96  0.11  0.04  0.07  0.08 
0.98  0.96  0.07  0.06  0.07  0.07 
Model 4  0.90  0.96  0.12  0.11  0.08  0.10 
0.98  0.96  0.07  0.05  0.07  0.09 