Measuring the strength of cointegration and Granger-causality

Abstract

This study uses Poskitt and Tremayne's (1987) posterior odds ratio test and the associated model portfolio approach to measure the strength of the evidence from cointegration and Granger-causality tests. As an illustration of the methodology, the bivariate relationship between money and income in Canada is re-examined using historical data.

Acknowledgements

I would like to thank Ronald G. Bodkin, R. Quentin, Grafton, and Regula Schaub Atukeren for their comments on earlier versions of this paper. Any remaining errors are, of course, my own.

Notes

Hsiao (1979) employed the Final Prediction Error criterion; FPE(p, q) = (RSS/T) [(T + p + q + 1)/(T − p − q − 1)], where p and q are lag orders, T is the sample size, and RSS is the residual-sum-of-squares.

The Schwarz BIC (SBIC) results from a Bayesian procedure of seeking the most probable (a posteriori) model. The formula is: SBIC(p, q) = (RSS/T) T(p + q + 1)/T, where p and q are lag orders, T is the sample size, and RSS is the residual-sum-of-squares. In practice, the natural logarithm of Schwarz BIC is reported. SBIC's properties are well documented in the literature. Mills and Prasad (1992), based on their Monte-Carlo experiments, recommend the Schwarz BIC as the first choice of applied researchers. See also Nickelsburg (1985), Lütkepohl (1985), and Yi and Judge (1988).

An estimated coefficient at lag k is considered to be ‘significant’, if the associated SBIC value is lower than the SBIC at lag k − 1. See also Penm and Terrell (1984) and Marin (1992).

See Mills (1990, pp. 140–2).

Chenoweth et al. (2004) examine Poskitt and Tremayne's (1987) R-ratio and demostrate that the proximity of competing models can be interpreted as a distance function in a Hilbert space.

For the example considered in this section (SBIC₀ = −5.793229 and SBIC₁ = −5.845061), the posterior odds ratio (R) is 7.75 for T = 79. That is, there is no decisive evidence for the model where ‘X Granger-causes Y’. The case where ‘X does not Granger-cause Y’ is a closely competing model (Table 3). Note that in order for the evidence for ‘X Granger-causes Y’ to become decisive, the sample size (T) would have to be at least 178, given the same SBIC values. At T = 79, the noise possibly prevents one to reach a decisive conclusion.

See Cagan (1989), Stock and Watson (1989), and Blanchard (1990) for surveys of the literature on the relationship between money and income.

Hsiao (1979, 1982) used second differencing on the natural logarithms (logs) of each variable. Lütkepohl (1982) and Penm and Terrell (1984) took first differences of the log-levels. Kim and Ro (1988) conducted their analyses on the log-levels of the variables.

The resulting modified t-statistics from a model where a constant and a trend term are included are −0.31, −1.05, and −0.26 for log Y, log M1, and log M2, respectively. Phillips and Perron's modified t-statistics on the first differences of the above variables are −7.41, −6.38, and −5.74, respectively. The truncation lag is set at 4 and the critical value is −3.13.

The five models are specified as follows: (1) no trend in the data with no trend and no constant term in the model; (2) no trend in the data with no trend but with a constant term in the model; (3) linear trend in the data with no trend but with a constant term in the model; (4) linear trend in the data with trend and a constant term in the model; (5) quadratic trend in the data with trend and a constant term in the model.

The first lag is included in the specification since its introduction led to a decrease in the Schwarz BIC over the specification where Y is regressed on a constant term only.

Dorfman (1995) and So and Li (1999) use a similar posterior odds ratio concept in the context of testing for unit roots and cointegration, whereas Cushman (2001) employs the PIC in the context of testing for the stationarity of real exchange rates.