Robust Estimation of GARMA Model Parameters and Application to Cointegration among Interest Rates of Industrialized Countries

Theoretical predictions for and empirical analysis of real economic time series indicate that there is dependence among distant observations. Empirical autocorrelation functions of such data show high persistence, decay slowly and show persistent cyclical patterns. Long-memory models take these stylized facts into account. The two long-memory models currently in vogue are Autoregressive Fractionally Integrated Moving Average process (ARFIMA) and Generalized Autoregressive Moving Average processes (GARMA). ARFIMA models take into account persistence in auto-correlations, whereas GARMA models characterize both the persistence and the cyclical patterns in the auto-correlation function. In the estimation of GARMA model parameters, the periodicity parameter, eta, converges faster than the others. This makes gradient-based methods unsuitable for estimating all parameters simultaneously. Chung (1996) and Gray et. al (1989) use a grid search on eta while estimating the other parameters by gradient methods. Here, we evaluate bisection techniques for eta combined with the Davidon-Fletcher-Powell gradient method for the other parameters. Based on these studies, we propose a robust and fast algorithm for GARMA model parameter estimation, including the difficult to recover parameter ranges. Such parameter ranges are of interest to economic data. As an application of our estimation procedure, we perform cointegration analysis on the interest rates of the industrialized countries by modeling them as GARMA processes. We look for evidence of cointegration among long-term interest rates of these countries. This study is motivated by the observed persistent, somewhat sinusoidal (rather than monotonic) decline of the autocorrelation function of the error correction term. We find that interest rates are better modelled as GARMA rather than ARFIMA processes and certain systems of interest rates are cointegrated. Using GARMA produces estimates with lower standard errors than other long-memory models. Further, we find that the difference between mean-reverting and nonstationary processes need not be considered in establishing cointegration. Rapid establishment of stationarity is useful in cointegration studies. Hence, we examined the power of the Dickie-Fuller test using generated GARMA data. We find that this test fails to identify stationarity correctly.