Comparison of predictions using generalized least squares, i.e. quasi-differencing on an interest rate model exhibiting first order serial correlation versus a model utilizing an error correction approach.

The data set obtained was rather small, 49 annual observations on interest rates, inflation and the U.S. annual deficit (as a percentage of GDP) for the years 1948 to 1996. A few regression iterations yielded a good working model for short term interest rates as being influenced by current inflation rate (as measured by the U.S. CPI), one lag of that inflation rate, and one lag of the annual U.S. deficit (the contemporaneus U.S, deficit being statistically insignificant, p-value=.53). However, a check for serial correlation in the error terms indicated a first order process, so a generalized least squares procedure was utilized to attenuate the serial correlation. A value of .6094 as the correlation estimate for p (t-value=5.15, with all other regressors exogenous) was obtained, and the quasi-differenced variables were estimated as

X(t)-p*x(t-1).

A check on the residuals from this gls model indicated no evidence of serial correlation after the transform, with a p-value of .2275. The mean squared error was 1.52545, with an rmse of 1.2351, which will be used to compare to the results from proc autoreg. The mape for the forecasts from the fgls model (after reversing the quasi-differencing operation) was a dismal 42%. Also the lag of U.S. annual deficit became insignificant, which from a free market perspective, does not make sense. It could be that such a small sample size is not adequate to train a useful model to make good predictions. However, we may still be able to gain useful insights on the impact of inflation on rates. Another approach might be to use the actual size of the annual debt, not as a pecentage of gdp.

Invoking proc autoreg with nlag=1 for an ar1 model yielded very similar values for mse(1.55259) and rmse(1.246). However, a comparison of the predicted values of autoreg versus genealized least squares (after reversing the quasi-differencing operation on the predicted values) indicates a substantial difference between the two approaches for forecasting, with fgls having 34% mean absolute percent difference with the error correction approach of proc autoreg. Autoreg had a mape of 27%, much better than 42% but still very poor – again, a much larger sample size would most likely yield better predicted values. The deficit variable was statistically significant in the autoreg model.