I am looking for those who use this series on their project (either still in progress or finished)--what model did you find best fit? Obviously we have to first-differenced the series, then I see a clear geometrically declining and oscillatory sample correlogram, which I infer an AR with order greater than 1 would work. I tried AR(1) first, of course the test stats came back significant; then I tried AR(2) in two ways--run the Excel add-in with 2 lag and using the textbook equations 4.3, 4.4, and 4.5--yet the test stats still came back significant.
Since this project is quite involved, I might have made some mistakes somewhere on my worksheets. All my correlograms stop at lag k=40, maybe that's not enough for 408 data points?
Any help is greatly appreciated.
Jacob:
If we have 400 observations, do we check 400 sample autocorrelations?
Rachel: The last 30 or 40 sample autocorrelations don’t have enough points. The first 4 or 5 sample autocorrelations can be distorted by other factors. Start with lags 6 through 55, for a total of 50 sample autocorrelations.
If only 2 or 3 sample autocorrelations are outside the 95% confidence interval, we presume the distribution is normal with the hypothesized standard deviation.
If 8 or 9 sample autocorrelations are outside the 95% confidence interval, we presume it is not a white noise process.
If 4 to 7 sample autocorrelations are outside the 95% confidence interval, we examine lags 56 to 105. We may have a white noise process with minor distortions. The ARIMA model may be reasonably good, even if it is not perfect.
Jacob: What do we expect from first differences of interest rates? Do we expect to find an oscillatory pattern?
Rachel: Some oscillatory patterns are real. Others stem from measurement error or rounding.
Illustration: Suppose the starting interest rate is 10.0%, and interest rates have a random walk with a drift of 0.04% a month. For simplicity, assume there is no stochasticity.
The interest rates are 10.00%, 10.04%, 10.08%, 10.12%, 10.16%, 10.20%, 10.24%, 10.28%, 10.32%, 10.36%, 10.40%, and so forth. The first differences are 0.04% each month.
If interest rates are rounded to one decimal place, the interest rates are 10.0%, 10.0%, 10.1%, 10.1%, 10.2%, 10.2%, 10.2%, 10.3%, 10.3%, 10.4%, 10.4%, and so forth. The first differences are 0.0%, 0.1%, 0.0%, 0.1%, 0.0%, 0.0%, 0.1%, 0.0%, 0.1%, 0.0%, and so forth.
The first differences have a spurious oscillatory pattern. It is caused by the rounding rule, and it has nothing to do with the time series process.
Jacob: Is this rounding problem common?
Rachel: This problem is very common, and it causes much of the apparent oscillatory patterns in stock prices and interest rates. The interest rates on the NEAS web site have two decimal places; we expect this type of spurious pattern.
Jacob: What if the drift is zero? Do we still see a spurious oscillatory pattern?
Rachel: A second reason for an oscillatory pattern is inaccurate measurement and other random errors.
Illustration: Suppose the interest rate is 8.0% each month and the stochasticity is small. The interest rate is measured with an error of –0.1% or +0.1% each month. The error need not be measurement error. Other causes of error are random fluctuations in demand for Treasury bills that affect the auction price but do not affect the underlying economics.
We examine the autocorrelation of the first differences. Each month the interest rate is 9.9% or 10.1%. The first difference is 0.0%, +0.2%, or –0.2%. For interest rates in any three month period, we have three possible scenarios:
| Month 1 | Month 2 | Month 3 | Difference 1 | Difference 2 |
1 | 7.9% | 7.9% | 7.9% | 0.0% | 0.0% |
2 | 7.9% | 7.9% | 8.1% | 0.0% | 0.2% |
3 | 7.9% | 8.1% | 7.9% | 0.2% | -0.2% |
4 | 7.9% | 8.1% | 8.1% | 0.2% | 0.0% |
5 | 8.1% | 7.9% | 7.9% | -0.2% | 0.0% |
6 | 8.1% | 7.9% | 8.1% | -0.2% | 0.2% |
7 | 8.1% | 8.1% | 7.9% | 0.0% | -0.2% |
8 | 8.1% | 8.1% | 8.1% | 0.0% | 0.0% |
The average first difference is zero, since the drift in interest rates is zero. Of the eight scenarios, six have at least one first difference of zero, and two scenarios have the opposite first differences. The autocorrelation is negative.
Jacob: What is the implication for model building?
Rachel: Many oscillatory patterns reflect rounding, measurement error, and random fluctuations, not the underlying time series process. We carefully examine apparent oscillatory patterns. We can not always determine the cause of the oscillatory pattern, but we can suggest possible causes.