1. An Empirical Comparison of Alternative Models of the Short-Term Interest Rate By K. C. CHAN, G. ANDREW KAROLYI, FRANCIS A. LONGSTAFF, and ANTHONY B. SANDERS THE JOURNAL OF FINANCE, VOL. XLVII, NO. 3, JULY 1992

2. The Framework

3. Parameter Restrictions Imposed on the Alternative Models
Alpha
Beta
Volatility
Gama
Merton
Vasicek
CIR SR
1/2
Dothan 1
GBM
Brennan-Schwartz
CIR VR
3/2
CEV

4. The Merton Model

5. The Vasicek Model

6. The CIR SR Model

7. The Dothan Model

8. The GMB Model

9. The Brennan-Schwartz Model

10. The CIR VR Model

11. The CEV Model

12. The Econometric Approach
They estimate the parameters of the continuous-time model using a discrete-time econometric specification.

13. The Econometric Approach
以Hansen(1982)所提出的Generalized Method of Moments (GMM)估計
GMM的優點是不須假設利率變動的分配作呈常態，而且GMM estimators and their standard deviations are consistent.
The χ2 measure provides provide a goodness-to-fit test for the model. A high value of this statistic means that the model is misspecified. ㄖ

14. The Data 美國國庫券收益率 1964/6 ~ 1989/12，共307筆月資料
The data are originally constructed by Fama (1984), and subsequently updated by the Center for Research in the Security Prices (CRSP)

15. The Empirical Results
The χ2 tests for goodness-to-fit suggest that the Merton, Vasicek, and CIR SR models are misspecified. All three models can be rejected at 95% confidence level.
The CEV model can be rejected at 90% confidence level.
The others, including Dothan, GBM, Brennan-Schwartz, and CIR VR, all have low χ2 values.

16. The Empirical Results (Continued)
The estimates of the unrestricted model.
There is a weak evidence of mean reversion in the short-term rate, since the parameter β is insignificant.
The unconstrained estimate of γ is 1.5, which indicates that the volatility of the process is highly sensitive to the level of r.

17. The Empirical Results (Continued)
The value of γ is the most important feature differentiating interest rate models.
Models which allow capture the dynamics of short-term interest rate better than those require
Those models assume are rejected, and those assume are not rejected.

18. Structure Break
To test whether the shift in Federal Reserve monetary policy in October 1979 resulted in a structural break in the interest rate process.
Introducing a dummy variable, which equals to one for all months after October 1979, and 0 otherwise.

19. Structure Break The Empirical Results
The χ2 tests statistic for the unrestricted, CIR VR, Brennan-Schwartz, and the CEV models are not significant at 95% confidence level.
作者因此下結論認為利率並未出現結構性的改變。
不過值得注意的是，除了上述模型外，其他模型包括 Merton, Vasicek, CIR SR, Dothan, GBM等模型的χ2 檢定均呈significant at 95% confidence level.

20. 結論 It is critical to model volatility correctly.
The models that best describe the dynamics of interest rates over time are those that allow the conditional volatility of interest rate changes to be highly dependent on the level of interest rates.
No evidence of a structural shift in the interest rate process in October 1979.