# 分布滯後

${\displaystyle y_{t}=a+w_{0}x_{t}+w_{1}x_{t-1}+w_{2}x_{t-2}+...+{\text{error term}}}$

${\displaystyle y_{t}=a+w_{0}x_{t}+w_{1}x_{t-1}+w_{2}x_{t-2}+...+w_{n}x_{t-n}+{\text{error term}},}$

## 結構化估計

### 有限落差分配

${\displaystyle w_{i}=\sum _{j=0}^{n}a_{j}i^{j}}$

### 無窮落差分配

${\displaystyle y_{t}=a+\lambda y_{t-1}+bx_{t}+{\text{error term}}.}$

${\displaystyle w_{i}=\sum _{j=2}^{n}{\frac {a_{j}}{(i+1)^{j}}},}$

Geometric combination lag[6]假設落差項的權重與下列式子當中線性的可估計參數 aj 有關

${\displaystyle w_{i}=\sum _{j=2}^{n}a_{j}(1/j)^{i},}$

${\displaystyle w_{i}=\sum _{j=1}^{n}a_{j}[j/(n+1)]^{i},}$

## 參考文獻

1. ^ Jeff B. Cromwell, et al., 1994. Multivariate Tests For Time Series Models. SAGE Publications, Inc. ISBN 0-8039-5440-9
2. ^ Judge, George, et al., 1980. The Theory and Practice of Econometrics. Wiley Publ.
3. ^ Almon, Shirley, "The distributed lag between capital appropriations and net expenditures," Econometrica 33, 1965, 178-196.
4. ^ Mitchell, Douglas W., and Speaker, Paul J., "A simple, flexible distributed lag technique: the polynomial inverse lag," Journal of Econometrics 31, 1986, 329-340.
5. ^ Gelles, Gregory M., and Mitchell, Douglas W., "An approximation theorem for the polynomial inverse lag," Economics Letters 30, 1989, 129-132.
6. ^ Speaker, Paul J., Mitchell, Douglas W., and Gelles, Gregory M., "Geometric combination lags as flexible infinite distributed lag estimators," Journal of Economic Dynamics and Control 13, 1989, 171-185.
7. ^ Schmidt, Peter, "A modification of the Almon distributed lag," Journal of the American Statistical Association 69, 1974, 679-681.
8. ^ Jorgenson, Dale W., "Rational distributed lag functions," Econometrica 34, 1966, 135-149.