# 分布滯後

$y_t = a + w_0x_t + w_1x_{t-1} + w_2x_{t-2} + ... + \text{error term}$

$y_t = a + w_0x_t + w_1x_{t-1} + w_2x_{t-2} + ... + w_nx_{t-n} + \text{error term},$

## 結構化估計

### 有限落差分配

$w_i = \sum_{j=0}^{n} a_j i^j$

### 無窮落差分配

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

$w_i = \sum_{j=2}^{n}\frac{a_j}{(i+1)^j},$

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

$w_i = \sum_{j=2}^{n} a_j(1/j)^i,$

$w_i = \sum_{j=1}^{n} a_j [j/(n+1)]^i,$

## 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.