古扎拉蒂计量经济学第四版讲义Ch10 Autoregression and Distribution Lag Model

第十章 自回归和分布滞后模型

Lecture Note 13 – Dynamic Econometric Models: Autoregressive and Distributed-Lag Models

1. Some concepts

Regression models that take into account time lags are known as dynamic or lagged regression models .

There are two types of lagged models: distributed-lag models and autoregressive models . In the former, the current and lagged values of regressors are explanatory variables. In the latter, the lagged value(s) of the regressand appears as explanatory variables.

2. The role of “lag” or “time” in economics

什么是lag ：

In economics the dependence of a variable y (the dependent variable) on another variable(s) x (the explanatory variable) is rarely instantaneous. Very often, y responds to x with a lapse of time. Such a lapse of time is called a lag .

The reasons for lag:

1. Psychological reasons.

2. Technological reasons.

3. Institutional reasons.

3. Estimation of distributed-lag models

假定含有一个解释变量及其滞后（这只是一种简化，当然可以推广到几个解释变量及其各自滞后）的分布滞后模型如下：

01122t t t t t y x x x αβββε−−=+++++ 17.3.1

这里没有定义滞后长度，即，how far back into the past we want to go ，这样的模型称为infinite (lag) model 。而

01122t t t t k t k t y x x x x αββββε−−−=++++++ 17.1.2

称为finite (lag) distributed-lag model 。

如何估计模型17.3.1，可以采用两种方法：(1) ad hoc estimation and (2) a priori restrictions on the β’s by assuming that the β’s follow some systematic pattern.

对分布滞后模型估计的描述：A purely distributed-lag model can be estimated by OLS, but in that case there is the problem of multicolliearity since successive lagged values of a regressor tend to be correlated. As a result, some shortcut methods have been devised. These include the Koyck,

the adaptive expectations, and partial adjustment mechanisms, the first being a purely algebraic approach and the other two being based on economic principles.

Ad hoc estimation of distribution-lag models

既然解释变量t x 被假定为非随机（或至少与干扰项t ε不相关），1t x −，2t x −等等也非随机，则原则上，OLS 方法可用来估计模型17.3.1，这是由Alt 和Tinbergen 提出的，他们建议对模型17.3.1进行序列（sequentially ）估计，即first regress t y on t x , then regress t y on t x and 1t x −, then regress t y on t x , 1t x − and 2t x −, and so on 。This sequential procedure stops when the regression coefficients of the lagged variables start becoming statistically insignificant and/or the coefficient of at least one of the variables changes signs from positive to negative or vice versa.

虽然seemingly straightward ，ad hoc estimation 也有很多缺陷，如：

1、对滞后的最大长度没有先验知识，假如滞后长度被错误设定，就会导致misspecification error 问题。

2、随着滞后变量的不断增加，自由度变少，会导致统计推断变得不可靠，因为我们研究的数据常常没有足够长度的观察值。

3、更重要的是，在经济时间序列中，连续的滞后变量趋向高度相关，因此会产生多重共线性问题（导致估计参数的标准误相对于参数估计量会变大，t 统计量变小，会给出统计不显著的结论）。

Clearly, some prior or theoretical considerations must be brought to bear upon the various β’s if we are to make headway with the estimation problem.

The Koyck approach to distributed-lag models

Koyck 提出了一个ingenious 方法来估计分布滞后模型。仍然以无限滞后模型17.3.1为例。 Assuming that the

β’s are all of the same sign, Koyck assumes that they decline geometrically as

follows, 00,1,k k k ββλ== 17.4.1

where λ (01λ<<) is known as the rate of decline , or decay , of the distributed lag and where 1λ− is known as the speed of adjustment .

有时候17.4.1式也被写成

()010,1,k k k ββλλ=−= 17.4.1-1

这个假设就是x 远期的滞后对y 的效应小于x 的近期滞后的效应。