# 经济代写|计量经济学代写Econometrics代考|EFN508

## 经济代写|计量经济学代写Econometrics代考|Infinite Distributed Lag

So far we have been dealing with a finite number of lags imposed on $X_t$. Some lags may be infinite. For example, the investment in building highways and roads several decades ago may still have an effect on today’s growth in GNP. In this case, we write Eq. (6.1) as
$$Y_t=\alpha+\sum_{i=0}^{\infty} \beta_i X_{t-i}+u_t \quad t=1,2, \ldots, T$$
There are an infinite number of $\beta_i$ ‘s to estimate with only $T$ observations. This can only be feasible if more structure is imposed on the $\beta_i$ ‘s. First, we normalize these $\beta_i$ ‘s hy their sum, i.e., let $w_i=\beta_i / \beta$ where $\beta=\sum_{i=0}^{\infty} \beta_i$. If all the $\beta_i$ ‘s have the same sign, then the $\beta_i$ ‘s take the sign of $\beta$ and $0 \leq w_i \leq 1$ for all $i$, with $\sum_{i=0}^{\infty} w_i=1$. This means that the $w_i$ ‘s can be interpreted as probabilities. In fact, Koyck (1954) imposed the geometric lag on the $w_i$ ‘s, i.e., $w_i=(1-\lambda) \lambda^i$ for $i=0,1, \ldots, \infty^1$. Substituting
$$\beta_i=\beta w_i=\beta(1-\lambda) \lambda^i$$

in (6.7) we get
$$Y_t=\alpha+\beta(1-\lambda) \sum_{i=0}^{\infty} \lambda^i X_{t-i}+u_t$$
Equation (6.8) is known as the infinite distributed lag form of the Koyck lag. The short-run effect of a unit change in $X_t$ on $Y_t$ is given by $\beta(1-\lambda)$, whereas the longrun effect of a unit change in $X_t$ on $Y_t$ is $\sum_{i=0}^{\infty} \beta_i=\beta \sum_{i=0}^{\infty} w_i=\beta$. Implicit in the Koyck lag structure is that the effect of a unit change in $X_t$ on $Y_t$ declines the further back we go in time. For example, if $\lambda=1 / 2$, then $\beta_0=\beta / 2, \beta_1=\beta / 4$, $\beta_2=\beta / 8$, etc. Defining $L X_t=X_{t-1}$, as the lag operator, we have $L^i X_t=X_{t-i}$, and (6.8) reduces to
$$Y_t=\alpha+\beta(1-\lambda) \sum_{i=0}^{\infty}(\lambda L)^i X_t+u_t=\alpha+\beta(1-\lambda) X_t /(1-\lambda L)+u_t$$

Suppose that output $Y_t$ is a function of expected sales $X_t^$ and that the latter is unobservable, i.e., $$Y_t=\alpha+\beta X_t^+u_t$$
where expected sales are updated according to the following method:
$$X_t^-X_{t-1}^=\delta\left(X_t-X_{t-1}^*\right)$$ that is, expected sales at time $t$ are a weighted combination of expected sales at time $t-1$ and actual sales at time $t$. In fact,
$$X_t^=\delta X_t+(1-\delta) X_{t-1}^$$
Equation (6.11) is also an error learning model, where one learns from past experience and adjusts expectations after observing current sales. Using the lag operator $L$, (6.12) can be rewritten as $X_t^*=\delta X_t /[1-(1-\delta) L]$. Substituting this last expression in the above relationship, we get
$$Y_t=\alpha+\beta \delta X_t /[1-(1-\delta) L]+u_t$$
Multiplying both sides of $(6.13)$ by $[1-(1-\delta) L]$, we get
$$Y_t-(1-\delta) Y_{t-1}=\alpha\left[(1-(1-\delta)]+\beta \delta X_t+u_t-(1-\delta) u_{t-1}\right.$$
(6.14) looks exactly like (6.10) with $\lambda=(1-\delta)$.

# 计量经济学代考

## 经济代写|计量经济学代写Econometrics代考|Infinite Distributed Lag

$$Y_t=\alpha+\sum_{i=0}^{\infty} \beta_i X_{t-i}+u_t \quad t=1,2, \ldots, T$$

$$\beta_i=\beta w_i=\beta(1-\lambda) \lambda^i$$

$$Y_t=\alpha+\beta(1-\lambda) \sum_{i=0}^{\infty} \lambda^i X_{t-i}+u_t$$

$$Y_t=\alpha+\beta(1-\lambda) \sum_{i=0}^{\infty}(\lambda L)^i X_t+u_t=\alpha+\beta(1-\lambda) X_t /(1-\lambda L)+u_t$$

$$Y_t=\alpha+\beta \delta X_t /[1-(1-\delta) L]+u_t$$

$$Y_t-(1-\delta) Y_{t-1}=\alpha\left[(1-(1-\delta)]+\beta \delta X_t+u_t-(1-\delta) u_{t-1}\right.$$
(6.14) 看起来和 $(6.10)$ 完全一样 $\lambda=(1-\delta)$.

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