Example 10.9 Klein’s Model I（英文文本内容）

A widely used example of a simultaneous equations model of the economy is Klein’s (1950) Model I. The model may be written as:

\[ \begin{align} C_{t}&=\alpha_{0}+\alpha_{1} P_{t}+\alpha_{2} P_{t-1}+\alpha_{3}\left(W_{t}^{p}+W_{t}^{g}\right)&+\epsilon_{1 t} & \quad \text { (Consumption) } \\ I_{t}&=\beta_{0}+\beta_{1} P_{t}+\beta_{2} P_{t-1}+\beta_{3} K_{t-1}&+\epsilon_{2 t} & \quad\text { (Investment) } \\ W_{t}^{p}&=\gamma_{0}+\gamma_{1} X_{t}+\gamma_{2} X_{t-1}+\gamma_{3} A_{t}&+\epsilon_{3 t} & \quad\text { (Private wages) } \\ X_{t}&=C_{t}+I_{t}+G_{t}& & \quad\text { (Equilibrium demand) } \\ P_{t}&=X_{t}-T_{t}-W_{t}^{p}& & \quad\text { (Private profits) } \\ K_{t}&=K_{t-1}+I_{t}& & \quad\text { (Capital stock) } \\ \end{align} \]

The endogenous variables are each on the left-hand side of an equation and are labeled on the right. The exogenous variables are \(G_t\) = government nonwage spending, \(T_t\) = indirect business taxes plus net exports, \(W_t^g\) = government wage bill, \(A_t\) = time trend measured as years from 1931, and the constant term. There are also three predetermined variables: the lagged values of the capital stock, private profits, and total demand. The model contains three behavioral equations, an equilibrium condition, and two accounting identities. This model provides an excellent example of a small, dynamic model of the economy. It has also been widely used as a test ground for simultaneous equations estimators. Klein estimated the parameters using yearly aggregate data for the U.S. for 1921 to 1941. The data are listed in Appendix Table F10.3. Table 10.8 presents limited and full information estimates for Klein’s Model I based on the original data.

It might seem, in light of the entire discussion, that one of the structural estimators described previously should always be preferred to ordinary least squares, which alone among the estimators considered here is inconsistent. Unfortunately, the issue is not so clear. First, it is often found that the OLS estimator is surprisingly close to the structural estimator. It can be shown that, at least in some cases, OLS has a smaller variance about its mean than does 2SLS about its mean, leading to the possibility that OLS might be more precise in a mean-squared-error sense. But this result must be tempered by the finding that the OLS standard errors are, in all likelihood, not useful for inference purposes. Obviously, this discussion is relevant only to finite samples. Asymptotically, 2SLS must dominate OLS, and in a correctly specified model, any full information estimator (3SLS) must dominate any limited information one (2SLS). The finite sample properties are of crucial importance. Most of what we know is asymptotic properties, but most applications are based on rather small or moderately sized samples.

Although the system methods of estimation are asymptotically better, they have two problems. First, any specification error in the structure of the model will be propagated throughout the system by 3SLS. The limited information estimators will, by and large, confine a problem to the particular equation in which it appears. Second, in the same fashion as the SUR model, the finite-sample variation of the estimated covariance matrix is transmitted throughout the system. Thus, the finite-sample variance of 3SLS may well be as large as or larger than that of 2SLS.

Example 10.9 Klein’s Model I（中文文本内容）

一个广泛使用的经济联立方程模型例子是Klein (1950)的模型I。该模型可以写成：

\[ \begin{align} C_{t}&=\alpha_{0}+\alpha_{1} P_{t}+\alpha_{2} P_{t-1}+\alpha_{3}\left(W_{t}^{p}+W_{t}^{g}\right)&+\epsilon_{1 t} & \quad \text { (消费) } \\ I_{t}&=\beta_{0}+\beta_{1} P_{t}+\beta_{2} P_{t-1}+\beta_{3} K_{t-1}&+\epsilon_{2 t} & \quad\text { (投资) } \\ W_{t}^{p}&=\gamma_{0}+\gamma_{1} X_{t}+\gamma_{2} X_{t-1}+\gamma_{3} A_{t}&+\epsilon_{3 t} & \quad\text { (私人工资) } \\ X_{t}&=C_{t}+I_{t}+G_{t}& & \quad\text { (均衡需求) } \\ P_{t}&=X_{t}-T_{t}-W_{t}^{p}& & \quad\text { (私人利润) } \\ K_{t}&=K_{t-1}+I_{t}& & \quad\text { (资本存量) } \\ \end{align} \]

内生变量都在方程的左边，并在右边标注。外生变量是 \(G_t\) = 政府非工资支出，\(T_t\) = 间接商业税加净出口，\(W_t^g\) = 政府工资支出，\(A_t\) = 以1931年为起点的年份时间趋势，以及常数项。还有三个前定变量：资本存量、私人利润和总需求的滞后值。该模型包含三个行为方程、一个均衡条件和两个会计恒等式。这个模型为小型动态经济模型提供了一个很好的例子。它也被广泛用作联立方程估计量的测试平台。Klein使用1921年至1941年美国年度汇总数据估计了参数。数据列在附录表F10.3中。表10.8基于原始数据给出了Klein模型I的有限信息和完全信息估计。

根据整个讨论，似乎应该总是优先选择之前描述的结构估计量之一，而不是普通最小二乘法，后者在本文考虑的估计量中是唯一不一致的。不幸的是，问题并不那么清楚。首先，经常发现OLS估计量出人意料地接近结构估计量。可以证明，至少在某些情况下，OLS关于其均值的方差小于2SLS关于其均值的方差，这导致OLS可能在均方误差意义上更精确。但这一结果必须与以下发现相权衡：OLS标准误差在大多数情况下对推断目的没有用处。显然，这一讨论仅与有限样本相关。渐近地，2SLS必须优于OLS，在正确设定的模型中，任何完全信息估计量（3SLS）必须优于任何有限信息估计量（2SLS）。有限样本性质至关重要。我们所知的大多是渐近性质，但大多数应用都基于相当小或中等大小的样本。

尽管系统估计方法在渐近上更好，但它们有两个问题。首先，模型结构中的任何设定误差都会通过3SLS在整个系统中传播。有限信息估计量大体上会将问题限制在出现问题的特定方程中。其次，与SUR模型一样，估计协方差矩阵的有限样本变异会在整个系统中传播。因此，3SLS的有限样本方差可能等于或大于2SLS的有限样本方差。