非线性模型极大似然估计方法

Tue, 13 Sep 2011 00:00:00 +0000

一、模型设定与概率框架

设非线性生产函数形式为： $$ y_i = aL_i^bK_i^c + \epsilon_i \quad (i=1,…,n) $$ 其中误差项 $\epsilon_i \sim N(0,\sigma^2)$，则观测值 $y_i$ 服从均值为 $aL_i^bK_i^c$、方差为 $\sigma^2$ 的正态分布。

二、似然函数构建

1. 联合概率密度函数

$$ L(a,b,c,\sigma) = \prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left[ -\frac{(y_i-aL_i^bK_i^c)^2}{2\sigma^2} \right] $$

2. 负对数似然函数（目标函数）

$$ \ell(a,b,c,\sigma) = -\ln L = \frac{n}{2}\ln(2\pi) + n\ln\sigma + \frac{1}{2\sigma^2}\sum_{i=1}^n (y_i - aL_i^bK_i^c)^2 $$

三、参数估计流程

1. 数据处理与准备

# 导入数据
data <- data.frame(
 y = c(76,47,97,107,123,139,159,152,191,201,207,200),
 L = c(0.02,0.02,0.06,0.06,0.11,0.11,0.22,0.22,0.56,0.56,1.10,1.10),
 K = c(1.1,0.9,2.3,2.1,3.4,3.6,4.5,4.3,5.8,6.0,7.2,7.0)
)

# 定义负对数似然函数
mloglik <- function(theta) {
 a <- theta[1]
 b <- theta[2]
 c <- theta[3]
 sigma <- theta[4]
 
 y_pred <- a * data$L^b * data$K^c
 residuals <- data$y - y_pred
 n <- length(data$y)
 
 # 避免\sigma 趋近于零
 if(sigma <= 0) return(1e10)
 
 (n/2)*log(2*pi) + n*log(sigma) + sum(residuals^2)/(2*sigma^2)
}

# 采用线性化方法获取初始值
log_model <- lm(log(y) ~ log(L) + log(K), data = data)
theta_init <- c(
 exp(coef(log_model)[1]),
 coef(log_model)[2],
 coef(log_model)[3],
 sd(resid(log_model))
)

# 优化求解
fit <- nlm(mloglik, theta_init, hessian = TRUE)

## Warning in nlm(mloglik, theta_init, hessian = TRUE): NA/NaN replaced by maximum
## positive value

## Warning in nlm(mloglik, theta_init, hessian = TRUE): Inf replaced by maximum
## positive value

# 提取估计结果
params <- fit$estimate
names(params) <- c("a", "b", "c", "sigma")
params

## a b c sigma 
## 55.45620820 -0.02273718 0.67704166 7.67493891