Question

我试着估计需求模型：

d_t^k = a_t - b^k p_t^k + e_t^k

指数t代表周数，k代表产品编号。每种产品d_t^k的需求取决于所有产品a_t共享的一般季节性，并且是该周p_t^k产品价格的仿射函数，加上一些正常随机错误e_t^k。

但是，如果我使用以下lm函数调用，它会为b提供单个系数price，而我想要的是每个产品的一个系数b^k price^k。

lm(demand ~ factor(week) + price, data = df)

表达模型的正确方法是什么？

lm(demand ~ factor(week) + factor(product) * price, data = df)

我猜这上面会有用，而且我找不到任何文件告诉我那里发生了什么。

作为一个具体的例子，我在一个略有不同的需求模型上运行以下代码 d_t ^ k = a_t + a ^ k - b ^ k p_t ^ k + e_t ^ k

# Generate fake prices and sales, and estimate the coefficients of
# the demand model.

number.of.items <- 20 # Must be a multiple of 4
number.of.weeks <- 5
coeff.item.min <- 300
coeff.item.max <- 500
coeff.price.min <- 1.4
coeff.price.max <- 2
normal.sd <- 40
set.seed(200)

# Generate random coefficients for the items
coeff.item <- runif(number.of.items, coeff.item.min, coeff.item.max)
coeff.price <- runif(number.of.items, coeff.price.min, coeff.price.max)
coeff.week <- 50 * 1:number.of.weeks

# Row is item, column is week
week.id.matrix <- outer(rep(1, number.of.items), 1:number.of.weeks)
item.id.matrix <- outer(1:number.of.items, rep(1, number.of.weeks))
price.matrix <- rbind(
  outer(rep(1, number.of.items / 4), c(100, 100, 90, 90, 80)),
  outer(rep(1, number.of.items / 4), c(100, 90, 90, 80, 60)),
  outer(rep(1, number.of.items / 4), c(100, 85, 85, 60, 60)),
  outer(rep(1, number.of.items / 4), c(100, 75, 60, 45, 45))
)
coeff.week.matrix <- outer(rep(1, number.of.items), coeff.week)
coeff.price.matrix <- outer(coeff.price, rep(1, number.of.weeks))
coeff.item.matrix <- outer(coeff.item, rep(1, number.of.weeks))
sales.matrix <- coeff.week.matrix +
  coeff.item.matrix -
  coeff.price.matrix * price.matrix +
  matrix(rnorm(number.of.weeks * number.of.items, 0, normal.sd),
         number.of.items, number.of.weeks)


df <- data.frame(item = factor(as.vector(item.id.matrix)),
                 week = factor(as.vector(week.id.matrix)),
                 price = as.vector(price.matrix),
                 sales = as.vector(sales.matrix))

model <- lm(sales ~ week + item + price, data = df)
model <- lm(sales ~ week + item + factor(item) * price, data = df)

print(summary(model))

Answer 1

经过一些实验，似乎

lm(demand ~ factor(week) + factor(product) * price, data = df)

确实有效。

我不知道为什么我认为它不会更早发挥作用。

具有因子和自变量的乘积的线性回归

1 个答案: