我试图估计一个需求模型：
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。每个产品的需求d_t^k取决于所有产品共有的一般季节性，是a_t该周产品价格的仿射函数p_t^k，加上一些正常的随机误差e_t^k。
但是，如果我使用以下lm函数调用，它会给我一个 的系数b，price而我想要的是每个产品b^k的一个系数price^k。
lm(demand ~ factor(week) + price, data = df)
模型的正确表达方式是什么？
lm(demand ~ factor(week) + factor(product) * price, data = df)
我猜上面的方法会起作用，但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，我有以下代码在稍微不同的需求模型上运行 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))