y = a + b*x + e
x = α+β̄ *M + ε
M(工具變數12.1) 預測 x
>兩階段TSLS
x再去預測y
香菸消費量有可能透過大幅提高香菸稅來減少。問題是必須提高多少稅才能達到減少一定程度的香菸消費量。經濟學家使用彈性來回答此類問題。由於香菸需求的價格彈性未知,因此必須估計。由於需求與供給之間存在同時因果關係,因此無法使用對數價格的對數數量OLS迴歸來估計感興趣的效應。故使用工具變數的回歸處理。
log(Qcigarettes)=β0+β1log(Pcigarettes)+ui,
log(Qcigarettes)=β0+β1 * log(Pcigarettes)+ui,
CigarettesSW$rprice <- with(CigarettesSW, price / cpi)
CigarettesSW$salestax <- with(CigarettesSW, (taxs – tax) / cpi)
#扣除聯邦、州,賣出香菸本身的稅(不受數量影響)
cor(CigarettesSW$salestax, CigarettesSW$price)
0.6141228,工具變量與x有關。
c1995 <- subset(CigarettesSW, year == “1995”)
log(Pcigarettes)=π0+π1 * SalesTax+νi.
cig_s1 <- lm(log(rprice) ~ salestax, data = c1995)
summary(cig_s1 )
names(cig_s1)
為顯著,x和工具變量有關
R^2 解釋力 0.471
lcigp_pred <- cig_s1$fitted.values
cig_s2 <- lm(log(c1995$packs) ~ lcigp_pred)
coeftest(cig_s2, vcov = vcovHC)
t test of coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 9.71988 1.70304 5.7074 7.932e-07 ***
lcigp_pred -1.08359 0.35563 -3.0469 0.003822 **
—
cig_ivreg <- ivreg(log(packs) ~ log(rprice) | salestax, data = c1995)
coeftest(cig_ivreg, vcov = vcovHC, type = “HC1”)
ivreg() <工具變量回歸 (~內生變數| 外生/工具變數)
t test of coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 9.71988 1.52832 6.3598 8.346e-08 ***
log(rprice) -1.08359 0.31892 -3.3977 0.001411 **
分段2次lm SE會比ivreg大
12.2
增加’所得’外生變數
Yi=β0+β1 * X1i+β2 * X2i+W1i+ui
y ~ x1 + x2 + w1 | w1 + z1 + z2 + z3 where w1 is “instrumenting itself
w1預測 w1
z1 + z2 + z3 預測 x1 + x2
cig_ivreg2 <- ivreg(log(packs) ~ log(rprice) + log(rincome) | log(rincome) +salestax, data = c1995)
coeftest(cig_ivreg2, vcov = vcovHC, type = “HC1”)
ivreg(y ~ x1 內生 + x2內生 + w1 外生+ w2外生| w1外生 + w2外生 +z1 工具'本身就是外生'+ z2工具 )
cig_ivreg3 <- ivreg(log(packs) ~ log(rprice) + log(rincome) | log(rincome) + salestax + cigtax, data = c1995)