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library(ivreg)
# Problem 1 part a
set.seed(123)
n <- 1000
x1 <- rnorm(n, 0, 1)
u <- rnorm(n, 0, 1)
x2 <- 2 + 0.5 * x1
y <- 1 + x1 + x2 + u
v1 <- rnorm(n, 0, 1)
x3 <- x2 + v1
# Problem 1 part c
ols_model_1 <- lm(y ~ x1 + x3)
summary(ols_model_1)##
## Call:
## lm(formula = y ~ x1 + x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.9801 -0.6834 0.0076 0.6871 3.2661
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.98345 0.07190 41.497 <2e-16 ***
## x1 1.57406 0.03572 44.061 <2e-16 ***
## x3 0.02909 0.03255 0.894 0.372
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.006 on 997 degrees of freedom
## Multiple R-squared: 0.7105, Adjusted R-squared: 0.7099
## F-statistic: 1223 on 2 and 997 DF, p-value: < 2.2e-16# Problem 1 part d
v2 <- rnorm(n, 0, 1)
x4 <- x2 + v2
iv_result_1 <- ivreg(y ~ x1 + x3 | x1 + x4)
summary(iv_result_1)##
## Call:
## ivreg(formula = y ~ x1 + x3 | x1 + x4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.25511 -0.71173 0.01997 0.70948 3.41019
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.3150 1.2934 2.563 0.0105 *
## x1 1.6546 0.3157 5.241 1.95e-07 ***
## x3 -0.1383 0.6530 -0.212 0.8323
##
## Diagnostic tests:
## df1 df2 statistic p-value
## Weak instruments 1 997 2.550 0.111
## Wu-Hausman 1 996 0.068 0.795
## Sargan 0 NA NA NA
##
## Residual standard error: 1.02 on 997 degrees of freedom
## Multiple R-Squared: 0.7028, Adjusted R-squared: 0.7022
## Wald test: 1191 on 2 and 997 DF, p-value: < 2.2e-16# Problem 1 part e
v3 <- rnorm(n, 0, 1)
x5 <- x2 + v3
iv_result_2 <- ivreg(y ~ x1 + x3 | x1 + x4 + x5)
summary(iv_result_2)##
## Call:
## ivreg(formula = y ~ x1 + x3 | x1 + x4 + x5)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.03148 -0.85777 0.02593 0.78733 3.97969
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 4.2510 1.4464 2.939 0.00337 **
## x1 1.8819 0.3532 5.328 1.23e-07 ***
## x3 -0.6110 0.7302 -0.837 0.40289
##
## Diagnostic tests:
## df1 df2 statistic p-value
## Weak instruments 2 996 1.378 0.2527
## Wu-Hausman 1 996 1.070 0.3013
## Sargan 1 NA 5.179 0.0229 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.186 on 997 degrees of freedom
## Multiple R-Squared: 0.5982, Adjusted R-squared: 0.5974
## Wald test: 881.5 on 2 and 997 DF, p-value: < 2.2e-16# Problem 2 part b
set.seed(123)
n <- 1000
x1 <- rnorm(n)
x2 <- rnorm(n)
x3 <- rnorm(n)
us <- rnorm(n)
ud <- rnorm(n)
p <- 0.5 * x3 + 0.5 * (ud - us)
y_eq <- 1 + x1 + x2 + 0.5 * x3 + 0.5 * us + 0.5 * ud
# Problem 2 part c
ols_supply <- lm(y_eq ~ p + x1 + x2)
summary(ols_supply)##
## Call:
## lm(formula = y_eq ~ p + x1 + x2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.41625 -0.54270 -0.01675 0.59500 2.26104
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.97498 0.02601 37.49 <2e-16 ***
## p 0.33220 0.03105 10.70 <2e-16 ***
## x1 0.99947 0.02630 38.00 <2e-16 ***
## x2 1.03462 0.02590 39.94 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.8213 on 996 degrees of freedom
## Multiple R-squared: 0.7797, Adjusted R-squared: 0.779
## F-statistic: 1175 on 3 and 996 DF, p-value: < 2.2e-16# Problem 2 part d
ols_demand <- lm(y_eq ~ p + x1 + x2 + x3)
summary(ols_demand)##
## Call:
## lm(formula = y_eq ~ p + x1 + x2 + x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.99490 -0.47029 -0.00389 0.47115 1.97312
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.977826 0.022304 43.840 <2e-16 ***
## p 0.004952 0.031739 0.156 0.876
## x1 1.011453 0.022564 44.826 <2e-16 ***
## x2 1.040707 0.022217 46.843 <2e-16 ***
## x3 0.514761 0.027162 18.952 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.7043 on 995 degrees of freedom
## Multiple R-squared: 0.8381, Adjusted R-squared: 0.8374
## F-statistic: 1288 on 4 and 995 DF, p-value: < 2.2e-16# Problem 2 part g
iv_supply <- ivreg(y_eq ~ p + x1 + x2 | x1 + x2 + x3)
summary(iv_supply)##
## Call:
## ivreg(formula = y_eq ~ p + x1 + x2 | x1 + x2 + x3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.257784 -0.670357 0.009886 0.701075 3.153253
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.99384 0.03324 29.89 <2e-16 ***
## p 1.11057 0.07289 15.24 <2e-16 ***
## x1 0.99728 0.03359 29.69 <2e-16 ***
## x2 0.98645 0.03330 29.63 <2e-16 ***
##
## Diagnostic tests:
## df1 df2 statistic p-value
## Weak instruments 1 996 418.7 <2e-16 ***
## Wu-Hausman 1 995 359.2 <2e-16 ***
## Sargan 0 NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.049 on 996 degrees of freedom
## Multiple R-Squared: 0.6406, Adjusted R-squared: 0.6396
## Wald test: 774.3 on 3 and 996 DF, p-value: < 2.2e-16# Problem 3 part a
set.seed(123)
x1 <- rnorm(1000)
x2 <- rnorm(1000)
u <- rnorm(1000)
v <- rnorm(1000)
for (t in 2:1000) {
u[t] <- 0.3 * u[t - 1] + v[t]
}
y <- rnorm(1000)
for (t in 2:1000) {
y[t] <- 1 + 0.5 * y[t - 1] + x1[t] + x2[t] + u[t]
}
# Problem 3 part b
ydep <- y[2:1000]
ylag <- y[1:999]
x1reg <- x1[2:1000]
x2reg <- x2[2:1000]
x1lag <- x1[1:999]
x2lag <- x2[1:999]
# Problem 3 part c
ols_ts <- lm(ydep ~ ylag + x1reg + x2reg)
summary(ols_ts)##
## Call:
## lm(formula = ydep ~ ylag + x1reg + x2reg)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.2882 -0.6671 0.0008 0.6769 3.1317
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.84989 0.04548 18.69 <2e-16 ***
## ylag 0.56528 0.01522 37.15 <2e-16 ***
## x1reg 1.00939 0.03285 30.73 <2e-16 ***
## x2reg 1.00098 0.03228 31.01 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.025 on 995 degrees of freedom
## Multiple R-squared: 0.7699, Adjusted R-squared: 0.7692
## F-statistic: 1109 on 3 and 995 DF, p-value: < 2.2e-16# Problem 3 part e
iv_ts <- ivreg(ydep ~ ylag + x1reg + x2reg | x1reg + x2reg + x1lag + x2lag)
summary(iv_ts)##
## Call:
## ivreg(formula = ydep ~ ylag + x1reg + x2reg | x1reg + x2reg +
## x1lag + x2lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.500429 -0.687554 -0.005943 0.692897 3.306406
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.04779 0.05860 17.88 <2e-16 ***
## ylag 0.47071 0.02311 20.37 <2e-16 ***
## x1reg 1.00395 0.03350 29.97 <2e-16 ***
## x2reg 0.99509 0.03292 30.23 <2e-16 ***
##
## Diagnostic tests:
## df1 df2 statistic p-value
## Weak instruments 2 994 407.304 < 2e-16 ***
## Wu-Hausman 1 994 32.669 1.44e-08 ***
## Sargan 1 NA 2.115 0.146
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.045 on 995 degrees of freedom
## Multiple R-Squared: 0.7609, Adjusted R-squared: 0.7602
## Wald test: 763.5 on 3 and 995 DF, p-value: < 2.2e-16