library(wooldridge)
## Warning: package 'wooldridge' was built under R version 4.4.1
data("barium")
# (C2 Part (i))
barium$trend <- barium$t # Use the variable 't' as the linear time trend
model1 <- lm(log(chnimp) ~ trend + log(chempi) + log(gas) + log(rtwex), data = barium)
summary(model1)
##
## Call:
## lm(formula = log(chnimp) ~ trend + log(chempi) + log(gas) + log(rtwex),
## data = barium)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.9462 -0.3095 0.0199 0.3774 1.1985
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.57512 20.67163 -0.125 0.901061
## trend 0.01356 0.00371 3.654 0.000377 ***
## log(chempi) -0.99233 1.19557 -0.830 0.408102
## log(gas) 0.57013 0.86725 0.657 0.512125
## log(rtwex) -0.08936 0.40007 -0.223 0.823613
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5731 on 126 degrees of freedom
## Multiple R-squared: 0.3498, Adjusted R-squared: 0.3292
## F-statistic: 16.95 on 4 and 126 DF, p-value: 3.838e-11
# (C2 Part (ii))
library(car)
## Loading required package: carData
linearHypothesis(model1, c("log(chempi) = 0", "log(gas) = 0", "log(rtwex) = 0"))
## Linear hypothesis test
##
## Hypothesis:
## log(chempi) = 0
## log(gas) = 0
## log(rtwex) = 0
##
## Model 1: restricted model
## Model 2: log(chnimp) ~ trend + log(chempi) + log(gas) + log(rtwex)
##
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 129 41.709
## 2 126 41.386 3 0.32274 0.3275 0.8054
# (C2 Part (iii))
# Include all monthly dummy variables except January (baseline)
model2 <- lm(log(chnimp) ~ trend + log(chempi) + log(gas) + log(rtwex) +
feb + mar + apr + may + jun + jul + aug + sep + oct + nov + dec, data = barium)
summary(model2)
##
## Call:
## lm(formula = log(chnimp) ~ trend + log(chempi) + log(gas) + log(rtwex) +
## feb + mar + apr + may + jun + jul + aug + sep + oct + nov +
## dec, data = barium)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.86409 -0.34931 0.01966 0.38943 1.06689
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 28.150788 30.924932 0.910 0.364571
## trend 0.012953 0.003771 3.435 0.000826 ***
## log(chempi) -0.684025 1.228814 -0.557 0.578845
## log(gas) -0.790531 1.336974 -0.591 0.555491
## log(rtwex) -0.300425 0.435624 -0.690 0.491809
## feb -0.351338 0.292753 -1.200 0.232560
## mar 0.062903 0.254089 0.248 0.804913
## apr -0.424580 0.256487 -1.655 0.100577
## may 0.057545 0.254148 0.226 0.821276
## jun -0.173419 0.253743 -0.683 0.495702
## jul 0.038939 0.262166 0.149 0.882187
## aug -0.097512 0.261111 -0.373 0.709500
## sep -0.043634 0.252288 -0.173 0.862993
## oct 0.093825 0.252192 0.372 0.710550
## nov -0.259692 0.252285 -1.029 0.305471
## dec 0.095602 0.260688 0.367 0.714495
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.577 on 115 degrees of freedom
## Multiple R-squared: 0.3984, Adjusted R-squared: 0.32
## F-statistic: 5.078 on 15 and 115 DF, p-value: 1.353e-07
# Compare Models With and Without Monthly Dummies
anova(model1, model2)
## Analysis of Variance Table
##
## Model 1: log(chnimp) ~ trend + log(chempi) + log(gas) + log(rtwex)
## Model 2: log(chnimp) ~ trend + log(chempi) + log(gas) + log(rtwex) + feb +
## mar + apr + may + jun + jul + aug + sep + oct + nov + dec
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 126 41.386
## 2 115 38.292 11 3.0945 0.8449 0.5959
# Check Individual Monthly Dummy Effects
coefficients <- summary(model2)$coefficients
monthly_dummies <- coefficients[grep("feb|mar|apr|may|jun|jul|aug|sep|oct|nov|dec", rownames(coefficients)), ]
monthly_dummies
## Estimate Std. Error t value Pr(>|t|)
## feb -0.35133811 0.2927528 -1.2001188 0.2325597
## mar 0.06290337 0.2540892 0.2475641 0.8049131
## apr -0.42457953 0.2564866 -1.6553673 0.1005765
## may 0.05754461 0.2541481 0.2264216 0.8212755
## jun -0.17341880 0.2537432 -0.6834421 0.4957022
## jul 0.03893868 0.2621663 0.1485267 0.8821873
## aug -0.09751238 0.2611115 -0.3734511 0.7095003
## sep -0.04363386 0.2522883 -0.1729524 0.8629929
## oct 0.09382470 0.2521916 0.3720374 0.7105497
## nov -0.25969185 0.2522849 -1.0293594 0.3054710
## dec 0.09560179 0.2606875 0.3667295 0.7144947
# (C9 Part (i))
data("volat")
# Hypothesize the Signs of β1 and β2
# β1 (pcip): Likely positive
# β2 (i3): Likely negative
# (C9 Part (ii))
model <- lm(rsp500 ~ pcip + i3, data = volat)
summary(model)
##
## Call:
## lm(formula = rsp500 ~ pcip + i3, data = volat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -157.871 -22.580 2.103 25.524 138.137
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 18.84306 3.27488 5.754 1.44e-08 ***
## pcip 0.03642 0.12940 0.281 0.7785
## i3 -1.36169 0.54072 -2.518 0.0121 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 40.13 on 554 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.01189, Adjusted R-squared: 0.008325
## F-statistic: 3.334 on 2 and 554 DF, p-value: 0.03637
# (C9 Part (iii))
coefficients <- summary(model)$coefficients
significance <- coefficients[, 4] < 0.05 # Check if p-value < 0.05
significant_vars <- rownames(coefficients)[significance]
significant_vars # Print significant variables
## [1] "(Intercept)" "i3"
# (C9 Part (iv))
# R-squared value
r_squared <- summary(model)$r.squared
r_squared
## [1] 0.01189219
# Interpretation:
# - Statistically significant coefficients suggest some level of predictability.
# - A low R-squared value indicates that the model explains only a small proportion of the variance.