#Packages list
library(AER)Loading required package: carLoading required package: carDataLoading required package: lmtestLoading required package: zoo
Attaching package: 'zoo'The following objects are masked from 'package:base':
as.Date, as.Date.numericLoading required package: sandwichLoading required package: survivallibrary(psych)
Attaching package: 'psych'The following object is masked from 'package:car':
logitdata(package = "AER")data1 <- data(CPSSW9298)
data2 <- data(CigarettesSW)This data set contains information from the Current Population Survey, showing demographic characteristics of different individuals across two different years, 1992 and 1998. The samples from each year seem to represent different individuals, rather than tracking the same individuals over time.
The characteristics of the population include earnings (income), gender, age, and degree (education level)
This data set contains information about cigarette consumption in the United States, showing certain variables of different states across two different years, 1985 and 1995. The data set follows the same 48 states over those two periods of time.
The variables are consumer price index, population, packs per capita, state personal income, tax (average state,federal,and average local excise taxes for fiscal year), price, and taxs (average excise taxes for fiscal year, including sales tax).
summary(CPSSW9298) year earnings degree gender age
1992:7590 Min. : 1.840 highschool:7936 male :7815 Min. :25.00
1998:5911 1st Qu.: 8.173 bachelor :5565 female:5686 1st Qu.:27.00
Median :11.538 Median :30.00
Mean :12.663 Mean :29.71
3rd Qu.:15.865 3rd Qu.:32.00
Max. :49.451 Max. :34.00 head(CPSSW9298) year earnings degree gender age
1 1992 11.188811 bachelor male 29
2 1992 10.000000 bachelor male 33
3 1992 5.769231 highschool male 30
4 1992 14.957265 bachelor male 31
5 1992 8.660096 bachelor female 26
6 1992 7.788462 highschool female 31tail(CPSSW9298) year earnings degree gender age
13496 1998 4.738461 highschool female 27
13497 1998 6.730769 highschool female 29
13498 1998 12.019231 highschool female 32
13499 1998 11.111111 bachelor male 28
13500 1998 16.826923 bachelor female 33
13501 1998 12.019231 bachelor female 34This data set is a pooled-cross sectional data.
summary(CigarettesSW) state year cpi population packs
AL : 2 1985:48 Min. :1.076 Min. : 478447 Min. : 49.27
AR : 2 1995:48 1st Qu.:1.076 1st Qu.: 1622606 1st Qu.: 92.45
AZ : 2 Median :1.300 Median : 3697472 Median :110.16
CA : 2 Mean :1.300 Mean : 5168866 Mean :109.18
CO : 2 3rd Qu.:1.524 3rd Qu.: 5901500 3rd Qu.:123.52
CT : 2 Max. :1.524 Max. :31493524 Max. :197.99
(Other):84
income tax price taxs
Min. : 6887097 Min. :18.00 Min. : 84.97 Min. : 21.27
1st Qu.: 25520384 1st Qu.:31.00 1st Qu.:102.71 1st Qu.: 34.77
Median : 61661644 Median :37.00 Median :137.72 Median : 41.05
Mean : 99878736 Mean :42.68 Mean :143.45 Mean : 48.33
3rd Qu.:127313964 3rd Qu.:50.88 3rd Qu.:176.15 3rd Qu.: 59.48
Max. :771470144 Max. :99.00 Max. :240.85 Max. :112.63
table(CigarettesSW$state)
AL AR AZ CA CO CT DE FL GA IA ID IL IN KS KY LA MA MD ME MI MN MO MS MT NC ND
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
NE NH NJ NM NV NY OH OK OR PA RI SC SD TN TX UT VA VT WA WI WV WY
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 This data set is panel data
Covariance is an indicator of the extend to which two variables are dependent on each other. The value of covariance lies between \(-\infty\) and \(+\infty\) . If it is positive, the two variables move in the same direction. If the correlation is 0, there is no relationship between the variables, and if it’s negative, the variables move in different directions.
Variance is a measure of dispersion. It measures how spread out points are with respect to their average value.
Dividing the covariance of \(cov(x,y)\) by the variance of \(var(x)\) gives us the slope because we are normalizing the relationship of \(x\) and \(y\) . It quantifies the change in \(y\) by a unit change in \(x\).
\[ Packs_i = \beta_0 + \beta_1Taxs_i + \epsilon_i \]
data1 <- subset(CigarettesSW, year == 1985)
#created a new cross-sectional data set of observations only in 1985
lm1 <- lm(data= data1,
formula = packs ~ taxs)
summary(lm1)
Call:
lm(formula = packs ~ taxs, data = data1)
Residuals:
Min 1Q Median 3Q Max
-54.683 -7.618 -1.726 8.589 73.700
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 166.0152 16.3061 10.18 2.29e-13 ***
taxs -1.2643 0.4615 -2.74 0.00872 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 19.88 on 46 degrees of freedom
Multiple R-squared: 0.1403, Adjusted R-squared: 0.1216
F-statistic: 7.507 on 1 and 46 DF, p-value: 0.00872plot(x = data1$taxs,
y = data1$packs,
xlim = c(0,60),
xlab = "Average excise taxes for fiscal year, including sales tax",
ylab = "Packs of cigarettes per capita",
main = "Cigarette consumption in the US in 1985")
abline(reg = lm1,
lwd = 2,
col = "red")
beta1 <- cov(data1$packs, data1$taxs)/var(data1$taxs)
round(beta1, digits = 4)[1] -1.2643beta0 <- mean(data1$packs) - beta1 * mean(data1$taxs)
beta0 [1] 166.0152\[ \beta_1 = \frac {\sum y_i x_i - n \sum y_i \sum x_i} {\sum x_i^2 - n(\sum x_i)^2} \]
\[ \beta_1 = \frac {\sum y_i x_i - \bar{y_i} \sum x_i} {\sum x_i^2 - \bar{x_i} \sum x_i} \]