Applied Econometrics
Last updated: 2020-12-22
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1 Setting Up
1.1 Packages
tidyverse - basic package for data wrangling
readxl - allows inputs of excel files
readr - allows inputs of text files
jtools -
summfunctions for better regression outputsbroom - result organization with tidy tibbles
stargazer - better organized regression outputs
1.2 Output Tables
Here is a basic example of linear regression. Age and Weight of an individual is used to predict the blood pressure.
model <- lm(bloodp~age+weight, data=bloodp)
summ(model)MODEL INFO:
Observations: 11
Dependent Variable: bloodp
Type: OLS linear regression
MODEL FIT:
F(2,8) = 168.76, p = 0.00
R² = 0.98
Adj. R² = 0.97
Standard errors: OLS
-------------------------------------------------
Est. S.E. t val. p
----------------- ------- ------- -------- ------
(Intercept) 30.99 11.94 2.59 0.03
age 0.86 0.25 3.47 0.01
weight 0.33 0.13 2.56 0.03
-------------------------------------------------The function summ is good and direct in printing output. The base r summary is okay to use, and the more intricate stargazer is good too.
summ will be used mostly here when printing regression outputs.
2 Initial Statistics
2.1 Correlation
2.1.1 One Variable
Correlation is the strength of linear assosciation. It can be sensitive to outliers. The pull function is used to print out the correlation value more.
cor <- corr_1 %>%
summarise(r=cor(X, Y)) %>%
pull(r)
cor[1] 0.8351438Correlations can also be visualized through scatterplots which important to any econometric analysis. Method = “lm” produces the blue linear regression line.
ggplot(corr_1, aes(x=X, y=Y))+
geom_point(alpha=0.5)+
geom_smooth(method = "lm", se=F)
2.1.2 Multiple Variables
ADD
3 Understanding Linear Regression
Linear regression can be performed by:
lm.model <- lm(Cancer_Diagnosis~Median_Income+Median_Age+Percent_Black, data=cancer_test)
lm.res <- augment(lm.model) # visualize all residuals in table form3.1 Least Square Lines
The following code is for two variables:
lm.ls <- lm.res %>%
summarize(x.sd=sd(Median_Age), y.sd=sd(Cancer_Diagnosis),
cor=cor(Cancer_Diagnosis, Median_Age)) %>%
mutate(slope=(x.sd/y.sd)*cor) # Slope = 0.015When we look at the lm model, the slope is also 0.015.
3.2 Visualizing Assumptions
a. Linearity (scatterplot + residual plot - residuals needs to be random)
b. Nearly normal residuals (histogram of residuals or QQ residual plot)
c. Constant variability (residual plot)
Link for interactive regression diagnostic test.
a <- ggplot(lm.res, aes(x=.fitted, y=.resid))+
geom_point()+
geom_hline(yintercept = 0, linetype="dashed", color="red")+
labs(title="Residuals vs Fitted Values", x="Fitted Values", y="Residuals")
b <- ggplot(lm.res, aes(x=.resid))+
geom_density()+
labs(title="Histogram of residuals", x="Residuals") #geom_density can also be added
c <- ggplot(lm.res, aes(sample=.resid))+
stat_qq()+
stat_qq_line()
4 Robust Standard Errors
5 Dummy Variables
Group 1, Group 2, and Group 3 are all dummy variables (0 or 1). When regressors are dummy variables you need to regress it without the intercept term.
reg <- lm(Earnings ~ -1+Group1+Group2+Group3, data=dummy)
summ(reg)MODEL INFO:
Observations: 4266
Dependent Variable: Earnings
Type: OLS linear regression
MODEL FIT:
F(3,4263) = 3842.49, p = 0.00
R² = 0.73
Adj. R² = 0.73
Standard errors: OLS
------------------------------------------------
Est. S.E. t val. p
------------ ---------- -------- -------- ------
Group1 22880.48 467.45 48.95 0.00
Group2 25080.17 380.02 66.00 0.00
Group3 27973.63 404.78 69.11 0.00
------------------------------------------------6 Lagged Regression
lag1 <- lag %>%
mutate(Work_1 = lag(Work, 1)) %>%
mutate(Work_2 = lag(Work, 2))
kable(head(lag1))| Year | Income | Work | Work_1 | Work_2 |
|---|---|---|---|---|
| 2000 | 55 | 30 | NA | NA |
| 2001 | 60 | 31 | 30 | NA |
| 2002 | 65 | 32 | 31 | 30 |
| 2003 | 70 | 33 | 32 | 31 |
| 2004 | 75 | 34 | 33 | 32 |
| 2005 | 80 | 35 | 34 | 33 |
Using this we can perform linear regression the normal way:
reg <- lm(Income ~ Work+Work_1+Work_2, data=dataset)7 Hypothesis Testing
7.1 t-test b=0
7.2 Confidence Interval
sessionInfo()R version 4.0.0 (2020-04-24)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRblas.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.0/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] gridExtra_2.3 jtools_2.1.1 stargazer_5.2.2 broom_0.5.6
[5] readxl_1.3.1 forcats_0.5.0 stringr_1.4.0 dplyr_0.8.5
[9] purrr_0.3.4 readr_1.3.1 tidyr_1.0.3 tibble_3.0.1
[13] ggplot2_3.3.0 tidyverse_1.3.0 knitr_1.28 workflowr_1.6.2
loaded via a namespace (and not attached):
[1] Rcpp_1.0.4.6 lubridate_1.7.8 lattice_0.20-41 assertthat_0.2.1
[5] rprojroot_1.3-2 digest_0.6.25 R6_2.4.1 cellranger_1.1.0
[9] backports_1.1.6 reprex_0.3.0 evaluate_0.14 highr_0.8
[13] httr_1.4.1 pillar_1.4.4 rlang_0.4.6 rstudioapi_0.11
[17] whisker_0.4 Matrix_1.2-18 rmarkdown_2.6 labeling_0.3
[21] splines_4.0.0 pander_0.6.3 munsell_0.5.0 compiler_4.0.0
[25] httpuv_1.5.2 modelr_0.1.7 xfun_0.19 pkgconfig_2.0.3
[29] mgcv_1.8-31 htmltools_0.5.0 tidyselect_1.1.0 fansi_0.4.1
[33] crayon_1.3.4 dbplyr_1.4.3 withr_2.2.0 later_1.0.0
[37] grid_4.0.0 nlme_3.1-147 jsonlite_1.6.1 gtable_0.3.0
[41] lifecycle_0.2.0 DBI_1.1.0 git2r_0.27.1 magrittr_1.5
[45] scales_1.1.1 cli_2.0.2 stringi_1.4.6 farver_2.0.3
[49] fs_1.4.1 promises_1.1.0 xml2_1.3.2 ellipsis_0.3.0
[53] generics_0.0.2 vctrs_0.3.0 tools_4.0.0 glue_1.4.1
[57] hms_0.5.3 yaml_2.2.1 colorspace_1.4-1 rvest_0.3.5
[61] haven_2.2.0