rm(list = ls()) # Clear all files from your environment
gc() # Clear unused memory used (Mb) gc trigger (Mb) limit (Mb) max used (Mb)
Ncells 581592 31.1 1326975 70.9 NA 669431 35.8
Vcells 1068585 8.2 8388608 64.0 16384 1852345 14.2 cat("\f") # Clear the console graphics.off() # Clear all graphsBias of an estimator is the difference between the expected value of the estimator and true value of the parameter being estimated (mean, median, etc). There would be no bias if expected value equals the true parameter value.
Measures how far the average of the estimates from repeated samples is from the actual parameter.
Negative bias means you’re consistently underestimating the true effect (your estimates are smaller than they should be).
Positive bias means you’re consistently overestimating the true effect (your estimates are larger than they should be).
What is OVB?
Does it go away as you increase the sample size?
Does it go away as you add more variables?
library("AER")Warning: package 'AER' was built under R version 4.3.3Loading 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: survivaldata("CigarettesSW")
cig <- CigarettesSW
# excluding taxes as price in data already includes taxes, want to isolate it
cig$price_ex_taxes <- cig$price - cig$taxs
# per pack metrics
cig$price_per_pack <- cig$price_ex_taxes / cig$packs
cig$tax_per_pack <- cig$taxs / cig$packs
#summary
head(cig) state year cpi population packs income tax price taxs
1 AL 1985 1.076 3973000 116.4863 46014968 32.5 102.18167 33.34834
2 AR 1985 1.076 2327000 128.5346 26210736 37.0 101.47500 37.00000
3 AZ 1985 1.076 3184000 104.5226 43956936 31.0 108.57875 36.17042
4 CA 1985 1.076 26444000 100.3630 447102816 26.0 107.83734 32.10400
5 CO 1985 1.076 3209000 112.9635 49466672 31.0 94.26666 31.00000
6 CT 1985 1.076 3201000 109.2784 60063368 42.0 128.02499 51.48333
price_ex_taxes price_per_pack tax_per_pack
1 68.83334 0.5909137 0.2862855
2 64.47500 0.5016159 0.2878603
3 72.40833 0.6927528 0.3460535
4 75.73334 0.7545940 0.3198787
5 63.26666 0.5600627 0.2744248
6 76.54166 0.7004284 0.4711211This is a panel data on cigarette consumption for the 48 continental US States from 1985–1995.
\(Cigarette \ pack \ consumption = \beta_0 + \beta_1 * income_i + \beta_2 * price \ per \ pack_i + \beta_3 * taxes \ per \ pack_i + \epsilon_i\)
\(Y_i\) = Number of packs consumed
\(\beta_0\) = Intercept (constant) term
\(\beta_1\) = Slope coefficient representing the change in pack consumption per unit change in income
\(X_1\) = State personal income
\(\beta_2\) = Slope coefficient representing the change in pack consumption per unit change in price per pack
\(X_2\) = Average price per pack during fiscal year (excluding taxes)
\(\beta_2\) = Slope coefficient representing the change in pack consumption per unit change in taxes per pack
\(X_2\) = Average excise taxes for fiscal year per pack, including sales tax.
\(\epsilon_i\) = Error term representing enexplainted variation
cig_lm <- lm(packs ~ income + price_per_pack + tax_per_pack,
data = cig)\(Cigarette \ pack \ consumption = \beta_0 + \beta_1 * income_i + \beta_2 * price \ per \ pack_i + \epsilon_i\)
ovb_cig_lm <- lm(packs ~ income + price_per_pack,
data = cig)Ommited variable bias has 2 conditions:
Based on the below graph we can see tax per pack has a strong correlation with both the dependent variable (packs -0.8) and the key independent variable (price per pack -0.92), satisfying both conditions.
library(ggcorrplot)Loading required package: ggplot2# excluding non numeric numbers
Cig_numeric <- cig[, sapply(cig, is.numeric)]
# Correlation matrix
Cig_cor_matrix <- cor(Cig_numeric,
use = "complete.obs")
# P value
p.mat <- ggcorrplot::cor_pmat(Cig_numeric)
# Correlation graph
ggcorrplot(corr = Cig_cor_matrix,
method = "square",
type = "full",
title = "Correlation Plot",
colors = c("red", "white", "green"),
lab = TRUE,
lab_size = 2,
p.mat = p.mat,
insig = "pch",
pch = 4,
hc.order = TRUE,
tl.cex = 8,
tl.col = "black",
digits = 2)
Given that we know:
The correlation between price per pack (A) and tax per pack (B) is positive
The effect of tax (B) on pack consumption (Y) is negative
We can predict the bias is likely to be negative.
Full model —> cig_lm
Short model —> ovb_cig_lm
library(stargazer)
Please cite as: Hlavac, Marek (2022). stargazer: Well-Formatted Regression and Summary Statistics Tables. R package version 5.2.3. https://CRAN.R-project.org/package=stargazer # side by side
stargazer(cig_lm, ovb_cig_lm,
type = "text")
==================================================================
Dependent variable:
----------------------------------------------
packs
(1) (2)
------------------------------------------------------------------
income 0.000 -0.000
(0.000) (0.000)
price_per_pack -34.218*** -44.707***
(7.689) (3.372)
tax_per_pack -18.064
(11.919)
Constant 150.979*** 152.410***
(3.445) (3.337)
------------------------------------------------------------------
Observations 96 96
R2 0.700 0.692
Adjusted R2 0.690 0.685
Residual Std. Error 14.409 (df = 92) 14.509 (df = 93)
F Statistic 71.422*** (df = 3; 92) 104.526*** (df = 2; 93)
==================================================================
Note: *p<0.1; **p<0.05; ***p<0.01As seen above you can see after we introduced the tax variable, the price coefficient becomes more negative when tax is omitted, meaning the effect of price on cigarette consumption appears stronger in the short model.
This indicates negative bias when tax is omitted from the model.
In this case tax is an important variables on the impact of consumption, cigarette companies have often tried to lower prices to offset taxes but it shows that taxes provides an important influence on overall pack consuption. Failing to control for it will misattribute some of its effect to price (as they are positively correlated), causing for a bias.