A researcher had a look at a newborn birthweight study where a think-tank was looking at the determinants of birthweight. They collected information at a hospital and obtained a large sample with newborns birthweight (bwght), mothers consumption of cigarettes per day (cigs), how many units of alcohol the mother drank per week (drink) and mothers age (mage). The regression model was given by:
After evaluating the study, the researcher claimed that the regression model was missing a key variable - mothers education. Based on that suggestion and your understanding of econometrics, read the following sentences:
I) If mothers education is a relevant variable in the model, then the estimates for the slope coefficients will be biased because E(u|cigs, drinks, mage)
II) If mothers education is a relevant variable in the model, then the estimates for the slope coefficients will be biased because
cigs, drinks, mage
III) Mothers education is expected to be negatively correlated with consumption of cigarettes and drinks.
IV) Considering only mothers education as an omitted variable and a positive correlation between mothers education and bwght, a simple linear regression model with only cigs as explanatory variable would underestimate the slope coefficient for cigs.
State the correct sentence below:
a) ( )I and IV are correct.
b) ( ) Only I is correct.
c) ( ) II, III and IV are correct.
d) ( ) I, III and IV are correct.
e) ( ) I, II and III are correct.
I) If mothers education is a relevant variable in the model, then the estimates for the slope coefficients will be biased because E(u|cigs, drinks, mage)
II) If mothers education is a relevant variable in the model, then the estimates for the slope coefficients will be biased because
III) Mothers education is expected to be negatively correlated with consumption of cigarettes and drinks.
IV) Considering only mothers education as an omitted variable and a positive correlation between mothers education and bwght, a simple linear regression model with only cigs as explanatory variable would underestimate the slope coefficient for cigs.
State the correct sentence below:
a) ( )I and IV are correct.
b) ( ) Only I is correct.
c) ( ) II, III and IV are correct.
d) ( ) I, III and IV are correct.
e) ( ) I, II and III are correct.
Passo 1
Beleza, o enunciado diz que tivemos um modelo de regressão linear pra explicar o peso ao nascer (bwght) a partir de três variáveis: cigarros por dia (cigs), bebidas por semana (drink) e idade da mãe (mage). Mas alguém falou: “Faltou a educação da mãe!” – e isso pode dar viés de variável omitida.
Conteúdo: viés de variável omitida em Mínimos Quadrados Ordinários (MQO).
O que vamos precisar:
Conteúdo: viés de variável omitida em Mínimos Quadrados Ordinários (MQO).
O que vamos precisar:
- Entender por que a omissão quebra a condição de “erro não-correlacionado” .
- Calcular o sinal do viés quando omitimos educação, sabendo como ela se relaciona com cigs e com bwght.
Passo 2
Sentença I:
“If mothers education is a relevant variable in the model, then the estimates for the slope coefficients will be biased because .”
Como a educação da mãe (digamos ) afeta o peso e está correlacionada com
, o termo de erro
absorve essa educação. Isso faz . Logo, I está certa.
“If mothers education is a relevant variable in the model, then the estimates for the slope coefficients will be biased because .”
Como a educação da mãe (digamos
Passo 3
Sentença II:
“If mothers education is a relevant variable in the model, then the estimates for the slope coefficients will be biased because .”
O viés de variável omitida não vem de heterocedasticidade (); vem da correlação entre e as regressoras. Então II está errada.
“If mothers education is a relevant variable in the model, then the estimates for the slope coefficients will be biased because .”
O viés de variável omitida não vem de heterocedasticidade (); vem da correlação entre
Passo 4
Sentença III:
“Mothers education is expected to be negatively correlated with consumption of cigarettes and drinks.”
Faz sentido: mães mais escolarizadas tendem a fumar e beber menos. Então a correlação entre e
/
é negativa. III está certa.
“Mothers education is expected to be negatively correlated with consumption of cigarettes and drinks.”
Faz sentido: mães mais escolarizadas tendem a fumar e beber menos. Então a correlação entre
Passo 5
Sentença IV:
“Considering only mothers education as an omitted variable and a positive correlation between mothers education and bwght, a simple linear regression model with only cigs as explanatory variable would underestimate the slope coefficient for cigs.”
Vamos ao viés formal: omitimos . Chamemos
e
. O viés em MQO é
“Considering only mothers education as an omitted variable and a positive correlation between mothers education and bwght, a simple linear regression model with only cigs as explanatory variable would underestimate the slope coefficient for cigs.”
Vamos ao viés formal: omitimos
Sabendo que
-
(mais educação aumenta ), - (mais educação, menos cigarro),
temos viés negativo. Ou seja, o coeficiente estimado de
sai menor (mais negativo) do que o real. Isso significa que a regressão simples “subestima” o valor do coeficiente—logo IV também está certa.
Resposta
d) I, III e IV são corretas.