OVB and IV

Paulo Fagandini

Nova SBE

Review

Review

• Corrigendum: Economics vs Statistical Significance
• OVB
• IV Estimator

Exercises

https://moodle.novasbe.pt/mod/folder/view.php?id=4021

Exercise 2.6

For each of the following situations, determine the sign of the expected bias due to variable omission:

1. In an equation for the demand for peanut butter, the impact on the coefficient of disposable income of omitting the price of peanut butter variable.
2. In an earnings equation, the impact on the coefficient of experience of omitting the variable age.
3. In a production function, the impact on the coefficient of labour of omitting the capital variable.
4. In an annual equation for corn yields per acre (in year \(t\)), the impact on the coefficient of rainfall in year \(t\) of omitting average temperature that year.
5. In an equation for annual consumption of apples in the US, the impact on the coefficient of the price of bananas of omitting the price of oranges.
6. In an equation for student grades ont he first midterm in this class, the impact on the coefficient of total hours studied (for the test) of omitting hours slept the night before the test.

1. \(Corr(Z,Y)<0\) \(Corr(X,Z)>0\), bias \(-\)
2. \(Corr(Z,Y)>0\) \(Corr(X,Z)>0\), bias \(+\)
3. \(Corr(Z,Y)>0\) \(Corr(X,Z)>0\), bias \(+\)
4. ? ? ?
5. \(Corr(Z,Y)>0\) \(Corr(X,Z)>0\), bias \(+\)
6. \(Corr(Z,Y)>0\) \(Corr(X,Z)<0\), bias \(-\)

Exercise 99.9

Suppose you want to estimate the effect of Price on Ice Cream sales. You collect 200 datapoints.

\[ sales = \beta_0 + \beta_{price} price + u \]

Exercise 99.9

ols <- lm(sales ~ price, data = data)
summary(ols)

Call:
lm(formula = sales ~ price, data = data)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.2422 -0.6196  0.0101  0.5171  2.9061 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 198.83877    0.35918   553.6   <2e-16 ***
price        -7.90647    0.02968  -266.4   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.9398 on 198 degrees of freedom
Multiple R-squared:  0.9972,    Adjusted R-squared:  0.9972 
F-statistic: 7.096e+04 on 1 and 198 DF,  p-value: < 2.2e-16

Can you trust this estimate \(\beta_1=-7.91\)?

Exercise 99.9

Let’s use \(temp\) as an instrument

library(AER)
iv <- ivreg(sales ~ price | temp, data = data)
summary(iv)

Call:
ivreg(formula = sales ~ price | temp, data = data)

Residuals:
      Min        1Q    Median        3Q       Max 
-2.642106 -0.622538  0.003097  0.609989  2.754196 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 199.92686    0.37612   531.6   <2e-16 ***
price        -7.99796    0.03111  -257.1   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.962 on 198 degrees of freedom
Multiple R-Squared: 0.9971, Adjusted R-squared: 0.9971 
Wald test: 6.611e+04 on 1 and 198 DF,  p-value: < 2.2e-16 

Exercise 99.9

Cov(price, U): 0.033

Cov(price, temp): 16.006

Cov(sales, temp): -128.015

So the instrument is exogenous and relevant!

\[\hat{\beta}_{iv}=\frac{-128.015}{16.006}=-7.998\]