1. Consider the following specification.

\(Doc.Visit=\alpha + \beta Education + \epsilon\) (eq. 1)

  1. In your own terms define \(\alpha\), \(\beta\) and \(\epsilon\).

  2. Now, say you estimate eq.1 and find that \(\hat \beta>0\). What can you and can you not infer from this?

  3. Why does the estimate of \(\beta\) not hold a causal interpretation. Explain precisely using eq. 1 above.

\(~\)

  1. Consider the following data between education of the household head and the number of doctor visits in the household.
    education doc_visit
    1 0
    2 0
    3 1
    4 0
    5 3
    6 4
    7 4
    8 5
    9 4
    10 6
    Table 1.

## 
## Call:
## lm(formula = doc_visit ~ education)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.6545 -0.1848  0.2091  0.5273  0.9515 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.13333    0.59280  -1.912   0.0923 .  
## education    0.69697    0.09554   7.295 8.43e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8678 on 8 degrees of freedom
## Multiple R-squared:  0.8693, Adjusted R-squared:  0.853 
## F-statistic: 53.22 on 1 and 8 DF,  p-value: 8.428e-05

What I have done above with the codes is estimated the specification given by eq. 1 using the data in Table 1.

Now, calculate the sum of the squared residual in Table 1. This is given by manipulating eq. 1 in the form \(\hat \epsilon^{2} = (Doc.Visit - \hat \alpha - \hat \beta \times Education)^2\), for every value of education and doctor visits such that the value of \(\hat \alpha = -1.33\) and value of \(\hat \beta = 0.697\). Add all of them at the end to find the sum of the squared of residual. Note that \(\hat \epsilon\) is called the residual term.

  1. Show your work to calculate the sum of the squared of residual and show the value.

  2. Now keeping value of \(\hat \alpha = -1.33\), change the value of \(\hat \beta\) such that \(\hat \beta=0.75\) and \(\hat \beta=0.60\).

  3. Comparing your answers from parts a. and b., what value of \(\hat \beta\) gives the minimum value of sum of the squared residuals.

  4. Convince yourself (with a certain level of confidence) that \(\hat \alpha = -1.33\) and value of \(\hat \beta = 0.697\) minimizes the sum of the squared residual.

\(~\)

  1. The natural experiment is one way that, if used carefully, allows a researcher to make causal interpretation. Following our in class lecture, consider the specification,

\(SunScreen_t = \alpha + \beta lemonade_t + \gamma temperature + \epsilon\) … eq. 2

  1. Say, you estimate the specification given by eq. 2 and find that \(\hat \beta>0\). What does it say?

  2. Does the estimate of \(\beta\) after estimating specification given by eq. 2 allow you to make causal assessments? Why or why not?

  3. Report an external or a natural factor that directly affects the consumption of lemonade but only affects the consumption of sun screen through consumption of lemonade (that is the external factor does not directly affect the consumption of lemonade). Note that this valid external factor (that meets the definition) is called ``instrument."

  4. Logically explain how the use of a valid instrument allows you to causally identify the effect of lemonade purchase on demand for sun screen.