1) What is a "natural experiment" and why are they of use to economists?

2.      Your friend ran a two-stage-least squares regression to perform an instrumental variables analysis. She spent some time looking at the output of her first stage regression: what might she be looking for? Why does she need to look at the first stage regression output before interpreting her second stage output?

3.      How do you interpret log-log, log-lin, and lin-log regressions? When are these appropriate?

1) Natural experiment is basically an observational study wherein the control and treatment variables are not adulterated by the researchers, that is the researchers do not make any changes to the naturally occurring phenomenon by artificially manipulating them rather they allow the control and treatment groups to change naturally without their interferences. The researchers just analyse and observe the changes in the variables. The results obtained are natural randomized results.

Natural experiments are of use to the economists as -

• No Sample limit - since these experiments are not controlled by the researchers, there is no sample limit of the experiment such that they capture the true picture providing a rare testing ground for the economist to undertake their research.
• They are useful when sampling is difficult, that is, in case of health experiments or to know the true impact of disease on a population with different characteristics, natural experiments make it possible to capture true impacts.
• They are also used when the subject is difficult to test, that is, when the study is defined by changes in laws, policies or is about a particular country. This is so because natural observation makes it easy for the researchers to get the true study. 

2) Two stage least square statistical(2SLS)  technique is an extension to ordinary least square (OLS) in which in order to deal with the deficiencies of the result obtained through OLS in the first stage, we make extensions to the OLS model such that the deficiencies can be removed.

After running a 2SLS, she must be observing the output of first stage regression to look for endogeneity in the model, that is to know that there is some significant relation between the error term and the independent variable due to which the results obtained in the first stage are biased and inconsistent.

This is important to observe first stage results before interpreting second stage results because it will help her to identify the requirement of introducing IV (instrumental variable) in the second stage as due to biases and inconsistency the results obtained are not BLUE. 

Also, because only after observing the results of the first stage,she will get to know some crucial things about the model which can badly affect the model in the second stage. Those are:

1. there must be some independent variables that are related to the error term in the model for which IV is required such that the correlation can be removed and  correlation between the independent variable and the error term of the dependent variable becomes zero.
2. There must be some omitted variables that can impact dependent variables but are missing in the model for which it is required to introduce an IV such that the IV can capture the impact of missing variables in the model and remove biases.

By knowing these deficiencies from the first stage model it will become easier for her to interpret the second stage results.

3) Log - log model is the one in which there is log transformation in both the dependent and independent variables. Such that -  

Log Y = B0 + B1logX + e

Here , we interpret this model as - one percentage change in an independent variable that is X, will bring about B1 Percent change in the dependent variable.

  Log lin model is the one in which there is log transformation in the dependent variables,       Such that -  

 logY = B0 + B1 X + e

Here , we interpret this model as - one unit change in an independent variable that is X, will bring about B1 percent change in the dependent variable. 

Lin log model is the one in which there is log transformation in independent variables, Such that -  

 Y = B0 + B1 log X + e

Here , we interpret this model as - one percentage change in an independent variable that is X, will bring about B1 unit change in the dependent variable.

Log - log model is appropriate when it is applied in the data set with only positive values. It is appropriate for the data which has  high skewness as the skewness can be reduced using the log log transformation. Also when we need to know about the elasticities of independent variables with respect to dependent variables applying log - log model is appropriate.  

Log lin and lin log model are mirror images of each other. Linlog models are appropriate in data sets where, as the value of x increases its impact on y is decreasing and basically used when change in x is greater than change in y. While a loglin model is appropriate in those data sets where as the value of an independent variable decreases, its impact on y is increasing and is basically used when change in y is greater than change in x.