1) What functional form is utilized for logistic (logit) regression? Why would one choose to use Logit rather than a linear probability model? Why would one choose to use a linear probability model rather than logit? Can one include fixed effects in a logit regression?

 

2.      What is the "difference in difference" estimator? Why is it useful? How does one estimate it econometrically?

 

3.      What is not captured by a difference in difference estimator? How can diff-in-diff-in-diff solve this problem?

 

[1] 

Functional form of Logistic regression is log log model, 

Logit (p) = log[(p)/(1-p)] = β0 + β1x1 + β2x2+.........+ βnxn

If an individual wants to find the solution in binary terms, such as yes/no and 0/1 and so on, they use logistic regression over linear regression.

If an individual wants to find the relationship between the independent and dependent variables, they use linear regression over logit regression.

Yes, any fixed effects can be included in logit regression.

[2]

The difference in the difference estimator is the estimator used to calculate the difference in mean outcome after and before treatment in the treatment group, minus the gap in mean outcome after and before treatment in the control group. 

Difference-in-difference is a beneficial method to use when it is not possible to randomize at the person level.

By finding the gap between two pre-versus-post estimators shown above in (D1), this estimator can be found econometrically by subtracting the control group's estimator, which includes the time trend γ, from the treatment group's estimator to obtain δ.

[3]

The major problem with (DID) difference in difference estimates is the failure of the assumption of parallel trend. 

The triple difference in difference in the estimator of the difference requires keeping just one parallel pattern assumption, in ratios. In fact, eliminating bias in the first is the sole aim of subtracting the second difference-in-differences.

Step-by-step explanation

[1]

Logistic regression is the right regression analysis to conduct, if the response variable is binary (dichotomous). It is a predictive analysis. To classify data and to illustrate the relationship between one or more independent ordinal, nominal,  ratio-level or interval variables and one dependent binary variable, logistic/logit regression is used. This is also recognized as Logit regression.

Functional form of logit regression is log log model,

Logit (p) = log[(p)/(1-p)] = β0 + β1x1 + β2x2+.........+ βnxn

The logistic model (or logit model) is used to model the likelihood of an actual class or occurrence, such as fail/pass, lose/win, dead/live or sick/healthy and so on.

An individual chooses Logit regression over the linear because:

The Logit regression model is used to use a given set of explanatory variables to estimate the categorical response variable. It is used to solve problems with classifications. In Logistic Regression, the linear relationship between the independent and dependent variable is not necessary. In Logistic Regression, the values of categorical variables are predicted. A categorical value such as 1 or 0, Yes or No, and so on must be the output of the Logit regression. But linear regression is supposed to use a given set of explanatory variables to estimate a continuous dependent variable and is used to solve problems with regression.A continuous value, such as price, age, etc., must be the result for Linear Regression. Thus, we can conclude that an individual chooses a logit model over linear regression if he wants the result in binary terms such as yes or no, 0 or 1 and so on.

An individual chooses Linear regression over Logit because:

It is used to use a given set of explanatory variables to estimate the continuous response variable and it is used to solve problems with regression. We estimate the significance of continuous variables in Linear regression. Through this, we determine the best fit line that predicts the output easily. A continuous value, such as price, age, etc., must be the result for Linear Regression. The relationship between the independent variable and the dependent variable must be linear in Linear regression. However, we used a given set of explanatory variables to estimate the categorical response variable in logit regression. Thus, if an individual wants to find the relationship between independent and dependent variables then linear regression will provide the best outcome. So, if there is no need to get only the binary value of the dependent variable, linear regression is chosen.

Yes,  fixed effects can be included in logit regression. Instead of fixed constants for sample individuals, prefer intercepts.

Pr (yit = 1) = [exp (αi + xitβ)] / [1 + exp (αi + xitβ)]

This is known as the fixed effect logit model.

This feature is used to monitor unobserved heterogeneity implicitly and reduces the issue of  omitted-variable bias and self-selection, but this is not helpful for panel results. In the logit model, the fixed effect can be included by the probability of realizing a sequence of results conditional on the number of outcome occurrences. Predicted odds and potential average marginal/discrete modifications.

[2]

The difference in difference (or "double difference" or DID) estimator is determined as the difference in average result after and before treatment in the treatment group subtract the gap in average outcome after and before treatment in the control group. 

It is helpful because this estimator depends on a less strict assumption of exchangeability such as the unnoticed variations between care and control groups are the same overtime in the absence of treatment. Hence, Difference-in-difference is a helpful approach to use when it is not possible to randomize at the individual  level.

To estimate the DID estimator the econometric function would be,

This estimator can be interpreted as taking the distinction between the two post versus pre estimators, subtracting the estimator of the control group. Here, δDD is the DID estimator, Y1T is a post estimator, Y0T is a pre-estimator after and before treatment. Y1c and Y0c are pre and post estimators of control groups. 

[3]

The failure of the assumption of parallel trend is one of the major issues with the DID estimates. In many program assessment studies, the failure of the assumption of parallel trend may in fact be a reasonably common problem, causing many variations in difference estimators to be biased. 

It is possible to compute the triple difference estimator as the gap between two estimators of DID. Despite this, we show that to have a causal interpretation, the triple difference estimator does not require two parallel trend assumptions. The theory is that the gap between two biased estimators of difference-in-differences would be impartial as long as the bias of both estimators is the same. In that case, as the triple difference is computed, the bias will be distinguished. This includes the keeping of only one parallel pattern assumption, in ratios. In fact, eliminating bias in the first is the sole aim of subtracting the second difference-in-differences.

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