A rant on estimation with binary dependent variables (technical) usd to cad exchange rate

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Suppose you are trying to explain some outcome [math]y[/math], where [math]y[/math] is equal to 0 or 1 (e.g., whether someone is a nonsmoker or a smoker) gbp usd rate today. You also have data on a vector of explanatory variables [math]x[/math] (e.g., someone’s age, their gender, their level of education, etc.) and on a treatment variable [math]D[/math], which we will also assume is binary, so that [math]D[/math] is equal to 0 or 1 (e.g., whether someone has attended an information session on the negative effects of smoking).

If you were interested in knowing what the effect of attending the information session on the likelihood that someone is a smoker, i.e., the impact of [math]D[/math] on [math]y[/math] The equation of interest in this case is

where [math]\alpha[/math] is a constant, [math]\beta[/math] is a vector of the coefficients attached to the explanatory variables, [math]\gamma[/math] is the parameter of interest, and [math]\epsilon[/math] is the error term.


This post is about why, in most cases, you should be estimating equation (1) by ordinary least squares, i.e., estimate a linear probability model (LPM) what is a binary system. I have heard and read so many arguments against the LPM and for the probit or logit that I wanted to write something on this usd to zmk. Arguments Typically Made against the LPM

• The error term of a binary variable has a Bernoulli structure, i.e., [math]Var(\epsilon_i)=p_i (1-p_i)[/math], where [math]p_i = Pr(y_i = 1)[/math] gbp to usd forecast 2016. This non-constant variance of the error term means that you have a heteroskedasticity problem and the LPM standard errors will be wrong.

• The LPM can yield values of [math]\hat{y}_i[/math], i.e., predicted values of the dependent variable, outside of the [math][0,1][/math] interval exchange rate usd inr. In other words, the LPM can yield predicted probabilities that are negative or greater than 100%.

• With robust standard errors, the standard errors are correct, and it is very easy it is to implement robust standard errors with most statistical packages euro dollar exchange rate. Indeed, in the package that I use the most, it is simply a matter of adding “, robust” at the end of my estimation command.

• This is only a concern if your reason for estimating equation (1) is to forecast probabilities python example programs. For most readers of this blog, that will not be why they are estimating equation (1) gdp to usd. Rather, they will be interested in knowing the precise value of [math]\gamma[/math].

• Sure, but who says an assumed nonlinear relationship is much better? On their Mostly Harmless Econometrics blog, Angrist and Pischke write:

If the conditional expectation function (CEF) is linear, as it is for a saturated model, regression gives the CEF – even for LPM exchange rate of indian rupee to us dollar. If the CEF is non-linear, regression approximates the CEF. Usually it does it pretty well. Obviously, the LPM won’t give the true marginal effects from the right nonlinear model. But then, the same is true for the “wrong” nonlinear model! The fact that we have a probit, a logit, and the LPM is just a statement to the fact that we don’t know what the “right” model is. Hence, there is a lot to be said for sticking to a linear regression function as compared to a fairly arbitrary choice of a non-linear one! Nonlinearity per se is a red herring.

People who dismiss the LPM, usually by invoking the two arguments above, usually argue in favor of estimating a probit or a logit instead. Here are some arguments one can make against the probit or logit:

• Both the probit and the logit can lead to identification by functional form exchange rate usd pound. If you are interested in identifying the causal relationship flowing from [math]D[/math] to [math]y[/math], i.e., in precisely estimating [math]\gamma[/math], you want to avoid this.

So when should you use an LPM, and when should you use a probit or a logit? If you have experimental data, i.e., if values of [math]D[/math] were randomly assigned, there is no harm in estimating a probit or a logit — your estimate of [math]\gamma[/math] is cleanly identified because of the random assignment. If you want to forecast the likelihood that something will happen, estimate a probit or a logit.

But if you are interested in estimating the causal impact of [math]D[/math] on [math]y[/math] and have any reason to believe that your identification is less than clean, if you want to use fixed effects, and if you are not interested in forecasting the value of [math]y[/math], you should prefer the LPM with robust standard errors. Conclusion

I have made the points above several times over the last few years, in conversations with colleagues, when advising students, in referee reports, etc. But every once in a while, I will get admonished by an anonymous reviewer for my use of the LPM, and so I wanted to write something about it.

Ultimately, I think the preference for one or the other is largely generational, with people who went to graduate school prior to the Credibility Revolution preferring the probit or logit to the LPM, and with people who went to graduate school during or after the Credibility Revolution preferring the LPM.

As always, the right way to approach things is probably to estimate all three if possible, to present your preferred specification, and to explain in a footnote (or show in an appendix) that your results are robust to the choice of estimator.


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