<< Chapter < Page Chapter >> Page >
This module reviews the basic concepts needed to estimate and understand logit and probit regressions using Stata. It is intended for advanced undergraduates.

Logit and probit models

Introduction

Consider a model that “explains” whether a wife enters the work force. It is straight forward to think of potential explanatory variables—her potential wage rate, the income of her partner, the number of children under the age of 6 in the household, and the number of children in the household between the ages of 6 and 18 are candidates to be independent variables used to explain the wife’s decision to enter the labor force. The dependent variable, Y , however, is a dummy variable because the wife chooses either to enter the labor force ( Y = 1 ) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaamywaiabg2da9iaaigdaaiaawIcacaGLPaaaaaa@3A10@ or not to enter the labor force ( Y = 0 ) . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaabmaabaGaamywaiabg2da9iaaicdaaiaawIcacaGLPaaacaGGUaaaaa@3AC1@ An OLS model of the form:

Y i = β 0 + β 1 x i + ε i MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadMfadaWgaaWcbaGaamyAaaqabaGccqGH9aqpcqaHYoGydaWgaaWcbaGaaGimaaqabaGccqGHRaWkcqaHYoGydaWgaaWcbaGaaGymaaqabaGccaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaeqyTdu2aaSbaaSqaaiaadMgaaeqaaaaa@44B9@

does not make sense. Figure 1 shows what the data of this model might look like when graphed against one of the explanatory variables. Figure 1 also includes the regression line that an OLS estimation of (1) will yield. It is easy to see one problem with this approach—the predicted values of Y that can be greater than 1 and less than 0. In addition, special properties must be attributed to the error term and it is the simple properties ascribed to the error term that make the OLS model so attractive. J. S. Cramer (2003) Logit Models from Economics and Other Fields (Cambridge: Cambridge University Press): 10.

Linear regression line for a discrete dependent variable

Linear representation of a discrete dependent variable
The linear regression line can be a poor representation of a discrete dependent variable.

The logit model

There does exist another approach to the modeling problem—assume that the dependent variable is the probability that the wife is in the labor force . For instance we might assume that we have a linear probability model of the form Pr ( x i ) = β 0 + β 1 x i + ε i . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiuaiaackhadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaacqGH9aqpcqaHYoGydaWgaaWcbaGaaGimaaqabaGccqGHRaWkcqaHYoGydaWgaaWcbaGaaGymaaqabaGccaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaeqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaaiOlaaaa@48F4@ This model can be estimated reasonably successfully if the observed frequencies are well away from their bounds of 0 and 1. For a full discussion of this model see Ladd, G. W. (1966) “Linear Probability Functions and Discriminant Functions,” Econometrica 34 : 873-888. However, is more appealing to assume that the probability varies monotonically with x and remains within the bounds of [0,1], as shown in Figure 2. This S-shaped curve is known as the sigmoid curve and can be represented algebraically for some variable z by: Pr ( z ) = e z 1 + e z . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiuaiaackhadaqadaqaaiaadQhaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaadwgadaahaaWcbeqaaiaadQhaaaaakeaacaaIXaGaey4kaSIaamyzamaaCaaaleqabaGaamOEaaaaaaGccaGGUaaaaa@41EC@

The signoid function.

The S-shaped graphical representation of a signoid function.
The signoid function forces the dependent variable to be between 0 and 1.

We can simplify our analysis by using a bit of algebra. First, the inverse probability is 1 Pr ( z ) = 1 e z 1 + e z = 1 1 + e z . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgkHiTiGaccfacaGGYbWaaeWaaeaacaWG6baacaGLOaGaayzkaaGaeyypa0JaaGymaiabgkHiTmaalaaabaGaamyzamaaCaaaleqabaGaamOEaaaaaOqaaiaaigdacqGHRaWkcaWGLbWaaWbaaSqabeaacaWG6baaaaaakiabg2da9maalaaabaGaaGymaaqaaiaaigdacqGHRaWkcaWGLbWaaWbaaSqabeaacaWG6baaaaaakiaac6caaaa@4ACA@ Thus,

Pr ( z ) 1 Pr ( z ) = e z 1 + e z 1 1 + e z = e z . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaciGGqbGaaiOCamaabmaabaGaamOEaaGaayjkaiaawMcaaaqaaiaaigdacqGHsislciGGqbGaaiOCamaabmaabaGaamOEaaGaayjkaiaawMcaaaaacqGH9aqpdaWcaaqaamaalaaabaGaamyzamaaCaaaleqabaGaamOEaaaaaOqaaiaaigdacqGHRaWkcaWGLbWaaWbaaSqabeaacaWG6baaaaaaaOqaamaalaaabaGaaGymaaqaaiaaigdacqGHRaWkcaWGLbWaaWbaaSqabeaacaWG6baaaaaaaaGccqGH9aqpcaWGLbWaaWbaaSqabeaacaWG6baaaOGaaiOlaaaa@4FB6@

Taking the natural logarithm of (2) gives ln ( Pr ( z ) 1 Pr ( z ) ) = z . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gadaqadaqaamaalaaabaGaciiuaiaackhadaqadaqaaiaadQhaaiaawIcacaGLPaaaaeaacaaIXaGaeyOeI0IaciiuaiaackhadaqadaqaaiaadQhaaiaawIcacaGLPaaaaaaacaGLOaGaayzkaaGaeyypa0JaamOEaiaac6caaaa@4677@ Assuming that z is a linear function of x (and, more generally, of other variables) gives the logit model:

ln ( Pr ( x i ) 1 Pr ( x i ) ) = β 0 + β 1 x i + ε i . MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gadaqadaqaamaalaaabaGaciiuaiaackhadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaeaacaaIXaGaeyOeI0IaciiuaiaackhadaqadaqaaiaadIhadaWgaaWcbaGaamyAaaqabaaakiaawIcacaGLPaaaaaaacaGLOaGaayzkaaGaeyypa0JaeqOSdi2aaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaeqOSdi2aaSbaaSqaaiaaigdaaeqaaOGaamiEamaaBaaaleaacaWGPbaabeaakiabgUcaRiabew7aLnaaBaaaleaacaWGPbaabeaaaaa@52D3@

We can estimate the parameters of this model using maximum likelihood methods . In the probit model the error term is assumed to be normally distributed with a mean of zero and a unit variance. The assumption that the variance is equal to 1 is due to technical considerations. See [Cramer, 22]. In the logit model the error term is assumed to have a standardized logistic distribution . This distribution has a mean of 0 and a variance of 1 and is very similar to a normal distribution with the same mean and variance. The pdf of a logistic distribution is f ( x ) = λ e λ x ( 1 + e λ x ) 2 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaa8aabaWdbiabeU7aSjaadwgapaWaaWbaaSqabeaapeGaeyOeI0Iaeq4UdWMaamiEaaaaaOWdaeaapeWaaeWaa8aabaWdbiaaigdacqGHRaWkcaWGLbWdamaaCaaaleqabaWdbiabgkHiTiabeU7aSjaadIhaaaaakiaawIcacaGLPaaapaWaaWbaaSqabeaapeGaaGOmaaaaaaaaaa@4A65@ , where λ = π 3 1.814 MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaeyypa0ZaaSaaaeaacqaHapaCaeaadaGcaaqaaiaaiodaaSqabaaaaOGaeyisISRaaGymaiaac6cacaaI4aGaaGymaiaaisdaaaa@40B6@ . See Cramer, 24-26 for a fuller discussion of the logistic distribution. While the choice of which model to use generally is personal, it should be noted that the ratio of the parameter of a logit model to the parameter of a probit model (using the same data set) usually varies between 1.6 and 2.0. We focus on the logit model in the balance of this discussion.

Questions & Answers

Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Econometrics for honors students. OpenStax CNX. Jul 20, 2010 Download for free at http://cnx.org/content/col11208/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Econometrics for honors students' conversation and receive update notifications?

Ask