Forum: open-discussion

RE: censReg with small number of observations [ Reply ] By: Arne Henningsen on 2013-11-12 08:56	[forum:40035]
Thank you, Ott, for your further explanations :-)

RE: censReg with small number of observations [ Reply ]
By: Ott Toomet on 2013-11-12 08:31

[forum:40034]

You are right. Think about sample mean in tiny samples. If you pick iid observations from a normal distribution, the result is normal, no central limit theorem needed (proof trivial with characteristic functions). Hence the asymptotic approximation is not approximation any more but the exact distribution.

The N/(N-1) term originates from the fact that variance (or other nuisance parameters) are typically estimated, not a priori known. These can be corrected for in certain circumstances.

RE: censReg with small number of observations [ Reply ]
By: Arne Henningsen on 2013-11-11 10:57

[forum:40025]

Hej Ott! Thank you for your clarifications. As far as I remember, in finite samples, if all standard Gauss-Markov assumptions are fulfilled and the (unknown true) disturbance terms are normally distributed, the estimated coefficients exactly follow a multivariate t-distribution around the (unknown) true coefficients with the estimated variance covariance matrix. However, as the variance covariance matrix of a vector of variables that is multivariate t-distributed with variance covariance matrix s and (finite) d degrees of freedom is not s but s * d / (d-2), the estimated variance covariance matrix of the OLS estimates is not an unbiased estimate of the "true" variance covariance matrix of the OLS estimates. Right?

RE: censReg with small number of observations [ Reply ]
By: Ott Toomet on 2013-11-11 08:47

[forum:40023]

Hey,
it is not just about small sample bias. Both OLS and ML estimate vcov matrix by _asymptotic_ approximation, in practice using the central limit theorem. Hence, although the estimators are unbiased asymptotically, they may not be correct small sample approximations. I stress here that this is not the same as small sample bias---the latter can be adjusted for in certain cases. However, as central limit theorem may not work for small samples, expect different "correct" variance-covariance estimators to give different results in small samples. See also

Calzolari, G. & Fiorentini, G. Alternative covariance estimators of the standard Tobit model Economics Letters, 1993, 42, 5-13

Ott

RE: censReg with small number of observations [ Reply ] By: Arne Henningsen on 2013-11-11 07:50	[forum:40022]
Yes, we are seldom in a perfect world ;-) In case of a significant number of observations with censoring, censored regression is usually less imperfect than OLS. The questions remains: what is a significant number of observations with censoring? 1%, 2%, 5%, 10%?

RE: censReg with small number of observations [ Reply ] By: Victor Bonami on 2013-11-10 15:21	[forum:40021]
Dear Arne, Thank you for your explanations. I understand that for estimating a censored regression model it is better to use a maximum likelihood method rather than OLS. However in my situation with a very limited number of observations it is still not perfect (at least for assessing the parameter significance). Thanks again, Victor

RE: censReg with small number of observations [ Reply ]
By: Arne Henningsen on 2013-11-09 23:10

[forum:40018]

Dear Victor

lm() estimates the model by the ordinary least squares method, while censReg() estimates the model by the maximum likelihood method -- no matter whether the dependent variable is censored or not. In case of no censoring, lm() and censReg() return the same estimates of the model parameters (except for the variance of the error term) but different variance-covariance matrices and hence, different standard errors, of the estimated parameters. Under certain assumptions, the variance-covariance matrix that is obtained from a correctly specified OLS estimation is unbiased and consistent, while the variance-covariance matrix that is obtained from a correctly specified maximum likelihood estimation is consistent but biased for finite samples. If the "true" data generating process is a censored regression model and you estimate the model by OLS, you get biased and inconsistent estimates, while the maximum likelihood estimation of a censored regression model returns consistent estimates. Thus, using (biased and inconsistent) OLS for estimating a censored regression model with few observations is not necessarily better than a (consistent) maximum likelihood estimation with biased standard errors of the parameters.

Please do not forget to cite the censReg package in your publications.

Best wishes,
Arne

censReg with small number of observations [ Reply ]
By: Victor Bonami on 2013-11-07 11:53

[forum:40002]

Hello,

I am interested in using the tobit model of the censReg package. However I only have a few number of observations and I noticed, doing simulations without censoring, that in this case the standard deviation of the residuals is different than the one obtained with the lm function.

Here is a sample of the code I used:

N<-20
myFactor<-factor(rep(c("A","B"),each=N/2))
myResponse<-rnorm(N)
myLm<-lm(myResponse~myFactor)
myCensReg<-censReg(myResponse~myFactor,left=-Inf)
summary(myLm)$sigma
exp(myCensReg$estimate[3])

On average, the difference between the sigma estimates seems to be inversely proportional to the number of observations.

By repeating this design, the distribution of the p-values obtained with lm is uniform as expected while it is not with censReg, which leads me to believe that I should not use the latter with small number of observations. However I am wondering whether this applies to the tobit model in general or if this is specifically related to its implementation in censReg (or me misusing it)?
In other words should I forget about using a tobit model with so few observations?

Thanks,

Victor