Restricted maximum likelihood estimate

To use the likelihood ratio method to test the hypothesis, we will require the restricted maximum likelihood estimate. Under the hypothesis,the model is the one in Section 15.2.2. The restricted estimate is given in (15-12) and the equations which follow. To obtain them, we make a small modification in our algorithm above. We replace step (3) with... 

FO and FOCE with maximum likelihood or FOCE with restricted maximum likelihood estimation provided as well as generalized least squares separate application that can connect with varied data sources (Enterprise edition)... 

Since the final form of a maximum likelihood estimator depends on the assumed error distribution, we partially answered the question why there are different criteria in use, but we have to go further. Maximum likelihood estimates are only guaranteed to have their expected properties if the error distribution behind the sample is the one assumed in the derivation of the method, but in many cases are relatively insensitive to deviations. Since the error distribution is known only in rare circumstances, this property of robustness is very desirable. The least squares method is relatively robust, and hence its use is not restricted to normally distributed errors. Thus, we can drop condition (vi) when talking about the least squares method, though then it is no more associated with the maximum likelihood principle. There exist, however, more robust criteria that are superior for errors with distributions significantly deviating from the normal one, as we will discuss... 

The system of three equations (cost and two shares) can be estimated as discussed in the text. Invariance is achieved by using a maximum likelihood estimator. The five parameters eliminated by the restrictions can be estimated after the others are obtained just by using the restrictions. The restrictions are linear, so the standard errors are also striaghtforward to obtain. 

Linear mixed effects (LME) models express the response variable as a linear function of both the fixed effects and the random effects, with an additive within-unit error, see Laird and Wase (1) or Searle et al. (2) for a good review of methodology. The frequentist approach to LME models is generally Ukelihood-based, with restricted maximum likelihood (REML) being the preferred method of estimation (3). 

One criticism of PL is that it does not take into account the loss of degrees of freedom in estimating 0f, which leads to a modification of the PL objective function called restricted maximum likelihood (REML)... 

Maximum Likelihood Estimates Model estimated Jul 31, 2002 Dependent variable Weighting variable Number of observations Iterations completed Log likelihood function Restricted log likelihood Chi squared Degrees of freedom Prob[ChiSqd > value] =... 

These assumptions are not overly restrictive. Since the value of u is due to many factors acting in opposite directions, it should be expected that small values of u occur more frequently than large values, and that is a variable with a probability distribution centered at zero and having a finite variance o. This is true when the form of Eq. (7.112) is close to the correct relationship. Because of the many factors involved, the central limit theorem would further suggest that u has a normal distribution, which gives the parameter estimates the desirable property of being maximum-likelihood estimates. Later on in the discussion, it will be shown that the regression method can handle cases where o is not constant, and where u is not independent of X. 

In the panel data models estimated in Example 21.5.1, neither the logit nor the probit model provides a framework for applying a Hausman test to determine whether fixed or random effects is preferred. Explain. (Hint Unlike our application in the linear model, the incidental parameters problem persists here.) Look at the two cases. Neither case has an estimator which is consistent in both cases. In both cases, the unconditional fixed effects effects estimator is inconsistent, so the rest of the analysis falls apart. This is the incidental parameters problem at work. Note that the fixed effects estimator is inconsistent because in both models, the estimator of the constant terms is a function of 1/T. Certainly in both cases, if the fixed effects model is appropriate, then the random effects estimator is inconsistent, whereas if the random effects model is appropriate, the maximum likelihood random effects estimator is both consistent and efficient. Thus, in this instance, the random effects satisfies the requirements of the test. In fact, there does exist a consistent estimator for the logit model with fixed effects - see the text. However, this estimator must be based on a restricted sample observations with the sum of the ys equal to zero or T muust be discarded, so the mechanics of the Hausman test are problematic. This does not fall into the template of computations for the Hausman test. 

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