Maximum likelihood estimation why it is used in finance

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Biometrika 43 , — Reid, N. Statistical Science 3 , — Sargan, J. Shoji, I. Journal of Time Series Analysis 18 , — Stochastic Analysis and Applications 16 , — The model assumes that each individual receives a wage offer one of the three potential wages and then decides on the number of hours of work.

Individuals are assumed to choose their preferred option by weighing the enjoyment from consuming their earnings against the disutility of work for each alternative level of hours. While the utility of consumption increases with consumption, the marginal disutility of work increases as the time spent at work increases. The observed differences in outcomes are thus the result of unobserved differences among individuals. The model depends on three parameters that measure the trade-off between consumption and work as well as the importance of the unobserved component in the distribution of individual choices.

Therefore, two independent probability parameters characterize the distribution of wages. The likelihood of the observed sample is then the product over all observations of the probabilities that the hours of work and the wage are observed. For non-participants, the wage is missing and zero hours are observed. The logarithm of the likelihood simplifies the analysis and the search for the parameter values that yield the largest sample likelihood.

In this example, it is the weighted sum of the logarithm of the probability of not working and the weighted sum of the summed logarithms of the probabilities of working positive hours for each possible level of hours and for all possible wages. The need to evaluate and code such an expression is an obvious disadvantage of the maximum likelihood method: in complex cases, it is difficult and time consuming to establish that the log-likelihood is correctly coded.

In the current, simple case, the log likelihood is easily maximized using conventional numerical methods both commercial statistical software and free, open-source math software like R or SageMath supply an optimizer.

The Illustration displays the observed distribution of outcomes and the predicted distribution using the maximum likelihood method for the five parameters of the model. The predicted distribution is similar, although not identical, to the observed distribution. The model for the preferences would yield a labor supply function that takes a simple log linear form, such that the logarithm of the hours of work is linearly related to the logarithm of the wage rate.

If the observations with zero hours are ignored, the wage elasticity can be estimated by a simple regression over the positive hours and positive wages only. The maximum likelihood estimates give an implied elasticity of 2.

The difference between the two estimates suggests that the parameter estimates are sensitive to the precise specification of the model and to the sample composition.

Since the wage offer distribution and the distribution of the unobserved taste components are independent, the difference between the two estimates arises because of sample selection the exclusion from the regression estimation sample of the observations for non-participating individuals and because the observed hours are not determined according to the labor supply function.

The shape of the log-likelihood describes how precisely the parameters are estimated. Figure 1 shows how the log-likelihood changes as the value of the implied wage elasticity varies, keeping all other parameters at their estimated values.

Values of the implied wage elasticity smaller than 2. Furthermore, the 0. Economic models and likelihood methods can be used together in evaluation studies. Suppose a policymaker wants to understand the labor supply response to the introduction of a minimum wage.

The minimum wage does not operate in the first region in either period 1 or period 2, and it operates in the second region only in period 2.

Assume, to simplify, that four independent random samples are observed, one for each region and time period, as shown in the upper part of Figure 2. The distribution of the random sample in the second region and in the second period across labor supply and wages after the introduction of the minimum wage is shown in the bottom right part of the table region 2 in period 2. Each cell in the table reports the number of observations for a given number of hours and wage rate. The model assumes that the preferences between work hours and consumption are unaffected by the introduction of the minimum wage; therefore, the preferences are common to all regions and periods.

Furthermore, the wage offer distribution with a minimum wage in region 2 is parametrized independently from the wage offer distribution without a minimum wage in region 1. This explains why the predicted values for region 1 are the same before and after the introduction of a minimum wage in region 2.

Relative to the estimates based on the data described in the Illustration , the behavioral parameters are almost unchanged, while the implied labor supply elasticity is slightly larger, at 2.

These distributional changes may not substantially affect the averages and may thus not be captured by a difference-in-differences DiD approach, which compares the average change in outcome over time in the case with and the case without the minimum wage, a method that attempts to mimic experimental conditions [7]. But the distributional changes are captured by the fitted economic model, as shown in the lower part of Figure 2 : the estimated model predicts a reduction in the number of people who are not working from 17 without the minimum wage to 8.

The predicted distribution implied by the model can be used to calculate the effect of the minimum wage, as measured by the DiD approach. The ability to reproduce the results of the less demanding DiD analysis provides a test of the robustness of the model and of the parameter estimates obtained using the maximum likelihood method.

The difference in the mean hours of work or mean positive hours or wage with or without the minimum wage can be predicted from the specified model. These predicted differences are complex mixtures of the parameters describing the behavioral responses and of the parameters describing the wage offers with and without a minimum wage. The predictions reported in Figure 3 from the fitted model and the point estimates from the DiD analysis are in close agreement, which supports the specification of the economic model.

The model in the example discussed above fits the data well because the data structure is relatively simple in the simplest case, the model needs to predict the probability across only ten cells and does not account for the possibility that the introduction of the minimum wage may reduce the number of job offers received and because the model depends on a large enough number of parameters five in the initial case and six when accounting for the effect of the minimum wage.

In more realistic and complex situations, it will be harder to understand the features that the model should contain in order to provide a faithful representation of the data and of the mechanisms that generate them. The specification search will include a search for a more flexible representation of the behavioral responses and a search for a parsimonious description of the unobserved variability among individual agents that the model seeks to represent.

This search is usually difficult to carry out successfully. More complex models bring further difficulties when a likelihood-based approach is used. Determining or evaluating the likelihood can be difficult, and numerical issues may arise in seeking to locate the maximum of the likelihood. Such numerical issues can require substantial effort to solve.

Often the choice is between a simpler model in which the likelihood estimates are easily obtained and a more complex model in which the estimates are more difficult to characterize or time consuming to obtain.

The specification of a model and the use of a likelihood-based estimation method are not enough to solve the common identification questions that trouble most empirical applications in labor economics. Researchers still need to provide careful arguments for or against the presence of a given explanatory variable in their model. The use of an economic model and the estimation of its parameters using maximum likelihood can be valuable tools for labor economists. That approach allows for the analysis of data from most sources and can provide insight into the determinants of the usual summary a parameter estimate in a regression that a less demanding method, like DiD, will obtain.

Drawing on the example presented in this article, the model could, for instance, be used to calculate a predicted distribution of the population, assuming that the wage distribution is very different, or to assess the effect of the introduction of a minimum wage or an income tax system, assuming that the estimated preferences remain constant.

This approach is useful for broader prospective policy analysis. Recent contributions to the literature provide more examples and discussions of the possibilities and difficulties [8] , [9] , [10] , [11] , [12].

Finally, the model and the estimates are useful for modeling and understanding the distributional effects of policy in different environments or under distinct scenarios.

The illustration of the analysis of the log-likelihood in Figure 1 suggests that such simulation exercises should be carried out over several nearly as likely alternative values for the model parameters.

Modern information technology and computer power make the analysis possible for models significantly more complex and valuable than the one presented here.

In summary, a methodological approach that uses a model plus maximum likelihood allows for the analysis of a more general class of data observational as well as experimental and for a more in-depth analysis of policy design given the parameter estimates. The author thanks an anonymous referee and the IZA World of Labor editors for many helpful suggestions on earlier drafts. The author declares to have observed these principles.

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