Jason Cook and Dr. Robert Burton, Computer Science
Economic models are built from economic theory for applications ranging from evaluating past policies to predicting future market behavior. Estimation of these models often requires the use of nonlinear optimization routines. As a result, the objective function being optimized is often characterized by local extrema, saddles, and relatively flat areas which make it difficult for search algorithms to converge at global extrema. Currently, only crude and subjective solutions are utilized in applied research, resembling simple guess and check strategies.
Rarely do models contain fewer than three variables; thus simply visualizing the objective function is not a viable option given past graphical capabilities. However, great strides have been made in higher-dimensional visualization in recent years which provides a valuable tool to overcome this dilemma. Professor Robert Burton along with members of his Hyperdimensional Research Group at Brigham Young University, have developed state of the art higherdimensional (nD) visualization software.
Utilizing this software, we propose to open the door for nD visualization of economic phenomena to allow economists to view data—here we focus on objective functions—with any number of explanatory variables. This new visual analytical approach to applied econometrics can be expected to provide a more systematic method to check the robustness of estimation techniques. Our research lays the groundwork for visualizing models with n variables using this sophisticated and developing graphics software facility. In essence, our research and development effort will pave the way for a whole new approach to graphical analysis in economics.
To understand the effect of local extrema on estimation techniques, consider maximum likelihood estimation (MLE). This estimation technique maximizes a likelihood function (Given a set of parameters that describe a line fit to data, a likelihood function assigns a nominal value for the likelihood that these parameters accurately fit the data). Figure 1 shows that optimization using the initial parameters, a1, a2, and a3, would lead to estimates of A, B, and C respectively. In this scenario, B is a global maximum, but MLE with either a1 or a3 as initial parameters would converge at a local minimum, thus biasing the estimate.
Sadly, though this problem is painfully common, very few systematic solutions exist and even fewer are actually utilized in applied work. The most common approach to this problem is to estimate the model using several different starting values. If each optimization routine converges on the same parameters, then the researcher can conclude with some confidence that he had found a global extremum. However, this method is subjective and thus conclusions are in a sense arbitrary.
One particularly problematic type of data is grouped data, where the dependent variable is reported only as an interval of values. For instance, the Census asks people to report their annual income in fairly large intervals, so maximum likelihood estimation techniques using these data are prone to these optimization difficulties.
As an illustration of the benefits of visualization in applied econometrics, we provide an example visualizing in 3D the aforementioned grouped data maximum likelihood estimator. Figure 2 displays the likelihood function for a grouped dependent variable with two explanatory variables. First note the presence of a local maximum at(β~1,β~2) and that optimization with any initial parameters in Area 2 would lead to convergence at this local maximum. Only initial parameters in Area 1 would lead to convergence at the global maximum. Convergence cannot be obtained for initial parameters in Area 3 since the likelihood function is non-concave and becomes flat as the estimates become negative.
Inspection of Figure 2 reveals where the optimizing algorithm would have difficulty locating the global extremum. This figure demonstrates how a graphical depiction can help researchers avoid using initial conditions in Areas 2 and 3. In addition, using initial conditions close to (β~1,β~2) would significantly decrease the iterations required to achieve convergence. This is extremely useful since many nonlinear estimators are very computationally intensive and take hours to run.
In this example, 3D visualization of the objective function is limited to depiction of a model with two exogenous variables. With nD visualizing capabilities, these aforementioned benefits would not be limited to three dimensions. Thus, models with any number of variables would be able to be first graphically depicted and estimation techniques validated prior to any empirical work.