Maize trade in Southern Africa
Maximum Likelihood
Of all the values these numbers could take, which set makes what I actually observed most likely to have happened?
This is how you find the numbers once the model is written down. It asks a single question: of all the possible values these coefficients could take, which set would make the data I actually observed most likely to have happened?
Ordinary regression fits by minimising the distance between its predictions and the observations, which works when the model is a straight line. The Tobit is not a straight line, so it needs this instead.
What it cannot do
It is worth knowing about mostly because of the caution attached to it. In non-linear models like the Tobit, maximum likelihood can produce biased estimates when the number of time periods is small. That is not a flaw you can inspect in the output. It is a property of the method, and the only defence is to meet the sample conditions in advance.
The equation
β̂ = argmaxβ Σi ln f(yi | xi, β)
- f(y | x, β)
- how likely this observation would be, if β were the truth
- Σ ln
- add that up across every observation, in logarithms so the multiplication becomes addition
- argmax
- pick the β that makes the total as large as it can be
Read it backwards and it is almost conversational: try every possible answer, and keep the one that makes what you actually saw the least surprising.
Further reading
- Greene, W. Econometric Analysis, the maximum likelihood chapters.Where the small-sample caution in this note comes from.
- Gujarati, D. Basic Econometrics.Written for people meeting this for the first time.
From An analysis of maize trade in the Southern African Development Community, MSc (Agricultural Economics), University of Pretoria, 2012. Read it in UPSpace.