Soc 712 Homework#3

Raven Shan

Explaining MLE results

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For both models on page 4.14 and 4.18, the objective is to examine the effect of education on income using maximum likelihood estimation (MLE). For each model, the regression parameters being estimated are different; for the example on page 4.14, sigma (standard deviation) is given, while the mu (population mean) is the parameter being estimated. Here, we are trying to introduce the independent variable education to explain the mean of the dependent variable, income. Its linear regression model would be as follows: Y ~ N(β0 + β1x, σ).

Contrarily, on page 4.18, sigma is the parameter being estimated, while the mu is known. This second model (where sigma is the parameter being estimated) is far less common but equally as informative as the first. While the first model is examining the effect of education on the mean income, this model is interested in the effect of education on the standard deviation/variation of income. Its regression model would appear as such: Y ~ N(µ, β0 + β1x).

The results of each example should be interpreted differently as well. In the first model, sigma is 2.54, beta 1 (intercept for mu) is -0.65, and beta 2 (slope for mu) is 0.38. A slope of 0.38 means that for every unit increase in education, there is a 0.38 unit increase in mean income; the negative intercept means that among those who have zero years of education, their mean income is expected to be -0.65.

In the second model on page 4.18, the mu (average income) is 3.52, theta 1 (intercept for sigma) is 1.46, and theta 2 (slope for sigma) is .109. A slope of .109 means that for every one unit increase in education, there is a .109 unit increase in the standard deviation of income. An intercept of 1.46 means that among those who have zero years of education, the standard deviation of income is 1.46.

Hypothetically, we could also introduce a second independent variable, age, to both models. In this case, for the first model, we would be interested in determining how age affects the population mean income. My prediction is that initially, as age increases, the average income increases as well. However, there is a threshold at which point the average income either becomes stagnant or decreases (perhaps, when people reach old age). As for the second model, here we would be interested in determining the effect of age on the standard deviation of income. In other words, how can age be used to explain the overall variance in earnings? One hypothesis could be that there is more variation for high income groups than for lower income. However, the only way to accurately determine the relationship between the variables is to test them.