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Parameter Estimation, Dummies, & Model Fit We know mechanically how to “run a regression”…but how are the parameters actually estimated? How can we handle.

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Presentation on theme: "Parameter Estimation, Dummies, & Model Fit We know mechanically how to “run a regression”…but how are the parameters actually estimated? How can we handle."— Presentation transcript:

1 Parameter Estimation, Dummies, & Model Fit We know mechanically how to “run a regression”…but how are the parameters actually estimated? How can we handle “categorical” explanatory (independent) variables? What is a measure of “goodness of fit” of a statistical model to data?

2 Example: Alien Species Exotic species cause economic and ecological damage Not all countries equally invaded Want to understand characteristics of country that make it more likely to be “invaded”.

3 Understanding Invasive Species Steps to improving our understanding: 1. Generate a set of hypotheses (so they can be “accepted” or “rejected”) 2. Develop a statistical model. Interpret hypotheses in context of statistical model. 3. Collect data. Estimate parameters of model. 4. Test hypotheses.

4 2 Hypotheses (in words) We’ll measure “invasiveness” as proportion of Alien/Native species (article by Dalmazzone). 1. Population density plays a role in a country’s invasiveness. 2. Island nations are more invaded than mainland nations.

5 Population Density

6 Island vs. Mainland

7 Variables Variables: Dependent: Proportion of number of alien species to native species in each country. Independent: Island? Population Density GDP per capita Agricultural activity

8 Computer Minimizes e i 2 Remember, OLS finds coefficients that minimize sum squared residuals Graphical representation Why is this appropriate? Can show that this criterion leads to estimates that are most precise unbiased estimates.

9 Dummy Variable Generally: Male/Female; Pre-regulation/Post-regulation; etc.. Use a “Dummy Variable”. Value = 1 if country is Island, 0 otherwise. More generally, if n categories, use n-1 dummies. E.g. if want to distinguish between 6 continents Problem: Lose “degrees of freedom”.

10 A Simple Model A simple linear model looks like this: Dummy changes intercept (explain). Interaction dummy variable? E.g. Invasions of island nations more strongly affected by agricultural activity.

11 Translating our Hypotheses 2 Hypotheses Hypothesis 1: Population: Focus on  3 Hypothesis 2: Island: Focus on  2 “Hypothesis Testing”… forthcoming in course. Parameter Estimates: Value Std.Error t value Pr(>|t|) (Intercept) Island Pop.dens GDP Agr

12 “Goodness of Fit”: R 2 “Coefficient of Determination” R 2 =Squared correlation between Y and OLS prediction of Y R 2 =% of total variation that is explained by regression, [0,1] OLS maximizes R 2. Adding independent cannot  R 2 Adjusted R 2 penalizes for # vars.

13 Answers Island nations are more heavily invaded (.0623) Not significant (p=.46) Population density has impact on invasions (.001) Significant (p=.0000) R 2 =.80; about 80% of variation in dependent variable explained by model. Also, corr(A,Ahat)=.89


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