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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 “categorical” explanatory (independent) variables? What is a measure of “goodness of fit” of a statistical model to data?

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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”.

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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.

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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.

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Population Density

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Island vs. Mainland

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

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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.

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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”.

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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.

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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) -0.0184 0.0859 -0.2141 0.8326 Island 0.0623 0.0821 0.7588 0.4564 Pop.dens 0.0010 0.0002 6.2330 0.0000 GDP 0.0000 0.0000 3.3249 0.0032 Agr -0.0014 0.0015 -0.9039 0.3763

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“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.

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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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