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# Parameter Estimation, Dummies, & Model Fit - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Parameter Estimation, Dummies, & Model Fit' - jorn

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Presentation Transcript

• 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?

• 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”.

Steps to improving our understanding:

• Generate a set of hypotheses (so they can be “accepted” or “rejected”)

• Develop a statistical model. Interpret hypotheses in context of statistical model.

• Collect data. Estimate parameters of model.

• Test hypotheses.

• We’ll measure “invasiveness” as proportion of Alien/Native species (article by Dalmazzone).

• Population density plays a role in a country’s invasiveness.

• Island nations are more invaded than mainland nations.

• Variables:

• Dependent: Proportion of number of alien species to native species in each country.

• Independent:

• Island?

• Population Density

• GDP per capita

• Agricultural activity

• 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.

• 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”.

• 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.

• 2 Hypotheses

• Hypothesis 1: Population: Focus on a3

• Hypothesis 2: Island: Focus on a2

• “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

• “Coefficient of Determination”

• R2=Squared correlation between Y and OLS prediction of Y

• R2=% of total variation that is explained by regression, [0,1]

• OLS maximizes R2.

• Adding independent cannot  R2

• Adjusted R2 penalizes for # vars.

• Island nations are more heavily invaded (.0623)

• Not significant (p=.46)

• Population density has impact on invasions (.001)

• Significant (p=.0000)

• R2=.80; about 80% of variation in dependent variable explained by model.

• Also, corr(A,Ahat)=.89