Parameter estimation dummies model fit
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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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Parameter Estimation, Dummies, & Model Fit

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Parameter estimation dummies model fit

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?


Example alien species

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”.


Understanding invasive species

Understanding Invasive Species

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.


2 hypotheses in words

2 Hypotheses (in words)

  • 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.


Population density

Population Density


Island vs mainland

Island vs. Mainland


Variables

Variables

  • Variables:

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

    • Independent:

      • Island?

      • Population Density

      • GDP per capita

      • Agricultural activity


Computer minimizes s e i 2

Computer Minimizes Sei2

  • 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.


Dummy variable

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”.


A simple model

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.


Translating our hypotheses

Translating our Hypotheses

  • 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


Goodness of fit r 2

“Goodness of Fit”: R2

  • “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.


Answers

Answers

  • 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


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