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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
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.
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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