Bootstrapping

1
Bootstrapping the neglected approach to
uncertainty
Paul Kershaw University of South Australia

• European Real Estate Society Conference
• Eindhoven, Nederlands, 15-18 June 2011

2
Overview

• The history of confidence intervals
• Pedagogical predilection to a parametric view
• Real estate research is NOT normal
• Do not provide a measure of probability
• Enter the Jackknife
• Monte Carlo simulation Bootstrapping
• The basic algorithms
• Real World Applications
• A better mousetrap

3
Introduction

• The origins of hypothesis tests is 1279
• Confidence intervals were derived in 1937
• A confidence interval estimates the uncertainty
  about the true value of some population parameter
• 50-year lag before medical journals for example
  advocated their use
• The lazy approach is to assume a normal
  distribution

4
Not Normal

• Very little about real estate can be considered
  to follow a Normal distribution
  includingPrices, Land area, building area, age,
  number of bedrooms, location, physical condition,
  construction, tenants covenant, heating, etc.
• Linear regression techniques are regularly
  applied, averages, standard errors and parametric
  confidence intervals proffered. Why?
• Is it because we are taught to do it that way
  or because we teach it that way gloss over the
  ignored assumptions just give me a number from
  the printout.

5
Not a measure of Probability

• This begs the question what is the confidence
  interval of a correlation coefficient? and leads
  to the second question why is it so rarely
  reported?
• What is a realistic confidence interval for a
  computer generated valuation using a linear
  regression model?
• Most proprietary AVMs provide their own, often
  ill defined, assessment of accuracy that is
  usually somewhat nebulous.

6
Enter the Jackknife

• Early efforts in the 1950s revolved around the
  Jackknife (Quenouille, M).
• The Jackknife provides a technique for
  estimating the bias and standard error of an
  estimate irrespective of the shape of the
  underlying distribution.
• The following example is based upon the work of
  Efron, B 1993. The datapoints are LSAT, the
  average score for the class on a national law
  test, and GPA, the average undergraduate
  grade-point average for the class.

7
Sample Data
8
The basic algorithms

• Compute sample statistics on n separate samples
  of size n-1. Each sample is the original data
  with a single observation omitted.
• Jackknife Heuristic
• Remove one data point only and calculate the
  statistic of interest to give estimate 1
• Repeat for each data points to give estimates 2,
  3, 4 n
• Calculate the percentiles of interest to obtain
  the confidence interval

9
Jackknife Calculations
10
Monte Carlo Bootstrapping

• Monte Carlo simulation caught the imagination of
  practitioners and researchers following Hertz,
  David 1964, Harvard Business Review
• Monte Carlo simulation uses repeated sampling to
  determine the properties of some result of
  interest
• The re-sampling is carried out with replacement
• If we apply this technique to the previous
  Jackknife data we would be Bootstrapping
  Adventures of Baron Munchausen
• Bootstrapping is repeatedly re-sampling with
  replacement, calculating the statistic of
  interest and recording its distribution.

11
Bootstrap Algorithm

• Remark to calculate the dispersion of the mean
• DataArray() n data points
• MeanResults(1000)
• For i 1 to 1000
• Sum0
• For j 1 to n
• Sum Sum DataArray(RandomBetween(1,n))
• Next j
• MeanResults(i) Sum / n
• Next i

12
Real World Application 1

• What annoys me most residential price change
  reporting and hot spotting
• Below are sale prices for Q4 2010 and Q1 2011 for
  Detached houses in Aberfoyle Park, South
  Australia

Median 382,500
Average 409,932
Median 385,000
Average 391,946
Change Median 0.65
Change Average -4.39
13
Bootstrap Results 1000 iterations

• The degree of uncertainty is clearly illustrated.
• The median has a 95 confidence interval of .

14
A better mousetrap

• The traditional approach is to select n from n
  with replacement and calculate statistic of
  interest and repeat m times
• This is inefficient for most statistics of
  interest including the mean, median, standard
  deviation or correlation coefficient
• For example the mean is sum/n
• If for each iteration we remove just one random
  element and replace it with another random
  element we can adjust the sum by subtracting the
  value of the removed element and adding the value
  of the ingoing element
• If n is say 50 we save 48 mathematical operations

15
Summary

• The bootstrap is simple to implement
• The results are meaningful and easy to interpret
• No specious assumptions regarding underlying
  distributions are required
• Widely accepted
• It should be embraced by all researchers and
  practitioners

16
Yesteryears Joys

• Bootstrap Methods Another Look at the Jackknife
• B. Efron
• Source Annals of Statistics Volume 7, Number 1
  (1979), 1-26.