Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson1-1 Lesson 1: Analysis of Economic Data is difficult but intuitive

Lesson1-1 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data

Lesson 1:

Analysis of Economic Data is difficult but intuitive


Outline

Capture-Recapture experiment

Estimator

Simulations

What is Statistics?

Sampling

How to estimate unemployment rate?


Capture/Re-captureCapture/Re-capture

Goal:1. Illustrate that how to estimate the population size

when the cost of counting all individuals is prohibitive.

2. Illustrate how intuitive statistics could be. Statistics need not be completely deep, murky, and mysterious. Our common sense can help us to negotiate our way through the course.


Counting the stones

We are interested in knowing the number of black stones in the box.

We only need to do to obtain a reasonable estimate of stones in the box – allowing for errors of counting or estimation.


Two examples

Example #1: The box contains only a small number of stones.


Two examples

Example #2: The box contains a lot of stones that will take days to count.


History and examples of capture / recapture method

Capture-recapture methods were originally developed in the wildlife biology to monitor the census of bird, fish, and insect populations (counting all individuals is prohibitive). Recently, these methods have been utilized considerably in the areas of disease and event monitoring.

http://www.pitt.edu/~yuc2/cr/history.htm


The fish example

Estimating the number (N) of fish in a lake or pond. 1. C fish is caught, tagged, and returned to the

lake. 2. Later on, R fish are caught and checked for tags.

Say T of them have tags. 3. The numbers C, R, and T are used to estimate

the fish population.


Stones in a box

The objective is to estimate the number (N) of fish (represented by black stones) in a pond.

Capture one handful of fish (black stones). Count them and call it C. Mark the fish by replacing the black stones with red stones. Put them back into the pond.

Capture another handful of fish (stones). Count the total number of fish or stones (R) and the number of marked fish or white stones (T).

Based on this information, How to obtain a reasonable estimate of

the number of fish in the pond or stones in the box?


Stones in a box

We know that C/N ≈ T/R Hence, a simple estimate is

N=CR/T C= the number of fish or stones captured

in the first round. R= the total number of fish or stones

captured in the second round. T= the number of marked fish or white

stones captured in the second round.


Stones in a box

N=CR/T is called an estimator. Give me any box of stones, I can use the same “method” or “procedure” or “formula” to estimate the number of stones.Give me a specific box of stones, I can give you an estimate of the number of stones in that box (using the estimator N=CR/T). For example, I estimate that there are 510 stones in the box. “510 stones” is an estimate of number of stones in the box.


Simulations to see the properties of this proposed estimator

How good is the proposed estimator? To see the properties of this proposed estimator, I

have used MATLAB to simulate our Capture-recapture experiment with different numbers of capture (C) and different numbers of recapture (R), relative to the total number of fish in the pond.

Throughout, N=500 and 1000 simulations

That is, I will give you 1000 boxes of stones, each having 500 stones. I am not telling you the number of stones in each box. You will have to produce 1000 estimates of stones in these 1000 boxes. I would like to see how good your estimator (“method” or “procedure” or “formula”) is.


Definition: Estimator

Estimator is a formula or a rule that takes a set of data and returns an estimate of the population quantity (also known as population parameter) we are interested in.

θ(x1,x2,...,xn)


Example: An estimator for the population mean

If we are interested in the population mean, a very intuitive estimator of the population mean based on a sample (x1,x2,...,xn) is

θ(x1,x2,...,xn)= (x1+x2+...+xn)/n

Suppose someone suggest θ(x1,x2,...,xn)= (x1+x2+...+xn+1)/n


Simulating the properties of a sample mean estimator

If we were to study the properties of the following two estimators for the population mean:

θ(x1,x2,...,xn)= (x1+x2+...+xn)/n

versus θ(x1,x2,...,xn)= (x1+x2+...+xn+1)/n

With some basic computing skills, we may perform Monte Carlo simulations to compare their properties.



1. We will need to define a population. Suppose the population consists of 10 balls numbered from 1 to 10 in a bag. We know that the population mean is (1+2+3+4+5+6+7+8+9+10)/10 = 5.5.

2. We will need to define the sampling process. Suppose we draw a sample of size 5 with replacement. For the sample, compute the two sample mean estimates of the population mean.

3. We will need to decide on the number of repetitions. Suppose we will repeat the process for 10,000 times.

4. After repeating the sampling process 10,000 times, we will have 10,000 sample means for each of the estimator, each of them are estimate of the population mean based on the respective samples.



The above simulation is performed using MATLAB. The means of the 10,000 sample means of the two estimators are

5.4990 and 5.6990.

The first estimator appears unbiased. That is, on average, the estimator correctly estimates the population mean.

5.4990 is very closed to 5.5. The second estimator appears biased. That is, on

average, the estimator does not correctly estimate the population mean.

5.6990 is not closed to 5.5.


Which estimator is more desirable?

00.050.10.150.20.250.30.350.40.45

0 5 10 15 20

The estimator with the blue distribution is preferred to the green one because the green one appears to have a mean different from the truth but the blue one does not.

The estimator with the purple distribution is preferred to the blue one because although both appear unbiased, the blue one is less precise, i.e., more likely to yield an estimate that is far from the truth.


Simulation design – via MATLAB

Individual simulation experiment: Create 500 “black” fish, labelled 1 to 500. Capture a random sample of C fish, mark them by

converting their label to zero (i.e., red fish). Capture another random sample of R fish. Count

the number of marked fish in the sample. Call it T.

Compute the estimate as CR/T. If T=0, we are in trouble. Such experiments

with T=0 are dropped. Repeat this experiment 1000 times. Hence, we have

1000 estimates. Compute the mean and standard deviation of these

1000 estimates.


Properties of our estimatorIncreasing C and R

N C R S Mean Std

500 40 40 971 640.76 401.57

500 60 60 1000 579.22 321.54

500 80 80 1000 533.61 154.67

500 100 100 1000 522.85 104.29

500 120 120 1000 513.82 77.41

500 140 140 1000 507.04 60.98

500 250 250 1000 500.64 22.93

500 500 500 1000 500.00 0.00

•N = Total number of fish in the pond.•C = number of captured fish.•R = number of re-captured fish.•S = number of simulation with at least one marked fish in recapture.


Properties of our estimatorConstant C and increasing R

N C R S Mean Std

500 120 40 1000 507.86 75.07

500 120 60 1000 513.40 79.55

500 120 80 1000 508.19 73.56

500 120 100 1000 511.24 74.55

500 120 120 1000 510.93 75.41

500 120 140 1000 511.21 75.63

500 120 250 1000 510.49 74.04

500 120 500 1000 507.47 77.32



Properties of our estimatorIncreasing C and constant R

N C R S Mean Std

500 40 120 961 646.59 405.72

500 60 120 1000 582.17 327.97

500 80 120 1000 533.28 142.23

500 100 120 1000 512.28 95.40

500 120 120 1000 508.78 78.75

500 140 120 1000 507.50 60.61

500 250 120 1000 500.86 22.38

500 500 120 1000 500.00 0.00



Conclusion from the simulations

The proposed estimator generally overestimate the number of fish in pond, i.e., estimate is larger than the true number of fish in pond.

That is, there is a bias. Holding R constant, increasing the number of

capture (C) helps: Bias is reduced, i.e., Mean is closer to the true

population The estimator is more precise, i.e., standard

deviation of the estimator is smaller. Holding C constant, increasing the number of

recapture (R) does not help: Bias is more or less unchanged. The precision of the estimator is more or less

unchanged.


Additional issues

Our proposed estimator is good enough but it can be better. Alternative estimators have been developed to reduce or eliminate the bias of estimating N.

For instance, Seber (1982, p.60) suggests an estimator of N

(C+1)(R+1)/(T+1) – 1(Note that our proposed formula is CR/T.)

Seber, G. (1982): The Estimation of Animal Abundance and Related Parameters, second edition, Charles.


Simulations to see the properties of this modified estimator

How good is the modified estimator? To see the properties of this modified estimator, we

repeat the above simulation exercise with this new formula.

(C+1)(R+1)/(T+1) – 1


Properties of modified estimatorIncreasing C and R

N C R S Mean Std

500 40 40 1000 488.60 271.05

500 60 60 1000 504.39 202.16

500 80 80 1000 498.88 121.47

500 100 100 1000 501.72 91.20

500 120 120 1000 498.10 72.01

500 140 140 1000 501.14 58.44

500 250 250 1000 498.60 21.72

500 500 500 1000 500.00 0.00

•N = Total number of fish in the pond.•C = number of captured fish.•R = number of re-captured fish.•S = number of simulation with non-zero marked fish in recapture.


Properties of modified estimatorConstant C and increasing R

N C R S Mean Std

500 120 40 1000 498.55 67.38

500 120 60 1000 500.05 71.54

500 120 80 1000 495.58 69.22

500 120 100 1000 497.01 71.14

500 120 120 1000 498.45 71.05

500 120 140 1000 495.17 67.46

500 120 250 1000 500.41 75.29

500 120 500 1000 496.73 74.27



Properties of modified estimatorIncreasing C and constant R

N C R S Mean Std

500 40 120 1000 491.84 291.00

500 60 120 1000 499.33 216.81

500 80 120 1000 496.51 117.05

500 100 120 1000 493.50 87.53

500 120 120 1000 503.24 73.65

500 140 120 1000 498.59 56.30

500 250 120 1000 499.76 22.58

500 500 120 1000 500.00 0.00



Conclusion from the simulations

The modified estimator performs better than the original estimator.

There is no apparent bias. The estimator is more precise.

Holding R constant, increasing the number of capture (C) helps:

The estimator is more precise, i.e., standard deviation of the estimator is smaller.

Holding C constant, increasing the number of recapture (R) does not help:

The precision of the estimator is more or less unchanged.


What is Meant by Statistics?

Statistics is the science of 1. collecting, 2. organizing, 3. presenting, 4. analyzing, and 5. interpreting numerical data to assist in making more effective decisions.


Who Uses Statistics?

Statistical techniques are used extensively by Economists, marketing, accounting, quality control, consumers, professional sports people, hospital administrators, educators, politicians, physicians, etc...



As economists, We must verifying our models with data. We need to provide forecast of the economy

(GDP growth). We need quantitative estimates of

How individual decisions are influenced by policy variables (such as unemployment benefits, education subsidy) in order to forecast the impact of public policies.

How macro policies (government expenditure) will affect output.



In the business community, managers must make decisions based on what

will happen to such things as demand, costs, and profits.

These decisions are an effort to shape the future of the organization.

If the managers make no effort to look at the past and extrapolate into the future, the likelihood of achieving success is slim.


Why do we need to understand Statistics?

We are constantly deluged with statistics in the media (newspapers, magazines, journals, text books, etc.).

We need to have a means to condense large quantities of information into a few facts or figures.

We need to predict what will likely occur given what has occurred in the past.

We need to generalize what we have learned in specific situations to the more general case.


We are users of statistics

We do not want to become professors of statistics. We do not want to develop advanced statistics

theory.

We are users of statistics To be effective users, we need to have a good

grip of basic statistics theory. We need to practice using the tools.

This course will give you the basic, enough for you to move on to your next Econometrics class.


Populations and Samples

A population is a collection of all possible individuals, objects, or measurements of interest.

A sample is a portion, or part, of the population of interest.


Populations and Samples

Population

Sample


Sampling a Population of Existing Units

Random Sampling A procedure for selecting a subset of the

population units in such a way that every unit in the population has an equal chance of selection

Sampling with replacement When a unit is selected as part of the sample,

its value is recorded and placed back into the population for possible reselection

Sampling without replacement Units are not placed back into the population

after selection


Approximate Random Samples

Frame

A list of all population units. Required for random sampling, but not for approximate random sampling methods like systematic and voluntary response sampling.

Systematic Sample

Every k-th element of the population is selected for the sample

Voluntary Response Sample

Sample units are self-selected (as in radio/TV surveys)


How to estimate the unemployment rate

First, survey a large number of individuals (say, 1000) Are you 15 and over? If not, you are definitely not

in the labor force. If you are 15 and over,

Have you work for pay or profit during the seven days before enumeration or have a formal job attachment?

If yes, you are counted as employed. If not employed,

Have you been available for work during the seven days before enumeration? And

Have you sought work during the 30 days before enumeration?

If yes to both questions, you are counted as unemployed.


How to estimate the unemployment rate

The unemployment rate is computed as#unemployed/ (#unemployed + #employed)

Note that the estimate of the unemployment rate is based on a random subset (which we call a sample) of the individuals of an economy -- not all individuals in an economy.


Simulate to understand the estimator: An estimation of the unemployment rate

A process of estimating unemployment rate may be simulated at home or in a classroom with a bag of black and white stones (as in a game of GO).

Suppose black stones stand for unemployed and white stones stand for employed individuals. A random selection of 20 individuals is like randomly grabbing 20 stones from the bag.

We ask each selected individuals whether they are white (employed) or black (unemployed). The unemployment rate may be computed using the formula


What to take away today

Statistics is difficult but could be intuitive. Statistics need not be completely deep, murky, and

mysterious. Our common sense can help us to negotiate our way

through the course.


- END -

Lesson 1: Lesson 1: Analysis of Economic Data is difficult but intuitive

Documents

Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson1-1 Lesson 1: Analysis of Economic Data is difficult but intuitive