Data and correlation

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E. Kusideł Demand Analysis

Conditions to apply statistical methods to demand analysis

1. Economic phenomena must be quantified

Most of them are naturally quantified: prices, income, GDP, level of demand, production, etc.

Some of them have not natural measure: properties of goods like quality, smell, color.

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Condition to apply statistical methods

2. Statistical data are accessible and long enough Sometimes we know that statistical data

do exists but we have no access to them Formally number of observations must

be larger that number of estimated parameters, but practically number of observations must be much larger that number of estimated parameters

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Kinds of statistical data

Time series data (TSD): xt for t=1,2,…,T.

Cross-section or spatial data (CSD): xi for i=1,2,…,N.

Panel data or cross-section time (PD): xit for i=1,2,…,N; t=1,2,…,T.

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An example of time series data: xt , t=1,2,…,T.

Number of employees in Poland in period:1st quarter 1995 – 4th quarter 2006

t=1,2,…48 (12 years with 4 quarters in every year: 4x12=48 observations).

7000

7500

8000

8500

9000

9500

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

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An example of cross-section data: xi, i=1,2,…,N.

Number of employees in 4th quarter of 2006 in 16 regions of Poland

0500

1000150020002500

i=1,…16 (number of regions in Poland).

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An example of panel data: xit i=1,2,…,N; t=1,2,…,T.

Number of employees in 6 regions of Poland measured in 4 quarters of 2006

dolnośląskie kujawsko-

pomorskie

lubelskie lubuskie łódzkie małopolskie mazowieckie

1Q2006

2Q2006

3Q2006

4Q2006

1151741 880

4411161 1305

2209

1116741 929

4421146 1378

2158

1064 725 929404

1111 1348

2133

1068702 920

3901096 1195

2076

i=1,…,6, t=1,…4. Number of observations: 4x6=24.

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How to measure relationships between economic phenomena

expressed in series of data?

1. Correlation coefficient2. Elasticities3. Regression analysis

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n

ii

n

ii

n

iii

yxxy

yyn

xxn

yyxxn

ss

yxr

1

2

1

2

1

)(1

)(1

))((1

),cov(

Formula 1Correlation coefficient

between two variables x and y

n- sample amount,cov(x,y)- covariance between x i y,sx, sy, - standard deviation of variable x i y .

yx, yx, yx,

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Properties of correlation coefficient -rxy

rxy is a measure which can differ (vary) between –1 and 1.

Module from rxy is a power of the relationship

Sign of rxy inform us about direction of relationship

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Sign of correlation coefficient

rxy <0 – negative correlation (if x grows then y falls or if x falls then y grows)

rxy >0 – positive correlation (if x grows then y grows or if x falls then y falls)

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Power of correlation coefficient

Module of rxy, value of which is between 0 and 1 informs us about power of relationship

rxy = 0 - variables are not connected (no connection, or no correlation). e.g. demand for woman bag is not correlated with demand for computers.

rxy =1 – very strong connection between two variables e.g. we can expect strong relationship between demand for computer and demand for computers screen.

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Scatter diagram

The first step in most correlation problems is to construct a graphic picture of the relationship between the two variables. Such a picture is best provided by the a so-called scatter diagram

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Example 1.Price and demand for a

product

price demand (sale)

2,40 136

2,30 151

2,20 157

2,20 157

2,10 185

2,00 199

1,90 235

1,80 246

1,70 271

1,60 294

priceA

0,00

10,00

20,00

30,00

40,00

1 2 3 4 5 6 7 8 9 10

demandA

050

100

150200250

1 2 3 4 5 6 7 8 9 10

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Scatter diagram:graph of relationships

between price and demand for

product A

demand (sale)

100150200250300

1,50 1,70 1,90 2,10 2,30 2,50

price

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Shape of scatter diagramSign of CC Growing line: positive correlation Falling line: negative correlationPower of CC The more straight scatter diagram is

the greater power of CC The more round the scatter diagram is

the less power of CC. Straight line, but perpendicular or

parallel to an axis: no correlation

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What does this scatter diagram say?

demand (sale)

100

200

300

1,50 1,70 1,90 2,10 2,30 2,50

price

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Task 1Draw a scatter diagram for priceB and demandB and income and demandA

(data from demandAB.xls)

And answer following question:

1. Is it the case of positive or negative correlation?

2. Is the correlation stronger then in case of product A?

priceA demandA wages priceB demandB

33,40 127 1001 2,27 136

32,30 134 1200 3,02 151

28,20 139 1355 1,11 157

27,10 135 1406 1,60 157

26,00 153 1800 4,03 185

25,90 170 2106 1,87 199

24,80 191 2653 0,63 235

23,70 197 3009 3,37 246

22,60 187 3504 1,44 271

21,50 195 4000 1,69 294

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Scatter for price and demand for product B

demandB

0

100

200

300

400

0,00 1,00 2,00 3,00 4,00 5,00

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Homework

Calculate correlation coefficient between price and demand for

product B using formula 1

Scatter diagram for price and demand for product B

demandB

0

100

200

300

400

0,00 1,00 2,00 3,00 4,00 5,00

1. Is it the case of positive or negative correlation?2. Is the correlation stronger then in case of product A?