Alternatively, dependent variable and independent variable.

Alternatively, endogenous variable and exogenous variable.

Association versus causation

Scatterplots

Weeks since beginning of semester

Per

cent

age

of c

ompu

ters

use

d in

com

pute

r la

bsfr

ee

Stata Exercise 1

Stata Exercise 2

Suppose we were considering the effect of hiring more people into the firm. On average, what total billings can we expect from a staff of 50? 150?

Stata Exercise 3

Stata Exercise 4

Stata Exercise 5

Adding Categorical Values to a Scatterplot

Often it is useful to have a way of distinguishing groups of data in a scatterplot

Stata Exercise 6

Transforming Data

Data analysts often look for a transformation of the data that simplifies the overall pattern.

The transformation typically involves turning a non-Normally distributed variable into a more-or-less Normally distributed variable.

Stata Exercise 7

Categorical Explanatory Variable

What if the explanation for the numbers is not another number but the category?

For example, investing in a particular sector of the economy might be great in some years or terrible in others.

Stata Exercise 8

More scatterplots

Relations between competitors

Stata Exercise 9

Correlation

Which one has the stronger correlation?

r = covariance(x,y) / [stdev(x)*stdev(y)]

r = (1/(n-1)) * sum of [(standardized values of x) (standardized values of] y)

week w - mean of w z-score of wprop of comps

p - mean of p z-score of pz-score * z-score

1 73.12 89.73 71.34 65.35 54.66 57.97 51.68 41.29 59.1

10 48.511 2412 4313 29.114 19.715 12.116 10.1

sum 0.00

8.5 4.8 46.9 23.1 count 16mean of w stdev of w mean of p stdev of p corr

Correlation

The r coefficient between measures of height and weight is positive because people who are of above-average height tend to be of above-average weight … so if the z-score for height is large, the z-score for weight tends to be large.

r = (1/(n-1)) * sum of [(standardized values of x) (standardized values of] y)

Correlation applet at www.whfreeman.com/pbs

Stata Exercise 11

Correlation

Correlation coefficients, as well as scatterplots can be used for comparisons.

For example, how well did Vanguard International Growth Fund (an investment vehicle) do compared to an average of the stocks in Europe, Australasia and the Far East?

Stata Exercise 12

Correlation

Doesn’t tell you anything about causality Variables must be numerical It is indifferent to units of measurement r>0 means positive association; r<0, negative -1 < r < 1. r = -1 means a perfectly straight

downward-sloping line. r=0 means no relation. r only measures linear relations r is not resistant to outliers

Stata Exercise 13

Regression

The Linear Regression Model

Errors have a mean 0 and a constant sd of and are independent of x.

iii errorbxay

05

0000

01

0000

00

1000 2000 3000 4000Square Footage of Homes

Linear prediction Price of Homes

05

0000

01

0000

00

1000 2000 3000 4000Square Footage of Homes

Price of Homes Linear prediction

05

01

001

50F

req

uenc

y

0 500000 1000000Price of Homes

1000<sqft<=1500

05

01

001

50F

req

uenc

y

0 500000 1000000Price of Homes

1500<sqft<=20000

50

100

150

Fre

que

ncy

0 500000 1000000Price of Homes

2000<sqft<=2500

05

01

001

502

00F

req

uenc

y

0 500000 1000000Price of Homes

2500<sqft<=3000

05

01

001

50F

req

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y

0 500000 1000000Price of Homes

3000<sqft<=35000

50

100

150

Fre

que

ncy

0 500000 1000000Price of Homes

3500<sqft<=4000

05

0000

01

0000

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1000 2000 3000 4000Square Footage of Homes

Price of Homes Linear prediction

05

0000

100

000

150

000

200

000

An

nual

ear

nin

gs (

dolla

rs)

55 60 65 70 75 80Height (inches)

earn Fitted values

(66.5’’, $20,000)

(76.5’’, $35,600)

(61.5’’, $12,200)

y – 20,000 = 1560 (x - 66.5)

y = – 84,000 + 1560 x

Sketch a scatterplot of the data consistent with this line

$37,694

95% of values

05

0000

100

000

150

000

200

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An

nual

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nin

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dolla

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55 60 65 70 75 80Height (inches)

earn Fitted values

01

23

y

0 1 2 3x

Draw the best-fitting line through the circles

Draw the best-fitting line through the circles

01

23

4y

0 1 2 3 4 5 6x

01

23

y

0 1 2 3x

Mark with an “X” the average “y” value for each “x” value. Then draw the best-fitting line through the Xs

01

23

4y

0 1 2 3 4 5 6x

Mark with an “X” the average “y” value for each “x” value. Then draw the best-fitting line through the Xs

Regression (unlike correlation) is sensitive to your determination of which variable is explanatory and which response.

Sales = a + b(item)Item = a + b(sales)

Fac

t 1

Stata Exercise 14

Facts 2 and 3

If x changes by one standard deviation ofx, y changes by r standard deviations of y.– E.g., sx = 1, sy = 2, and r = 0.61.

If x changes by 1, y will change by 2*0.61 = 1.22

The regression line goes through the point– The point-slope form of the line requires only the

information on this slide to draw a line.

),( yx

Fact 4

Correlation r is related to the slope of the regression line and therefore to the relation between x and y.

Actually, the square of r, that is, R2 is the fraction of the variation in y that is explained by the variation in x.

),( yx

y

xyR

of valuesobservedin variationtotal

line thealongit pulls as ˆin variation 2

Because most of the variation in gas consumption is explained by temperature, the R2 of this regression is very high.

tbill98 tbill98_hat residuals

11.5 10.84649  

12.6 12.19961  

13.8 14.81564  

6.4 5.975251  

5.3 6.336083  

Excel Exercise 1

Stata Exercises 15 and 16

With influential observations

Without influential observation 21

Stata Exercise 17

Cautions about Correlation and Regression

Don’t extrapolate too far Correlations are stronger for averages than

for individuals Beware of lurking (latent, hidden, excluded,

neglected) variables Association is not causation

– Establishing causation takes a lot of work (see p. 139).