Covariance

1
Covariance
(x,y)
x and y axes
2
Covariance
(x,y)
x and y axes
3
Covariance
Below average values of x are with above average
values of y
Above average values of x are also above average
values of y
So what happens on balance?
Below average values of x are also below average
values of y
Above average values of x are with below average
values of y
4
Covariance
What happens on balance?
Calculate the average of the squared deviations.
5
Covariance
What happens on balance?
Calculate the average of the squared deviations.
6
Covariance Example
Sxy 1.999
Wage
Aptitude
7
Correlation
rxy 0.476
Wage
Aptitude
8
Perfect Correlation
9
Fit That Line !
y2,5001,800x
y10,0001,000x
y13,000750x
10
Fit That Line !
y8,135 1,233xminimizes the squared errors
11
Word Problem

• Students in a small class were polled by a
  researcher attempting to establish a relationship
  between hours of study in a week preceding a test
  and the result of the test.
• If you get data on hours studied and exam
  results, which variable is the dependent
  variable? why?

12
Word Problem
y39.406 2.122x
13
Word Problem
Excel Regression Output (Data Analysis Add-In)
14
Word Problem
Excel Regression Output (StatPad Add-In)
15
The Nine Lives of Goldfish
16
Predicting Job Performance
Simple Regression Perform 3.956 0.022 age
17
Predicting Job Performance
Perform 4.865 0.037 age 0.011 seniority -
0.032 cognitive
Note importance of ceteris paribus (all else
constant)
18
Predicting Job Performance
Perform 4.865 0.037 age 0.011 seniority -
0.032 cognitive And holding seniority constant at
10 and cognitive constant at 1
19
Predicting Job Performance
Perform 4.865 0.037 age 0.011 seniority -
0.032 cognitive And holding seniority constant at
20 and cognitive constant at -1
With linear models, other values dont matter
just all else constant
20
Predicting Job Perf. With a Dummy Variable
Structured Interview Dummy Variable 1yes, 0no
21
Predicting Job Perf. With a Dummy Variable
Perform 4.820 0.037 age 0.010 seniority -
0.025 cognitive 2.850 structured interview
Dummy variable turns on and off with all else
constant.
22
Predicting Job Perf. With a Dummy Variable
Perform 4.865 0.037 age 0.010 seniority -
0.025 cognitive 2.850 structured interview And
holding seniority constant at 10 and cognitive
constant at 1
23
Predicting Job Perf. With a Dummy Variable
Note new y-intercept
Seniority20, Cognitive0
24
Multiple Dummy Variables

• Source SS df MS
  Number of obs 3525
• ---------------------------------------
  F( 14, 3510) 125.63
• Model 5035.58483 14 359.684631
  Prob gt F 0.0000
• Residual 10049.2032 3510 2.86302087
  R-squared 0.3338
• ---------------------------------------
  Adj R-squared 0.3312
• Total 15084.7881 3524 4.28058685
  Root MSE 1.692
• --------------------------------------------------
  ----------------------------
• perform Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• age -.0301543 .0016933 -17.808
  0.000 -.0334742 -.0268344
• seniorty .0016888 .002762 0.611
  0.541 -.0037265 .007104
• cognitve .0119113 .0286362 0.416
  0.677 -.0442339 .0680565
• strucint 3.665569 .7995184 4.585
  0.000 2.098001 5.233137
• job1 1.928286 .1277788 15.091
  0.000 1.677758 2.178814
• job2 .426524 .1260009 3.385
  0.001 .1794815 .6735664
• job3 .1407506 .1306411 1.077
  0.281 -.1153896 .3968908
• job4 .2921016 .1347211 2.168
  0.030 .0279621 .5562411
• job5 -1.069262 .1331017 -8.033
  0.000 -1.330227 -.8082974

25
Interaction Variables

• Source SS df MS
  Number of obs 3525
• ---------------------------------------
  F( 6, 3518) 121.08
• Model 2581.89927 6 430.316544
  Prob gt F 0.0000
• Residual 12502.8888 3518 3.55397635
  R-squared 0.1712
• ---------------------------------------
  Adj R-squared 0.1697
• Total 15084.7881 3524 4.28058685
  Root MSE 1.8852
• --------------------------------------------------
  ----------------------------
• perform Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• age -.006 .0034204 -1.705
  0.088 -.0125379 .0008743
• seniorty .011 .0030589 3.559
  0.000 .0048879 .0168827
• cognitve -.005 .0318774 -0.167
  0.867 -.0678283 .0571719
• strucint 2.129 .8937022 2.383
  0.017 .3770909 3.881545
• manual -1.513 .2391962 -6.327
  0.000 -1.982442 -1.044488
• manl_age -.042 .004011 -10.439
  0.000 -.0497349 -.0340066
• _cons 6.009 .2354444 25.526
  0.000 5.548275 6.471517
• --------------------------------------------------
  ----------------------------
• Note manual is a dummy variable indicating a
  manual occupation manl_age is age interacted
  with manual (i.e. manl_age manualage)

26
Interaction Variables
Note different slopes, too.
Seniority20, Cognitive0, StrucInt0
27
Another Interaction Variable Example

• Source SS df MS
  Number of obs 15321
• -------------------------------------------
  F( 5, 15315) 800.50
• Model 804247599 5 160849520
  Prob gt F 0.0000
• Residual 3.0773e09 15315 200936.252
  R-squared 0.2072
• -------------------------------------------
  Adj R-squared 0.2069
• Total 3.8816e09 15320 253367.252
  Root MSE 448.26
• --------------------------------------------------
  ----------------------------
• earnwkly Coef.
• -------------------------------------------------
  ----------------------------
• married 136.003
• female -169.837
• exper 2.946
• parttime -227.716
• exp_pt -1.896
• _cons 700.802
• --------------------------------------------------
  ----------------------------
• exper is potential labor market experience
  (age-educ-6)

28
Interaction Variables
Married1, Female1
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Adjusted R2

• Source SS df MS
  Number of obs 3525
• ---------------------------------------
  F( 14, 3510) 125.63
• Model 5035.58483 14 359.684631
  Prob gt F 0.0000
• Residual 10049.2032 3510 2.86302087
  R-squared 0.3338
• ---------------------------------------
  Adj R-squared 0.3312
• Total 15084.7881 3524 4.28058685
  Root MSE 1.692
• --------------------------------------------------
  ----------------------------
• perform Coef. Std. Err. t
  Pgtt 95 Conf. Interval
• -------------------------------------------------
  ----------------------------
• age -.0301543 .0016933 -17.808
  0.000 -.0334742 -.0268344
• seniorty .0016888 .002762 0.611
  0.541 -.0037265 .007104
• cognitve .0119113 .0286362 0.416
  0.677 -.0442339 .0680565
• strucint 3.665569 .7995184 4.585
  0.000 2.098001 5.233137
• job1 1.928286 .1277788 15.091
  0.000 1.677758 2.178814
• job2 .426524 .1260009 3.385
  0.001 .1794815 .6735664
• job3 .1407506 .1306411 1.077
  0.281 -.1153896 .3968908
• job4 .2921016 .1347211 2.168
  0.030 .0279621 .5562411
• job5 -1.069262 .1331017 -8.033
  0.000 -1.330227 -.8082974

30
Causality ?
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31
Causality

• Reasons for an estimated statistical relationship
  
• The explanatory variable is the direct cause of
  the response (dependent) variable
• The response variable is causing a change in the
  explanatory variable (reverse causality)
• The explanatory variable is a contributing, but
  not sole, cause of the response variable
• Confounding variables may exist
• Both variables may stem from a common cause
• Both variables are changing over time
• Coincidence

Source Jessica M. Utts (1999) Seeing Through
Statistics, 2nd ed., Pacific Grove, CA Duxbury,
p. 186.