spurious correlation spurious regression
xt=xt-1+n(0,σ^2) yt=yt-1+n(0,σ^2)
n=20 type1error(5%)=0.4703 no trend 0 1000 2000 3000 4000 p for r
xt=xt-1+n(0,σ^2) random walk random walk variable -5 0 5 variable -4-2 0 2 4 6 8 10 0 20 40 60 80 100 0 20 40 60 80 100 time time i.i.d. normal variable -2-1 0 1 2 3 xt=n(0,σ^2)
Granger & Newbold, 1974 Phillips, 1986
n=20 type1error(5%)=0.4703 no trend histgram of p value for r resource output type error for r=0 0 1000 2000 3000 4000 0 100 200 300 400 500 0 500 1000 1500 2000 p for r p (difference) p for r
n=20 type1error(5%)=0.4703 no trend 0 1000 2000 3000 4000 p for r
histgram of correlation coefficient r histgram of correlation coefficient r 0 100 200 300 400 500 600 0 500 1000 1500-1.0-0.5 0.0 0.5 1.0-0.4-0.2 0.0 0.2 0.4 r r (difference)
n=20 type1error(5%)=0.4173 no trend 0 1000 2000 3000 4000 0 1000 2000 3000 4000 n=20 type1error(5%)=0.4703 no trend p for r p for tau
sd=1and5 n=100 type1error=0.7610 0 2000 4000 6000 p value for r
type1error=0.7593 unif vs unif type1error=0.7695 unif vs normal 0 2000 4000 6000 0 2000 4000 6000 8000 p value for r p value for r
type1error=0.0497 RW vs iid 0 100 200 300 400 500 p value for r
sine curve 4cycles sine curve 2cycles sine curve 1 cycle variable -1.0-0.5 0.0 0.5 1.0 variable -1.0-0.5 0.0 0.5 1.0 variable -1.0-0.5 0.0 0.5 1.0 0 20 40 60 80 100 time 0 20 40 60 80 100 time 0 20 40 60 80 100 time
sine curve 4cycles sine curve 2cycles sine curve 1 cycle type1error=0.2764 RW vs sine4cycles type1error=0.6461 RW vs sine2cycles type1error=0.8541 RW vs sine1cycles variable -1.0-0.5 0.0 0.5 1.0 variable -1.0-0.5 0.0 0.5 1.0 variable -1.0-0.5 0.0 0.5 1.0 0 20 40 60 80 100 time 0 20 40 60 80 100 time 0 20 40 60 80 100 time 0 500 1000 1500 2000 2500 0 1000 3000 5000 0 2000 4000 6000 8000 p value for r p value for r p value for r
sine curve quarter cycle type1error=0.8693 RW vs sine1/4cycle variable 0 2000 4000 6000 8000 0 20 40 60 80 100 time p value for r
xt=xt-1+n(0,σ^2) random walk random walk variable -5 0 5 variable -4-2 0 2 4 6 8 10 0 20 40 60 80 100 0 20 40 60 80 100 time time i.i.d. normal variable -2-1 0 1 2 3 xt=n(0,σ^2)
n=20 n=10000 n=20 type1error(5%)=0.4703 no trend n=10000 type1error(5%)=0.9755 no trend 0 1000 2000 3000 4000 0 2000 4000 6000 8000 10000 p for r p for r
1.0 0.8 type1 error rate (5%) 0.05 0 2000 4000 6000 8000 10000 sample size (n)
n=10000 n=10000 type1error(5%)=0.9755 no trend 0 2000 4000 6000 8000 10000 p for r
n=10000 n=10000 type1 error rate 0.00 0.01 0.02 0.03 0.04 0.05 level of significance
Granger & Newbold, 1974 Phillips, 1986
xt=θx xt-1+n(0,σ^2) θx 1 θx 1
i.i.d. normal coef=0.95 coef=0.98 θx 0.00 θx 0.95 θx 0.98 variable -2-1 0 1 2 3 variable -4-2 0 2 variable -6-4 -2 0 2 4 0 20 40 60 80 100 time 0 20 40 60 80 100 time 0 20 40 60 80 100 time random walk coef=1.01 coef=1.02 θx 1.00 θx 1.01 θx 1.02 variable -4-2 0 2 4 6 8 10 variable 0 5 10 15 variable -30-25 -20-15 -10-5 0 0 20 40 60 80 100 time 0 20 40 60 80 100 time 0 20 40 60 80 100 time
Granger et al (2001) Applied Economics, 33: 899-904.
xt=θx xt-1+n(0,σ^2) θx 1 θx 1
θx=0.98 θx=0.95 type1error=0.674 theta=0.98 type1error=0.6028 theta=0.95 0 1000 3000 5000 7000 0 1000 2000 3000 4000 5000 6000 p value for r p value for r θx=0.90 type1error=0.5029 theta=0.90 0 1000 2000 3000 4000 5000 p value for r
yt=α+β xt yt=α+β xt+εt εt 0
yt=α+β xt yt=α+β xt+εt εt 0
n=100 n=1000 Distribution of b n=100 RW Distribution of b n=1000 RW 0 100 200 300 400 500 600 700 0 200 400 600-4 -2 0 2 4 b -4-2 0 2 4 b n=2000 Distribution of b n=2000 RW 0 200 400 600-4 -2 0 2 4 b
n=100 n=1000 Distribution of a n=100 RW Distribution of a n=1000 RW 0 200 400 600 800 1000 0 50 100 150 200 250 300-40 -20 0 20 40-20 0 50 100 150 200 a n=2000 +20 Distribution of a n=2000 RW -50 0 50-50 -100 +100 a +50-150 -100-50 0 50 100 150
var(b) 0.0 0.1 0.2 0.3 0.4 0.5 500 1000 1500 2000 sample size (n) var(a) 0 100 200 300 400 500 500 1000 1500 2000 sample size (n)
resource output time series resource output time series y output (resource) 0 5 10 15 output (resource) 0 5 10 15 0 20 40 60 80 100 time 0 20 40 60 80 100 time
resource output type error for r=0 resource output correlation 0 500 1000 1500 2000 0 500 1000 1500 0 p for r 1-1.0-0.5 0.0 0.5 1.0-1.0 r r +1.0