February 7, 2017
Dow Jones Industrial Average
Log(Dow)
Take logs to adjust for exponential growth and level-dependent volatility.
Stylized Fact #1
Many datasets exhibit exponential growth and level-dependent volatility.
Regress Log(Dow) on Time
Call:
lm(formula = log_dow ~ time)
Residuals:
Min 1Q Median 3Q Max
-1.08022 -0.40173 -0.02597 0.42103 1.31975
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.708e+00 5.667e-03 654.5 <2e-16 ***
time 2.079e-04 3.586e-07 579.6 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.4687 on 27367 degrees of freedom
Multiple R-squared: 0.9247, Adjusted R-squared: 0.9247
F-statistic: 3.36e+05 on 1 and 27367 DF, p-value: < 2.2e-16
Log(Dow) vs. Time, with Fit
Today's vs. Yesterday's Log(Dow)
Regress Today on Yesterday
Call:
lm(formula = log_dow ~ lag1_log_dow)
Residuals:
Min 1Q Median 3Q Max
-0.25654574 -0.00470490 0.00025780 0.00508635 0.14253642
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.57303e-04 2.63292e-04 0.59745 0.55021
lag1_log_dow 1.00001e+00 3.88805e-05 25720.06238 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.0109845 on 27366 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.999959, Adjusted R-squared: 0.999959
F-statistic: 6.61522e+08 on 1 and 27366 DF, p-value: < 2.22e-16
Slope significantly different from 1?
Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0001573034 0.0002632925 5.974476e-01 0.5502135 lag1_log_dow 1.0000089250 0.0000388805 2.572006e+04 0.0000000
T statistic:
[1] 0.2295497
95% Confidence Interval:
2.5 % 97.5 % lag1_log_dow 0.999933 1.00009
We probably shouldn't trust these results. (Why?)
Stylized Fact #2
Nearby observations are highly correlated with each other.
Log(Dow) vs. Time (again)