R Notebook

rm(list=ls())
library(corrplot)

## corrplot 0.84 loaded

df <- read.csv(file="./volatility.csv")
head(df)

AAPL.PVol = df$AAPL.PVol
GOOG.PVol = df$GOOG.PVol
AMZN.PVol = df$AMZN.PVol
GSPC.PVol = df$GSPC.PVol
VIX.Close = df$VIX.Close
AAPL.Close = df$AAPL.Close
VXAPLCLS = df$VXAPLCLS
VXAZNCLS = df$VXAZNCLS
VXGOGCLS = df$VXGOGCLS 
GOOG.Close = df$GOOG.Close
AMZN.Close = df$AMZN.Close
GSPC.Close = df$GSPC.Close 
Date = df$Date
AAPL.Ret = df$AAPL.Ret
AMZN.Ret = df$AMZN.Ret
GOOG.Ret = df$GOOG.Ret
GSPC.Ret = df$GSPC.Ret
AAPL.Vol = df$AAPL.Vol

library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(ggplot2)

m = cov(cbind(AAPL.PVol,GOOG.PVol,AMZN.PVol,GSPC.PVol, 
          VIX.Close, AAPL.Close, VXAPLCLS, VXAZNCLS, 
          VXGOGCLS, GOOG.Close, AMZN.Close, GSPC.Close,
          Date, AAPL.Ret, AMZN.Ret, GOOG.Ret, GSPC.Ret, AAPL.Vol))
ans <- data.frame(row=rownames(m)[row(m)], col=colnames(m)[col(m)], corr=c(m)) %>% filter(col == 'AAPL.Vol')
write.csv(ans, "./cov.csv", row.names = FALSE)
ans

Looking at correlation between variables

corr_matrix = cbind(AAPL.PVol, GOOG.PVol,AMZN.PVol,GSPC.PVol,  AAPL.Close, GOOG.Close, AMZN.Close, GSPC.Close, AAPL.Ret, GOOG.Ret, AMZN.Ret, GSPC.Ret, VXAPLCLS, VXGOGCLS, VXAZNCLS, VIX.Close, AAPL.Vol, Date)
corr_matrix = cor(corr_matrix, method = c("pearson"))
corrplot(corr_matrix, method="color")

Descriptive Statistics:

# Getting descriptive statistics
summary(AAPL.PVol)

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## 0.006714 0.011094 0.014523 0.016165 0.020046 0.034721

summary(GOOG.PVol)

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## 0.006429 0.010943 0.013973 0.015663 0.020833 0.028110

summary(AMZN.PVol)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## 0.00610 0.01079 0.01394 0.01699 0.02175 0.04279

summary(GSPC.PVol)

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## 0.003286 0.005072 0.007195 0.008519 0.011868 0.019970

summary(VXAPLCLS) 

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   16.25   23.51   27.16   27.58   30.69   52.58

summary(VXAZNCLS)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   15.98   24.60   28.35   30.35   34.66   60.07

summary(VXGOGCLS) 

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   14.11   20.73   23.90   24.76   27.64   42.72

summary(VIX.Close)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    9.15   12.94   14.95   16.01   18.07   37.32

summary(AAPL.Close)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   35.55   43.88   48.15   49.67   53.87   73.41

summary(GOOG.Close) 

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   976.2  1085.2  1146.2  1150.9  1205.0  1361.2

summary(AMZN.Close)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    1189    1606    1750    1716    1836    2040

summary(GSPC.Close)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    2351    2724    2818    2830    2924    3240

summary(AAPL.Ret) 

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -0.104924 -0.006761  0.001644  0.001096  0.010182  0.068053

summary(AMZN.Ret) 

##       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
## -0.0814235 -0.0069531  0.0015277  0.0009095  0.0108153  0.0902540

summary(GOOG.Ret) 

##       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
## -0.0800893 -0.0065340  0.0006584  0.0004872  0.0098441  0.0993795

summary(GSPC.Ret) 

##       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
## -0.0418425 -0.0031881  0.0008367  0.0003763  0.0057131  0.0484032

summary(AAPL.Vol)

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## 0.006714 0.011161 0.014546 0.016221 0.020046 0.034721

Looking at Apple.Volatility vs Apple.PVolatility

# Fitting a simple linear regression model for AAPL.Vol and 
# each independent variable, and checking the five key
# assumptions: Linear relationship, Multivariate normality, 
# No multicollinearity, No auto-correlation and Homoscedasticity, 
# the assumptions of the simple linear regression are acceptable.
summary(lm(AAPL.Vol~AAPL.PVol))

## 
## Call:
## lm(formula = AAPL.Vol ~ AAPL.PVol)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.012131 -0.003853 -0.001856  0.003688  0.015031 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.0103326  0.0007178  14.395   <2e-16 ***
## AAPL.PVol   0.3642590  0.0413424   8.811   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.005874 on 501 degrees of freedom
## Multiple R-squared:  0.1342, Adjusted R-squared:  0.1324 
## F-statistic: 77.63 on 1 and 501 DF,  p-value: < 2.2e-16

plot(AAPL.PVol,AAPL.Vol)
abline(lm(AAPL.Vol~AAPL.PVol))

plot(lm(AAPL.Vol ~ AAPL.PVol))

X = AAPL.PVol
Y = AAPL.Vol^1.3
summary(lm(Y ~ X))

## 
## Call:
## lm(formula = Y ~ X)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0047056 -0.0015163 -0.0007553  0.0013152  0.0063993 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.0024677  0.0002809   8.786   <2e-16 ***
## X           0.1469616  0.0161769   9.085   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.002298 on 501 degrees of freedom
## Multiple R-squared:  0.1414, Adjusted R-squared:  0.1397 
## F-statistic: 82.53 on 1 and 501 DF,  p-value: < 2.2e-16

mod = lm(formula = Y ~ X)
plot(X, Y)
abline(mod)

plot(lm(Y~X))

Looking at Apple.Volatility vs Google.PVolatility

summary(lm(AAPL.Vol~GOOG.PVol))

## 
## Call:
## lm(formula = AAPL.Vol ~ GOOG.PVol)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.010577 -0.004010 -0.001332  0.003647  0.016917 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.0095353  0.0007591  12.561   <2e-16 ***
## GOOG.PVol   0.4268321  0.0455419   9.372   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.005823 on 501 degrees of freedom
## Multiple R-squared:  0.1492, Adjusted R-squared:  0.1475 
## F-statistic: 87.84 on 1 and 501 DF,  p-value: < 2.2e-16

plot(GOOG.PVol, AAPL.Vol)
abline(lm(AAPL.Vol~GOOG.PVol))

plot(lm(AAPL.Vol~GOOG.PVol))

WOrking to transform data

# plot(AAPL.PVol, resid(lm(AAPL.Vol~exp(AAPL.PVol))))
# plot(lm(AAPL.Vol ~ log(AAPL.PVol)))
X = GOOG.PVol^3
Y = AAPL.Vol
summary(lm(Y ~ X))

## 
## Call:
## lm(formula = Y ~ X)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.011667 -0.004118 -0.001034  0.003294  0.017549 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1.340e-02  3.636e-04   36.87   <2e-16 ***
## X           5.183e+02  4.794e+01   10.81   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.005685 on 501 degrees of freedom
## Multiple R-squared:  0.1891, Adjusted R-squared:  0.1875 
## F-statistic: 116.8 on 1 and 501 DF,  p-value: < 2.2e-16

mod = lm(formula = Y ~ X)
plot(X, Y)
abline(mod)

plot(lm(Y~X))

# abline(lm(AAPL.Vol~(AAPL.PVol^2)))

Looking at Apple.Volatility vs Amazon.PVolatility

summary(lm(AAPL.Vol~AMZN.PVol))

## 
## Call:
## lm(formula = AAPL.Vol ~ AMZN.PVol)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.011690 -0.003388 -0.001062  0.003772  0.012970 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.0097048  0.0005251   18.48   <2e-16 ***
## AMZN.PVol   0.3836141  0.0275303   13.93   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.005359 on 501 degrees of freedom
## Multiple R-squared:  0.2793, Adjusted R-squared:  0.2779 
## F-statistic: 194.2 on 1 and 501 DF,  p-value: < 2.2e-16

plot(AMZN.PVol,AAPL.Vol)
abline(lm(AAPL.Vol~AMZN.PVol))

plot(lm(AAPL.Vol~AMZN.PVol))

X = AMZN.PVol^2
Y = AAPL.Vol^1.3
summary(lm(Y ~ X))

## 
## Call:
## lm(formula = Y ~ X)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0041362 -0.0013751 -0.0004717  0.0013413  0.0056002 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.0035150  0.0001229    28.6   <2e-16 ***
## X           3.6507349  0.2295454    15.9   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.002022 on 501 degrees of freedom
## Multiple R-squared:  0.3355, Adjusted R-squared:  0.3342 
## F-statistic: 252.9 on 1 and 501 DF,  p-value: < 2.2e-16

mod = lm(formula = Y ~ X)
plot(X, Y)
abline(mod)

plot(lm(Y~X))

Looking at Apple.Volatility vs GSPC.PVol

summary(lm(AAPL.Vol~GSPC.PVol))

## 
## Call:
## lm(formula = AAPL.Vol ~ GSPC.PVol)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.010067 -0.004458 -0.001580  0.004240  0.016591 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 0.0113798  0.0005934  19.177   <2e-16 ***
## GSPC.PVol   0.5683058  0.0625725   9.082   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.00585 on 501 degrees of freedom
## Multiple R-squared:  0.1414, Adjusted R-squared:  0.1397 
## F-statistic: 82.49 on 1 and 501 DF,  p-value: < 2.2e-16

plot(GSPC.PVol, AAPL.Vol)
abline(lm(AAPL.Vol~GSPC.PVol))

plot(lm(AAPL.Vol~GSPC.PVol))

X = GSPC.PVol^2
Y = AAPL.Vol
summary(lm(Y ~ X))

## 
## Call:
## lm(formula = Y ~ X)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.010616 -0.004461 -0.001523  0.004048  0.017018 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1.362e-02  3.729e-04  36.512   <2e-16 ***
## X           2.897e+01  2.990e+00   9.687   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.005794 on 501 degrees of freedom
## Multiple R-squared:  0.1578, Adjusted R-squared:  0.1561 
## F-statistic: 93.84 on 1 and 501 DF,  p-value: < 2.2e-16

mod = lm(formula = Y ~ X)
plot(X, Y)
abline(mod)

plot(lm(Y~X))

Closing Prices

A. Appl.Vol vs AAPL.Close

summary(lm(AAPL.Vol~AAPL.Close))

## 
## Call:
## lm(formula = AAPL.Vol ~ AAPL.Close)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.010130 -0.003786 -0.001369  0.003280  0.017794 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.635e-02  1.831e-03  14.391  < 2e-16 ***
## AAPL.Close  -2.038e-04  3.645e-05  -5.593 3.68e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.006125 on 501 degrees of freedom
## Multiple R-squared:  0.05877,    Adjusted R-squared:  0.05689 
## F-statistic: 31.28 on 1 and 501 DF,  p-value: 3.678e-08

plot(AAPL.Vol~AAPL.Close)
abline(lm(AAPL.Vol~AAPL.Close))

# plot(AAPL.Close, lm(AAPL.Vol~AAPL.Close)$residuals)
plot(lm(AAPL.Vol~AAPL.Close))

B. Appl.Vol vs GOOG.Close

summary(lm(AAPL.Vol~GOOG.Close))

## 
## Call:
## lm(formula = AAPL.Vol ~ GOOG.Close)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0117719 -0.0033752 -0.0005929  0.0032927  0.0172550 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  4.845e-02  3.603e-03  13.446   <2e-16 ***
## GOOG.Close  -2.801e-05  3.123e-06  -8.968   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.00586 on 501 degrees of freedom
## Multiple R-squared:  0.1383, Adjusted R-squared:  0.1366 
## F-statistic: 80.43 on 1 and 501 DF,  p-value: < 2.2e-16

plot(AAPL.Vol~GOOG.Close)
abline(lm(AAPL.Vol~GOOG.Close))

#plot(GOOG.Close, lm(AAPL.Vol~GOOG.Close)$residuals)
plot(lm(AAPL.Vol~GOOG.Close))

C. Appl.Vol vs AMZN.Close

summary(lm(AAPL.Vol~AMZN.Close))

## 
## Call:
## lm(formula = AAPL.Vol ~ AMZN.Close)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.010157 -0.004862 -0.001207  0.003546  0.018764 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.421e-02  2.784e-03   8.698  < 2e-16 ***
## AMZN.Close  -4.658e-06  1.614e-06  -2.885  0.00408 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.006261 on 501 degrees of freedom
## Multiple R-squared:  0.01635,    Adjusted R-squared:  0.01438 
## F-statistic: 8.325 on 1 and 501 DF,  p-value: 0.004079

plot(AAPL.Vol~AMZN.Close)
abline(lm(AAPL.Vol~AMZN.Close))

# plot(AMZN.Close, lm(AAPL.Vol~AMZN.Close)$residuals)
plot(lm(AAPL.Vol~AMZN.Close))

D. Appl.Vol vs GSPC.Close

summary(lm(AAPL.Vol~GSPC.Close))

## 
## Call:
## lm(formula = AAPL.Vol ~ GSPC.Close)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.011231 -0.003555 -0.001053  0.004014  0.017945 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.590e-02  4.913e-03  11.378  < 2e-16 ***
## GSPC.Close  -1.402e-05  1.733e-06  -8.088 4.59e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.005937 on 501 degrees of freedom
## Multiple R-squared:  0.1155, Adjusted R-squared:  0.1137 
## F-statistic: 65.42 on 1 and 501 DF,  p-value: 4.594e-15

plot(AAPL.Vol~GSPC.Close)
abline(lm(AAPL.Vol~GSPC.Close))

# plot(GSPC.Close, lm(AAPL.Vol~GSPC.Close)$residuals)
plot(lm(AAPL.Vol~GSPC.Close))

RETURNS

A. Appl.Vol vs AAPL.Ret

##returns
summary(lm(AAPL.Vol~AAPL.Ret))

## 
## Call:
## lm(formula = AAPL.Vol ~ AAPL.Ret)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.010232 -0.004573 -0.001546  0.003899  0.020859 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.0162987  0.0002766  58.932  < 2e-16 ***
## AAPL.Ret    -0.0709453  0.0158672  -4.471 9.63e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.006191 on 501 degrees of freedom
## Multiple R-squared:  0.03837,    Adjusted R-squared:  0.03645 
## F-statistic: 19.99 on 1 and 501 DF,  p-value: 9.629e-06

plot(AAPL.Vol~AAPL.Ret)
abline(lm(AAPL.Vol~AAPL.Ret))

# plot(AAPL.Ret, lm(AAPL.Vol~AAPL.Ret)$residuals)
plot(lm(AAPL.Vol~AAPL.Ret))

B. Appl.Vol vs GOOG.Ret

summary(lm(AAPL.Vol~GOOG.Ret))

## 
## Call:
## lm(formula = AAPL.Vol ~ GOOG.Ret)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.010034 -0.005098 -0.001675  0.003929  0.018778 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.0162340  0.0002809   57.79   <2e-16 ***
## GOOG.Ret    -0.0267257  0.0170239   -1.57    0.117    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.006297 on 501 degrees of freedom
## Multiple R-squared:  0.004895,   Adjusted R-squared:  0.002909 
## F-statistic: 2.465 on 1 and 501 DF,  p-value: 0.1171

plot(AAPL.Vol~GOOG.Ret)
abline(lm(AAPL.Vol~GOOG.Ret))

#plot(AAPL.Ret, lm(AAPL.Vol~GOOG.Ret)$residuals)
plot(lm(AAPL.Vol~GOOG.Ret))

C. Appl.Vol vs. AMZN.Ret

summary(lm(AAPL.Vol~AMZN.Ret))

## 
## Call:
## lm(formula = AAPL.Vol ~ AMZN.Ret)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.010004 -0.005099 -0.001600  0.003971  0.019868 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.0162477  0.0002807  57.886   <2e-16 ***
## AMZN.Ret    -0.0293726  0.0147407  -1.993   0.0468 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.006288 on 501 degrees of freedom
## Multiple R-squared:  0.007863,   Adjusted R-squared:  0.005883 
## F-statistic: 3.971 on 1 and 501 DF,  p-value: 0.04684

plot(AAPL.Vol~AMZN.Ret)
abline(lm(AAPL.Vol~AMZN.Ret))

# plot(AMZN.Ret, lm(AAPL.Vol~AMZN.Ret)$residuals)
plot(lm(AAPL.Vol~AMZN.Ret))

D. Appl.Vol vs GSPC.Ret

summary(lm(AAPL.Vol~GSPC.Ret))

## 
## Call:
## lm(formula = AAPL.Vol ~ GSPC.Ret)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.010180 -0.004948 -0.001665  0.004012  0.019477 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  0.016256   0.000279  58.274  < 2e-16 ***
## GSPC.Ret    -0.092991   0.029554  -3.147  0.00175 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.006251 on 501 degrees of freedom
## Multiple R-squared:  0.01938,    Adjusted R-squared:  0.01742 
## F-statistic:   9.9 on 1 and 501 DF,  p-value: 0.001751

plot(AAPL.Vol~GSPC.Ret)
abline(lm(AAPL.Vol~GSPC.Ret))

# plot(GSPC.Ret, lm(AAPL.Vol~GSPC.Ret)$residuals)
plot(lm(AAPL.Vol~GSPC.Ret))

VIX

A. Appl.Vol vs. VXAPLCLS

summary(lm(AAPL.Vol~VXAPLCLS))

## 
## Call:
## lm(formula = AAPL.Vol ~ VXAPLCLS)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0114745 -0.0036975 -0.0005408  0.0032929  0.0189228 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -3.550e-03  1.025e-03  -3.465 0.000576 ***
## VXAPLCLS     7.169e-04  3.635e-05  19.719  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.004737 on 501 degrees of freedom
## Multiple R-squared:  0.437,  Adjusted R-squared:  0.4359 
## F-statistic: 388.9 on 1 and 501 DF,  p-value: < 2.2e-16

plot(AAPL.Vol~VXAPLCLS)
abline(lm(AAPL.Vol~VXAPLCLS))

# plot(VXAPLCLS, lm(AAPL.Vol~VXAPLCLS)$residuals)
plot(lm(AAPL.Vol~VXAPLCLS))

B. Appl.Vol vs. VXGOGCLS

summary(lm(AAPL.Vol~VXGOGCLS))

## 
## Call:
## lm(formula = AAPL.Vol ~ VXGOGCLS)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0117181 -0.0033167 -0.0002358  0.0028568  0.0172900 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -2.997e-03  9.787e-04  -3.062  0.00232 ** 
## VXGOGCLS     7.761e-04  3.861e-05  20.102  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.004697 on 501 degrees of freedom
## Multiple R-squared:  0.4465, Adjusted R-squared:  0.4453 
## F-statistic: 404.1 on 1 and 501 DF,  p-value: < 2.2e-16

plot(AAPL.Vol~VXGOGCLS)
abline(lm(AAPL.Vol~VXGOGCLS))

# plot(VXGOGCLS, lm(AAPL.Vol~VXGOGCLS)$residuals)
plot(lm(AAPL.Vol~VXGOGCLS))

C. Appl.Vol vs. VXAZNCLS

summary(lm(AAPL.Vol~VXAZNCLS))

## 
## Call:
## lm(formula = AAPL.Vol ~ VXAZNCLS)
## 
## Residuals:
##        Min         1Q     Median         3Q        Max 
## -0.0112543 -0.0029578 -0.0002878  0.0026793  0.0170604 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -5.448e-04  7.839e-04  -0.695    0.487    
## VXAZNCLS     5.523e-04  2.497e-05  22.120   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.00449 on 501 degrees of freedom
## Multiple R-squared:  0.4941, Adjusted R-squared:  0.4931 
## F-statistic: 489.3 on 1 and 501 DF,  p-value: < 2.2e-16

plot(AAPL.Vol~VXAZNCLS)
abline(lm(AAPL.Vol~VXAZNCLS))

# plot(VXAZNCLS, lm(AAPL.Vol~VXAZNCLS)$residuals)
plot(lm(AAPL.Vol~VXAZNCLS))

D. Appl.Vol vs. VIX.Close

summary(lm(AAPL.Vol~VIX.Close))

## 
## Call:
## lm(formula = AAPL.Vol ~ VIX.Close)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.015174 -0.004049 -0.001003  0.003370  0.018191 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 4.658e-03  1.007e-03   4.627 4.74e-06 ***
## VIX.Close   7.221e-04  6.093e-05  11.851  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.005579 on 501 degrees of freedom
## Multiple R-squared:  0.2189, Adjusted R-squared:  0.2174 
## F-statistic: 140.4 on 1 and 501 DF,  p-value: < 2.2e-16

plot(AAPL.Vol~VIX.Close)
abline(lm(AAPL.Vol~VIX.Close))

# plot(VIX.Close, lm(AAPL.Vol~VIX.Close)$residuals)
plot(lm(AAPL.Vol~VIX.Close))

Date

summary(lm(AAPL.Vol~Date))

## 
## Call:
## lm(formula = AAPL.Vol ~ Date)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.010509 -0.004040 -0.001550  0.003462  0.018375 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.780e-02  5.556e-04  32.038  < 2e-16 ***
## Date        -4.342e-06  1.322e-06  -3.285  0.00109 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.006246 on 501 degrees of freedom
## Multiple R-squared:  0.02109,    Adjusted R-squared:  0.01914 
## F-statistic: 10.79 on 1 and 501 DF,  p-value: 0.00109

plot(AAPL.Vol~Date)
abline(lm(AAPL.Vol~Date))

# plot(VXAZNCLS, lm(AAPL.Vol~VXAZNCLS)$residuals)
plot(lm(AAPL.Vol~Date))

X = Date^2
Y = AAPL.Vol
summary(lm(Y ~ X))

## 
## Call:
## lm(formula = Y ~ X)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.010907 -0.003706 -0.001486  0.003309  0.017932 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.778e-02  4.115e-04  43.199  < 2e-16 ***
## X           -8.802e-09  1.735e-09  -5.075 5.48e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.006157 on 501 degrees of freedom
## Multiple R-squared:  0.04889,    Adjusted R-squared:  0.04699 
## F-statistic: 25.75 on 1 and 501 DF,  p-value: 5.482e-07

mod = lm(formula = Y ~ X)
plot(X, Y)
abline(mod)

plot(lm(Y~X))

It’s very hard to transform the data between

Francis’ Closing & VIX Transformation Analysis

AAPL vs Closing prices: For most of the transformations in regards to Apple closing values, Transformations do not seem to make quite an impact on our data. One exception may be the first one of Apple.vol vs Apple.Close, which does seem to make the residuals a bit more normal, however, are skewed heavily towards one side or the other.

c <- lm(AAPL.Vol ~ AAPL.Close)

d <- lm(log(AAPL.Vol) ~ AAPL.Close)

### Some transforms of Appl Volitility vs Appl Close

par(mfrow=c(2,2)) # <- Remove this line if you want graphs not grouped

plot(fitted(c), resid(c), col = alpha("grey36", 0.5), pch = 20,
     xlab = "AAPL.Close", ylab = "Residuals", main = "Apple Close versus Apple vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(fitted(d), resid(d), col = alpha("grey36", 0.5), pch = 20,
     xlab = "AAPL.Close", ylab = "Log of Residuals", main = "Apple Close versus Apple vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)


plot(AAPL.Close, rstandard(c), col = alpha("grey36", 0.5), pch = 20,
     xlab = "AAPL.Close", ylab = "Standardized Residuals", main = "Apple close versus Apple vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(AAPL.Close, rstandard(d), col = alpha("grey36", 0.5), pch = 20,
     xlab = "AAPL.Close", ylab = "Standardized Residuals", main = "Apple close versus Apple vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(c, which = 1)
plot(d, which = 1)

c <- lm(AAPL.Vol ~ AMZN.Close)

d <- lm(sqrt(AAPL.Vol) ~ AMZN.Close)

### Some transforms of Appl Volitility vs AMZN Close

#par(mfrow=c(2,2)) # <- Remove this line if you want graphs not grouped
plot(AMZN.Close, resid(c), col = alpha("grey36", 0.5), pch = 20,
     xlab = "AMZN.Close", ylab = "Residuals", main = "AMZN Close versus Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(AMZN.Close, resid(d), col = alpha("grey36", 0.5), pch = 20,
     xlab = "AMZN.Close", ylab = "Squared root of Residuals", main = "AMZN Close versus Sqrt Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(AMZN.Close, rstandard(c), col = alpha("grey36", 0.5), pch = 20,
     xlab = "AMZN.Close", ylab = "Standardized Residuals", main = "AMZN Close versus Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(AMZN.Close, rstandard(d), col = alpha("grey36", 0.5), pch = 20,
     xlab = "AMZN.Close", ylab = "Standardized Residuals", main = "AMZN Close versus Sqrt Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(c, which = 1)

plot(d, which = 1)

For Transformation on google, seems that it is not neccesary as it barely makes any difference upon analyzing the regressions as the graphs barely change.

c <- lm(AAPL.Vol ~ GOOG.Close)

d <- lm(sqrt(AAPL.Vol) ~ GOOG.Close)

### Some transforms of Appl Volitility vs Google Close

par(mfrow=c(2,2)) # <- Remove this line if you want graphs not grouped
plot(GOOG.Close, resid(c), col = alpha("grey36", 0.5), pch = 20,
     xlab = "GOOG.Close", ylab = "Residuals", main = "Google Close versus Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(GOOG.Close, resid(d), col = alpha("grey36", 0.5), pch = 20,
     xlab = "GOOG.Close", ylab = "Squared root of Residuals", main = "Google Close versus Sqrt Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(GOOG.Close, rstandard(c), col = alpha("grey36", 0.5), pch = 20,
     xlab = "GOOG.Close", ylab = "Standardized Residuals", main = "Google Close versus Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(GOOG.Close, rstandard(d), col = alpha("grey36", 0.5), pch = 20,
     xlab = "GOOG.Close", ylab = "Standardized Residuals", main = "Google Close versus Sqrt Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(c, which = 1)
plot(d, which = 1)

For the S&P 500, not much seems to suggest that a transform changes much. Standardized residuals also reveal only a few outliers very close to 3. With a transform on the x-axis (to make up for the curve in residual trendline) and y-axis in attempt to make the residuals linear,there abnormality suggests its not any kind of linear relationship

c <- lm(AAPL.Vol ~ GSPC.Close)

d <- lm(sqrt(AAPL.Vol) ~ GSPC.Close^2)

### Some transforms of Appl Volitility vs Google Close

#par(mfrow=c(2,2)) # <- Remove this line if you want graphs not grouped
plot(GSPC.Close, resid(c), col = alpha("grey36", 0.5), pch = 20,
     xlab = "GSPC.Close", ylab = "Residuals", main = "S&P 500 Close versus Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(GSPC.Close, resid(d), col = alpha("grey36", 0.5), pch = 20,
     xlab = "GSPC.Close", ylab = "Squared root of Residuals", main = "S&P 500^2 Close versus Sqrt Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(GSPC.Close, rstandard(c), col = alpha("grey36", 0.5), pch = 20,
     xlab = "GSPC.Close", ylab = "Standardized Residuals", main = "S&P 500 Close versus Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(GSPC.Close, rstandard(d), col = alpha("grey36", 0.5), pch = 20,
     xlab = "GSPC.Close", ylab = "Standardized Residuals", main = "S&P 500^2 Close versus Sqrt Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(c, which = 1)

plot(d, which = 1)

The more interesting variables we have are the VIX variables, as they have the most correlation with the predicted APPL.vol. And it makes sense because the VIX is calculated to help predict stock volatility using options and other macroeconomic factors other than by directly a stock’s return or close price

c <- lm(AAPL.Vol ~ VIX.Close)

d <- lm(log(AAPL.Vol) ~ VIX.Close)

### Some transforms of Appl Volitility vs Google Close

#par(mfrow=c(2,2)) # <- Remove this line if you want graphs not grouped
plot(VIX.Close, resid(c), col = alpha("grey36", 0.5), pch = 20,
     xlab = "VIX.Close", ylab = "Residuals", main = "VIX Close versus Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(VIX.Close, resid(d), col = alpha("grey36", 0.5), pch = 20,
     xlab = "VIX.Close", ylab = "Log of Residuals", main = "VIX Close versus Sqrt Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(c, which = 1)

plot(d, which = 1)

After the Transform of Log(Appl.vol), the residual plots became a lot more distributed like a normal distribution, though not uniformly distributed throughout the graph. However, a Logarithmic transformation on the Y-value certainly made an improvement to our model, and brings the possibility of using a transform to analyze our correlations to perhaps get a better model.

c <- lm(AAPL.Vol ~ VXAPLCLS)

d <- lm(log(AAPL.Vol) ~ VXAPLCLS)

### Some transforms of Appl Volitility vs Google Close

#par(mfrow=c(2,2)) # <- Remove this line if you want graphs not grouped
plot(VXAPLCLS, resid(c), col = alpha("grey36", 0.5), pch = 20,
     xlab = "VXAPLCLS", ylab = "Residuals", main = "VIX Apple versus Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(VXAPLCLS, resid(d), col = alpha("grey36", 0.5), pch = 20,
     xlab = "VXAPLCLS", ylab = "Log of Residuals", main = "VIX Apple versus Sqrt Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(c, which = 1)

plot(d, which = 1)

The Log transformation of Appl.vol with Appl.vix is an even better result of transformation. The values arent as left skewed, and better reflects the linear relationship between the two variables.

c <- lm(AAPL.Vol ~ VXAZNCLS)

d <- lm(log(AAPL.Vol) ~ VXAZNCLS)

### Some transforms of Appl Volitility vs Google Close

#par(mfrow=c(2,2)) # <- Remove this line if you want graphs not grouped
plot(VXAZNCLS, resid(c), col = alpha("grey36", 0.5), pch = 20,
     xlab = "VXAZNCLS", ylab = "Residuals", main = "VIX AMZN versus Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(VXAZNCLS, resid(d), col = alpha("grey36", 0.5), pch = 20,
     xlab = "VXAZNCLS", ylab = "Log of Residuals", main = "VIX AMZN versus Sqrt Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(c, which = 1)

plot(d, which = 1)

The same transformations of using logarithmic transform also works very well to better serve our linear model. the results are much more normal than without a transform.

c <- lm(AAPL.Vol ~ VXGOGCLS)

d <- lm(log(AAPL.Vol) ~ VXGOGCLS)

### Some transforms of Appl Volitility vs Google Close

#par(mfrow=c(2,2)) # <- Remove this line if you want graphs not grouped
plot(VXGOGCLS, resid(c), col = alpha("grey36", 0.5), pch = 20,
     xlab = "VXGOGCLS", ylab = "Residuals", main = "VIX GOOG versus Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(VXGOGCLS, resid(d), col = alpha("grey36", 0.5), pch = 20,
     xlab = "VXGOGCLS", ylab = "Log of Residuals", main = "VIX GOOG versus Sqrt Appl Vol Residuals")
abline(h = 0, col = "darkorange", lwd = 2)

plot(c, which = 1)

plot(d, which = 1)

Conclusion: Using a logarithmic transformation upon our VIX variables in our models would be better to show a much more linear relationship between our data.