Part 3: Stocks Project
Chapter 3: Single Hypothesis
library(readr)
df <- read_csv("volatility.csv")##
## -- Column specification --------------------------------------------------------
## cols(
## VIX.Close = col_double(),
## AAPL.Close = col_double(),
## VXAPLCLS = col_double(),
## VXAZNCLS = col_double(),
## VXGOGCLS = col_double(),
## GOOG.Close = col_double(),
## AMZN.Close = col_double(),
## GSPC.Close = col_double(),
## Date = col_double(),
## AAPL.Ret = col_double(),
## AMZN.Ret = col_double(),
## GOOG.Ret = col_double(),
## GSPC.Ret = col_double(),
## AAPL.Vol = col_double(),
## AAPL.PVol = col_double(),
## AMZN.PVol = col_double(),
## GOOG.PVol = col_double(),
## GSPC.PVol = col_double()
## )head(df)attach(df)Beta1 hypothesis tests for linearity
Null Hypothesis: Beta1 = 0; no relation between X and Y
Construct a 95% Confidence Interval for the value of Beta1
If the interval contains zero, then there is no linear relation
mod <- lm(AAPL.Vol ~ AAPL.PVol)
CI_b11 = confint(mod)[2,]
CI_b11## 2.5 % 97.5 %
## 0.2830332 0.4454848mod <- lm(AAPL.Vol ~ AMZN.PVol)
CI_b12 = confint(mod)[2,]
CI_b12## 2.5 % 97.5 %
## 0.3295250 0.4377032mod <- lm(AAPL.Vol ~ GOOG.PVol)
CI_b13 = confint(mod)[2,]
CI_b13## 2.5 % 97.5 %
## 0.3373554 0.5163088mod <- lm(AAPL.Vol ~ GSPC.PVol)
CI_b14 = confint(mod)[2,]
CI_b14## 2.5 % 97.5 %
## 0.4453690 0.6912425mod <- lm(AAPL.Vol ~ AAPL.Close)
CI_b15 = confint(mod)[2,]
CI_b15## 2.5 % 97.5 %
## -0.0002754410 -0.0001322303mod <- lm(AAPL.Vol ~ AMZN.Close)
CI_b16 = confint(mod)[2,]
CI_b16## 2.5 % 97.5 %
## -7.829995e-06 -1.486254e-06mod <- lm(AAPL.Vol ~ GOOG.Close)
CI_b17 = confint(mod)[2,]
CI_b17## 2.5 % 97.5 %
## -3.414044e-05 -2.186995e-05mod <- lm(AAPL.Vol ~ GSPC.Close)
CI_b18 = confint(mod)[2,]
CI_b18## 2.5 % 97.5 %
## -1.742576e-05 -1.061455e-05mod <- lm(AAPL.Vol ~ VXAPLCLS)
CI_b19 = confint(mod)[2,]
CI_b19## 2.5 % 97.5 %
## 0.0006454469 0.0007882949mod <- lm(AAPL.Vol ~ VXAZNCLS)
CI_b110 = confint(mod)[2,]
CI_b110## 2.5 % 97.5 %
## 0.0005032842 0.0006014044mod <- lm(AAPL.Vol ~ VXGOGCLS)
CI_b111 = confint(mod)[2,]
CI_b111## 2.5 % 97.5 %
## 0.0007002769 0.0008519949mod <- lm(AAPL.Vol ~ VIX.Close)
CI_b112 = confint(mod)[2,]
CI_b112## 2.5 % 97.5 %
## 0.0006023719 0.0008418008mod <- lm(AAPL.Vol ~ AAPL.Ret)
CI_b113 = confint(mod)[2,]
CI_b113## 2.5 % 97.5 %
## -0.10211976 -0.03977091mod <- lm(AAPL.Vol ~ AMZN.Ret)
CI_b114 = confint(mod)[2,]
CI_b114## 2.5 % 97.5 %
## -0.0583337486 -0.0004114984mod <- lm(AAPL.Vol ~ GOOG.Ret)
CI_b115 = confint(mod)[2,]
CI_b115## 2.5 % 97.5 %
## -0.060172750 0.006721432mod <- lm(AAPL.Vol ~ GSPC.Ret)
CI_b116 = confint(mod)[2,]
CI_b116## 2.5 % 97.5 %
## -0.1510560 -0.0349265Although the beta1 value for several of the linear regressions was very close to zero, the only variable for which we can be statistically certain has NO linear relation with AAPL.Vol is GOOG.Ret.
Pvol predictors have positive linear relation. Close predictors have negative linear relation with slope very close to zero. VIX predictors have a positive linear relation with slope very close to zero. Return variables have mostly negative relations, but GOOG.Ret has no linear relation with AAPL.Vol.
Chapter 9
rm(list=ls())
library(corrplot)## corrplot 0.84 loadeddf <- 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#install.packages("olsrr")
library("olsrr")##
## Attaching package: 'olsrr'## The following object is masked from 'package:datasets':
##
## riversmod <- lm(AAPL.Vol ~ VXAPLCLS+VXAZNCLS+VXGOGCLS+VIX.Close+AMZN.PVol+AAPL.PVol+AMZN.PVol+GOOG.PVol+GSPC.PVol, data=df)
k <- ols_step_all_possible(mod)
plot(k)
ksub <- ols_step_best_subset(mod)
subplot(sub)
Forward Regression:
ols_step_forward_p(mod, details = TRUE)## Forward Selection Method
## ---------------------------
##
## Candidate Terms:
##
## 1. VXAPLCLS
## 2. VXAZNCLS
## 3. VXGOGCLS
## 4. VIX.Close
## 5. AMZN.PVol
## 6. AAPL.PVol
## 7. GOOG.PVol
## 8. GSPC.PVol
##
## We are selecting variables based on p value...
##
##
## Forward Selection: Step 1
##
## - VXAZNCLS
##
## Model Summary
## --------------------------------------------------------------
## R 0.703 RMSE 0.004
## R-Squared 0.494 Coef. Var 27.681
## Adj. R-Squared 0.493 MSE 0.000
## Pred R-Squared 0.490 MAE 0.004
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.010 1 0.010 489.282 0.0000
## Residual 0.010 501 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ---------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ---------------------------------------------------------------------------------------
## (Intercept) -0.001 0.001 -0.695 0.487 -0.002 0.001
## VXAZNCLS 0.001 0.000 0.703 22.120 0.000 0.001 0.001
## ---------------------------------------------------------------------------------------
##
##
##
## Forward Selection: Step 2
##
## - VXAPLCLS
##
## Model Summary
## --------------------------------------------------------------
## R 0.724 RMSE 0.004
## R-Squared 0.524 Coef. Var 26.877
## Adj. R-Squared 0.522 MSE 0.000
## Pred R-Squared 0.518 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.010 2 0.005 275.235 0.0000
## Residual 0.010 500 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -3.889 0.000 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.480 9.562 0.000 0.000 0.000
## VXAPLCLS 0.000 0.000 0.282 5.608 0.000 0.000 0.000
## ----------------------------------------------------------------------------------------
##
##
##
## Forward Selection: Step 3
##
## - GSPC.PVol
##
## Model Summary
## --------------------------------------------------------------
## R 0.733 RMSE 0.004
## R-Squared 0.537 Coef. Var 26.543
## Adj. R-Squared 0.534 MSE 0.000
## Pred R-Squared 0.530 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 3 0.004 192.67 0.0000
## Residual 0.009 499 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -4.539 0.000 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.549 10.362 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.311 6.188 0.000 0.000 0.000
## GSPC.PVol -0.221 0.060 -0.146 -3.692 0.000 -0.339 -0.104
## ----------------------------------------------------------------------------------------
##
##
##
## Forward Selection: Step 4
##
## - GOOG.PVol
##
## Model Summary
## --------------------------------------------------------------
## R 0.743 RMSE 0.004
## R-Squared 0.552 Coef. Var 26.114
## Adj. R-Squared 0.549 MSE 0.000
## Pred R-Squared 0.544 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 4 0.003 153.684 0.0000
## Residual 0.009 498 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.006 0.001 -5.729 0.000 -0.008 -0.004
## VXAZNCLS 0.000 0.000 0.556 10.650 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.301 6.075 0.000 0.000 0.000
## GSPC.PVol -0.485 0.086 -0.321 -5.623 0.000 -0.654 -0.315
## GOOG.PVol 0.239 0.057 0.217 4.190 0.000 0.127 0.351
## ----------------------------------------------------------------------------------------
##
##
##
## Forward Selection: Step 5
##
## - VIX.Close
##
## Model Summary
## --------------------------------------------------------------
## R 0.748 RMSE 0.004
## R-Squared 0.560 Coef. Var 25.916
## Adj. R-Squared 0.556 MSE 0.000
## Pred R-Squared 0.550 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 5 0.002 126.556 0.0000
## Residual 0.009 497 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.005 0.001 -4.946 0.000 -0.007 -0.003
## VXAZNCLS 0.000 0.000 0.591 11.117 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.342 6.691 0.000 0.000 0.000
## GSPC.PVol -0.420 0.088 -0.278 -4.754 0.000 -0.594 -0.246
## GOOG.PVol 0.278 0.058 0.252 4.781 0.000 0.164 0.393
## VIX.Close 0.000 0.000 -0.158 -2.937 0.003 0.000 0.000
## ----------------------------------------------------------------------------------------
##
##
##
## Forward Selection: Step 6
##
## - AMZN.PVol
##
## Model Summary
## --------------------------------------------------------------
## R 0.751 RMSE 0.004
## R-Squared 0.564 Coef. Var 25.835
## Adj. R-Squared 0.558 MSE 0.000
## Pred R-Squared 0.551 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 6 0.002 106.815 0.0000
## Residual 0.009 496 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -4.322 0.000 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.527 8.541 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.350 6.859 0.000 0.000 0.000
## GSPC.PVol -0.495 0.095 -0.327 -5.184 0.000 -0.682 -0.307
## GOOG.PVol 0.233 0.062 0.210 3.734 0.000 0.110 0.355
## VIX.Close 0.000 0.000 -0.136 -2.481 0.013 0.000 0.000
## AMZN.PVol 0.092 0.045 0.127 2.032 0.043 0.003 0.181
## ----------------------------------------------------------------------------------------
##
##
##
## Forward Selection: Step 7
##
## - AAPL.PVol
##
## Model Summary
## --------------------------------------------------------------
## R 0.754 RMSE 0.004
## R-Squared 0.569 Coef. Var 25.696
## Adj. R-Squared 0.563 MSE 0.000
## Pred R-Squared 0.555 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## --------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## --------------------------------------------------------------------
## Regression 0.011 7 0.002 93.453 0.0000
## Residual 0.009 495 0.000
## Total 0.020 502
## --------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -3.406 0.001 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.488 7.705 0.000 0.000 0.000
## VXAPLCLS 0.000 0.000 0.379 7.282 0.000 0.000 0.001
## GSPC.PVol -0.398 0.102 -0.264 -3.893 0.000 -0.599 -0.197
## GOOG.PVol 0.258 0.063 0.234 4.112 0.000 0.135 0.381
## VIX.Close 0.000 0.000 -0.154 -2.807 0.005 0.000 0.000
## AMZN.PVol 0.162 0.053 0.223 3.059 0.002 0.058 0.266
## AAPL.PVol -0.166 0.066 -0.167 -2.522 0.012 -0.296 -0.037
## ----------------------------------------------------------------------------------------
##
##
##
## No more variables to be added.
##
## Variables Entered:
##
## + VXAZNCLS
## + VXAPLCLS
## + GSPC.PVol
## + GOOG.PVol
## + VIX.Close
## + AMZN.PVol
## + AAPL.PVol
##
##
## Final Model Output
## ------------------
##
## Model Summary
## --------------------------------------------------------------
## R 0.754 RMSE 0.004
## R-Squared 0.569 Coef. Var 25.696
## Adj. R-Squared 0.563 MSE 0.000
## Pred R-Squared 0.555 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## --------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## --------------------------------------------------------------------
## Regression 0.011 7 0.002 93.453 0.0000
## Residual 0.009 495 0.000
## Total 0.020 502
## --------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -3.406 0.001 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.488 7.705 0.000 0.000 0.000
## VXAPLCLS 0.000 0.000 0.379 7.282 0.000 0.000 0.001
## GSPC.PVol -0.398 0.102 -0.264 -3.893 0.000 -0.599 -0.197
## GOOG.PVol 0.258 0.063 0.234 4.112 0.000 0.135 0.381
## VIX.Close 0.000 0.000 -0.154 -2.807 0.005 0.000 0.000
## AMZN.PVol 0.162 0.053 0.223 3.059 0.002 0.058 0.266
## AAPL.PVol -0.166 0.066 -0.167 -2.522 0.012 -0.296 -0.037
## ----------------------------------------------------------------------------------------##
## Selection Summary
## ----------------------------------------------------------------------------
## Variable Adj.
## Step Entered R-Square R-Square C(p) AIC RMSE
## ----------------------------------------------------------------------------
## 1 VXAZNCLS 0.4941 0.4931 81.2384 -4006.8538 0.0045
## 2 VXAPLCLS 0.5240 0.5221 48.9014 -4035.5372 0.0044
## 3 GSPC.PVol 0.5367 0.5339 36.3838 -4047.0950 0.0043
## 4 GOOG.PVol 0.5525 0.5489 20.2916 -4062.5191 0.0042
## 5 VIX.Close 0.5601 0.5557 13.5341 -4069.1750 0.0042
## 6 AMZN.PVol 0.5637 0.5584 11.3705 -4071.3432 0.0042
## 7 AAPL.PVol 0.5693 0.5632 7.0234 -4075.7645 0.0042
## ----------------------------------------------------------------------------Stepwise Backward Regression
k <- ols_step_backward_p(mod, details=TRUE)## Backward Elimination Method
## ---------------------------
##
## Candidate Terms:
##
## 1 . VXAPLCLS
## 2 . VXAZNCLS
## 3 . VXGOGCLS
## 4 . VIX.Close
## 5 . AMZN.PVol
## 6 . AAPL.PVol
## 7 . GOOG.PVol
## 8 . GSPC.PVol
##
## We are eliminating variables based on p value...
##
## - VXGOGCLS
##
## Backward Elimination: Step 1
##
## Variable VXGOGCLS Removed
##
## Model Summary
## --------------------------------------------------------------
## R 0.754 RMSE 0.004
## R-Squared 0.569 Coef. Var 25.696
## Adj. R-Squared 0.563 MSE 0.000
## Pred R-Squared 0.555 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## --------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## --------------------------------------------------------------------
## Regression 0.011 7 0.002 93.453 0.0000
## Residual 0.009 495 0.000
## Total 0.020 502
## --------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -3.406 0.001 -0.006 -0.002
## VXAPLCLS 0.000 0.000 0.379 7.282 0.000 0.000 0.001
## VXAZNCLS 0.000 0.000 0.488 7.705 0.000 0.000 0.000
## VIX.Close 0.000 0.000 -0.154 -2.807 0.005 0.000 0.000
## AMZN.PVol 0.162 0.053 0.223 3.059 0.002 0.058 0.266
## AAPL.PVol -0.166 0.066 -0.167 -2.522 0.012 -0.296 -0.037
## GOOG.PVol 0.258 0.063 0.234 4.112 0.000 0.135 0.381
## GSPC.PVol -0.398 0.102 -0.264 -3.893 0.000 -0.599 -0.197
## ----------------------------------------------------------------------------------------
##
##
##
## No more variables satisfy the condition of p value = 0.3
##
##
## Variables Removed:
##
## - VXGOGCLS
##
##
## Final Model Output
## ------------------
##
## Model Summary
## --------------------------------------------------------------
## R 0.754 RMSE 0.004
## R-Squared 0.569 Coef. Var 25.696
## Adj. R-Squared 0.563 MSE 0.000
## Pred R-Squared 0.555 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## --------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## --------------------------------------------------------------------
## Regression 0.011 7 0.002 93.453 0.0000
## Residual 0.009 495 0.000
## Total 0.020 502
## --------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -3.406 0.001 -0.006 -0.002
## VXAPLCLS 0.000 0.000 0.379 7.282 0.000 0.000 0.001
## VXAZNCLS 0.000 0.000 0.488 7.705 0.000 0.000 0.000
## VIX.Close 0.000 0.000 -0.154 -2.807 0.005 0.000 0.000
## AMZN.PVol 0.162 0.053 0.223 3.059 0.002 0.058 0.266
## AAPL.PVol -0.166 0.066 -0.167 -2.522 0.012 -0.296 -0.037
## GOOG.PVol 0.258 0.063 0.234 4.112 0.000 0.135 0.381
## GSPC.PVol -0.398 0.102 -0.264 -3.893 0.000 -0.599 -0.197
## ----------------------------------------------------------------------------------------k##
##
## Elimination Summary
## --------------------------------------------------------------------------
## Variable Adj.
## Step Removed R-Square R-Square C(p) AIC RMSE
## --------------------------------------------------------------------------
## 1 VXGOGCLS 0.5693 0.5632 7.0234 -4075.7645 0.0042
## --------------------------------------------------------------------------Stepwise Regression
Build regression model from a set of candidate predictor variables by entering and removing predictors based on p values, in a stepwise manner until there is no variable left to enter or remove any more. The model should include all the candidate predictor variables. If details is set to TRUE, each step is displayed.
ols_step_both_p(mod, details = TRUE)## Stepwise Selection Method
## ---------------------------
##
## Candidate Terms:
##
## 1. VXAPLCLS
## 2. VXAZNCLS
## 3. VXGOGCLS
## 4. VIX.Close
## 5. AMZN.PVol
## 6. AAPL.PVol
## 7. GOOG.PVol
## 8. GSPC.PVol
##
## We are selecting variables based on p value...
##
##
## Stepwise Selection: Step 1
##
## - VXAZNCLS added
##
## Model Summary
## --------------------------------------------------------------
## R 0.703 RMSE 0.004
## R-Squared 0.494 Coef. Var 27.681
## Adj. R-Squared 0.493 MSE 0.000
## Pred R-Squared 0.490 MAE 0.004
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.010 1 0.010 489.282 0.0000
## Residual 0.010 501 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ---------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ---------------------------------------------------------------------------------------
## (Intercept) -0.001 0.001 -0.695 0.487 -0.002 0.001
## VXAZNCLS 0.001 0.000 0.703 22.120 0.000 0.001 0.001
## ---------------------------------------------------------------------------------------
##
##
##
## Stepwise Selection: Step 2
##
## - VXAPLCLS added
##
## Model Summary
## --------------------------------------------------------------
## R 0.724 RMSE 0.004
## R-Squared 0.524 Coef. Var 26.877
## Adj. R-Squared 0.522 MSE 0.000
## Pred R-Squared 0.518 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.010 2 0.005 275.235 0.0000
## Residual 0.010 500 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -3.889 0.000 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.480 9.562 0.000 0.000 0.000
## VXAPLCLS 0.000 0.000 0.282 5.608 0.000 0.000 0.000
## ----------------------------------------------------------------------------------------
##
##
##
## Model Summary
## --------------------------------------------------------------
## R 0.724 RMSE 0.004
## R-Squared 0.524 Coef. Var 26.877
## Adj. R-Squared 0.522 MSE 0.000
## Pred R-Squared 0.518 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.010 2 0.005 275.235 0.0000
## Residual 0.010 500 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -3.889 0.000 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.480 9.562 0.000 0.000 0.000
## VXAPLCLS 0.000 0.000 0.282 5.608 0.000 0.000 0.000
## ----------------------------------------------------------------------------------------
##
##
##
## Stepwise Selection: Step 3
##
## - GSPC.PVol added
##
## Model Summary
## --------------------------------------------------------------
## R 0.733 RMSE 0.004
## R-Squared 0.537 Coef. Var 26.543
## Adj. R-Squared 0.534 MSE 0.000
## Pred R-Squared 0.530 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 3 0.004 192.67 0.0000
## Residual 0.009 499 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -4.539 0.000 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.549 10.362 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.311 6.188 0.000 0.000 0.000
## GSPC.PVol -0.221 0.060 -0.146 -3.692 0.000 -0.339 -0.104
## ----------------------------------------------------------------------------------------
##
##
##
## Model Summary
## --------------------------------------------------------------
## R 0.733 RMSE 0.004
## R-Squared 0.537 Coef. Var 26.543
## Adj. R-Squared 0.534 MSE 0.000
## Pred R-Squared 0.530 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 3 0.004 192.67 0.0000
## Residual 0.009 499 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -4.539 0.000 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.549 10.362 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.311 6.188 0.000 0.000 0.000
## GSPC.PVol -0.221 0.060 -0.146 -3.692 0.000 -0.339 -0.104
## ----------------------------------------------------------------------------------------
##
##
##
## Stepwise Selection: Step 4
##
## - GOOG.PVol added
##
## Model Summary
## --------------------------------------------------------------
## R 0.743 RMSE 0.004
## R-Squared 0.552 Coef. Var 26.114
## Adj. R-Squared 0.549 MSE 0.000
## Pred R-Squared 0.544 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 4 0.003 153.684 0.0000
## Residual 0.009 498 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.006 0.001 -5.729 0.000 -0.008 -0.004
## VXAZNCLS 0.000 0.000 0.556 10.650 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.301 6.075 0.000 0.000 0.000
## GSPC.PVol -0.485 0.086 -0.321 -5.623 0.000 -0.654 -0.315
## GOOG.PVol 0.239 0.057 0.217 4.190 0.000 0.127 0.351
## ----------------------------------------------------------------------------------------
##
##
##
## Model Summary
## --------------------------------------------------------------
## R 0.743 RMSE 0.004
## R-Squared 0.552 Coef. Var 26.114
## Adj. R-Squared 0.549 MSE 0.000
## Pred R-Squared 0.544 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 4 0.003 153.684 0.0000
## Residual 0.009 498 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.006 0.001 -5.729 0.000 -0.008 -0.004
## VXAZNCLS 0.000 0.000 0.556 10.650 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.301 6.075 0.000 0.000 0.000
## GSPC.PVol -0.485 0.086 -0.321 -5.623 0.000 -0.654 -0.315
## GOOG.PVol 0.239 0.057 0.217 4.190 0.000 0.127 0.351
## ----------------------------------------------------------------------------------------
##
##
##
## Stepwise Selection: Step 5
##
## - VIX.Close added
##
## Model Summary
## --------------------------------------------------------------
## R 0.748 RMSE 0.004
## R-Squared 0.560 Coef. Var 25.916
## Adj. R-Squared 0.556 MSE 0.000
## Pred R-Squared 0.550 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 5 0.002 126.556 0.0000
## Residual 0.009 497 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.005 0.001 -4.946 0.000 -0.007 -0.003
## VXAZNCLS 0.000 0.000 0.591 11.117 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.342 6.691 0.000 0.000 0.000
## GSPC.PVol -0.420 0.088 -0.278 -4.754 0.000 -0.594 -0.246
## GOOG.PVol 0.278 0.058 0.252 4.781 0.000 0.164 0.393
## VIX.Close 0.000 0.000 -0.158 -2.937 0.003 0.000 0.000
## ----------------------------------------------------------------------------------------
##
##
##
## Model Summary
## --------------------------------------------------------------
## R 0.748 RMSE 0.004
## R-Squared 0.560 Coef. Var 25.916
## Adj. R-Squared 0.556 MSE 0.000
## Pred R-Squared 0.550 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 5 0.002 126.556 0.0000
## Residual 0.009 497 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.005 0.001 -4.946 0.000 -0.007 -0.003
## VXAZNCLS 0.000 0.000 0.591 11.117 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.342 6.691 0.000 0.000 0.000
## GSPC.PVol -0.420 0.088 -0.278 -4.754 0.000 -0.594 -0.246
## GOOG.PVol 0.278 0.058 0.252 4.781 0.000 0.164 0.393
## VIX.Close 0.000 0.000 -0.158 -2.937 0.003 0.000 0.000
## ----------------------------------------------------------------------------------------
##
##
##
## Stepwise Selection: Step 6
##
## - AMZN.PVol added
##
## Model Summary
## --------------------------------------------------------------
## R 0.751 RMSE 0.004
## R-Squared 0.564 Coef. Var 25.835
## Adj. R-Squared 0.558 MSE 0.000
## Pred R-Squared 0.551 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 6 0.002 106.815 0.0000
## Residual 0.009 496 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -4.322 0.000 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.527 8.541 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.350 6.859 0.000 0.000 0.000
## GSPC.PVol -0.495 0.095 -0.327 -5.184 0.000 -0.682 -0.307
## GOOG.PVol 0.233 0.062 0.210 3.734 0.000 0.110 0.355
## VIX.Close 0.000 0.000 -0.136 -2.481 0.013 0.000 0.000
## AMZN.PVol 0.092 0.045 0.127 2.032 0.043 0.003 0.181
## ----------------------------------------------------------------------------------------
##
##
##
## Model Summary
## --------------------------------------------------------------
## R 0.751 RMSE 0.004
## R-Squared 0.564 Coef. Var 25.835
## Adj. R-Squared 0.558 MSE 0.000
## Pred R-Squared 0.551 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 6 0.002 106.815 0.0000
## Residual 0.009 496 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -4.322 0.000 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.527 8.541 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.350 6.859 0.000 0.000 0.000
## GSPC.PVol -0.495 0.095 -0.327 -5.184 0.000 -0.682 -0.307
## GOOG.PVol 0.233 0.062 0.210 3.734 0.000 0.110 0.355
## VIX.Close 0.000 0.000 -0.136 -2.481 0.013 0.000 0.000
## AMZN.PVol 0.092 0.045 0.127 2.032 0.043 0.003 0.181
## ----------------------------------------------------------------------------------------
##
##
##
## Stepwise Selection: Step 7
##
## - AAPL.PVol added
##
## Model Summary
## --------------------------------------------------------------
## R 0.754 RMSE 0.004
## R-Squared 0.569 Coef. Var 25.696
## Adj. R-Squared 0.563 MSE 0.000
## Pred R-Squared 0.555 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## --------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## --------------------------------------------------------------------
## Regression 0.011 7 0.002 93.453 0.0000
## Residual 0.009 495 0.000
## Total 0.020 502
## --------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -3.406 0.001 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.488 7.705 0.000 0.000 0.000
## VXAPLCLS 0.000 0.000 0.379 7.282 0.000 0.000 0.001
## GSPC.PVol -0.398 0.102 -0.264 -3.893 0.000 -0.599 -0.197
## GOOG.PVol 0.258 0.063 0.234 4.112 0.000 0.135 0.381
## VIX.Close 0.000 0.000 -0.154 -2.807 0.005 0.000 0.000
## AMZN.PVol 0.162 0.053 0.223 3.059 0.002 0.058 0.266
## AAPL.PVol -0.166 0.066 -0.167 -2.522 0.012 -0.296 -0.037
## ----------------------------------------------------------------------------------------
##
##
##
## Model Summary
## --------------------------------------------------------------
## R 0.754 RMSE 0.004
## R-Squared 0.569 Coef. Var 25.696
## Adj. R-Squared 0.563 MSE 0.000
## Pred R-Squared 0.555 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## --------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## --------------------------------------------------------------------
## Regression 0.011 7 0.002 93.453 0.0000
## Residual 0.009 495 0.000
## Total 0.020 502
## --------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -3.406 0.001 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.488 7.705 0.000 0.000 0.000
## VXAPLCLS 0.000 0.000 0.379 7.282 0.000 0.000 0.001
## GSPC.PVol -0.398 0.102 -0.264 -3.893 0.000 -0.599 -0.197
## GOOG.PVol 0.258 0.063 0.234 4.112 0.000 0.135 0.381
## VIX.Close 0.000 0.000 -0.154 -2.807 0.005 0.000 0.000
## AMZN.PVol 0.162 0.053 0.223 3.059 0.002 0.058 0.266
## AAPL.PVol -0.166 0.066 -0.167 -2.522 0.012 -0.296 -0.037
## ----------------------------------------------------------------------------------------
##
##
##
## No more variables to be added/removed.
##
##
## Final Model Output
## ------------------
##
## Model Summary
## --------------------------------------------------------------
## R 0.754 RMSE 0.004
## R-Squared 0.569 Coef. Var 25.696
## Adj. R-Squared 0.563 MSE 0.000
## Pred R-Squared 0.555 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## --------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## --------------------------------------------------------------------
## Regression 0.011 7 0.002 93.453 0.0000
## Residual 0.009 495 0.000
## Total 0.020 502
## --------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -3.406 0.001 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.488 7.705 0.000 0.000 0.000
## VXAPLCLS 0.000 0.000 0.379 7.282 0.000 0.000 0.001
## GSPC.PVol -0.398 0.102 -0.264 -3.893 0.000 -0.599 -0.197
## GOOG.PVol 0.258 0.063 0.234 4.112 0.000 0.135 0.381
## VIX.Close 0.000 0.000 -0.154 -2.807 0.005 0.000 0.000
## AMZN.PVol 0.162 0.053 0.223 3.059 0.002 0.058 0.266
## AAPL.PVol -0.166 0.066 -0.167 -2.522 0.012 -0.296 -0.037
## ----------------------------------------------------------------------------------------##
## Stepwise Selection Summary
## ----------------------------------------------------------------------------------------
## Added/ Adj.
## Step Variable Removed R-Square R-Square C(p) AIC RMSE
## ----------------------------------------------------------------------------------------
## 1 VXAZNCLS addition 0.494 0.493 81.2380 -4006.8538 0.0045
## 2 VXAPLCLS addition 0.524 0.522 48.9010 -4035.5372 0.0044
## 3 GSPC.PVol addition 0.537 0.534 36.3840 -4047.0950 0.0043
## 4 GOOG.PVol addition 0.552 0.549 20.2920 -4062.5191 0.0042
## 5 VIX.Close addition 0.560 0.556 13.5340 -4069.1750 0.0042
## 6 AMZN.PVol addition 0.564 0.558 11.3700 -4071.3432 0.0042
## 7 AAPL.PVol addition 0.569 0.563 7.0230 -4075.7645 0.0042
## ----------------------------------------------------------------------------------------new_mod <- lm(AAPL.Vol ~ VXAPLCLS+VXAZNCLS+VXGOGCLS+VIX.Close+AMZN.PVol, data=df)
k <- ols_step_both_p(new_mod, details = TRUE)## Stepwise Selection Method
## ---------------------------
##
## Candidate Terms:
##
## 1. VXAPLCLS
## 2. VXAZNCLS
## 3. VXGOGCLS
## 4. VIX.Close
## 5. AMZN.PVol
##
## We are selecting variables based on p value...
##
##
## Stepwise Selection: Step 1
##
## - VXAZNCLS added
##
## Model Summary
## --------------------------------------------------------------
## R 0.703 RMSE 0.004
## R-Squared 0.494 Coef. Var 27.681
## Adj. R-Squared 0.493 MSE 0.000
## Pred R-Squared 0.490 MAE 0.004
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.010 1 0.010 489.282 0.0000
## Residual 0.010 501 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ---------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ---------------------------------------------------------------------------------------
## (Intercept) -0.001 0.001 -0.695 0.487 -0.002 0.001
## VXAZNCLS 0.001 0.000 0.703 22.120 0.000 0.001 0.001
## ---------------------------------------------------------------------------------------
##
##
##
## Stepwise Selection: Step 2
##
## - VXAPLCLS added
##
## Model Summary
## --------------------------------------------------------------
## R 0.724 RMSE 0.004
## R-Squared 0.524 Coef. Var 26.877
## Adj. R-Squared 0.522 MSE 0.000
## Pred R-Squared 0.518 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.010 2 0.005 275.235 0.0000
## Residual 0.010 500 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -3.889 0.000 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.480 9.562 0.000 0.000 0.000
## VXAPLCLS 0.000 0.000 0.282 5.608 0.000 0.000 0.000
## ----------------------------------------------------------------------------------------
##
##
##
## Model Summary
## --------------------------------------------------------------
## R 0.724 RMSE 0.004
## R-Squared 0.524 Coef. Var 26.877
## Adj. R-Squared 0.522 MSE 0.000
## Pred R-Squared 0.518 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.010 2 0.005 275.235 0.0000
## Residual 0.010 500 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.004 0.001 -3.889 0.000 -0.006 -0.002
## VXAZNCLS 0.000 0.000 0.480 9.562 0.000 0.000 0.000
## VXAPLCLS 0.000 0.000 0.282 5.608 0.000 0.000 0.000
## ----------------------------------------------------------------------------------------
##
##
##
## Stepwise Selection: Step 3
##
## - VIX.Close added
##
## Model Summary
## --------------------------------------------------------------
## R 0.732 RMSE 0.004
## R-Squared 0.535 Coef. Var 26.579
## Adj. R-Squared 0.533 MSE 0.000
## Pred R-Squared 0.528 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 3 0.004 191.694 0.0000
## Residual 0.009 499 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.003 0.001 -3.237 0.001 -0.005 -0.001
## VXAZNCLS 0.000 0.000 0.549 10.278 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.339 6.483 0.000 0.000 0.000
## VIX.Close 0.000 0.000 -0.160 -3.498 0.001 0.000 0.000
## ----------------------------------------------------------------------------------------
##
##
##
## Model Summary
## --------------------------------------------------------------
## R 0.732 RMSE 0.004
## R-Squared 0.535 Coef. Var 26.579
## Adj. R-Squared 0.533 MSE 0.000
## Pred R-Squared 0.528 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 3 0.004 191.694 0.0000
## Residual 0.009 499 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.003 0.001 -3.237 0.001 -0.005 -0.001
## VXAZNCLS 0.000 0.000 0.549 10.278 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.339 6.483 0.000 0.000 0.000
## VIX.Close 0.000 0.000 -0.160 -3.498 0.001 0.000 0.000
## ----------------------------------------------------------------------------------------
##
##
##
## No more variables to be added/removed.
##
##
## Final Model Output
## ------------------
##
## Model Summary
## --------------------------------------------------------------
## R 0.732 RMSE 0.004
## R-Squared 0.535 Coef. Var 26.579
## Adj. R-Squared 0.533 MSE 0.000
## Pred R-Squared 0.528 MAE 0.003
## --------------------------------------------------------------
## RMSE: Root Mean Square Error
## MSE: Mean Square Error
## MAE: Mean Absolute Error
##
## ANOVA
## ---------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ---------------------------------------------------------------------
## Regression 0.011 3 0.004 191.694 0.0000
## Residual 0.009 499 0.000
## Total 0.020 502
## ---------------------------------------------------------------------
##
## Parameter Estimates
## ----------------------------------------------------------------------------------------
## model Beta Std. Error Std. Beta t Sig lower upper
## ----------------------------------------------------------------------------------------
## (Intercept) -0.003 0.001 -3.237 0.001 -0.005 -0.001
## VXAZNCLS 0.000 0.000 0.549 10.278 0.000 0.000 0.001
## VXAPLCLS 0.000 0.000 0.339 6.483 0.000 0.000 0.000
## VIX.Close 0.000 0.000 -0.160 -3.498 0.001 0.000 0.000
## ----------------------------------------------------------------------------------------plot(k)
k##
## Stepwise Selection Summary
## ----------------------------------------------------------------------------------------
## Added/ Adj.
## Step Variable Removed R-Square R-Square C(p) AIC RMSE
## ----------------------------------------------------------------------------------------
## 1 VXAZNCLS addition 0.494 0.493 44.5640 -4006.8538 0.0045
## 2 VXAPLCLS addition 0.524 0.522 14.3970 -4035.5372 0.0044
## 3 VIX.Close addition 0.535 0.533 4.1540 -4045.7258 0.0043
## ----------------------------------------------------------------------------------------m <- new_mod
inf_df <- data.frame(
"dffits" = dffits(m),
"cooks" = cooks.distance(m),
"dfb1" = dfbetas(m)[,1],
"dfb2" = dfbetas(m)[,2],
"dfb3" = dfbetas(m)[,3],
"dfb4" = dfbetas(m)[,4],
"dfb5" = dfbetas(m)[,5]
)
plot(inf_df$cooks, type = "l",col="red")
Chapter 8:
For the full regression,the adjusted R-squared of the full regression was 0.5836, the p-value of variables VIX.Close,AAPL.Close,VXGOGCLS,GOOG.Close,AAPL.Ret,GOOG.Ret and GSPC.Ret are all less than 0.05, so we may wish to know whether these variables can be dropped from the full model. For level of significance alpha = 0.05, we require F(0.95; 486, 493) = 1.1604. Since F* = 1.5935 >= 1.1604and the p-value of F is 0.1349 > 0.05, we conclude H0 , that variables VIX.Close, AAPL.Close, VXGOGCLS, GOOG.Close, AAPL.Ret, GOOG.Ret and GSPC.Ret should be dropped from the full regression model.
volatility_df <- read.csv("volatility.csv")
volatility_df$Date<-NULL
fit1<-lm(AAPL.Vol~.,volatility_df)
summary(fit1)##
## Call:
## lm(formula = AAPL.Vol ~ ., data = volatility_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0088397 -0.0028762 -0.0002416 0.0022632 0.0165948
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.744e-02 9.141e-03 -4.096 4.93e-05 ***
## VIX.Close -1.497e-04 9.404e-05 -1.592 0.112078
## AAPL.Close -6.918e-05 6.127e-05 -1.129 0.259388
## VXAPLCLS 2.560e-04 6.928e-05 3.695 0.000245 ***
## VXAZNCLS 4.842e-04 8.190e-05 5.912 6.38e-09 ***
## VXGOGCLS 6.316e-05 1.129e-04 0.560 0.576046
## GOOG.Close 5.568e-06 5.328e-06 1.045 0.296552
## AMZN.Close 3.077e-06 1.506e-06 2.043 0.041623 *
## GSPC.Close 8.004e-06 3.878e-06 2.064 0.039535 *
## AAPL.Ret -1.439e-02 1.656e-02 -0.869 0.385400
## AMZN.Ret 3.271e-02 1.531e-02 2.137 0.033104 *
## GOOG.Ret 2.615e-02 1.818e-02 1.439 0.150882
## GSPC.Ret -6.291e-02 4.008e-02 -1.570 0.117152
## AAPL.PVol -2.104e-01 6.722e-02 -3.130 0.001851 **
## AMZN.PVol 2.110e-01 5.787e-02 3.645 0.000296 ***
## GOOG.PVol 2.831e-01 6.789e-02 4.169 3.62e-05 ***
## GSPC.PVol -3.270e-01 1.129e-01 -2.896 0.003949 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.00407 on 486 degrees of freedom
## Multiple R-squared: 0.5968, Adjusted R-squared: 0.5836
## F-statistic: 44.97 on 16 and 486 DF, p-value: < 2.2e-16fit2<-lm(AAPL.Vol~VXAPLCLS+VXAZNCLS+AMZN.Close+GSPC.Close+AMZN.Ret+AAPL.PVol+AMZN.PVol+GOOG.PVol+GSPC.PVol,volatility_df)
summary(fit2)##
## Call:
## lm(formula = AAPL.Vol ~ VXAPLCLS + VXAZNCLS + AMZN.Close + GSPC.Close +
## AMZN.Ret + AAPL.PVol + AMZN.PVol + GOOG.PVol + GSPC.PVol,
## data = volatility_df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0083519 -0.0026370 -0.0004731 0.0023915 0.0164127
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -3.092e-02 6.389e-03 -4.840 1.74e-06 ***
## VXAPLCLS 2.454e-04 6.220e-05 3.946 9.11e-05 ***
## VXAZNCLS 5.043e-04 5.766e-05 8.746 < 2e-16 ***
## AMZN.Close 3.328e-06 1.463e-06 2.275 0.023343 *
## GSPC.Close 6.559e-06 2.262e-06 2.900 0.003901 **
## AMZN.Ret 2.376e-02 1.006e-02 2.361 0.018618 *
## AAPL.PVol -1.893e-01 6.550e-02 -2.890 0.004027 **
## AMZN.PVol 1.875e-01 5.209e-02 3.599 0.000352 ***
## GOOG.PVol 2.480e-01 6.195e-02 4.002 7.24e-05 ***
## GSPC.PVol -3.798e-01 1.033e-01 -3.678 0.000261 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.004087 on 493 degrees of freedom
## Multiple R-squared: 0.5876, Adjusted R-squared: 0.5801
## F-statistic: 78.05 on 9 and 493 DF, p-value: < 2.2e-16anova(fit1,fit2)par(mfrow=c(1,2))
plot(fit1,which=1)
plot(fit2,which=1)
Lekha - Chapter 8 (Transformation Part)
Importing required data:
rm(list=ls())
library(corrplot)
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.Vollibrary(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, unionlibrary(ggplot2)We will check for higher-order terms and whether they’re necessary to add or not by centering the predictor variables for each case, and performing required tests on these centered predictor variables and Y. For each one, we considered \(\alpha = 0.05\).
Y = AAPL.VolStarting with:
AAPL.Vol vs AAPL.PVol:
currX = AAPL.PVol
n = length(AAPL.Vol)
Xbar = sum(AAPL.PVol)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.013281 -0.004299 -0.001671 0.003755 0.015198
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0158368 0.0003359 47.150 < 2e-16 ***
## smallx 0.3092554 0.0511319 6.048 2.87e-09 ***
## xsquare 9.5705312 5.2578866 1.820 0.0693 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.005861 on 500 degrees of freedom
## Multiple R-squared: 0.1399, Adjusted R-squared: 0.1364
## F-statistic: 40.65 on 2 and 500 DF, p-value: < 2.2e-16Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between AAPL.PVol and AAPL.Vol, but now doing the same when AAPL.PVol is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = 0.00011380 / (0.0171732/dfsse)
#or could be taken from ANOVA table
Fobs = curranova[2,4]
Fobs## [1] 3.313215Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is less than critical value, so we conclude that we fail to reject the null hypothesis.
Hence, in the case of predictor variable AAPL.Vol, we can use linear model of order 1, and not add its square.
Now, to de-center our function, we replace x by \(X - \bar{X}\). Since we anyway will not consider the quadratic term, we can just state the linear regression equation as it is for Y and X:
mod = lm(Y ~ currX)
mod$coefficients## (Intercept) currX
## 0.0103326 0.3642590\[\hat{Y} = 0.01033 + 0.3643X_1\]
AAPL.Vol vs AMZN.PVol higher order:
currX = AMZN.PVol
n = length(AAPL.Vol)
Xbar = sum(AMZN.PVol)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0101734 -0.0039287 -0.0008275 0.0034527 0.0135901
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.486e-02 3.151e-04 47.155 < 2e-16 ***
## smallx 1.970e-01 3.965e-02 4.969 9.25e-07 ***
## xsquare 1.808e+01 2.857e+00 6.330 5.46e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.005162 on 500 degrees of freedom
## Multiple R-squared: 0.3328, Adjusted R-squared: 0.3301
## F-statistic: 124.7 on 2 and 500 DF, p-value: < 2.2e-16Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between AMZN.PVol and AAPL.Vol, but now doing the same when AMZN.PVol is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 40.06772#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 40.06772Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is greater than critical value, so we conclude that we reject the null hypothesis.
Hence, the quadratic term of AMZN.PVol is significant.
AAPL.Vol vs GOOG.PVol higher order:
currX = GOOG.PVol
n = length(AAPL.Vol)
Xbar = sum(GOOG.PVol)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.013832 -0.004297 -0.000841 0.003069 0.018441
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.434e-02 3.818e-04 37.554 < 2e-16 ***
## smallx 3.085e-01 4.739e-02 6.508 1.85e-10 ***
## xsquare 5.792e+01 8.892e+00 6.513 1.79e-10 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.005596 on 500 degrees of freedom
## Multiple R-squared: 0.2157, Adjusted R-squared: 0.2126
## F-statistic: 68.76 on 2 and 500 DF, p-value: < 2.2e-16Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between GOOG.PVol and AAPL.Vol, but now doing the same when GOOG.PVol is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 42.42305#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 42.42305Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is greater than critical value, so we conclude that we reject the null hypothesis.
Hence, the quadratic term \(X^2\) in the case of GOOG.PVol is significant.
AAPL.Vol vs GSPC.PVol higher order:
currX = GSPC.PVol
n = length(AAPL.Vol)
Xbar = sum(GSPC.PVol)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.010839 -0.004529 -0.001325 0.003793 0.017576
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.523e-02 3.766e-04 40.434 < 2e-16 ***
## smallx 3.787e-01 8.104e-02 4.673 3.82e-06 ***
## xsquare 5.719e+01 1.580e+01 3.619 0.000326 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.00578 on 500 degrees of freedom
## Multiple R-squared: 0.1633, Adjusted R-squared: 0.1599
## F-statistic: 48.79 on 2 and 500 DF, p-value: < 2.2e-16Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between GSPC.PVol and AAPL.Vol, but now doing the same when GSPC.PVol is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 13.09402#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 13.09402Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is greater than critical value, so we conclude that we reject the null hypothesis.
Hence, the quadratic term for GSPC.PVol is significant.
AAPL.Vol vs AAPL.Close higher order:
currX = AAPL.Close
n = length(AAPL.Vol)
Xbar = sum(AAPL.Close)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.010134 -0.003786 -0.001354 0.003274 0.017791
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.623e-02 3.412e-04 47.563 < 2e-16 ***
## smallx -2.030e-04 4.408e-05 -4.605 5.23e-06 ***
## xsquare -1.232e-07 3.637e-06 -0.034 0.973
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.006131 on 500 degrees of freedom
## Multiple R-squared: 0.05877, Adjusted R-squared: 0.055
## F-statistic: 15.61 on 2 and 500 DF, p-value: 2.656e-07Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between AAPL.Close and AAPL.Vol, but now doing the same when AAPL.Close is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 0.001147359#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 0.001147359Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is less than critical value, so we conclude that we fail to reject the null hypothesis.
Hence, in the case of AAPL.Close, its quadratic term isn’t significant.
AAPL.Vol vs AMZN.Close higher order:
currX = AMZN.Close
n = length(AAPL.Vol)
Xbar = sum(AMZN.Close)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.014740 -0.004550 -0.001449 0.003090 0.019341
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.539e-02 3.507e-04 43.901 < 2e-16 ***
## smallx -1.827e-06 1.758e-06 -1.039 0.299267
## xsquare 2.764e-08 7.256e-09 3.809 0.000157 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.006178 on 500 degrees of freedom
## Multiple R-squared: 0.04409, Adjusted R-squared: 0.04026
## F-statistic: 11.53 on 2 and 500 DF, p-value: 1.272e-05Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between AMZN.Close and AAPL.Vol, but now doing the same when AMZN.Close is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 14.51073#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 14.51073Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is greater than critical value, so we conclude that we fail to reject the null hypothesis.
Hence, the quadratic term \(X^2\) for AMZN.Close is significant
AAPL.Vol vs GOOG.Close higher order:
currX = GOOG.Close
n = length(AAPL.Vol)
Xbar = sum(GOOG.Close)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0121719 -0.0033757 -0.0008007 0.0032752 0.0174526
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.583e-02 3.388e-04 46.705 <2e-16 ***
## smallx -2.995e-05 3.292e-06 -9.098 <2e-16 ***
## xsquare 5.653e-08 3.092e-08 1.828 0.0681 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.005846 on 500 degrees of freedom
## Multiple R-squared: 0.1441, Adjusted R-squared: 0.1406
## F-statistic: 42.07 on 2 and 500 DF, p-value: < 2.2e-16Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between GOOG.Close and AAPL.Vol, but now doing the same when GOOG.Close is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 3.343177#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 3.343177Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is less than critical value, so we conclude that we fail to reject the null hypothesis.
Hence, the quadratic term \(X^2\) for GOOG.Close is not significant.
AAPL.Vol vs GSPC.Close higher order:
currX = GSPC.Close
n = length(AAPL.Vol)
Xbar = sum(GSPC.Close)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0115261 -0.0036555 -0.0009736 0.0040549 0.0184977
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.558e-02 3.199e-04 48.713 < 2e-16 ***
## smallx -1.507e-05 1.741e-06 -8.654 < 2e-16 ***
## xsquare 2.731e-08 7.880e-09 3.465 0.000575 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.005873 on 500 degrees of freedom
## Multiple R-squared: 0.1362, Adjusted R-squared: 0.1328
## F-statistic: 39.43 on 2 and 500 DF, p-value: < 2.2e-16Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between GSPC.Close and AAPL.Vol, but now doing the same when GSPC.Close is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 12.009#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 12.009Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is greater than critical value, so we conclude that we reject the null hypothesis.
Hence, the quadratic term \(X^2\) for GSPC.Close is significant.
AAPL.Vol vs VXAPLCLS higher order:
currX = VXAPLCLS
n = length(AAPL.Vol)
Xbar = sum(VXAPLCLS)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0114239 -0.0036964 -0.0007178 0.0033604 0.0190810
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.605e-02 2.497e-04 64.259 <2e-16 ***
## smallx 6.943e-04 4.028e-05 17.236 <2e-16 ***
## xsquare 5.135e-06 3.955e-06 1.298 0.195
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.004734 on 500 degrees of freedom
## Multiple R-squared: 0.4389, Adjusted R-squared: 0.4366
## F-statistic: 195.5 on 2 and 500 DF, p-value: < 2.2e-16Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between VXAPLCLS and AAPL.Vol, but now doing the same when VXAPLCLS is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 1.685429#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 1.685429Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is less than critical value, so we conclude that we fail to reject the null hypothesis.
Hence, the quadratic term \(X^2\) for VXAPLCLS is not significant.
AAPL.Vol vs VXAZNCLS higher order:
currX = VXAZNCLS
n = length(AAPL.Vol)
Xbar = sum(VXAZNCLS)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0110639 -0.0030523 -0.0003204 0.0026780 0.0173869
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.594e-02 2.630e-04 60.616 <2e-16 ***
## smallx 5.234e-04 3.052e-05 17.152 <2e-16 ***
## xsquare 4.367e-06 2.659e-06 1.643 0.101
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.004483 on 500 degrees of freedom
## Multiple R-squared: 0.4968, Adjusted R-squared: 0.4948
## F-statistic: 246.8 on 2 and 500 DF, p-value: < 2.2e-16Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between VXAZNCLS and AAPL.Vol, but now doing the same when VXAZNCLS is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 2.698467#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 2.698467Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is less than critical value, so we conclude that we fail to reject the null hypothesis.
Hence, the quadratic term \(X^2\) for VXAZNCLS is not significant.
AAPL.Vol vs VXGOGCLS higher order:
currX = VXGOGCLS
n = length(AAPL.Vol)
Xbar = sum(VXGOGCLS)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0116470 -0.0032753 -0.0002388 0.0029779 0.0176801
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.587e-02 2.701e-04 58.739 <2e-16 ***
## smallx 7.340e-04 4.358e-05 16.844 <2e-16 ***
## xsquare 1.202e-05 5.829e-06 2.062 0.0397 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.004682 on 500 degrees of freedom
## Multiple R-squared: 0.4511, Adjusted R-squared: 0.4489
## F-statistic: 205.5 on 2 and 500 DF, p-value: < 2.2e-16Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between VXGOGCLS and AAPL.Vol, but now doing the same when VXGOGCLS is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 4.25168#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 4.25168Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is greater than critical value, so we conclude that we fail to reject the null hypothesis.
Hence, the quadratic term \(X^2\) for VXGOGCLS is significant.
AAPL.Vol vs VIX.Close higher order:
currX = VIX.Close
n = length(AAPL.Vol)
Xbar = sum(VIX.Close)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.015370 -0.004054 -0.000990 0.003382 0.018212
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.620e-02 2.857e-04 56.717 <2e-16 ***
## smallx 7.153e-04 8.025e-05 8.913 <2e-16 ***
## xsquare 1.095e-06 8.401e-06 0.130 0.896
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.005585 on 500 degrees of freedom
## Multiple R-squared: 0.219, Adjusted R-squared: 0.2158
## F-statistic: 70.09 on 2 and 500 DF, p-value: < 2.2e-16Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between VIX.Close and AAPL.Vol, but now doing the same when VIX.Close is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 0.01698776#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 0.01698776Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is less than critical value, so we conclude that we fail to reject the null hypothesis.
Hence, the quadratic term \(X^2\) for VIX.Close is not significant.
AAPL.Vol vs AAPL.Ret higher order:
currX = AAPL.Ret
n = length(AAPL.Vol)
Xbar = sum(AAPL.Ret)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.014072 -0.004654 -0.001502 0.003897 0.019557
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0158752 0.0002959 53.646 < 2e-16 ***
## smallx -0.0593904 0.0161778 -3.671 0.000267 ***
## xsquare 1.1425422 0.3716630 3.074 0.002226 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.006139 on 500 degrees of freedom
## Multiple R-squared: 0.05621, Adjusted R-squared: 0.05244
## F-statistic: 14.89 on 2 and 500 DF, p-value: 5.233e-07Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between AAPL.Ret and AAPL.Vol, but now doing the same when AAPL.Ret is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 9.450306#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 9.450306Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is greater than critical value, so we conclude that we fail to reject the null hypothesis.
Hence, the quadratic term \(X^2\) for AAPL.Ret is significant
AAPL.Vol vs AMZN.Ret higher order:
currX = AMZN.Ret
n = length(AAPL.Vol)
Xbar = sum(AMZN.Ret)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.014725 -0.004525 -0.001245 0.003449 0.018966
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.015252 0.000289 52.778 < 2e-16 ***
## smallx -0.016177 0.013952 -1.159 0.247
## xsquare 2.677416 0.327522 8.175 2.46e-15 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.005912 on 500 degrees of freedom
## Multiple R-squared: 0.1248, Adjusted R-squared: 0.1213
## F-statistic: 35.66 on 2 and 500 DF, p-value: 3.333e-15Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between AMZN.Ret and AAPL.Vol, but now doing the same when AMZN.Ret is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 66.82684#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 66.82684Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is much greater than critical value, so we conclude that we reject the null hypothesis.
Hence, the quadratic term \(X^2\) for AMZN.Ret is quite significant.
AAPL.Vol vs GSPC.Ret higher order:
currX = GSPC.Ret
n = length(AAPL.Vol)
Xbar = sum(GSPC.Ret)/n
#centering predictor variable
smallx = currX - Xbar
xsquare = smallx*smallx
currmod = lm(Y ~ smallx+xsquare)
summary(currmod)##
## Call:
## lm(formula = Y ~ smallx + xsquare)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.014347 -0.004640 -0.001417 0.003814 0.018906
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0156042 0.0002964 52.638 < 2e-16 ***
## smallx -0.0527817 0.0298301 -1.769 0.0774 .
## xsquare 6.9330205 1.3317649 5.206 2.82e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.006095 on 500 degrees of freedom
## Multiple R-squared: 0.0698, Adjusted R-squared: 0.06608
## F-statistic: 18.76 on 2 and 500 DF, p-value: 1.394e-08Now plotting Y against X only:
par(lwd = 3, cex.axis = 1.5, cex.lab = 1.5, mar = c(5,5,2,2))
plot(Y~smallx, pch = 20, bty = "n", lwd = 5)
The graph doesn’t seem to be parabolic, so it doesn’t indicate interaction term at first glance.
We found from Part 2, that there is a relation between GSPC.Ret and AAPL.Vol, but now doing the same when GSPC.Ret is centered, we will test if there is any relation at all: * Null hypothesis \(H_0: \beta_2 = 0\) * Alternate hypothesis \(H_A: \beta_2 \neq 0\)
Test statistic is \(F_{obs} = \frac{MSR}{MSE} = \frac{SSR(x^2|x)/1}{SSE(x,x^2)/n-3}\)
curranova = anova(currmod)
curranovadfsse = n-3
## SSR is second value in Mean Sq column of the anova table
## SSE is bottom most value in Sum Sq column
Fobs = curranova[2,3] / ((curranova[3,2])/dfsse)
Fobs## [1] 27.10128#or could be taken from ANOVA table
Fobs2 = curranova[2,4]
Fobs2## [1] 27.10128Our critical value is:
alpha = 0.05
Fstar = qf(1 - alpha, 1, n-3)
Fstar## [1] 3.860124Our test statistic value is quite greater than critical value, so we conclude that we reject the null hypothesis.
Hence, the quadratic term \(X^2\) for GSPC.Ret is significant.