If you followed the notes from last week, you should have an object called BAC.stats in the workspace. You can get the previous notes from this link. The ‘BAC.stats’ object contains the calculation of the mean, standard deviation, skewness and kurtosis, maximum and minimum.

The code is

BAC <- read.csv("../Data/BAC.csv")
BAC$Date <- as.Date(BAC$Date, format = "%d/%m/%Y")
BAC <- BAC[,c(1, 7)]
colnames(BAC) <- c("Date", "Close")
BAC <- BAC[rev(rownames(BAC)),]
BACR <- diff(BAC$Close) / BAC$Close[-length(BAC$Close)] 
BACr <- diff(log(BAC$Close))
BAC <- cbind(BAC[-1,], BACR, BACr)
head(BAC)
##           Date    Close        BACR        BACr
## 167 2000-02-01 14.73385 -0.05032254 -0.05163287
## 166 2000-03-01 16.98035  0.15247242  0.14190956
## 165 2000-04-03 15.86722 -0.06555424 -0.06780170
## 164 2000-05-01 18.11670  0.14176910  0.13257891
## 163 2000-06-01 14.05218 -0.22435177 -0.25405617
## 162 2000-07-03 15.48191  0.10174426  0.09689461
require(moments)
statNames <- c("mean", "std dev", "skewness", "kurtosis", "max", "min")
BAC.stats <- c(mean(BACr), sd(BACr), skewness(BACr), kurtosis(BACr), 
               max(BACr), min(BACr))
names(BAC.stats) <- statNames
round(BAC.stats, 4)
##     mean  std dev skewness kurtosis      max      min 
##  -0.0001   0.1323  -1.3077  11.8954   0.5489  -0.7607

Therefore, the standard errors of the sampling distribution should be easy to calculate. For example, the standard error of the mean is

\[ SE_\hat{\mu} = \frac{s}{\sqrt{T}}}\]

where, \(SE_\hat{\mu}\) is the standard error of the mean; \(s\) is the estimated standard deviation; \(T\) is the number of variables.

The standard error and the 95% confidence intervals can be calculated using the following

SEmeanBAC <- BAC.stats[1]/length(BACr)^0.5
lowerconfBAC <- BAC.stats[1] - 2 * SEmeanBAC
upperConfBAC <- BAC.stats[1] + 2 * SEmeanBAC

The other confidence intervals for standard deviation, skew and Kurtosis can be calculated in the same way.

Autocorrelation

The covariance or correlation coefficient can be calculated with the acf function. Take a look at the documentation with ?afc.

acf(BACr)

The plot is automatic (unless turned off with the argument plot = FALSE). The confidence intervals for the estimate of the correlation coefficient are the blue dashed lines. The first correlation is with itself. This is of course 1. The function acf will return a number of values. To find out what is available type names(acf(BACr)). These are called Value in the documentation. It lets you know the calculations that have been returned. They can be accessed by using the $ key. To get the correlation coefficients for each of the lags

names(acf(BACr, plot = FALSE))
## [1] "acf"    "type"   "n.used" "lag"    "series" "snames"
acf(BACr, plot = FALSE)$acf
## , , 1
## 
##               [,1]
##  [1,]  1.000000000
##  [2,]  0.169952718
##  [3,]  0.042480048
##  [4,]  0.040962280
##  [5,] -0.150726957
##  [6,] -0.092506445
##  [7,] -0.192226651
##  [8,] -0.106231040
##  [9,]  0.221902901
## [10,] -0.044545803
## [11,]  0.030791710
## [12,]  0.113212097
## [13,] -0.030161143
## [14,] -0.032657273
## [15,] -0.046622596
## [16,] -0.071234876
## [17,]  0.036839168
## [18,] -0.077613283
## [19,]  0.024389485
## [20,]  0.038937806
## [21,] -0.001167009
## [22,]  0.089261313
## [23,] -0.020748271

Bootstrap

If you want to know more about this method.

n <- 1000 # sample size
MeanSample <- numeric(n)
for(i in 1:n){
  temp <- BACr[sample(length(BACr), length(BACr), replace = TRUE)]
  MeanSample[i] <- mean(temp, na.rm = TRUE)
}
hist(MeanSample)

quantile(MeanSample, 0.025)
##       2.5% 
## -0.0190047
quantile(MeanSample, 0.975)
##    97.5% 
## 0.020226