Dow Jones DJI & TS

Time Series and Financial Data:

library(quantmod)
library(tseries)
library(forecast)

DJI = read.csv("^DJI.csv")
DJI$Date <- as.Date(DJI$Date,format="%m/%d/%Y")
rownames(DJI) <- DJI$Date
DJI$Date <- NULL 
head(DJI)

##               Open    High     Low   Close Adj.Close   Volume
## 1985-01-29 1277.72 1295.49 1266.89 1292.62   1292.62 13560000
## 1985-01-30 1297.37 1305.10 1278.93 1287.88   1287.88 16820000
## 1985-01-31 1283.24 1293.40 1272.64 1286.77   1286.77 14070000
## 1985-02-01 1276.94 1286.11 1269.77 1277.72   1277.72 10980000
## 1985-02-04 1272.08 1294.94 1268.99 1290.08   1290.08 11630000
## 1985-02-05 1294.06 1301.13 1278.60 1285.23   1285.23 13800000

d = DJI$Adj.Close
summary(d)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    1242    3398    9334    8613   11523   21116

chartSeries(DJI)

adf.test(d)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  d
## Dickey-Fuller = -1.8305, Lag order = 20, p-value = 0.6502
## alternative hypothesis: stationary

The dataset is not stationary. We will difference and log:

dow = 100 * diff(log(d))[-1]
summary(dow)

##      Min.   1st Qu.    Median      Mean   3rd Qu.      Max. 
## -25.63151  -0.44699   0.05353   0.03431   0.56239  10.50835

adf.test(dow)

## 
##  Augmented Dickey-Fuller Test
## 
## data:  dow
## Dickey-Fuller = -20.777, Lag order = 20, p-value = 0.01
## alternative hypothesis: stationary

Now it is stationary.

We can fit a model using the Yule-Walker equations

summary(arma(dow, order = c(2,2)))

## 
## Call:
## arma(x = dow, order = c(2, 2))
## 
## Model:
## ARMA(2,2)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -25.97387  -0.47220   0.02879   0.53071   9.93999 
## 
## Coefficient(s):
##            Estimate  Std. Error  t value Pr(>|t|)  
## ar1        -0.12861     0.19944   -0.645   0.5190  
## ar2         0.12745     0.18311    0.696   0.4864  
## ma1         0.09203     0.19737    0.466   0.6410  
## ma2        -0.18879     0.17934   -1.053   0.2925  
## intercept   0.03440     0.01592    2.161   0.0307 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Fit:
## sigma^2 estimated as 1.228,  Conditional Sum-of-Squares = 9997.75,  AIC = 24800.96

So both the AR and MA coefficients are significant.

We can divide instead the data into training and testing sets (\(90\,%\) v \(10\%\)), fit the model in the training data, and then assess the performance on the testing set:

train = d[1 : (0.9 * length(dow))]
test  = d[(0.9 * length(dow) + 1):  length(dow)]

fit = arima(log(train), c(2, 1, 2), seasonal=list(order = c(0, 1, 1), period=12))

predi = predict(fit, n.ahead = (length(dow) - (0.9*length(dow))))$pred
fore  = forecast(fit, h = 365 * 5) # Five years ahead

plot(fore)

Well, it is an illustration. However, the accuracy assessment is not great:

accuracy(predi, test)

##                ME    RMSE      MAE      MPE     MAPE
## Test set 17855.26 17899.3 17855.26 99.94482 99.94482

If we calculate without using the i=1 middle value in arima:

dow = 100 * diff(log(d))[-1] # The difference taken outside the arima() call. 100 probably aides in computing since the numbers are very close to zero.
train = dow[1 : (0.9 * length(dow))]
test  = dow[(0.9 * length(dow) + 1):  length(dow)]

fit = arima(train, order = c(2, 0, 2), seasonal=list(order = c(0, 1, 1), period=12))

predi = predict(fit, n.ahead = (length(dow) - (0.9*length(dow))))$pred
accuracy(predi, test)

##                   ME      RMSE       MAE MPE MAPE
## Test set 0.003215416 0.7946501 0.5670157 NaN  Inf

At this point I’m not sure how to evaluate these differences in output between both apparently equivalent systems.

Home Page

NOTE: These are tentative notes on different topics for personal use - expect mistakes and misunderstandings.

Dow Jones DJI & TS

NOTES ON STATISTICS, PROBABILITY and MATHEMATICS

Time Series and Financial Data: