Forecasting Time Series
Analysis of the Google Series
Patrick Perry
March 7, 2017
library("forecast")Loading required package: zoo
Attaching package: 'zoo'The following objects are masked from 'package:base':
as.Date, as.Date.numericLoading required package: timeDateLoading required package: methodsThis is forecast 7.3 Google Series (GOOG)
Daily closing price: May 2, 2005 to December 29, 2006.
data <- read.csv("http://ptrckprry.com/course/forecasting/data/google.csv")
GOOG <- data$google
n <- length(GOOG)
time <- 1:n
plot(time, GOOG, type="l", col=2)
Log GOOG
log.GOOG <- log(GOOG)
plot(time, log.GOOG, type="l", col=2)
Acf(log.GOOG)
Pacf(log.GOOG)
Diff Log GOOG
diff.log.GOOG <- c(NA, diff(log.GOOG))
plot(time, diff.log.GOOG, type="l", col=2)
Acf(diff.log.GOOG)
Pacf(diff.log.GOOG)
2nd Diff Log GOOG
diff2.log.GOOG <- c(NA, diff(diff.log.GOOG))
plot(time, diff2.log.GOOG, type="l", col=2)
Acf(diff2.log.GOOG)
Pacf(diff2.log.GOOG)
ARIMA(0, 1, 0)
fit.00 <- Arima(log.GOOG, c(0, 1, 0), include.constant=FALSE)
print(fit.00)Series: log.GOOG
ARIMA(0,1,0)
sigma^2 estimated as 0.0004203: log likelihood=1036.72
AIC=-2071.45 AICc=-2071.44 BIC=-2067.41fit.00$aicc[1] -2071.437AICC for Candidate Models (Buggy Version)
# Without constant:
fit.00 <- Arima(log.GOOG, c(0, 1, 0), include.constant=FALSE)
fit.01 <- Arima(log.GOOG, c(0, 1, 1), include.constant=FALSE)
fit.02 <- Arima(log.GOOG, c(0, 1, 2), include.constant=FALSE)
fit.10 <- Arima(log.GOOG, c(1, 1, 0), include.constant=FALSE)
fit.11 <- Arima(log.GOOG, c(1, 1, 1), include.constant=FALSE)
fit.12 <- Arima(log.GOOG, c(1, 1, 2), include.constant=FALSE)
fit.20 <- Arima(log.GOOG, c(2, 1, 0), include.constant=FALSE)
fit.21 <- Arima(log.GOOG, c(2, 1, 1), include.constant=FALSE)
fit.22 <- Arima(log.GOOG, c(2, 1, 2), include.constant=FALSE)
# With constant:
fit.00c <- Arima(log.GOOG, c(0, 1, 0), include.constant=TRUE)
fit.01c <- Arima(log.GOOG, c(0, 1, 1), include.constant=TRUE)
fit.02c <- Arima(log.GOOG, c(0, 1, 2), include.constant=TRUE)
fit.10c <- Arima(log.GOOG, c(1, 1, 0), include.constant=TRUE)
fit.11c <- Arima(log.GOOG, c(1, 1, 1), include.constant=TRUE)
fit.12c <- Arima(log.GOOG, c(1, 1, 2), include.constant=TRUE)
fit.20c <- Arima(log.GOOG, c(2, 1, 0), include.constant=TRUE)
fit.21c <- Arima(log.GOOG, c(2, 1, 1), include.constant=TRUE)
fit.22c <- Arima(log.GOOG, c(2, 1, 2), include.constant=TRUE)
# Summarize Results
models <- data.frame(p = rep(c(0, 0, 0, 1, 1, 1, 2, 2, 2), 2),
d = rep(1, 18),
q = rep(c(0, 1, 2), 6),
include.constant = c(rep(FALSE, 9), rep(TRUE, 9)),
loglik = c(fit.00$loglik, fit.01$loglik, fit.02$loglik,
fit.10$loglik, fit.11$loglik, fit.12$loglik,
fit.20$loglik, fit.21$loglik, fit.22$loglik,
fit.00c$loglik, fit.01c$loglik, fit.02c$loglik,
fit.10c$loglik, fit.11c$loglik, fit.12c$loglik,
fit.20c$loglik, fit.21c$loglik, fit.22c$loglik),
aicc = c(fit.00$aicc, fit.01$aicc, fit.02$aicc,
fit.10$aicc, fit.11$aicc, fit.12$aicc,
fit.20$aicc, fit.21$aicc, fit.22$aicc,
fit.00c$aicc, fit.01c$aicc, fit.02c$aicc,
fit.10c$aicc, fit.11c$aicc, fit.12c$aicc,
fit.20c$aicc, fit.21c$aicc, fit.22c$aicc)
)
print(models, digits=6) p d q include.constant loglik aicc
1 0 1 0 FALSE 1036.72 -2071.44
2 0 1 1 FALSE 1036.94 -2069.84
3 0 1 2 FALSE 1037.05 -2068.05
4 1 1 0 FALSE 1036.95 -2069.86
5 1 1 1 FALSE 1037.63 -2069.19
6 1 1 2 FALSE 1037.63 -2067.16
7 2 1 0 FALSE 1037.06 -2068.07
8 2 1 1 FALSE 1039.76 -2071.42
9 2 1 2 FALSE 1039.64 -2069.14
10 0 1 0 TRUE 1038.23 -2072.43
11 0 1 1 TRUE 1038.37 -2070.67
12 0 1 2 TRUE 1038.43 -2068.76
13 1 1 0 TRUE 1038.37 -2070.68
14 1 1 1 TRUE 1038.67 -2069.25
15 1 1 2 TRUE 1041.25 -2072.35
16 2 1 0 TRUE 1038.43 -2068.77
17 2 1 1 TRUE 1041.10 -2072.06
18 2 1 2 TRUE 1043.53 -2074.87These results are unreliable. Use the method described in the next seciton instead.
AICC for Candidate Models (Correct Version)
# Without constant:
fit.00 <- Arima(diff.log.GOOG, c(0, 0, 0), include.constant=FALSE)
fit.01 <- Arima(diff.log.GOOG, c(0, 0, 1), include.constant=FALSE)
fit.02 <- Arima(diff.log.GOOG, c(0, 0, 2), include.constant=FALSE)
fit.10 <- Arima(diff.log.GOOG, c(1, 0, 0), include.constant=FALSE)
fit.11 <- Arima(diff.log.GOOG, c(1, 0, 1), include.constant=FALSE)
fit.12 <- Arima(diff.log.GOOG, c(1, 0, 2), include.constant=FALSE)
fit.20 <- Arima(diff.log.GOOG, c(2, 0, 0), include.constant=FALSE)
fit.21 <- Arima(diff.log.GOOG, c(2, 0, 1), include.constant=FALSE)
fit.22 <- Arima(diff.log.GOOG, c(2, 0, 2), include.constant=FALSE)
# With constant:
fit.00c <- Arima(diff.log.GOOG, c(0, 0, 0), include.constant=TRUE)
fit.01c <- Arima(diff.log.GOOG, c(0, 0, 1), include.constant=TRUE)
fit.02c <- Arima(diff.log.GOOG, c(0, 0, 2), include.constant=TRUE)
fit.10c <- Arima(diff.log.GOOG, c(1, 0, 0), include.constant=TRUE)
fit.11c <- Arima(diff.log.GOOG, c(1, 0, 1), include.constant=TRUE)
fit.12c <- Arima(diff.log.GOOG, c(1, 0, 2), include.constant=TRUE)
fit.20c <- Arima(diff.log.GOOG, c(2, 0, 0), include.constant=TRUE)
fit.21c <- Arima(diff.log.GOOG, c(2, 0, 1), include.constant=TRUE)
fit.22c <- Arima(diff.log.GOOG, c(2, 0, 2), include.constant=TRUE)
# Summarize Results
models <- data.frame(p = rep(c(0, 0, 0, 1, 1, 1, 2, 2, 2), 2),
d = rep(1, 18),
q = rep(c(0, 1, 2), 6),
include.constant = c(rep(FALSE, 9), rep(TRUE, 9)),
loglik = c(fit.00$loglik, fit.01$loglik, fit.02$loglik,
fit.10$loglik, fit.11$loglik, fit.12$loglik,
fit.20$loglik, fit.21$loglik, fit.22$loglik,
fit.00c$loglik, fit.01c$loglik, fit.02c$loglik,
fit.10c$loglik, fit.11c$loglik, fit.12c$loglik,
fit.20c$loglik, fit.21c$loglik, fit.22c$loglik),
aicc = c(fit.00$aicc, fit.01$aicc, fit.02$aicc,
fit.10$aicc, fit.11$aicc, fit.12$aicc,
fit.20$aicc, fit.21$aicc, fit.22$aicc,
fit.00c$aicc, fit.01c$aicc, fit.02c$aicc,
fit.10c$aicc, fit.11c$aicc, fit.12c$aicc,
fit.20c$aicc, fit.21c$aicc, fit.22c$aicc)
)
print(models, digits=6) p d q include.constant loglik aicc
1 0 1 0 FALSE 1036.72 -2071.44
2 0 1 1 FALSE 1036.94 -2069.84
3 0 1 2 FALSE 1037.05 -2068.05
4 1 1 0 FALSE 1036.95 -2069.86
5 1 1 1 FALSE 1037.63 -2069.19
6 1 1 2 FALSE 1039.91 -2071.72
7 2 1 0 FALSE 1037.06 -2068.07
8 2 1 1 FALSE 1039.76 -2071.42
9 2 1 2 FALSE 1039.64 -2069.14
10 0 1 0 TRUE 1038.23 -2072.43
11 0 1 1 TRUE 1038.37 -2070.67
12 0 1 2 TRUE 1038.43 -2068.76
13 1 1 0 TRUE 1038.37 -2070.68
14 1 1 1 TRUE 1038.67 -2069.24
15 1 1 2 TRUE 1041.25 -2072.35
16 2 1 0 TRUE 1038.43 -2068.77
17 2 1 1 TRUE 1041.10 -2072.06
18 2 1 2 TRUE 1040.73 -2069.26Selected Model
fit.best <- Arima(log.GOOG, c(0, 1, 0), include.constant=TRUE)
print(fit.best)Series: log.GOOG
ARIMA(0,1,0) with drift
Coefficients:
drift
0.0017
s.e. 0.0010
sigma^2 estimated as 0.0004183: log likelihood=1038.23
AIC=-2072.46 AICc=-2072.43 BIC=-2064.38Note: constant (drift) is not significant. However, the hypothesis tests relies on the model being correct, so the z-statistic and corresponding p-value are probably not reliable. It’s better to use AICC to determine whether to include a constant.
Residuals
resid <- residuals(fit.best)
plot(time, resid, type="l", col=2)
Acf(resid)
Pacf(resid)
Ljung-Box Goodness-of-Fit Test
# fitdf = 1 (p = 0, q = 0, model includes constant)
Box.test(resid, lag=12, type = "Ljung-Box", fitdf=1)
Box-Ljung test
data: resid
X-squared = 21.904, df = 11, p-value = 0.02513Box.test(resid, lag=24, type = "Ljung-Box", fitdf=1)
Box-Ljung test
data: resid
X-squared = 27.885, df = 23, p-value = 0.2202Box.test(resid, lag=36, type = "Ljung-Box", fitdf=1)
Box-Ljung test
data: resid
X-squared = 40.881, df = 35, p-value = 0.2279Box.test(resid, lag=48, type = "Ljung-Box", fitdf=1)
Box-Ljung test
data: resid
X-squared = 54.659, df = 47, p-value = 0.2065Forecasts From Selected Model
forecast(fit.best, h=10, level=95) Point Forecast Lo 95 Hi 95
422 6.134003 6.093917 6.174090
423 6.135737 6.079047 6.192428
424 6.137471 6.068040 6.206903
425 6.139205 6.059033 6.219378
426 6.140940 6.051303 6.230576
427 6.142674 6.044482 6.240865
428 6.144408 6.038349 6.250466
429 6.146142 6.032760 6.259523
430 6.147876 6.027616 6.268135
431 6.149610 6.022845 6.276374plot(forecast(fit.best, h=100, level=95), col=2)
Forecasts From Model Without Constant
fit.00 <- Arima(log.GOOG, c(0, 1, 0), include.constant=FALSE)
plot(forecast(fit.00, h=100, level=95), col=2)
Don’t Use p-values for Model Selection
fit.11 <- Arima(log.GOOG, c(1, 1, 1), include.constant=FALSE)
print(fit.11)Series: log.GOOG
ARIMA(1,1,1)
Coefficients:
ar1 ma1
0.8953 -0.8659
s.e. 0.1402 0.1571
sigma^2 estimated as 0.0004205: log likelihood=1037.63
AIC=-2069.25 AICc=-2069.19 BIC=-2057.13(zstat.ar1 <- 0.895 / 0.140)[1] 6.392857(pval.ar1 <- 2*pnorm(-abs(zstat.ar1)))[1] 1.628144e-10(zstat.ma1 <- -0.866 / 0.157)[1] -5.515924(pval.ma1 <- 2*pnorm(-abs(zstat.ma1)))[1] 3.46953e-08