Assign10

Q1. (1)Taking the expected value of both sides of the AR(1) equation, we get:

E[x_t] = E[δ + φ_1 x_{t-1} + w_t]

= E[δ] + E[φ_1 x_{t-1}] + E[w_t]

Since w_t has a mean of 0 (w_t ~ N(0, σ_w^2)), we have:E[w_t] = 0

Also, due to the stationarity of x_t, we have: E[x_t] = E[x_{t-1}]

Substituting these conditions, we get: E[x_t] = δ + φ_1 E[x_t]

Rearranging the terms, we get: (1 - φ_1) E[x_t] = δ

Therefore, the expected value of x_t is given by: E[x_t] = δ / (1 - φ_1)

(2)Substituting these values, we get: x_t = 11,000,000,000 + 0.001 × 10,000,000,000 + w_t

Since we are forecasting for the next year, we need to set w_t = 0 (assuming no error term for the forecast).

x_t = 11,000,000,000 + 0.001 × 10,000,000,000 + 0

= 11,000,000,000 + 10,000,000

= 11,010,000,000

Q2.

goo = tq_get("GOOG", get="stock.prices", from="1997-01-01") %>% tq_transmute(mutate_fun=to.period,period="months")

ggplot(goo, aes(date,close)) + geom_line()

adf.test(diff(log(goo$close)), alternative="stationary", k=0)

## Warning in adf.test(diff(log(goo$close)), alternative = "stationary", k = 0):
## p-value smaller than printed p-value

## 
##  Augmented Dickey-Fuller Test
## 
## data:  diff(log(goo$close))
## Dickey-Fuller = -14.781, Lag order = 0, p-value = 0.01
## alternative hypothesis: stationary

acf(diff(log(goo$close)))

pacf(diff(log(goo$close)))

(fit <- arima(diff(log(goo$close)), c(3,0,1)))

## 
## Call:
## arima(x = diff(log(goo$close)), order = c(3, 0, 1))
## 
## Coefficients:
##           ar1      ar2      ar3     ma1  intercept
##       -0.7440  -0.0149  -0.0676  0.7866     0.0178
## s.e.   0.3758   0.0881   0.0691  0.3716     0.0054
## 
## sigma^2 estimated as 0.007079:  log likelihood = 249.3,  aic = -486.6

fitARIMA <- auto.arima(diff(log(goo$close)), trace=TRUE)

## 
##  Fitting models using approximations to speed things up...
## 
##  ARIMA(2,0,2) with non-zero mean : -511.4437
##  ARIMA(0,0,0) with non-zero mean : -493.1642
##  ARIMA(1,0,0) with non-zero mean : -497.314
##  ARIMA(0,0,1) with non-zero mean : -491.5451
##  ARIMA(0,0,0) with zero mean     : -484.8265
##  ARIMA(1,0,2) with non-zero mean : -505.8596
##  ARIMA(2,0,1) with non-zero mean : -513.3808
##  ARIMA(1,0,1) with non-zero mean : -506.5986
##  ARIMA(2,0,0) with non-zero mean : -514.9837
##  ARIMA(3,0,0) with non-zero mean : -512.6675
##  ARIMA(3,0,1) with non-zero mean : -511.0328
##  ARIMA(2,0,0) with zero mean     : -508.1801
## 
##  Now re-fitting the best model(s) without approximations...
## 
##  ARIMA(2,0,0) with non-zero mean : -489.9322
## 
##  Best model: ARIMA(2,0,0) with non-zero mean

plot(as.ts(diff(log(goo$close))))
lines(fitted(fitARIMA), col="red")

futurVal <- forecast(fitARIMA,h=25, level=c(99))
plot(forecast(futurVal))

futurVal$mean

## Time Series:
## Start = 237 
## End = 261 
## Frequency = 1 
##  [1] 0.01927305 0.01238502 0.01749310 0.01803598 0.01781342 0.01777811
##  [7] 0.01778733 0.01778940 0.01778905 0.01778893 0.01778895 0.01778895
## [13] 0.01778895 0.01778895 0.01778895 0.01778895 0.01778895 0.01778895
## [19] 0.01778895 0.01778895 0.01778895 0.01778895 0.01778895 0.01778895
## [25] 0.01778895