Homework 5
==========
*Firstname Lastname (Replace this part with your name)*


For this assignment, we will again be using the `forecast` library.

```{r}
library("forecast")
```

The dataset is the unemployment rate, monthly, seasonally adjusted.

```{r}
data <- read.csv("http://ptrckprry.com/course/forecasting/data/unemployment.csv")
date <- as.Date(data$date)
time <- 1:length(date)
unemp <- data$unemployment
```


Part A
------

We first construct the log unemployment series, and its first two differences:
```{r}
# edit and uncomment the following lines
# log.unemp <- ??
# diff.log.unemp <- ??
# diff2.log.unemp <- ??
```

Now, we use time series plots, ACFs and PACFs to determine the appropriate
level of differencing.

Here are the plots for the original series:

```{r}
# add TS plot, ACF, and Pacf for log.unemp
```

Here are the plots for the first difference:

```{r}
# add TS plot, ACF, and Pacf for diff.log.unemp
```

Here are the plots for the second difference:

```{r}
# add TS plot, ACF, and Pacf for diff2.log.unemp
```


**Choose d for an ARIMA model.  Explain your reasoning.**



Part B
------

Here are the AICC values for the models under consideration:

```{r}
# Given values for p and q, to compute the AICC for an ARMA(p,q) model,
# use the following command:
#
#     Arima(x, c(p, 0, q), include.constant=FALSE)$aicc
#
# This command will do the same, but with a constant in the model:
#
#     Arima(x, c(p, 0, q), include.constant=TRUE)$aicc
#
#
# IMPORTANT NOTE:
# 
# The Arima command accepts other values for d, but if d is nonzero, the
# reported value of AICC is incorrect when the model includes a constant
# (there is a bug in the R implementation).
#
# As a workaround, to get the AICC for an ARIMA(p,1,q)  model, use the commands
#
#     Arimia(diff(x), c(p, 0, q), include.constant=FALSE)$aicc
#     Arimia(diff(x), c(p, 0, q), include.constant=TRUE)$aicc
#
# For an ARIMA(p,2,q), use 
#
#     Arimia(diff(diff(x)), c(p, 0, q), include.constant=FALSE)$aicc
#     Arimia(diff(diff(x)), c(p, 0, q), include.constant=TRUE)$aicc
#


# Use these commands to get the AICC values for all 18 models under
# consideration.  You will need to call the Arima funciton 18 times.
```

**What (p,d,q) is selected by AICC?  Does the model include a constant?**


Part C
------

Here is the estimated model:
```{r}
# edit and uncomment the following two lines
# fit <- Arima(log.unemp, ???, include.constant=???)
# print(fit)
```


**Are all of the parameters statistically significant?**


Part D
------

**Write the complete form of the fitted model. (No code required.)**



Part E
------

Here are the Ljung-Box statistics for lack of fit:

```{r}
# Hint: use Box.test function.  You must call this function four times,
# for lags 12, 24, 36, and 48.
```

**Does the model seem to be adequate?**


Part F
------

Here is a plot of the residuals, along with the ACF and the PACF of the
residuals:

```{r}
# Add TS plot, ACF, and PACF
```

**Do these plots indicate any inadequacies in the model?**


Part G
------

Here is the original data, along with the forecasts and 95% forecast intervals
for lead times 1 to 30:

```{r}
# Hint: use the forecast function
```

**Do the forecasts seem reasonable?**


**Do the forecast intervals seem excessively wide?**

