HW1: Problem 1
==============
*Your name goes here*


In this problem we will investigate the United States Gross Domestic Product 
(GDP).  The data file `gdp.csv` contains the real US GDP, quarterly, from 1946, 
Q4 to 2016 Q3, seasonally adjusted and inflation adjusted, in billions of 2009
dollars.  



Part A
------

We read the data file and extract the `date` and `gdp` columns with 
the following commands:

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

Here is a plot of the GDP series versus date:

```{r}
# Replace this comment with code to make the plot.
#
# Hint: Use the plot command.
```

**Does GDP seem to grow linearly over time?**



Part B
------

Here is a plot of log(GDP).

```{r}
log_gdp <- log(gdp) # create the log(GDP) variable

# Replace this comment with the code to plot log(GDP) vs. Date
```

**Does log GDP appear to grow linearly over time?  If so, then what does this
imply about the growth of GDP itself?**


Part C
------

We fit an ordinary linear regression model for log(GDP), using time as the 
predictor variable.

```{r}
time <- 1:length(gdp) # create the time variable

# Replace this comment with code to fit a linear model and print the
# fitted coefficients.
#
# Hint: Use the "lm" and "summary" commands; use log_gdp as your response
# and use time as your predictor (*not* date).
```

Based on the fitted model, we forecast the log(GDP) for the fourth quarter
of the year 2016 (time = 281).

```{r}
# Replace with code to predict the value of gdp when time = 272
#
# Hint: use the "coef" or the "predict" function
```

We also construct a 95% prediction interval to go with the
point forecast:

```{r}
# Replace with code to compute a 95% prediction interval
#
# Hint: use the "predict" function.
```

**Do you think this interval forecast (prediction interval) is valid?  Does it
seem too wide, or too precise?  Explain.**


Part D
------

Next is a plot of the log(GDP) series with the fitted line superimposed.

```{r}
plot(time, log_gdp, type="l")

# Replace this comment with code to add a line to the plot.
#
# Hint: use the "abline" function.
```

**Does the line fit well?**


Part E
------

Here is a plot the residuals from the fitted line versus time.

```{r}
# Replace with code to add the plot.
#
# Hint: use the "residuals" function.
```

**What potential problems with the linear model are indicated by this plot?
Do you think these problems could spoil the validity of the forecast
interval?**
