---
title: Massey University --- School of Mathematical and Computational Sciences
subtitle: 161.251 Regression Modelling
author: "Staff member responsible for this workshop: Jonathan Marshall j.c.marshall@massey.ac.nz"
date: "last updated `r format(lubridate::today(), '%d %B %Y')`"
output:
  html_document:
    code_download: true
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(comment="", dev="png")
library(tidyverse)
library(broom)
library(nlme)
library(tsibble)
library(fable)
library(feasts)
```

# Computer Laboratory Exercise 12B

## Modelling House Prices in Perth

In this exercise you will:

  - gain experience in fitting linear models with autocorrelated errors
    in R;

  - use linear modelling on house price data from Australia.

This Exercise concerns quarterly data on house prices in Australian
state and territory capital cities from the second quarter of 1986 to
the second quarter 1997 (inclusive). The following variables are
available:

| Variable      | Description                                    |
| :------------ | :--------------------------------------------- |
|  `Quarter`    | Quarter, including year                        |
|  `Sydney`     | Mean relative house price in Sydney            |
|  `Melbourne`  | Mean relative house price in Melbourne         |
|  `Brisbane`   | Mean relative house price in Brisbane          |
|  `Adelaide`   | Mean relative house price in Adelaide          |
|  `Perth`      | Mean relative house price in Perth             |
|  `Hobart`     | Mean relative house price in Hobart            |
|  `Darwin`     | Mean relative house price in Darwin            |
|  `Canberra`   | Mean relative house price in Canberra          |
|  `WtAve`      | Weighted average price over all eight capitals |

The house prices are expressed as a percentage, relative to the average
house price in Darwin in the second quarter of 1986 (which is therefore
recorded as 100). The data source is the Australian Bureau of
Statistics.

Perth in Western Australia is rather isolated from the other cities
(indeed, it claims to be the most isolated city in the world). As such,
it is interesting to examine whether house prices in Perth can be
predicted in terms of the overall Australian housing market. We will
therefore seek to model `Perth` as the response, in terms of `WtAve`, taking account of any temporal trend.

1. The data we need for this exercise can be read in and assigned to the data frame `hp_raw`, with the following:

```{r, message=FALSE}
hp_raw <- read_csv(file="https://R-Resources.massey.ac.nz/data/161251/hp.csv")
hp_raw
```

2. We'll start by tidying up the (silly, but not atypical!) `Quarter` column to a friendly date object for charting etc. `lubridate` to the rescue!

```{r}
hp <- hp_raw |>
  mutate(Quarter = my(Quarter)) # my is "month year" format. lubridate magically figures out how to split things up
```

3. Produce a time plot of the Perth prices. Comment as to whether there is seasonality,
trend etc.

```{r}
```

4. Create a variable `Time` recording the number of quarters from the start of the
observational period. Use the code from Lab 12A or similar, noting that we have quarters
rather than months (you could measure in months and then divide by 3).

```{r}
```

5. Use the method of least squares to fit a linear regression (i.e. with `lm` rather than `gls`) with `Perth` as response and `WtAve` and `Time` as predictors

```{r}
```

6. Produce a plot of residuals against `Time` or `Date`, and an ACF plot for the residuals for this regression model.

    - Do you think that there is evidence of autocorrelation in the error terms?

```{r}
```

7. Load the `nlme` library, and then use the `gls()` function to fit a linear
regression in which the error terms are assumed to follow an *AR1* process.

```{r}
```

8. Look at the model summaries for the two models that you have fitted. Are
their parameter estimates and/or standard errors markedly different?

9. You should be able to see from the summary for the second model that the
estimated lag one correlation for the residuals is
$\hat{\phi}$ = 0.914. This suggests strong autocorrelation, but
estimates of the parameter $\phi$ can be very imprecise unless the
time series is very long. To get an idea of the uncertainty in our
estimate, generate a confidence interval with `intervals()`.

```{r}
```

10. Finally, look at the fitted GLS model.

    - Is the Australian average house price a useful predictor of Perth house prices? 
    - Is there evidence that the secular trend in Perth is different to that across the rest of the nation?