---
title: "Introduction to _'OrthoPanels'_"
author: Davor Cubranic and Mark Pickup
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    fig_width: 7
    fig_height: 5
vignette: >
  %\VignetteIndexEntry{Introduction to _OrthoPanels_}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---


```{r knitr_setup, echo = FALSE}
knitr::opts_chunk$set(tidy = TRUE, warning = FALSE)
library(OrthoPanels)
```

We demonstrate how to use the OrthoPanels package with an analysis of
a subset of data
from the 2010 British Election Study. The data consists of 3 waves of
panel survey data with 1845 respondents. Our dependent variable is
government approval ("Approve"), while independent variables in the
model are evaluations of each party's leader ("Brown", "Cameron", and
"Clegg" for Labour, Conservative, and Liberal Party respectively) and
government health and security policies ("NHS" and "Terror");
assesment of the economic situation ("Econ"); preference for taxing
and spending policy ("Tax"); and personal identification with the
governing Labour Party ("PID").

This subset of BES 2010 data is included with the OrthoPanels package
as dataset
`BES_panel`. It is a data frame that includes all of the survey
variables as described above, plus the variables for the case number
("n") and the wave ("t") of each observation:

```{r}
head(BES_panel)
```

We begin by specifying our model using a formula, indicating the data
to be used, and the case and time variables:
```{r}
BES.opm <- opm(Approve~Econ+Clegg+Brown+Cameron-NHS+Terror+PID-Tax,
data=BES_panel, index = c('n', 't'), n.samp=1000, add.time.indicators=TRUE)
```

The first argument is a formula specifying the model symbolically:
`response ~ term1 + term2`. This is consistent with the `lm()` function.
It is not necessary to include the lagged dependent variable or the
fixed effects in the model specification. This is done automatically.

The other arguments are: `data` which specifies the data frame, list
or environment containing the variables in the model; `n.samp` which
specifies the number of samples (MC iterations) to use to estimate the
parameters; `index` which is a two-element vector containing the index
or name of the case and time variables, respectively; and
`add.time.indicators` which is a logical argument. If
`add.time.indicators` is `TRUE`, the model includes dummy variables for
each wave (time point). The default is `FALSE`. The `data` and `index`
arguments are optional; if `data` is missing, variables are looked up
in the environment, while the default value of `index` is the first
two variables in the data frame. An additional optional argument is
`subset`. This is a vector specifying a subset of observations to be
used in the estimation.

The function `opm()` returns an object of class `opm` which includes a
list,` samples`, with the following elements: `rho`, a vector of `n`.samp
parameter samples of $\rho$; `sig2`, a vector of n.samp parameter samples of $\sigma^2$;
and `beta`, an `n.samp x variable` matrix of parameters samples of $\beta$. If
included in the model, the parameters for the time dummies are
included in this matrix:
```{r}
str(BES.opm)
```

The summary of the object provides us with the median, 68% equal
tailed credible intervals and 95% equal tailed credible intervals for
each parameter:

```{r}
summary(BES.opm)
```

The package includes functions that may be of use in exploring the
results. Method `confint` for objects of `opm` class computes
equal-tailed credible intervals for one or more parameters in the
fitted opm model. We can calculate 90% equal-tailed credible intervals
as follows:

```{r}
confint(BES.opm, level=0.9)
```

The function `caterplot` creates side-by-side plots of credible
intervals of the opm model parameters. The intervals are displayed as
horizontal lines, with the 90% interval using a thicker line width and
the 95% interval a thinner one. The posterior median is indicated with
a dot.

```{r}
caterplot(BES.opm, main = "BES 2010 opm parameter estimates")
abline(v=0)
```


We can use the function `plot()` to obtain the posterior density of
each parameter:


```{r}
plot(BES.opm, 'rho')
```

In a dynamic model, the $\beta$ coefficients are the immediate effects
of a change in each of the covariates. This is known as the short run
effect. However, in a dynamic model, the short run effect is not the
full effect of a change in a covariate. As the future value of the
dependent variable depends on the current value, the effect of a
change in a covariate has a knock on effect into the future. The
effect builds at a declining rate until it asymptotically reaches its
full effect. This is known as the long-run effect and is estimated as
$\frac{\beta}{1-\rho}$, where $\rho$ is the autoregressive parameter
(Wooldridge, 2013; Pickup, 2014). We can calculate the median and 95%
equal tailed credible intervals for the long-run effects from the
parameter samples. For example, for the first independent variable
(the economy):

```{r}
quantile(BES.opm$samples$beta[,1]/(1-BES.opm$samples$rho), probs=c(0.025, 0.5, 0.975))
```



The function `longRunEffects` Computes long term effects and confidence intervals of \code{opm} Model Parameters, using the formula as per explanation above.
By default, it computes the median and 95%
equal tailed credible intervals for the long-run effects from the
parameter samples.

```{r}
longRunEffects(BES.opm,prob=c(0.025,0.5,0.975))
```



The function `caterplot_longRun` creates side-by-side plots of credible
intervals of the opm model long-run effects, which are computed using the 
function The function `longRunEffects`. The intervals are displayed as
horizontal lines, with the 90% interval using a thicker line width and
the 95% interval a thinner one. The posterior median is indicated with
a dot.


```{r}
caterplot_longRun(BES.opm)
```
