```{r setup, include=FALSE}
  knitr::opts_chunk$set(echo = TRUE)
  knitr::opts_chunk$set(dev = 'pdf')
  def.chunk.hook  <- knitr::knit_hooks$get("chunk")
  knitr::knit_hooks$set(chunk = function(x, options) {
    x <- def.chunk.hook(x, options)
    ifelse(options$size != "normalsize", paste0("\n \\", options$size,"\n\n", x, "\n\n \\normalsize"), x)
  })
```


## Recall...
\framesubtitle{Conditions for validity of inference in linear regression}


 - The relationship between $X$ and $Y$, if there is one, is actually \underline{\textbf{L}inear}
   - e.g. not quadratic, exponential, etc.

\vspace{2mm}

 - \textbf{I}ndependence of observations

\vspace{2mm}

 - \textbf{N}ormality of $\epsilon_i$
   - Note that this can also be achieved, due to the central limit theorem, with a large sample size even if $\epsilon_i$ does not follow a normal distribution

\vspace{2mm}

 - \textbf{E}qual variance across all values of $X$
   - Also known as homoskedasticity
   
   
## Time Series Data {.t}

Time Series data violate the independence condition in a very particular way. 

Simply put, time series data are data that are collected over time.

### Examples:
\vspace{-2mm}
 - Stock market prices
 - Temperature of a particular location over time
 - Quiz scores within individual students over time


## Time Series Data {.t}
\framesubtitle{Example \#1: TSM Stock Price} 
TSM is the stock symbol for Taiwan Semiconductor Manufacturing Company. The R package `quantmod` lets you grab pricing information for stocks:

```{r, eval=FALSE}
install.packages("quantmod")
library(quantmod)
getSymbols("TSM")
```

```{r, echo=FALSE, message=FALSE, size="footnotesize"}
library(quantmod)
getSymbols("TSM", from="2025-05-22", to="2026-05-22")
knitr::kable(TSM[1:5])
```


## Time Series Data {.t}
\framesubtitle{Example \#1: TSM Stock Price} 
```{r, echo=FALSE, message=FALSE}
tsm_df <- TSM
tsm_df$day_num <- 1:dim(tsm_df)[1]
library(ggplot2)
theme_update(text=element_text(size=20))
ggplot(tsm_df, aes(x=day_num, y=TSM.Open)) + 
  geom_line() + labs(title="TSM Opening Stock Price for the past year", y="Opening Price ($)",
                     x="Day Number")

y <- as.numeric(tsm_df$TSM.Open)
```

## Time Series Data {.t}
\framesubtitle{Example \#1: TSM Stock Price} 
To assess autocorrelation, the standard approach is to look at correlograms of two things:

 - autocorrelation function (ACF)
   - This shows the correlation between $x_t$ and $x_{t+k}$ for any $k$, where $k$ is the number of steps in the future. For example, for $k=1$, this would just be looking at the correlation between subsequent days
   
\vspace{5mm}

 - partial autocorrelation function (PACF)
   - Even if the data generating process only has e.g. one-step correlations, it will show up as correlations between steps that are further apart as well due to carry-over. The PACF controls for these intermediate steps in an attempt to isolate only the correlations that are actually driving any relationship.
   
## Time Series Data {.t}
\framesubtitle{Example \#1: TSM Stock Price} 
```{r, out.width="85%"}
acf(y)
```


   
## Time Series Data {.t}
\framesubtitle{Example \#1: TSM Stock Price} 
```{r, out.width="85%"}
pacf(y)
```



## Time Series Data {.t}
\framesubtitle{Example \#1: TSM Stock Price} 
So, let's try an AR(1) model
```{r}
library(forecast)
t <- 1:length(y)
stock_model <- Arima(y, order=c(1,0,0), xreg=t)
stock_model
```



## Time Series Data {.t}
\framesubtitle{Example \#1: TSM Stock Price} 
```{r}
library(lmtest)
coeftest(stock_model)
```


## Your Turn \#1 {.t}
\framesubtitle{Example \#2: Poker data from Lecture \#5}
```{r, echo=FALSE, message=FALSE, warning=FALSE}
library(dplyr)
library(readr)
library(ggplot2)
export_all <- read_csv("ReportExport.csv")
won_all <- gsub("\\$", "", export_all$`My C Won`)
won_all <- gsub("\\(", "-", won_all)
won_all <- gsub("\\)", "", won_all)
won_all <- as.numeric(won_all)

df_all <- data.frame(won_all = won_all,
                     hand = 1:length(won_all))

df_all <- df_all %>% mutate(won_cum = cumsum(won_all))

ggplot(data=df_all, aes(x=hand, y=won_cum)) + 
  geom_line() + 
  labs(title="Running total of money won starting from May 7, 2020", y = "Cumulative amount won") + 
  scale_y_continuous(labels = scales::label_dollar()) +
  theme(text = element_text(size=20))
```



## Your Turn \#1 {.t}
\framesubtitle{Example \#2: Poker data from Lecture \#5}

 - Load the data into R (`PokerHands1NL.csv` file on course website)

\vspace{5mm}

 - Using the `won_cumul` column in that dataframe, recreate the graph on the previous slide
 
\vspace{5mm}

 - Create ACF and PACF plots with the `won_cumul` column
 
\vspace{5mm}

 - Run the appropriate Arima model and find the p-value for whether there is a significant directional trend, adjusting for any autocorrelation.