Non-Stationary Time Series

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.title[
# Non-Stationary Time Series
]
.subtitle[
## EC 421, Set 9
]
.author[
### Edward Rubin
]

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# Prologue

---
name: schedule

# Schedule

## Last Time

Autocorrelation

## Today

A brief introduction to nonstationarity

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# Nonstationarity

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name: intro
# Nonstationarity

---

## Intro

Let's go back to our assumption of .hi[weak dependence/persistence]

> 1. **Weakly persistent outcomes**—essentially, `$x_{t+k}$` in the distant period `$t+k$` weakly correlates with `$x_t$` (when `$k$` is "big").

We're essentially saying we need the time series `$x$` to behave.

We'll define this *good behavior* as .hi[stationarity].

---

## Stationarity

Requirements for .hi[stationarity] (a *stationary* time-series process):

1. The .hi[mean] of the distribution is independent of time, _i.e._,
.center[
`$\mathop{\boldsymbol{E}}\left[ x_t \right]  = \mathop{\boldsymbol{E}}\left[ x_{t-k} \right]$` for all `$k$`
]
--

2. The .hi[variance] of the distribution is independent of time, _i.e._,
.center[
`$\mathop{\text{Var}} \left( x_t \right) = \mathop{\text{Var}} \left( x_{t-k} \right)$` for all `$k$`
]

--
3. The .hi[covariance] between `$x_t$` and `$x_{t-k}$` depends only on `$k$`—.pink[not on] `$\color{#e64173}{t}$`, _i.e._,
.center[
`$\mathop{\text{Cov}} \left( x_t,\,x_{t-k} \right) = \mathop{\text{Cov}} \left( x_s,\, x_{s-k} \right)$` for all `$t$` and `$s$`
]

---
name: walks

## Random walks

.hi[Random walks] are a famous example of a nonstationary process:

$$
`\begin{align}
  x_t = x_{t-1} + \varepsilon_t
\end{align}`
$$

Why?
--
 `$\mathop{\text{Var}} \left( x_t \right) = t \sigma_\varepsilon^2$`, which .pink[violates stationary variance].

$$
`\begin{align}
   \mathop{\text{Var}} \left( x_t \right)
   &= \mathop{\text{Var}} \left( x_{t-1} + \varepsilon_t \right) \\
   &= \mathop{\text{Var}} \left( x_{t-2} + \varepsilon_{t-1} + \varepsilon_t \right) \\
   &= \mathop{\text{Var}} \left( x_{t-3} + \varepsilon_{t-2} + \varepsilon_{t-1} + \varepsilon_t \right) \\
   &\cdots \\
   &= \mathop{\text{Var}} \left( x_0 + \varepsilon_1 + \cdots + \varepsilon_{t_2} + \varepsilon_{t-1} + \varepsilon_t \right) \\
   &= \sigma^2_\varepsilon + \cdots + \sigma^2_\varepsilon + \sigma^2_\varepsilon + \sigma^2_\varepsilon \\
   &= t \sigma^2_\varepsilon
\end{align}`
$$

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**Q:** What's the big deal with this violation?

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.hi-slate[One 100-period random walk]
<img src="slides_files/figure-html/walk1-1.svg" style="display: block; margin: auto;" />

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.hi-slate[Two 100-period random walks]
<img src="slides_files/figure-html/walk2-1.svg" style="display: block; margin: auto;" />

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.hi-slate[Three 100-period random walks]
<img src="slides_files/figure-html/walk3-1.svg" style="display: block; margin: auto;" />

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.hi-slate[Four 100-period random walks]
<img src="slides_files/figure-html/walk4-1.svg" style="display: block; margin: auto;" />

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.hi-slate[Five 100-period random walks]
<img src="slides_files/figure-html/walk5-1.svg" style="display: block; margin: auto;" />

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.hi-slate[Fifty 100-period random walks]
<img src="slides_files/figure-html/walk50-1.svg" style="display: block; margin: auto;" />

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.hi-slate[1,000 100-period random walks]
<img src="slides_files/figure-html/walk1000-1.svg" style="display: block; margin: auto;" />

---
# Nonstationarity
## Problem

*One* problem is that nonstationary processes can lead to .hi[spurious] results.

>**Defintion:** .hi[Spurious]
>- not being what it purports to be; false or fake
>- apparently but not actually valid

Back in 1974, Granger and Newbold showed that when they **generated random walks** and **regressed the random walks on each other**, .hi[77/100 regressions were statistically significant] at the 5% level (should have been approximately 5/100).

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.hi-slate[Granger and Newbold simulation example:] _t_ statistic ≈ -10.58

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.hi-slate[Granger and Newbold simulation example:] _t_ statistic ≈ -8.92

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.hi-slate[Granger and Newbold simulation example:] _t_ statistic ≈ -7.23

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# Nonstationarity

---
name: problem

## Problem

In our data, 74.6 percent of (independently generated) pairs reject the null hypothesis at the 5% level.

**The point?**
--
 If our disturbance is nonstationary, we cannot trust plain OLS.

Random walks are only one example of .pink[nonstationary processes]...

.hi[Random walk:] `$u_t = u_{t-1} + \varepsilon_t$`

.hi[Random walk with drift:] `$u_t = \alpha_0 + u_{t-1} + \varepsilon_t$`

.hi[Deterministic trend:] `$u_t = \alpha_0 + \beta_1 t + \varepsilon_t$`

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# Nonstationarity
## A potential solution

---
name: solution

Some processes are .hi[difference stationary], which means we can get back to our stationarity (good behavior) requirement by taking the difference between `$u_t$` and `$u_{t-1}$`.

.hi-slate[Nonstationary:] `$u_t = u_{t-1} + \varepsilon_t$` .slate[(a random walk)]
--
<br>.hi[Stationary:] `$u_t - u_{t-1} = u_{t-1} + \varepsilon_t - u_{t-1} = \color{#e64173}{\varepsilon_t}$`

So if we have good reason to believe that our disturbances follow a random walk, we can use OLS on the differences, _i.e._,
--
$$
`\begin{align}
  y_t &= \beta_0 + \beta_1 x_t + u_t \\
  y_{t-1} &= \beta_0 + \beta_1 x_{t-1} + u_{t-1} \\
  y_t - y_{t-1} &= \beta_1 \left( x_t - x_{t-1} \right) + \left( u_t - u_{t-1} \right) \\
  \Delta y_t &= \beta_1 \Delta x_t + \Delta u_t
\end{align}`
$$

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name: test
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# Nonstationarity
## Testing

.pink[Dickey-Fuller] and .pink[augmented Dickey-Fuller] tests are popular ways to test of random walks and other forms of nonstationarity.

.hi[Dickey-Fuller tests] compare

H.sub[o]: `$y_t = \beta_0 + \beta_1 y_{t-1} + u_t$` with `$|\beta_1|<1$` (.hi[stationarity])
<br>
H.sub[a]: `$y_t = y_{t-1} + \varepsilon_t$` (.hi[random walk])

using a *t* test that `$|\beta_1|<1$`.<sup>.pink[†]</sup>

.footnote[
.pink[†] People often just test `$\beta_1<1$`.
]

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# Table of contents

.pull-left[
### Admin
.smallest[

1. [Schedule](#schedule)
]
]

.pull-right[
### Nonstationarity
.smallest[

1. [Introduction](#intro)
1. [Random walks](#walks)
1. [The actual problem](#problem)
1. [A potential solution](#solution)
1. [Dickey-Fuller tests](#test)
]
]

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