# Introduction to Autoregressive models

This tutorial presents introduction of autoregressive models, and theoir implementation in R.

### 1  AR(1) model

#### 1.1  Model definition

[r_t=phi_0+phi_1r_{t-1}+a_t]

where ({a_t},) is a white noise series of mean zero and variance (sigma_a^2).

Notes:

• AR(1) model is widely used not only for returns, as shown with (r_t,) here, but also for volatility with (r_t,) replaced with (sigma_t).
• Conditional on past return (r_{t-1}), we have conditional mean and variance as following[begin{gather*}mathbb{E}[r_t|r_{t-1}]=phi_0+phi_1r_{t-1}\text{Var}[r_t|r_{t-1}]=sigma_a^2end{gather*}]This is a Markov property in that, conditional on (r_{t-1}), the return (r_t,) is not correlated with (r_{t-i},) for i > 1.

# Stationarity, Autocorrelation, White Noise, and Linear Time Series

This tutorial introduces basic concepts about stationarity, autocorrelation, white noise, and linear time series.

### 1  Stationarity

#### 1.1  Strict stationarity

A time series {$$r_t$$} is said to be strictly stationary if the joint distribution of $$(t_{t_1},\cdots,r_{t_k})\,$$ is identical to  $$(t_{t_1+l},\cdots,r_{t_k+l})\,$$ for all t, where k is an arbitrary positive integer and ($$t_1,\cdots,r_k$$) is a collection of k positive integers.

In other words, strict stationarity requires that the joint distribution of ($$r_{t_1},\cdots,r_{t_k}$$) is invariant under time shift. This is a very hard condition that is hard to verify empirically. Read more Stationarity, Autocorrelation, White Noise, and Linear Time Series

# Asset Return and Distributions

This post talks about asset return and distributions, covering various definitions of returns and the relationship among them, return distributions and tests of returns.

### 1  Asset returns

Most financial studies involves returns, instead of prices, for two reasons:

1. Return is a complete and scale-free summary of investment opportunity;
2. Return has more attractive statistical properties than price.