From fbe4c82f07c835efd1bf558c07308bf364e7d252 Mon Sep 17 00:00:00 2001 From: JingkunZhao <155940781+SylviaZhaooo@users.noreply.github.com> Date: Mon, 8 Jul 2024 13:44:23 +1000 Subject: [PATCH] [AR1] Update editorial suggestions (#458) * [AR1] Update editorial suggestions * Update ar1_processes.md * misc * misc --------- Co-authored-by: John Stachurski Co-authored-by: Matt McKay --- lectures/ar1_processes.md | 101 +++++++++++++++++++++++++------------- 1 file changed, 68 insertions(+), 33 deletions(-) diff --git a/lectures/ar1_processes.md b/lectures/ar1_processes.md index c12d5f18..3df2c113 100644 --- a/lectures/ar1_processes.md +++ b/lectures/ar1_processes.md @@ -19,7 +19,7 @@ kernelspec: ``` (ar1_processes)= -# AR1 Processes +# AR(1) Processes ```{index} single: Autoregressive processes ``` @@ -35,10 +35,8 @@ These simple models are used again and again in economic research to represent t * dividends * productivity, etc. -AR(1) processes can take negative values but are easily converted into positive processes when necessary by a transformation such as exponentiation. - We are going to study AR(1) processes partly because they are useful and -partly because they help us understand important concepts. +partly because they help us understand important concepts. Let's start with some imports: @@ -48,9 +46,9 @@ import matplotlib.pyplot as plt plt.rcParams["figure.figsize"] = (11, 5) #set default figure size ``` -## The AR(1) Model +## The AR(1) model -The **AR(1) model** (autoregressive model of order 1) takes the form +The *AR(1) model* (autoregressive model of order 1) takes the form ```{math} :label: can_ar1 @@ -60,18 +58,30 @@ X_{t+1} = a X_t + b + c W_{t+1} where $a, b, c$ are scalar-valued parameters. -This law of motion generates a time series $\{ X_t\}$ as soon as we -specify an initial condition $X_0$. +For example, $X_t$ might be + +* the log of labor income for a given household, or +* the log of money demand in a given economy. + +In either case, {eq}`can_ar1` shows that the current value evolves as a linear function +of the previous value and an IID shock $W_{t+1}$. -This is called the **state process** and the state space is $\mathbb R$. +(We use $t+1$ for the subscript of $W_{t+1}$ because this random variable is not +observed at time $t$.) + +The specification {eq}`can_ar1` generates a time series $\{ X_t\}$ as soon as we +specify an initial condition $X_0$. To make things even simpler, we will assume that -* the process $\{ W_t \}$ is IID and standard normal, +* the process $\{ W_t \}$ is {ref}`IID ` and standard normal, * the initial condition $X_0$ is drawn from the normal distribution $N(\mu_0, v_0)$ and * the initial condition $X_0$ is independent of $\{ W_t \}$. -### Moving Average Representation + + + +### Moving average representation Iterating backwards from time $t$, we obtain @@ -99,7 +109,7 @@ Equation {eq}`ar1_ma` shows that $X_t$ is a well defined random variable, the va Throughout, the symbol $\psi_t$ will be used to refer to the density of this random variable $X_t$. -### Distribution Dynamics +### Distribution dynamics One of the nice things about this model is that it's so easy to trace out the sequence of distributions $\{ \psi_t \}$ corresponding to the time series $\{ X_t\}$. @@ -110,10 +120,9 @@ This is immediate from {eq}`ar1_ma`, since linear combinations of independent normal random variables are normal. Given that $X_t$ is normally distributed, we will know the full distribution -$\psi_t$ if we can pin down its first two moments. +$\psi_t$ if we can pin down its first two [moments](https://en.wikipedia.org/wiki/Moment_(mathematics)). -Let $\mu_t$ and $v_t$ denote the mean and variance -of $X_t$ respectively. +Let $\mu_t$ and $v_t$ denote the mean and variance of $X_t$ respectively. We can pin down these values from {eq}`ar1_ma` or we can use the following recursive expressions: @@ -140,8 +149,7 @@ $$ \psi_t = N(\mu_t, v_t) $$ -The following code uses these facts to track the sequence of marginal -distributions $\{ \psi_t \}$. +The following code uses these facts to track the sequence of marginal distributions $\{ \psi_t \}$. The parameters are @@ -173,9 +181,21 @@ ax.legend(bbox_to_anchor=[1.05,1],loc=2,borderaxespad=1) plt.show() ``` -## Stationarity and Asymptotic Stability -Notice that, in the figure above, the sequence $\{ \psi_t \}$ seems to be converging to a limiting distribution. + +## Stationarity and asymptotic stability + +When we use models to study the real world, it is generally preferable that our +models have clear, sharp predictions. + +For dynamic problems, sharp predictions are related to stability. + +For example, if a dynamic model predicts that inflation always converges to some +kind of steady state, then the model gives a sharp prediction. + +(The prediction might be wrong, but even this is helpful, because we can judge the quality of the model.) + +Notice that, in the figure above, the sequence $\{ \psi_t \}$ seems to be converging to a limiting distribution, suggesting some kind of stability. This is even clearer if we project forward further into the future: @@ -248,16 +268,21 @@ plt.show() As claimed, the sequence $\{ \psi_t \}$ converges to $\psi^*$. -### Stationary Distributions +We see that, at least for these parameters, the AR(1) model has strong stability +properties. + + + -A stationary distribution is a distribution that is a fixed -point of the update rule for distributions. +### Stationary distributions -In other words, if $\psi_t$ is stationary, then $\psi_{t+j} = -\psi_t$ for all $j$ in $\mathbb N$. +Let's try to better understand the limiting distribution $\psi^*$. -A different way to put this, specialized to the current setting, is as follows: a -density $\psi$ on $\mathbb R$ is **stationary** for the AR(1) process if +A stationary distribution is a distribution that is a "fixed point" of the update rule for the AR(1) process. + +In other words, if $\psi_t$ is stationary, then $\psi_{t+j} = \psi_t$ for all $j$ in $\mathbb N$. + +A different way to put this, specialized to the current setting, is as follows: a density $\psi$ on $\mathbb R$ is **stationary** for the AR(1) process if $$ X_t \sim \psi @@ -279,8 +304,8 @@ Thus, when $|a| < 1$, the AR(1) model has exactly one stationary density and tha The concept of ergodicity is used in different ways by different authors. -One way to understand it in the present setting is that a version of the Law -of Large Numbers is valid for $\{X_t\}$, even though it is not IID. +One way to understand it in the present setting is that a version of the law +of large numbers is valid for $\{X_t\}$, even though it is not IID. In particular, averages over time series converge to expectations under the stationary distribution. @@ -310,11 +335,21 @@ $$ \quad \text{as } m \to \infty $$ -In other words, the time series sample mean converges to the mean of the -stationary distribution. +In other words, the time series sample mean converges to the mean of the stationary distribution. + + +Ergodicity is important for a range of reasons. + +For example, {eq}`ar1_ergo` can be used to test theory. + +In this equation, we can use observed data to evaluate the left hand side of {eq}`ar1_ergo`. + +And we can use a theoretical AR(1) model to calculate the right hand side. + +If $\frac{1}{m} \sum_{t = 1}^m X_t$ is not close to $\psi^(x)$, even for many +observations, then our theory seems to be incorrect and we will need to revise +it. -As will become clear over the next few lectures, ergodicity is a very -important concept for statistics and simulation. ## Exercises @@ -339,7 +374,7 @@ M_k = \end{cases} $$ -Here $n!!$ is the double factorial. +Here $n!!$ is the [double factorial](https://en.wikipedia.org/wiki/Double_factorial). According to {eq}`ar1_ergo`, we should have, for any $k \in \mathbb N$,