Reverse Engineering a la Muth¶

Contents¶

• Reverse Engineering a la Muth
  • Friedman (1956) and Muth (1960)
  • A Process for Which Adaptive Expectations are Optimal
  • Some Useful State-Space Math
  • Estimates of Unobservables
  • Relationship of Unobservables to Observables
  • MA and AR Representations

In addition to what’s in Anaconda, this lecture uses the quantecon library.

We’ll also need the following imports:

This lecture uses the Kalman filter to reformulate John F. Muth’s first paper [Mut60] about rational expectations.

Muth used classical prediction methods to reverse engineer a stochastic process that renders optimal Milton Friedman’s [Fri56] “adaptive expectations” scheme.

Friedman (1956) and Muth (1960)¶

Milton Friedman [Fri56] (1956) posited that consumer’s forecast their future disposable income with the adaptive expectations scheme

$$ y_{t+i,t}^* = K \sum_{j=0}^\infty (1 - K)^j y_{t-j} \tag{3.1} $$

where $ K \in (0,1) $ and $ y_{t+i,t}^* $ is a forecast of future $ y $ over horizon $ i $.

Milton Friedman justified the exponential smoothing forecasting scheme (3.1) informally, noting that it seemed a plausible way to use past income to forecast future income.

In his first paper about rational expectations, John F. Muth [Mut60] reverse-engineered a univariate stochastic process $ \{y_t\}_{t=- \infty}^\infty $ for which Milton Friedman’s adaptive expectations scheme gives linear least forecasts of $ y_{t+j} $ for any horizon $ i $.

Muth sought a setting and a sense in which Friedman’s forecasting scheme is optimal.

That is, Muth asked for what optimal forecasting question is Milton Friedman’s adaptive expectation scheme the answer.

Muth (1960) used classical prediction methods based on lag-operators and $ z $-transforms to find the answer to his question.

Please see lectures Classical Control with Linear Algebra and Classical Filtering and Prediction with Linear Algebra for an introduction to the classical tools that Muth used.

Rather than using those classical tools, in this lecture we apply the Kalman filter to express the heart of Muth’s analysis concisely.

The lecture First Look at Kalman Filter describes the Kalman filter.

We’ll use limiting versions of the Kalman filter corresponding to what are called stationary values in that lecture.

A Process for Which Adaptive Expectations are Optimal¶

Suppose that an observable $ y_t $ is the sum of an unobserved random walk $ x_t $ and an IID shock $ \epsilon_{2,t} $:

$$ \begin{aligned} x_{t+1} & = x_t + \sigma_x \epsilon_{1,t+1} \cr y_t & = x_t + \sigma_y \epsilon_{2,t} \end{aligned} \tag{3.2} $$

where

$$ \begin{bmatrix} \epsilon_{1,t+1} \cr \epsilon_{2,t} \end{bmatrix} \sim {\mathcal N} (0, I) $$

is an IID process.

Note

A property of the state-space representation (3.2) is that in

general neither $ \epsilon_{1,t} $ nor $ \epsilon_{2,t} $ is in the space spanned by square-summable linear combinations of $ y_t, y_{t-1}, \ldots $.

In general $ \begin{bmatrix} \epsilon_{1,t} \cr \epsilon_{2t} \end{bmatrix} $ has more information about future $ y_{t+j} $’s than is contained in $ y_t, y_{t-1}, \ldots $.

We can use the asymptotic or stationary values of the Kalman gain and the one-step-ahead conditional state covariance matrix to compute a time-invariant innovations representation

$$ \begin{aligned} \hat x_{t+1} & = \hat x_t + K a_t \cr y_t & = \hat x_t + a_t \end{aligned} \tag{3.3} $$

where $ \hat x_t = E [x_t | y_{t-1}, y_{t-2}, \ldots ] $ and $ a_t = y_t - E[y_t |y_{t-1}, y_{t-2}, \ldots ] $.

Note

A key property about an innovations representation is that

$ a_t $ is in the space spanned by square summable linear combinations of $ y_t, y_{t-1}, \ldots $.

For more ramifications of this property, see the lectures Shock Non-Invertibility and Recursive Models of Dynamic Linear Economies.

Later we’ll stack these state-space systems (3.2) and (3.3) to display some classic findings of Muth.

But first, let’s create an instance of the state-space system (3.2) then apply the quantecon Kalman class, then uses it to construct the associated “innovations representation”

Some Useful State-Space Math¶

Now we want to map the time-invariant innovations representation (3.3) and the original state-space system (3.2) into a convenient form for deducing the impulse responses from the original shocks to the $ x_t $ and $ \hat x_t $.

Putting both of these representations into a single state-space system is yet another application of the insight that “finding the state is an art”.

We’ll define a state vector and appropriate state-space matrices that allow us to represent both systems in one fell swoop.

Note that

$$ a_t = x_t + \sigma_y \epsilon_{2,t} - \hat x_t $$

so that

$$ \begin{aligned} \hat x_{t+1} & = \hat x_t + K (x_t + \sigma_y \epsilon_{2,t} - \hat x_t) \cr & = (1-K) \hat x_t + K x_t + K \sigma_y \epsilon_{2,t} \end{aligned} $$

The stacked system

$$ \begin{bmatrix} x_{t+1} \cr \hat x_{t+1} \cr \epsilon_{2,t+1} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \cr K & (1-K) & K \sigma_y \cr 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_{t} \cr \hat x_t \cr \epsilon_{2,t} \end{bmatrix}+ \begin{bmatrix} \sigma_x & 0 \cr 0 & 0 \cr 0 & 1 \end{bmatrix} \begin{bmatrix} \epsilon_{1,t+1} \cr \epsilon_{2,t+1} \end{bmatrix} $$$$ \begin{bmatrix} y_t \cr a_t \end{bmatrix} = \begin{bmatrix} 1 & 0 & \sigma_y \cr 1 & -1 & \sigma_y \end{bmatrix} \begin{bmatrix} x_{t} \cr \hat x_t \cr \epsilon_{2,t} \end{bmatrix} $$

is a state-space system that tells us how the shocks $ \begin{bmatrix} \epsilon_{1,t+1} \cr \epsilon_{2,t+1} \end{bmatrix} $ affect states $ \hat x_{t+1}, x_t $, the observable $ y_t $, and the innovation $ a_t $.

With this tool at our disposal, let’s form the composite system and simulate it

Now that we have simulated our joint system, we have $ x_t $, $ \hat{x_t} $, and $ y_t $.

We can now investigate how these variables are related by plotting some key objects.

Estimates of Unobservables¶

First, let’s plot the hidden state $ x_t $ and the filtered version $ \hat x_t $ that is linear-least squares projection of $ x_t $ on the history $ y_{t-1}, y_{t-2}, \ldots $

Note how $ x_t $ and $ \hat{x_t} $ differ.

For Friedman, $ \hat x_t $ and not $ x_t $ is the consumer’s idea about her/his permanent income.

Relationship of Unobservables to Observables¶

Now let’s plot $ x_t $ and $ y_t $.

Recall that $ y_t $ is just $ x_t $ plus white noise

We see above that $ y $ seems to look like white noise around the values of $ x $.

Innovations¶

Recall that we wrote down the innovation representation that depended on $ a_t $. We now plot the innovations $ \{a_t\} $:

MA and AR Representations¶

Now we shall extract from the Kalman instance kmuth coefficients of

• a fundamental moving average representation that represents $ y_t $ as a one-sided moving sum of current and past $ a_t $s that are square summable linear combinations of $ y_t, y_{t-1}, \ldots $.
• a univariate autoregression representation that depicts the coefficients in a linear least square projection of $ y_t $ on the semi-infinite history $ y_{t-1}, y_{t-2}, \ldots $.

Then we’ll plot each of them

The moving average coefficients in the top panel show tell-tale signs of $ y_t $ being a process whose first difference is a first-order autoregression.

The autoregressive coefficients decline geometrically with decay rate $ (1-K) $.

These are exactly the target outcomes that Muth (1960) aimed to reverse engineer