.ipynb .pdf

Binder

Contents

import matplotlib.pyplot as plt
import pandas as pd
from fredapi import Fred
fred = Fred(api_key="your_api_key")

start = '1948-01'
end = '2022-01'

unr = fred.get_series('UNRATE',  observation_start=start)
unr.index.freq = unr.index.inferred_freq
unr.name = 'Unemployment'
recessions = fred.get_series('USREC',  observation_start=start, observation_end=end)
recessions.name='recessions'
df = pd.merge(unr, recessions, how='outer', left_index=True, right_index=True)
df = df.fillna(value=0)
df.index.freq = df.index.inferred_freq
dates = df.index._mpl_repr()


indpr = fred.get_series('INDPRO',  observation_start=start, observation_end=end)
indpr.index.freq = indpr.index.inferred_freq
indpr.name = 'Industrial production'
indpr = indpr.to_frame()

Regime-switching models

Motivation

• empirically, economic contractions are steeper and more short-lived than expansions

fig, ax = plt.subplots(figsize=(12,6))

df[['Unemployment']].plot(ax=ax)
ylim = ax.get_ylim()
ax.fill_between(dates, ylim[0], ylim[1], df[['recessions']].values.flatten(), facecolor='k', alpha=0.1);

fig2, ax = plt.subplots(figsize=(12,6))

indpr.plot(ax=ax)
ylim = ax.get_ylim()
ax.fill_between(dates, ylim[0], ylim[1], df[['recessions']].values.flatten(), facecolor='k', alpha=0.1);

• asymmetry between different phases of the business cycle

• linear models rule out different dynamics of variables during contractions and expansions

• contractions are much less common part of the sample

  • trend-cycle decompositions based on linear models are better at representing dynamics during expansions.

• non-linear models may be better at capturing the business cycle asymmetry

• asymmetry may be explained by different propagation mechanism in contractions vs expansions, or by asymmetry in the shocks

  • capacity constraints, monopoly power, downward vs upward price/wage stickiness, etc

Regime-switching AR§ model

$$\begin{array}{r}\begin{array}{rl}\alpha (L)(z_t-{\mu }_t)& ={\epsilon }_t,{\epsilon }_t\sim \mathcal{N}(0,{\sigma }_t^2)\\ \\ {\mu }_t={\mu }_0& +{\mu }_{1}I(S_t=1)\\ \\ {\sigma }_t^2={\sigma }_0^2& +{\sigma }_1^{2}I(S_t=1)\\ \\ S_t& =0,1(\text{expansion, contraction})\end{array}\end{array}$$

• if $S_t$ is known (observed) this model can be estimated in the usual way by MLE or OLS using dummies

$$\begin{array}{r}\begin{array}{rl}\ell (\mathit{\theta }|\mathbf{z},\mathbf{S})& =\sum _{t=1}^T\ln (p(z_t|Z_{t-1},S_t,S_{t-1},...,S_{t-p}))\\ \\ & \text{if p = 1}\\ \\ \ln (p(z_t|Z_{t-1},S_t,S_{t-1}))& =-\frac{1}{2}\ln (2\pi {\sigma }_t^2)-\frac{1}{2{\sigma }_t^2}((z_t-{\mu }_t)-{\alpha }_1(y_{t-1}-{\mu }_{t-1}))\end{array}\end{array}$$

• not very appealing as this assumes regime switches are deterministic - can be predicted

• if $S_t$ is unobserved, we cannot condition on $S_t,S_{t-1},...,S_{t-p}$, and have to work with

$$\begin{array}{rl}\ell (\mathit{\theta }|\mathbf{z})& =\sum _{t=1}^T\ln (p(z_t|Z_{t-1}))\end{array}$$

• integrate out $S_t,S_{t-1}$ from the joint density $p(z_t,S_t,S_{t-1}|Z_{t-1})$ by summing over all possible values of $S_t$ and $S_{t-1}$

$$\begin{array}{rl}p(z_t|Z_{t-1})& =\sum _{S_t=0}^1\sum _{S_{t-1}=0}^{1}p(z_t,S_t,S_{t-1}|Z_{t-1})\end{array}$$

using

$$p(z_t,S_t,S_{t-1}|Z_{t-1})=p(z_t|S_t,S_{t-1},Z_{t-1})\mathrm{Pr}(S_t,S_{t-1}|Z_{t-1})$$

we have

$$\begin{array}{rl}p(z_t|Z_{t-1})& =\sum _{S_t=0}^1\sum _{S_{t-1}=0}^{1}p(z_t|S_t,S_{t-1},Z_{t-1})\mathrm{Pr}(S_t,S_{t-1}|Z_{t-1})\end{array}$$

• $p(z_t|S_t,S_{t-1},Z_{t-1})$ computed as before (for known $S_t$)

Hamilton filter (Hamilton (1989)):

• assumes time-invariant Markov chain

$$\begin{array}{r}\begin{array}{rl}\mathrm{Pr}[S_t=i|S_{t-1}=j,S_{t-2}=k,...,Z_{t-1}]& =\mathrm{Pr}[S_t=i|S_{t-1}=j]\\ & =p_{ij}\end{array}\end{array}$$

• what if $p_{1,1}=1?$

• restrictions?

$$p_{0,j}+p_{1,j}=1$$

• computes $\mathrm{Pr}(S_t=i,S_{t-1}=j|Z_{t-1})$ iteratively starting with the unconditional probabilities

$$\begin{array}{r}\begin{array}{c}\mathrm{Pr}(S_1=0)=\frac{1-p_{11}}{2-p_{11}-p_{00}},\mathrm{Pr}(S_1=1)=\frac{1-p_{00}}{2-p_{11}-p_{00}}\end{array}\end{array}$$

Markov-switching models in Statsmodels

Two classes of models:

• markov switching dynamic regression models

• markov switching autoregression models

markov-switching dynamic regression models

$$z_t={\mu }_t+x_t^{\prime }\gamma +y_t^{\prime }{\beta }_t+{\epsilon }_t,{\epsilon }_t\sim \mathcal{N}(0,{\sigma }_t^2)$$

where both $x_t$ and $y_t$ may include lags of $z_t$

• $z_t$ depends only on the current state (the intercept has 2 possible values for a model with 2 states)

markov-switching autoregression models

$$\begin{array}{r}\begin{array}{rl}{z}_t& ={\mu }_t+x_t^{\prime }\gamma +y_t^{\prime }{\beta }_t+\sum _{i=1}^p{\alpha }_{i,t}(z_{t-i}-{\mu }_{t-1}+x_{t-i}^{\prime }\gamma +y_{t-i}^{\prime }{\beta }_{t-i}),{\epsilon }_t\sim \mathcal{N}(0,{\sigma }_t^2)\\ \\ & {\alpha }_t(L)(z_t-{\mu }_t-x_t^{\prime }\gamma -y_t^{\prime }{\beta }_t)={\epsilon }_t,{\epsilon }_t\sim \mathcal{N}(0,{\sigma }_t^2)\end{array}\end{array}$$

• $z_t$ depends on the current and past states (the intercept has 4 possible values for a model with 2 states when p=1)

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Granger causality