VAR Example Fed

by Professor Throckmorton
for Time Series Econometrics
W&M ECON 408/PUBP 616

Summary

• [CEE05]: Christiano, Eichenbaum, and Evans “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy” (Journal of Polical Economy, 2005) has over 9,000 citations (astronomical!) according to google scholar.

• Let \(\mathbf{Y}_t\) denote variables observed between 1965Q3 to 1995Q3,

  \[\begin{gather*} \mathbf{Y}_t = [\mathbf{Y}_{1t}, R_t, \mathbf{Y}_{2t}]' \end{gather*}\]

  \(\mathbf{Y}_{1t}\): real GDP, real consumption, GDP deflator, real investment, real wages, and labor productivity

  \(R_t\): federal funds rate

  \(\mathbf{Y}_{2t}\): real profits, M2 growth rate

Assumptions

• \(R_t\) appears after \(\mathbf{Y}_{1t}\), i.e., the assumption is that none of the variables in \(\mathbf{Y}_{1t}\) respond contemporaneously to \(R_t\), so the real economy lags behind monetary policy (citing Friedman)

• “All these variables, except money growth, have been natural logged.”

• “With one exception (the growth rate of money), all the variables in are included in levels.”

• “The VAR contains four lags of each variable, and the sample period is 1965:3–1995:3.”

Read Data

# Libraries
from fredapi import Fred
import pandas as pd
# Setup acccess to FRED
fred_api_key = pd.read_csv('fred_api_key.txt', header=None)
fred = Fred(api_key=fred_api_key.iloc[0,0])
# Series to get
series = ['GDP','PCE','GDPDEF','GPDI','COMPRNFB','OPHNFB','FEDFUNDS','CP','M2SL']
rename = ['gdp','cons','price','inv','wage','prod','ffr','profit','money']
# Get and append data to list
dl = []
for idx, string in enumerate(series):
    var = fred.get_series(string).to_frame(name=rename[idx])
    dl.append(var)
    print(var.head(2)); print(var.tail(2))

            gdp
1946-01-01  NaN
1946-04-01  NaN
                  gdp
2025-01-01  29962.047
2025-04-01  30331.117
             cons
1959-01-01  306.1
1959-02-01  309.6
               cons
2025-05-01  20615.2
2025-06-01  20685.2

             price
1947-01-01  11.141
1947-04-01  11.299
              price
2025-01-01  127.429
2025-04-01  128.059
            inv
1939-01-01  NaN
1939-04-01  NaN
                 inv
2025-01-01  5595.694
2025-04-01  5372.966

              wage
1947-01-01  34.386
1947-04-01  34.670
               wage
2024-10-01  108.178
2025-01-01  108.496
              prod
1947-01-01  22.256
1947-04-01  22.762
               prod
2024-10-01  115.725
2025-01-01  115.303

             ffr
1954-07-01  0.80
1954-08-01  1.22
             ffr
2025-06-01  4.33
2025-07-01  4.33
            profit
1946-01-01     NaN
1946-04-01     NaN
              profit
2024-10-01  3631.383
2025-01-01  3602.551

            money
1959-01-01  286.6
1959-02-01  287.7
              money
2025-05-01  21883.6
2025-06-01  22020.8

# Concatenate data to create data frame (time-series table)
raw = pd.concat(dl, axis=1).sort_index()
# Resample/reindex to quarterly frequency
raw = raw.resample('QE').last().dropna()
# Display dataframe
display(raw)

	gdp	cons	price	inv	wage	prod	ffr	profit	money
1959-03-31	510.330	312.7	15.224	83.166	49.137	31.034	2.80	32.706	289.2
1959-06-30	522.653	318.2	15.248	89.381	49.471	31.336	3.39	35.446	294.1
1959-09-30	525.034	324.2	15.307	83.606	49.502	31.379	3.76	32.441	296.7
1959-12-31	528.600	322.9	15.367	86.524	49.672	31.277	3.99	31.960	297.8
1960-03-31	542.648	330.2	15.428	96.476	50.690	32.007	3.84	34.614	299.3
...	...	...	...	...	...	...	...	...	...
2024-03-31	28624.069	19553.2	124.163	5159.903	107.989	113.845	5.33	3314.945	20901.1
2024-06-30	29016.714	19747.5	124.943	5297.844	107.562	114.430	5.33	3413.018	21065.6
2024-09-30	29374.914	20044.1	125.532	5345.165	107.543	115.241	5.13	3402.982	21272.1
2024-12-31	29723.864	20408.1	126.257	5288.763	108.178	115.725	4.48	3631.383	21441.8
2025-03-31	29962.047	20578.5	127.429	5595.694	108.496	115.303	4.33	3602.551	21656.9

265 rows × 9 columns

Transform Data

# Scientific computing
import numpy as np
data = pd.DataFrame()
# log real GDP
data['gdp'] = 100*(np.log(raw['gdp']/raw['price']))
# log real Consumption
data['cons'] = 100*(np.log(raw['cons']/raw['price']))
# log price level
data['price'] = 100*np.log(raw['price'])
# log real Investment
data['inv'] = 100*(np.log(raw['inv']/raw['price']))
# log real Wage
data['wage'] = 100*(np.log(raw['wage']/raw['price']))
# log labor productivity
data['prod'] = 100*np.log(raw['prod'])
# log federal funds rate
data['ffr'] = 100*np.log(raw['ffr'])
# log real Profits
data['profit'] = 100*(np.log(raw['profit']/raw['price']))
# M2 growth rate
data['money'] = 100*(raw['money']/raw['money'].shift(1)-1)

# Christiano et al. (2005) sample
sample = data['09-30-1965':'09-30-1995']
display(sample)

	gdp	cons	price	inv	wage	prod	ffr	profit	money
1965-09-30	381.006869	330.458073	280.910145	206.790340	122.363364	362.846575	139.128190	117.089792	2.135878
1965-12-31	383.285180	331.896783	281.594750	208.003984	122.626131	364.565875	146.325540	121.965505	2.157953
1966-03-31	385.693415	334.098399	282.233081	214.887041	122.783543	365.919314	153.901545	124.014461	1.742160
1966-06-30	386.031488	333.386309	283.050379	213.583821	122.650228	365.224140	164.287269	123.982126	0.856164
1966-09-30	386.873224	335.264086	284.013054	212.406981	122.249973	365.552947	168.639895	122.713398	0.891341
...	...	...	...	...	...	...	...	...	...
1994-09-30	471.421551	428.548732	418.566193	294.663512	16.380277	412.626304	155.392520	192.242943	0.189682
1994-12-31	472.561602	429.393768	419.106284	298.485983	16.161222	413.679723	169.561561	194.694279	0.008606
1995-03-31	472.914588	429.752656	419.647988	299.463765	15.592482	413.742002	178.842057	197.218061	0.134810
1995-06-30	473.213115	431.141418	420.126885	297.191369	15.181804	414.211834	179.175947	201.596380	1.655639
1995-09-30	474.059760	431.510077	420.618404	296.643085	14.784170	414.202300	175.785792	202.686497	1.499056

121 rows × 9 columns

Non-stationarity

# Plotting
import matplotlib.pyplot as plt
# Plot GDP
fig, ax = plt.subplots(figsize=(6.5,2.5))
ax.plot(sample['gdp'])
ax.set_title('100 log(Real GDP)')
ax.grid()

• Q: Do these transformations make the data stationary? A: Clearly not.

• [SSW90]: Sims, Stock, and Watson “Inference in linear time series models with some unit roots” (Econometrica, 1990) find that VARs in levels can yield consistent impulse response functions even if the variables are non-stationary, I(\(1\)), and cointegrated.

• Fortunately, Christiano et al. (2005) is only concerned with estimating IRFs.

Cointegration

• Suppose you have two non-stationary time series: \(y_t\) and \(x_t\), both I(\(1\)).

• Typically, any linear combination of them, \(z_t = y_t - \beta x_t\), would also be non-stationary.

• But if some linear combination is stationary, then the variables are cointegrated.

• The coefficients (like \([1, -\beta]\)) form a cointegrating vector.

• This implies the variables share a stable long-run relationship despite short-run fluctuations.

Example

• Let \(y_t\) be \(\log(GDP)\) (i.e., income) and \(x_t\) be \(\log(PCE)\) (i.e., consumption)

• Both series are non-stationary, I(\(1\)), because they have a stochastic time trend.

• There might exist a linear combination \(z_t = y_t - \beta x_t\) that is stationary, I(\(0\)), meaning

  • Income and consumption have a long-run linear relationship, i.e., they are cointegrated.

  • Deviations from the long-run are mean-reverting.

Estimation

# VAR model
from statsmodels.tsa.api import VAR
# make the VAR model
model = VAR(sample)
# Estimate VAR(4)
results = model.fit(4)
# Assign impulse response functions (IRFs)
irf = results.irf(20)

# Plot IRFs
plt = irf.plot(orth=True,impulse='ffr',figsize=(6.5,22.5));
plt.suptitle('');