Notebook

Vector Autoregressions: inflation-unemployment-interest rate¶

Vector Autoregression (VAR), introduced by Nobel laureate Christopher Sims in 1980, is a powerful statistical tool in the macroeconomist's toolkit.

Formally a VAR model is

$$Y_t = A_1 Y_{t-1} + \ldots + A_p Y_{t-p} + u_t$$$$u_t \sim {\sf Normal}(0, \Sigma_u)$$

where $Y_t$ is of dimension $K$ and $A_i$ is a $K \times K$ coefficient matrix.

In [1]:

dta = sm.datasets.macrodata.load_pandas().data
endog = dta[["infl", "unemp", "tbilrate"]]

In [2]:

index = sm.tsa.datetools.dates_from_range('1959Q1', '2009Q3')
dta.index = pandas.Index(index)
del dta['year']
del dta['quarter']
endog.index = pandas.Index(index) # DatetimeIndex or PeriodIndex in 0.8.0
print endog.head(10)

            infl  unemp  tbilrate
1959-03-31  0.00    5.8      2.82
1959-06-30  2.34    5.1      3.08
1959-09-30  2.74    5.3      3.82
1959-12-31  0.27    5.6      4.33
1960-03-31  2.31    5.2      3.50
1960-06-30  0.14    5.2      2.68
1960-09-30  2.70    5.6      2.36
1960-12-31  1.21    6.3      2.29
1961-03-31 -0.40    6.8      2.37
1961-06-30  1.47    7.0      2.29

In [3]:

endog.plot(subplots=True, figsize=(14,18));

/usr/local/lib/python2.7/dist-packages/matplotlib/font_manager.py:1210: UserWarning: findfont: Font family ['monospace'] not found. Falling back to Bitstream Vera Sans
  (prop.get_family(), self.defaultFamily[fontext]))

In [4]:

# model only after Volcker appointment
var_mod = sm.tsa.VAR(endog.ix['1979-12-31':]).fit(maxlags=4, ic=None)
print var_mod.summary()

  Summary of Regression Results   
==================================
Model:                         VAR
Method:                        OLS
Date:           Tue, 17, Jul, 2012
Time:                     12:29:31
--------------------------------------------------------------------
No. of Equations:         3.00000    BIC:                   -1.38214
Nobs:                     116.000    HQIC:                  -1.93210
Log likelihood:          -320.932    FPE:                  0.0997509
AIC:                     -2.30792    Det(Omega_mle):       0.0725307
--------------------------------------------------------------------
Results for equation infl
==============================================================================
                 coefficient       std. error           t-stat            prob
------------------------------------------------------------------------------
const               1.861765         0.975933            1.908           0.059
L1.infl             0.043747         0.099137            0.441           0.660
L1.unemp           -1.772916         1.068556           -1.659           0.100
L1.tbilrate         1.068492         0.308544            3.463           0.001
L2.infl            -0.047131         0.094792           -0.497           0.620
L2.unemp            3.350585         1.947357            1.721           0.088
L2.tbilrate        -0.801488         0.364698           -2.198           0.030
L3.infl             0.130702         0.096459            1.355           0.178
L3.unemp            0.483236         1.915270            0.252           0.801
L3.tbilrate         0.767990         0.323361            2.375           0.019
L4.infl            -0.118506         0.105590           -1.122           0.264
L4.unemp           -2.109414         1.046155           -2.016           0.046
L4.tbilrate        -0.709641         0.246646           -2.877           0.005
==============================================================================

Results for equation unemp
==============================================================================
                 coefficient       std. error           t-stat            prob
------------------------------------------------------------------------------
const               0.196149         0.091636            2.141           0.035
L1.infl            -0.006733         0.009309           -0.723           0.471
L1.unemp            1.543511         0.100333           15.384           0.000
L1.tbilrate        -0.101861         0.028971           -3.516           0.001
L2.infl             0.004712         0.008901            0.529           0.598
L2.unemp           -0.434913         0.182848           -2.379           0.019
L2.tbilrate         0.154581         0.034244            4.514           0.000
L3.infl             0.009677         0.009057            1.068           0.288
L3.unemp           -0.239875         0.179835           -1.334           0.185
L3.tbilrate        -0.092268         0.030362           -3.039           0.003
L4.infl             0.026235         0.009914            2.646           0.009
L4.unemp            0.082838         0.098229            0.843           0.401
L4.tbilrate         0.035650         0.023159            1.539           0.127
==============================================================================

Results for equation tbilrate
==============================================================================
                 coefficient       std. error           t-stat            prob
------------------------------------------------------------------------------
const              -0.210139         0.313916           -0.669           0.505
L1.infl            -0.034879         0.031888           -1.094           0.277
L1.unemp           -0.679575         0.343709           -1.977           0.051
L1.tbilrate         1.151534         0.099246           11.603           0.000
L2.infl             0.043524         0.030491            1.427           0.156
L2.unemp            0.601872         0.626382            0.961           0.339
L2.tbilrate        -0.448569         0.117308           -3.824           0.000
L3.infl             0.023667         0.031027            0.763           0.447
L3.unemp            0.748752         0.616061            1.215           0.227
L3.tbilrate         0.558119         0.104012            5.366           0.000
L4.infl             0.041376         0.033964            1.218           0.226
L4.unemp           -0.586963         0.336504           -1.744           0.084
L4.tbilrate        -0.373850         0.079336           -4.712           0.000
==============================================================================

Correlation matrix of residuals
                infl     unemp  tbilrate
infl        1.000000 -0.150786  0.258308
unemp      -0.150786  1.000000 -0.389991
tbilrate    0.258308 -0.389991  1.000000

Diagnostics¶

In [5]:

np.abs(var_mod.roots)

Out[5]:

array([ 2.7821,  2.7821,  1.5472,  1.5472,  1.3908,  1.3908,  1.3806,
        1.3806,  1.1395,  1.1395,  1.104 ,  1.104 ])

var_mod.test_normality() and var_mod.test_whiteness() are also available. There are problems with this model...

Granger-Causality tests¶

In [6]:

var_mod.test_causality('unemp', 'tbilrate', kind='Wald')

Granger causality Wald-test
=======================================================
   Test statistic   Critical Value          p-value  df
-------------------------------------------------------
        22.700855         9.487729            0.000   4
=======================================================
H_0: ['tbilrate'] do not Granger-cause unemp
Conclusion: reject H_0 at 5.00% significance level

Out[6]:

{'conclusion': 'reject',
 'crit_value': 9.487729036781154,
 'df': 4,
 'pvalue': 0.00014529611487641404,
 'signif': 0.05,
 'statistic': 22.700854791605487}

In [7]:

table = pandas.DataFrame(np.zeros((9,3)), columns=['chi2', 'df', 'prob(>chi2)'])
index = []
variables = set(endog.columns.tolist())
i = 0
for vari in variables:
    others = []
    for j,ex_vari in enumerate(variables):
        if vari == ex_vari: # don't want to test this
            continue
        others.append(ex_vari)
        res = var_mod.test_causality(vari, ex_vari, kind='Wald', verbose=False)
        table.ix[[i], ['chi2', 'df', 'prob(>chi2)']] = (res['statistic'], res['df'], res['pvalue'])
        i += 1
        index.append([vari, ex_vari])
    res = var_mod.test_causality(vari, others, kind='Wald', verbose=False)
    table.ix[[i], ['chi2', 'df', 'prob(>chi2)']] = res['statistic'], res['df'], res['pvalue']
    index.append([vari, 'ALL'])
    i += 1
table.index = pandas.MultiIndex.from_tuples(index, names=['Equation', 'Excluded'])

In [8]:

print table

                     chi2  df  prob(>chi2)
Equation Excluded                         
unemp    infl      10.866   4        0.028
         tbilrate  22.701   4        0.000
         ALL       35.853   8        0.000
infl     unemp     10.497   4        0.033
         tbilrate  21.629   4        0.000
         ALL       29.597   8        0.000
tbilrate unemp     12.447   4        0.014
         infl       6.119   4        0.190
         ALL       16.362   8        0.037

From this we reject the null that these variables do not Granger cause for all cases except for infl -> tbilrate. In other words, in almost all cases we can reject the null hypothesis that the lags of the excluded variable are jointly zero in Equation.

Order Selection¶

In [9]:

var_mod.model.select_order()

                 VAR Order Selection                  
======================================================
            aic          bic          fpe         hqic
------------------------------------------------------
0         3.821        3.895        45.63        3.851
1        -2.693       -2.393      0.06769       -2.571
2       -3.424*      -2.900*     0.03259*      -3.212*
3        -3.335       -2.586      0.03567       -3.031
4        -3.404       -2.429      0.03338       -3.009
5        -3.289       -2.090      0.03754       -2.803
6        -3.239       -1.815      0.03967       -2.661
7        -3.199       -1.550      0.04156       -2.530
8        -3.213       -1.340      0.04131       -2.454
9        -3.255       -1.156      0.04006       -2.404
10       -3.278      -0.9547      0.03969       -2.336
11       -3.206      -0.6580      0.04339       -2.173
12       -3.144      -0.3716      0.04711       -2.020
13       -3.065     -0.06718      0.05231       -1.850
======================================================
* Minimum

Out[9]:

{'aic': 2, 'bic': 2, 'fpe': 2, 'hqic': 2}

Impulse Response Functions¶

Suppose we want to examine what happens to each of the variables when a 1 unit increase in the current value of one of the VAR errors occurs (a "shock"). To isolate the effects of only one error while holding the others constant, we need the model to be in a form so that the contemporaneous errors are uncorrelated across equations. One such way to achieve this is the so-called recursive VAR. In the recursive VAR, the order of the variables is determined by how the econometrician views the economic processes as ocurring. Given this order, inflation is determined by the contemporaneous unemployment rate and tbilrate is determined by the contemporaneous inflation and unemployment rates. Unemployment is a function of only the past values of itself, inflation, and the T-bill rate. We achieve such a structure by using the Choleski decomposition.

In [10]:

irf = var_mod.irf(24)

In [16]:

irf.plot(orth=True, signif=.33, subplot_params = {'fontsize' : 18})

Note that inflation dynamics are not very persistent, but do appear to have a significant and immediate impact on interest rates and on unemployment in the medium run.

Forecast Error Decompositions¶

In [12]:

var_mod.fevd(24).summary()

FEVD for infl
          infl     unemp  tbilrate
0     1.000000  0.000000  0.000000
1     0.852378  0.068321  0.079300
2     0.821960  0.072742  0.105298
3     0.800366  0.071008  0.128626
4     0.782810  0.069296  0.147895
5     0.778382  0.073392  0.148226
6     0.774765  0.077698  0.147538
7     0.773304  0.078759  0.147937
8     0.772711  0.079996  0.147293
9     0.771720  0.081095  0.147185
10    0.770519  0.081963  0.147518
11    0.769590  0.082942  0.147467
12    0.768802  0.083723  0.147474
13    0.768341  0.084163  0.147496
14    0.768186  0.084338  0.147476
15    0.768135  0.084354  0.147511
16    0.768094  0.084314  0.147592
17    0.768034  0.084282  0.147684
18    0.767934  0.084280  0.147786
19    0.767816  0.084301  0.147883
20    0.767702  0.084342  0.147955
21    0.767601  0.084394  0.148005
22    0.767518  0.084444  0.148038
23    0.767455  0.084490  0.148056

FEVD for unemp
          infl     unemp  tbilrate
0     0.022736  0.977264  0.000000
1     0.044880  0.930399  0.024721
2     0.041784  0.930260  0.027956
3     0.037486  0.930276  0.032238
4     0.027721  0.931560  0.040719
5     0.022468  0.934437  0.043095
6     0.021990  0.936267  0.041743
7     0.025581  0.935272  0.039147
8     0.031058  0.931753  0.037189
9     0.037851  0.924482  0.037667
10    0.045527  0.913280  0.041193
11    0.053402  0.898884  0.047714
12    0.061321  0.882177  0.056503
13    0.068823  0.864635  0.066542
14    0.075418  0.847729  0.076853
15    0.080988  0.832408  0.086604
16    0.085516  0.819282  0.095202
17    0.089098  0.808583  0.102320
18    0.091923  0.800184  0.107893
19    0.094151  0.793811  0.112038
20    0.095907  0.789125  0.114968
21    0.097291  0.785765  0.116944
22    0.098382  0.783403  0.118215
23    0.099239  0.781764  0.118996

FEVD for tbilrate
          infl     unemp  tbilrate
0     0.066723  0.126097  0.807180
1     0.046150  0.198100  0.755749
2     0.056621  0.251998  0.691381
3     0.081954  0.248844  0.669202
4     0.099227  0.238387  0.662386
5     0.108544  0.228719  0.662736
6     0.113163  0.212939  0.673899
7     0.117658  0.199234  0.683108
8     0.120920  0.193892  0.685188
9     0.122879  0.197398  0.679723
10    0.124824  0.208597  0.666578
11    0.126204  0.224548  0.649248
12    0.126978  0.242303  0.630718
13    0.127918  0.259257  0.612825
14    0.129091  0.273537  0.597373
15    0.130529  0.284485  0.584986
16    0.132460  0.292021  0.575519
17    0.134787  0.296476  0.568738
18    0.137301  0.298526  0.564173
19    0.139874  0.298875  0.561251
20    0.142334  0.298162  0.559504
21    0.144535  0.296922  0.558544
22    0.146418  0.295511  0.558070
23    0.147973  0.294149  0.557878

In [13]:

var_mod.fevd(24).plot(figsize=(12,12))

There is some amount of interaction between the variables. For instance, at the 12 quarter horizon, 40% of the error in the forecast of the T-bill rate is attributed to the inflation and unemployment shocks in the recursive VAR.To make structural inferences - e.g., what is the effect on the rate of inflation and unemployment of an unexpected 100 basis point increase in the Federal Funds rate (proxied by the T-bill rate here), we might want to fit a structural VAR model based on economic theory of monetary policy. For instance, we might replace the VAR equation for the T-bill rate with a policy equation such as a Taylor rule and restrict coefficients. You can do so with the sm.tsa.SVAR class.

Exercises¶

Experiment with different VAR models. You can try to adjust the number of lags in the VAR model calculated above or the ordering of the variables and see how it affects the model.

You might also try adding variables to the VAR, say M1 measure of money supply, or estimating a different model using measures of consumption (realcons), government spending (realgovt), or GDP (realgdp).