Ordinary Least Squares

In [1]:
%matplotlib inline

from __future__ import print_function
import numpy as np
import statsmodels.api as sm
import matplotlib.pyplot as plt
from statsmodels.sandbox.regression.predstd import wls_prediction_std

np.random.seed(9876789)

OLS estimation

Artificial data:

In [2]:
nsample = 100
x = np.linspace(0, 10, 100)
X = np.column_stack((x, x**2))
beta = np.array([1, 0.1, 10])
e = np.random.normal(size=nsample)

Our model needs an intercept so we add a column of 1s:

In [3]:
X = sm.add_constant(X)
y = np.dot(X, beta) + e

Fit and summary:

In [4]:
model = sm.OLS(y, X)
results = model.fit()
print(results.summary())
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       1.000
Model:                            OLS   Adj. R-squared:                  1.000
Method:                 Least Squares   F-statistic:                 4.020e+06
Date:                Sat, 18 Feb 2017   Prob (F-statistic):          2.83e-239
Time:                        09:41:09   Log-Likelihood:                -146.51
No. Observations:                 100   AIC:                             299.0
Df Residuals:                      97   BIC:                             306.8
Df Model:                           2                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const          1.3423      0.313      4.292      0.000       0.722       1.963
x1            -0.0402      0.145     -0.278      0.781      -0.327       0.247
x2            10.0103      0.014    715.745      0.000       9.982      10.038
==============================================================================
Omnibus:                        2.042   Durbin-Watson:                   2.274
Prob(Omnibus):                  0.360   Jarque-Bera (JB):                1.875
Skew:                           0.234   Prob(JB):                        0.392
Kurtosis:                       2.519   Cond. No.                         144.
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Quantities of interest can be extracted directly from the fitted model. Type dir(results) for a full list. Here are some examples:

In [5]:
print('Parameters: ', results.params)
print('R2: ', results.rsquared)
Parameters:  [  1.34233516  -0.04024948  10.01025357]
R2:  0.999987936503

OLS non-linear curve but linear in parameters

We simulate artificial data with a non-linear relationship between x and y:

In [6]:
nsample = 50
sig = 0.5
x = np.linspace(0, 20, nsample)
X = np.column_stack((x, np.sin(x), (x-5)**2, np.ones(nsample)))
beta = [0.5, 0.5, -0.02, 5.]

y_true = np.dot(X, beta)
y = y_true + sig * np.random.normal(size=nsample)

Fit and summary:

In [7]:
res = sm.OLS(y, X).fit()
print(res.summary())
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.933
Model:                            OLS   Adj. R-squared:                  0.928
Method:                 Least Squares   F-statistic:                     211.8
Date:                Sat, 18 Feb 2017   Prob (F-statistic):           6.30e-27
Time:                        09:41:10   Log-Likelihood:                -34.438
No. Observations:                  50   AIC:                             76.88
Df Residuals:                      46   BIC:                             84.52
Df Model:                           3                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
x1             0.4687      0.026     17.751      0.000       0.416       0.522
x2             0.4836      0.104      4.659      0.000       0.275       0.693
x3            -0.0174      0.002     -7.507      0.000      -0.022      -0.013
const          5.2058      0.171     30.405      0.000       4.861       5.550
==============================================================================
Omnibus:                        0.655   Durbin-Watson:                   2.896
Prob(Omnibus):                  0.721   Jarque-Bera (JB):                0.360
Skew:                           0.207   Prob(JB):                        0.835
Kurtosis:                       3.026   Cond. No.                         221.
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Extract other quantities of interest:

In [8]:
print('Parameters: ', res.params)
print('Standard errors: ', res.bse)
print('Predicted values: ', res.predict())
Parameters:  [ 0.46872448  0.48360119 -0.01740479  5.20584496]
Standard errors:  [ 0.02640602  0.10380518  0.00231847  0.17121765]
Predicted values:  [  4.77072516   5.22213464   5.63620761   5.98658823   6.25643234
   6.44117491   6.54928009   6.60085051   6.62432454   6.6518039
   6.71377946   6.83412169   7.02615877   7.29048685   7.61487206
   7.97626054   8.34456611   8.68761335   8.97642389   9.18997755
   9.31866582   9.36587056   9.34740836   9.28893189   9.22171529
   9.17751587   9.1833565    9.25708583   9.40444579   9.61812821
   9.87897556  10.15912843  10.42660281  10.65054491  10.8063004
  10.87946503  10.86825119  10.78378163  10.64826203  10.49133265
  10.34519853  10.23933827  10.19566084  10.22490593  10.32487947
  10.48081414  10.66779556  10.85485568  11.01006072  11.10575781]

Draw a plot to compare the true relationship to OLS predictions. Confidence intervals around the predictions are built using the wls_prediction_std command.

In [9]:
prstd, iv_l, iv_u = wls_prediction_std(res)

fig, ax = plt.subplots(figsize=(8,6))

ax.plot(x, y, 'o', label="data")
ax.plot(x, y_true, 'b-', label="True")
ax.plot(x, res.fittedvalues, 'r--.', label="OLS")
ax.plot(x, iv_u, 'r--')
ax.plot(x, iv_l, 'r--')
ax.legend(loc='best');

OLS with dummy variables

We generate some artificial data. There are 3 groups which will be modelled using dummy variables. Group 0 is the omitted/benchmark category.

In [10]:
nsample = 50
groups = np.zeros(nsample, int)
groups[20:40] = 1
groups[40:] = 2
#dummy = (groups[:,None] == np.unique(groups)).astype(float)

dummy = sm.categorical(groups, drop=True)
x = np.linspace(0, 20, nsample)
# drop reference category
X = np.column_stack((x, dummy[:,1:]))
X = sm.add_constant(X, prepend=False)

beta = [1., 3, -3, 10]
y_true = np.dot(X, beta)
e = np.random.normal(size=nsample)
y = y_true + e

Inspect the data:

In [11]:
print(X[:5,:])
print(y[:5])
print(groups)
print(dummy[:5,:])
[[ 0.          0.          0.          1.        ]
 [ 0.40816327  0.          0.          1.        ]
 [ 0.81632653  0.          0.          1.        ]
 [ 1.2244898   0.          0.          1.        ]
 [ 1.63265306  0.          0.          1.        ]]
[  9.28223335  10.50481865  11.84389206  10.38508408  12.37941998]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 1 1 1 2 2 2 2 2 2 2 2 2 2]
[[ 1.  0.  0.]
 [ 1.  0.  0.]
 [ 1.  0.  0.]
 [ 1.  0.  0.]
 [ 1.  0.  0.]]

Fit and summary:

In [12]:
res2 = sm.OLS(y, X).fit()
print(res2.summary())
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.978
Model:                            OLS   Adj. R-squared:                  0.976
Method:                 Least Squares   F-statistic:                     671.7
Date:                Sat, 18 Feb 2017   Prob (F-statistic):           5.69e-38
Time:                        09:41:11   Log-Likelihood:                -64.643
No. Observations:                  50   AIC:                             137.3
Df Residuals:                      46   BIC:                             144.9
Df Model:                           3                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
x1             0.9999      0.060     16.689      0.000       0.879       1.121
x2             2.8909      0.569      5.081      0.000       1.746       4.036
x3            -3.2232      0.927     -3.477      0.001      -5.089      -1.357
const         10.1031      0.310     32.573      0.000       9.479      10.727
==============================================================================
Omnibus:                        2.831   Durbin-Watson:                   1.998
Prob(Omnibus):                  0.243   Jarque-Bera (JB):                1.927
Skew:                          -0.279   Prob(JB):                        0.382
Kurtosis:                       2.217   Cond. No.                         96.3
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Draw a plot to compare the true relationship to OLS predictions:

In [13]:
prstd, iv_l, iv_u = wls_prediction_std(res2)

fig, ax = plt.subplots(figsize=(8,6))

ax.plot(x, y, 'o', label="Data")
ax.plot(x, y_true, 'b-', label="True")
ax.plot(x, res2.fittedvalues, 'r--.', label="Predicted")
ax.plot(x, iv_u, 'r--')
ax.plot(x, iv_l, 'r--')
legend = ax.legend(loc="best")

Joint hypothesis test

F test

We want to test the hypothesis that both coefficients on the dummy variables are equal to zero, that is, $R \times \beta = 0$. An F test leads us to strongly reject the null hypothesis of identical constant in the 3 groups:

In [14]:
R = [[0, 1, 0, 0], [0, 0, 1, 0]]
print(np.array(R))
print(res2.f_test(R))
[[0 1 0 0]
 [0 0 1 0]]
<F test: F=array([[ 145.49268198]]), p=1.2834419617287716e-20, df_denom=46, df_num=2>

You can also use formula-like syntax to test hypotheses

In [15]:
print(res2.f_test("x2 = x3 = 0"))
<F test: F=array([[ 145.49268198]]), p=1.2834419617287596e-20, df_denom=46, df_num=2>

Small group effects

If we generate artificial data with smaller group effects, the T test can no longer reject the Null hypothesis:

In [16]:
beta = [1., 0.3, -0.0, 10]
y_true = np.dot(X, beta)
y = y_true + np.random.normal(size=nsample)

res3 = sm.OLS(y, X).fit()
In [17]:
print(res3.f_test(R))
<F test: F=array([[ 1.22491119]]), p=0.3031864410631761, df_denom=46, df_num=2>
In [18]:
print(res3.f_test("x2 = x3 = 0"))
<F test: F=array([[ 1.22491119]]), p=0.3031864410631761, df_denom=46, df_num=2>

Multicollinearity

The Longley dataset is well known to have high multicollinearity. That is, the exogenous predictors are highly correlated. This is problematic because it can affect the stability of our coefficient estimates as we make minor changes to model specification.

In [19]:
from statsmodels.datasets.longley import load_pandas
y = load_pandas().endog
X = load_pandas().exog
X = sm.add_constant(X)

Fit and summary:

In [20]:
ols_model = sm.OLS(y, X)
ols_results = ols_model.fit()
print(ols_results.summary())
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                 TOTEMP   R-squared:                       0.995
Model:                            OLS   Adj. R-squared:                  0.992
Method:                 Least Squares   F-statistic:                     330.3
Date:                Sat, 18 Feb 2017   Prob (F-statistic):           4.98e-10
Time:                        09:41:12   Log-Likelihood:                -109.62
No. Observations:                  16   AIC:                             233.2
Df Residuals:                       9   BIC:                             238.6
Df Model:                           6                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const      -3.482e+06    8.9e+05     -3.911      0.004    -5.5e+06   -1.47e+06
GNPDEFL       15.0619     84.915      0.177      0.863    -177.029     207.153
GNP           -0.0358      0.033     -1.070      0.313      -0.112       0.040
UNEMP         -2.0202      0.488     -4.136      0.003      -3.125      -0.915
ARMED         -1.0332      0.214     -4.822      0.001      -1.518      -0.549
POP           -0.0511      0.226     -0.226      0.826      -0.563       0.460
YEAR        1829.1515    455.478      4.016      0.003     798.788    2859.515
==============================================================================
Omnibus:                        0.749   Durbin-Watson:                   2.559
Prob(Omnibus):                  0.688   Jarque-Bera (JB):                0.684
Skew:                           0.420   Prob(JB):                        0.710
Kurtosis:                       2.434   Cond. No.                     4.86e+09
==============================================================================

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 4.86e+09. This might indicate that there are
strong multicollinearity or other numerical problems.
/private/tmp/sm/lib/python3.6/site-packages/scipy/stats/stats.py:1327: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=16
  "anyway, n=%i" % int(n))

Condition number

One way to assess multicollinearity is to compute the condition number. Values over 20 are worrisome (see Greene 4.9). The first step is to normalize the independent variables to have unit length:

In [21]:
norm_x = X.values
for i, name in enumerate(X):
    if name == "const":
        continue
    norm_x[:,i] = X[name]/np.linalg.norm(X[name])
norm_xtx = np.dot(norm_x.T,norm_x)

Then, we take the square root of the ratio of the biggest to the smallest eigen values.

In [22]:
eigs = np.linalg.eigvals(norm_xtx)
condition_number = np.sqrt(eigs.max() / eigs.min())
print(condition_number)
56240.8714071

Dropping an observation

Greene also points out that dropping a single observation can have a dramatic effect on the coefficient estimates:

In [23]:
ols_results2 = sm.OLS(y.ix[:14], X.ix[:14]).fit()
print("Percentage change %4.2f%%\n"*7 % tuple([i for i in (ols_results2.params - ols_results.params)/ols_results.params*100]))
Percentage change -13.35%
Percentage change -55778.29%
Percentage change 121935204.58%
Percentage change 1282863.00%
Percentage change 998687.27%
Percentage change -47263797.15%
Percentage change 677415.26%

We can also look at formal statistics for this such as the DFBETAS -- a standardized measure of how much each coefficient changes when that observation is left out.

In [24]:
infl = ols_results.get_influence()

In general we may consider DBETAS in absolute value greater than $2/\sqrt{N}$ to be influential observations

In [25]:
2./len(X)**.5
Out[25]:
0.5
In [26]:
print(infl.summary_frame().filter(regex="dfb"))
    dfb_const  dfb_GNPDEFL       dfb_GNP     dfb_UNEMP     dfb_ARMED  \
0   -0.016406  -169.822675  1.673981e+06  54490.318088  51447.824036   
1   -0.020608  -187.251727  1.829990e+06  54495.312977  52659.808664   
2   -0.008382   -65.417834  1.587601e+06  52002.330476  49078.352379   
3    0.018093   288.503914  1.155359e+06  56211.331922  60350.723082   
4    1.871260  -171.109595  4.498197e+06  82532.785818  71034.429294   
..        ...          ...           ...           ...           ...   
11   0.005987  -100.926843  1.662425e+06  51744.606934  48968.560299   
12  -0.135883   -32.093127  1.245487e+06  50203.467593  51148.376274   
13   0.032736   -78.513866  1.648417e+06  52509.194459  50212.844641   
14   0.305868   -16.833121  1.829996e+06  60975.868083  58263.878679   
15  -0.538323   102.027105  1.344844e+06  54721.897640  49660.474568   

          dfb_POP      dfb_YEAR  
0   207954.113588 -31969.158503  
1    25343.938289 -29760.155888  
2   107465.770565 -29593.195253  
3   456190.215133 -36213.129569  
4  -389122.401699 -49905.782854  
..            ...           ...  
11   95414.217290 -29690.904188  
12  258559.048569 -29296.334617  
13  104434.061226 -30025.564763  
14  275103.677859 -36060.612522  
15 -110176.960671 -28053.834556  

[16 rows x 7 columns]
/private/tmp/sm/lib/python3.6/site-packages/statsmodels/statsmodels/stats/outliers_influence.py:309: RuntimeWarning: invalid value encountered in sqrt
  return  self.results.resid / sigma / np.sqrt(1 - hii)
/private/tmp/sm/lib/python3.6/site-packages/scipy/stats/_distn_infrastructure.py:875: RuntimeWarning: invalid value encountered in greater
  return (self.a < x) & (x < self.b)
/private/tmp/sm/lib/python3.6/site-packages/scipy/stats/_distn_infrastructure.py:875: RuntimeWarning: invalid value encountered in less
  return (self.a < x) & (x < self.b)
/private/tmp/sm/lib/python3.6/site-packages/scipy/stats/_distn_infrastructure.py:1814: RuntimeWarning: invalid value encountered in less_equal
  cond2 = cond0 & (x <= self.a)
/private/tmp/sm/lib/python3.6/site-packages/statsmodels/statsmodels/stats/outliers_influence.py:323: RuntimeWarning: invalid value encountered in sqrt
  dffits_ = self.resid_studentized_internal * np.sqrt(hii / (1 - hii))
/private/tmp/sm/lib/python3.6/site-packages/statsmodels/statsmodels/stats/outliers_influence.py:352: RuntimeWarning: invalid value encountered in sqrt
  dffits_ = self.resid_studentized_external * np.sqrt(hii / (1 - hii))