networkinference.OLS

class networkinference.OLS(Y, X, A=None)

Bases: object

OLS estimator.

Parameters

Ynumpy float array

n-dimensional array of outcomes.

Xnumpy float array

n x k array of regressors (not including intercept) or n-dimensional array.

ANetworkX graph

Graph on n nodes. NOTE: Assumes nodes are labeled as integers 0, …, n-1 in A, so that the outcome of node i is given by the ith component of Y. Network can be weighted or directed, although weights and directions are ignored when computing network SEs. Argument not used for dependence robust test or CI. Default value: None.

Examples

>>> import networkinference as ni
>>> from networkinference.utils import FakeData
>>> Y, X, A = FakeData.ols()
>>> ols_model = ni.OLS(Y, X, A)
>>> print(ols_model.estimate)

Attributes

datadictionary

Stores all input data, adding a column of ones to X.

summandsnumpy array

n-dimensional array of intermediate products used to compute OLS estimator.

estimatefloat

OLS estimator.

residnumpy array

Regression residuals.

invhessiannumpy array

Inverse hessian matrix.

scoresnumpy array

Regression scores.

Methods

arand_ci(grid_start, grid_stop[, dimension, ...])	Returns confidence interval (CI) obtained by inverting an approximate randomization test [1].
arand_test(mu[, dimension, num_clusters, ...])	Returns p-value of approximate randomization test [1].
cluster_se([num_clusters, decimals, verbose])	Returns clustered standard errors.
drobust_ci(grid_start, grid_stop[, ...])	Returns confidence interval (CI) derived from the dependence-robust test due to [1].
drobust_test(mu[, dimension, alpha, beta, ...])	Returns conclusion of dependence-robust test due to [1].
est_by_cluster(dimension)	Returns array of OLS estimates, one for each cluster.
get_clusters(num_clusters[, clusters, seed, ...])	Returns network clusters obtained from normalized spectral clustering algorithm due to [2] (also see [3]).
network_se([b, decimals, verbose, PD_alert])	Returns standard errors derived from network HAC variance estimator due to [1] using bandwidth suggested by [2].
trobust_ci([dimension, num_clusters, ...])	Returns confidence interval (CI) from the t-statistic based cluster-robust procedure due to [1].

arand_ci(grid_start, grid_stop, dimension=None, grid_size=151, coverage=0.95, num_clusters=5, decimals=3, seed=None, verbose=True)

Returns confidence interval (CI) obtained by inverting an approximate randomization test [1]. If the result is a trivial interval, try increasing grid_size. The CI is narrower with more clusters. However, since the test computes estimates cluster by cluster, the output can be more unstable with a larger number of clusters since the sample size within each cluster can be small.

Parameters

grid_startfloat

Need to specify a grid of values over which to invert the test. This is the leftmost point of the grid.

grid_stopfloat

Rightmost point of the grid.

dimensionint

Dimension of the estimand for which you want the CI. To generate a table of CIs for all dimensions, set dimension=None. Ignored if estimand is scalar. Default value: None.

grid_sizeint

Number of points in the grid. Default value: 151.

coveragefloat

Desired coverage. Default value: 0.95.

num_clustersint

Ignored if get_clusters() was already run on this object. If it wasn’t, this calls the get_cluster() method, asking for this many clusters. Default value: 5.

decimalsint

Number of decimals to which to round the output table.

seedint

Seed for drawing permutations, which is only relevant when there are more than 12 clusters. Set to None to not set a seed. Default value: None.

verboseboolean

If True, calling this method prints out the results. Default value: True.

References

1(1,2)

Canay, I., J. Romano, and A. Shaikh, “Randomization Tests Under an Approximate Symmetry Assumption,” Econometrica, 2017, 85 (3), 1013-1030.

Examples

>>> import networkinference as ni
>>> from networkinference.utils import FakeData
>>> Y, X, A = FakeData.ols(network='RGG')
>>> ols_model = ni.OLS(Y, X, A)
>>> ols.get_clusters(10)
>>> ols.arand_ci(-5, 5)

Attributes

arand_ci_resultlist

Confidence interval.

arand_test(mu, dimension=0, num_clusters=5, seed=None, verbose=True)

Returns p-value of approximate randomization test [1]. The test is more powerful with more clusters. However, since the test computes estimates cluster by cluster, the output can be more unstable with a larger number of clusters since the sample size within each cluster can be small.

Parameters

dimensionint

Dimension of estimand being tested. Ignored if estimand is scalar. Default value: 0.

mufloat

Null value of the estimand in the specified dimension.

num_clustersint

Ignored if get_clusters() was already run on this object. If it wasn’t, this calls the get_cluster() method, asking for this many clusters. Default value: 5.

seedint

Seed for drawing permutations, which is only relevant when there are more than 12 clusters. Set to None to not set a seed. Default value: None.

verboseboolean

If True, calling this method prints out the results. Default value: True.

References

1(1,2)

Canay, I., J. Romano, and A. Shaikh, “Randomization Tests Under an Approximate Symmetry Assumption,” Econometrica, 2017, 85 (3), 1013-1030.

Examples

>>> import networkinference as ni
>>> from networkinference.utils import FakeData
>>> Y, X, A = FakeData.ols(network='RGG')
>>> ols_model = ni.OLS(Y, X, A)
>>> ols.get_clusters(10)
>>> ols.arand_test(1, dimension=1)

Attributes

arand_test_resultfloat

P-value.

arand_test_statfloat

Test statistic.

cluster_se(num_clusters=30, decimals=3, verbose=True)

Returns clustered standard errors.

Parameters

num_clustersint

Ignored if get_clusters() was already run on this object. If it wasn’t, this calls the get_cluster() method, asking for this many clusters. Default value: 30.

decimalsint

Number of decimals to which to round the output table.

verboseboolean

If True, calling this method prints out the results. Default value: True.

Examples

>>> import networkinference as ni
>>> from networkinference.utils import FakeData
>>> Y, X, A = FakeData.ols(network='RGG')
>>> ols_model = ni.OLS(Y, X, A)
>>> ols.get_clusters(30)
>>> ols.cluster_se()

Attributes

cluster_se_vcovfloat

Cluster-robust variance estimate.

cluster_se_resultfloat

Clustered standard errors.

drobust_ci(grid_start, grid_stop, dimension=None, grid_size=151, coverage=0.95, beta=0.01, R=None, L=1000, seed=None, decimals=3, verbose=True)

Returns confidence interval (CI) derived from the dependence-robust test due to [1]. Note that the output of the test is random by nature. L is the number of simulation draws, and larger values reduce the random variation of the test. If the result is a trivial interval, try increasing grid_size.

Test is implemented using the U-type statistic and randomized confidence function approach due to [2] discussed in Remark 2 of [1].

Parameters

grid_startfloat

Need to specify a grid of values to test for inclusion in the CI. This is the leftmost point of the grid.

grid_stopfloat

Rightmost point of the grid.

dimensionint

Dimension of the estimand for which you want the CI. Ignored if estimand is scalar. To generate a table of CIs for all dimensions, set dimension=None. Default value: None.

grid_sizeint

Number of points in the grid. Default value: 151.

coveragefloat

Desired coverage. Default value: 0.95.

betafloat

beta in Remark 2 of Leung (2021). The closer this is to 1-coverage, the more conservative the CI. Default value: 0.01.

Rint

Number of resampling draws for test statistic. Uses default if R=None. Default value: None.

Lint

Number of resampling draws for randomized confidence function. The larger the value, the less random the output. Default value: 1000.

seedint

seed for resampling draws. Set to None to not set a seed. Default value: None.

decimalsint

Number of decimals to which to round the output table.

verboseboolean

If True, calling this method prints out the results. Default value: True.

References

1(1,2,3)

Leung, M. “Dependence-Robust Inference Using Resampled Statistics,” Journal of Applied Econometrics (forthcoming), 2021.

2

Song, K. “Ordering-Free Inference from Locally Dependent Data,” UBC working paper, 2016.

Examples

>>> import networkinference as ni
>>> from networkinference.utils import FakeData
>>> Y, X, A = FakeData.ols()
>>> ols_model = ni.OLS(Y, X, A)
>>> ols.drobust_ci(-5, 5)

Attributes

drobust_ci_resultlist

Confidence interval.

drobust_test(mu, dimension=0, alpha=0.05, beta=0.01, R=None, L=1000, seed=None, verbose=True)

Returns conclusion of dependence-robust test due to [1]. Note that the output of the test is random by nature. L is the number of simulation draws, and larger values reduce the random variation of the test.

Test is implemented using the U-type statistic and randomized confidence function approach due to [2] discussed in Remark 2 of [1].

Parameters

mufloat

Null value of the estimand in the specified dimension.

dimensionint

Dimension of the estimand being tested. Ignored if estimand is scalar. Default value: 0.

alphafloat

Significance level. Default value: 0.05.

betafloat

beta in Remark 2 of Leung (2021). The closer this is to alpha, the more conservative the test. Default value: 0.01.

Rint

Number of resampling draws for test statistic. Uses default if R=None. Default value: None.

Lint

Number of resampling draws for randomized confidence function. The larger the value, the less random the output. Default value: 1000.

seedint

seed for resampling draws. Set to None to not set a seed. Default value: None.

verboseboolean

If True, calling this method prints out the results. Default value: True.

References

1(1,2,3)

Leung, M. “Dependence-Robust Inference Using Resampled Statistics,” Journal of Applied Econometrics (forthcoming), 2021.

2

Song, K. “Ordering-Free Inference from Locally Dependent Data,” UBC working paper, 2016.

Examples

>>> import networkinference as ni
>>> from networkinference.utils import FakeData
>>> Y, X, A = FakeData.ols()
>>> ols_model = ni.OLS(Y, X, A)
>>> ols.drobust_test(1, dimension=1)

Attributes

drobust_test_resultstring

Reject or not reject.

est_by_cluster(dimension)

Returns array of OLS estimates, one for each cluster. This is a helper method used by arand_test() and arand_ci().

Parameters

dimensionint

Dimension of estimand being tested. Ignore if estimand is scalar. Default value: 0.

Returns

thetahatnumpy array

L-dimensional array of OLS estimates, one for each of the L clusters.

get_clusters(num_clusters, clusters=None, seed=None, weight=None, verbose=True)

Returns network clusters obtained from normalized spectral clustering algorithm due to [2] (also see [3]). Returns maximal conductance of clusters, a [0,1]-measure of cluster quality that should be at most 0.1 for cluster-robust methods to have good performance (see [1]). All nodes not in the giant component are grouped into a single cluster.

Parameters

num_clustersint

Number of desired clusters in the giant component.

seedint

Seed for k-means clustering initialization. Set to None to not set a seed. Default value: None.

clustersnumpy array

Optional array of cluster memberships obtained from the output of this method or spectral_clustering() in the core class. The only purpose of this argument is to load clusters obtained elsewhere into the current object.

weightstring

Specifies how edge weights are labeled in A, if A is a weighted graph. Default value: None.

verboseboolean

When set to True, the method prints the maximal conductance of the clusters. Default value: True.

References

1

Leung, M., “Network Cluster-Robust Inference,” arXiv preprint arXiv:2103.01470, 2021.

2(1,2)

Ng, A., M. Jordan, Y. Weiss, “On Spectral Clustering: Analysis and an Algorithm.” Advances in Neural Information Processing Systems, 2002, 849-856.

3(1,2)

von Luxburg, U., “A Tutorial on Spectral Clustering,” Statistics and Computing, 2007, 17 (4), 395-416.

Examples

>>> import networkinference as ni
>>> from networkinference.utils import FakeData
>>> Y, X, A = FakeData.ols(network='RGG')
>>> ols_model = ni.OLS(Y, X, A)
>>> ols.get_clusters(10)

Attributes

clustersnumpy array

n-dimensional array of cluster labels from 0 to num_clusters-1, where n is the number of nodes.

conductancefloat

Maximal conductance of the clusters.

network_se(b=None, decimals=3, verbose=True, PD_alert=False)

Returns standard errors derived from network HAC variance estimator due to [1] using bandwidth suggested by [2]. Setting b=0 outputs the conventional heteroskedasticity-robust variance estimator for i.i.d. data. Network is converted to an unweighted, undirected version by dropping edge weights and directionality of links.

The default output uses a uniform kernel. If the result is not positive definite, the output is an estimator guaranteed to be positive definite due to the first working paper version of [2].

Parameters

bfloat

HAC bandwidth. Recommend keeping b=None, which uses the bandwidth choice recommended by [2]. Default value: None.

decimalsint

Number of decimals to which to round the output table.

verboseboolean

If True, calling this method prints out the results. Default value: True.

PD_alertboolean

If True, method will print an alert whenever the default estimator is not positive definite.

References

1(1,2)

Kojevnikov, D., V. Marmer, and K. Song, “Limit Theorems for Network Dependent Random Variables,” Journal of Econometrics, 2021, 222 (2), 882-908.

2(1,2,3,4)

Leung, M. “Causal Inference Under Approximate Neighborhood Interference,” Econometrica (forthcoming), 2021.

Examples

>>> import networkinference as ni
>>> from networkinference.utils import FakeData
>>> Y, X, A = FakeData.ols()
>>> ols_model = ni.OLS(Y, X, A)
>>> ols.network_se()

Attributes

network_se_vcovfloat

Estimate of variance-covariance matrix.

network_se_resultfloat

Standard errors.

trobust_ci(dimension=None, num_clusters=5, coverage=0.95, decimals=3, verbose=True)

Returns confidence interval (CI) from the t-statistic based cluster-robust procedure due to [1]. The more clusters, the more powerful the test. However, since the test computes estimates cluster by cluster, the output can be more unstable with a larger number of clusters since the sample size within each cluster can be small.

Parameters

dimensionint

Dimension of the estimand for which you want the CI. Ignored if estimand is scalar. To generate a table of CIs for all dimensions, set dimension=None. Default value: None.

num_clustersint

Ignored if get_clusters() was already run on this object. If it wasn’t, this calls the get_cluster() method, asking for this many clusters. Default value: 5.

coveragefloat

Desired coverage. Default value: 0.95.

decimalsint

Number of decimals to which to round the output table.

verboseboolean

If True, calling this method prints out the results. Default value: True.

References

1(1,2)

Ibragimov, R. and U. Mueller, “t-Statistic Based Correlation and Heterogeneity Robust Inference,” Journal of Business and Economic Statistics, 2010, 28 (4), 453-468.

Examples

>>> import networkinference as ni
>>> from networkinference.utils import FakeData
>>> Y, X, A = FakeData.ols(network='RGG')
>>> ols_model = ni.OLS(Y, X, A)
>>> ols.get_clusters(10)
>>> ols.trobust_ci()

Attributes

trobust_ci_resultlist

Confidence interval.