code: def test_formulas(noise):
      np.random.seed(8789)
  
      f, df = run_formula(1000, model1, noise)
      mod = f.model
  
      f.summary()  # Smoke test
  
      # Compare the parameter estimates to population values.
      epar = np.concatenate(model1(noise))
      assert_allclose(f.params, epar, atol=0.1, rtol=1)
  
      # Test the fitted covariance matrix
      exog_scale = pd.DataFrame(
          mod.exog_scale[0:5, :], columns=["xsc1", "xsc2"]
      )
      exog_smooth = pd.DataFrame(
          mod.exog_smooth[0:5, :], columns=["xsm1", "xsm2"]
      )
      cv = f.covariance(mod.time[0:5], exog_scale, exog_smooth)
      assert_allclose(cv, cv.T)
      a, _ = np.linalg.eig(cv)
      assert_equal(a > 0, True)
  
      # Test predict
      yhat = f.predict()
      assert_equal(np.corrcoef(yhat, mod.endog)[0, 1] > 0.2, True)
      yhatm = f.predict(exog=df)
      assert_equal(yhat, yhatm)
      yhat0 = mod.predict(params=f.params, exog=df)
      assert_equal(yhat, yhat0)
  
      # Smoke test t-test
      f.t_test(np.eye(len(f.params)))
  
file_path: statsmodels/statsmodels/regression/tests/test_processreg.py
imports: from statsmodels.compat.platform import PLATFORM_OSX
  from statsmodels.regression.process_regression import ProcessMLE, GaussianCovariance
  import numpy
  import pandas
  import pytest
  import collections
  import statsmodels.tools.numdiff
  from numpy.testing import assert_allclose, assert_equal
label_desc: Use image node_id 6 for calling a global function with example usage: test_formulas(noise) without return types