def test_ols(): # More comprehensive tests against OLS estimates mod = RecursiveLS(endog, dta['m1']) res = mod.fit() mod_ols = OLS(endog, dta['m1']) res_ols = mod_ols.fit() # Regression coefficients, standard errors, and estimated scale assert_allclose(res.params, res_ols.params) assert_allclose(res.bse, res_ols.bse) # Note: scale here is computed according to Harvey, 1989, 4.2.5, and is # the called the ML estimator and sometimes (e.g. later in section 5) # denoted \tilde \sigma_*^2 assert_allclose(res.filter_results.obs_cov[0, 0], res_ols.scale) # OLS residuals are equivalent to smoothed forecast errors # (the latter are defined as e_t|T by Harvey, 1989, 5.4.5) # (this follows since the smoothed state simply contains the # full-information estimates of the regression coefficients) actual = (mod.endog[:, 0] - np.sum(mod['design', 0, :, :] * res.smoothed_state, axis=0)) assert_allclose(actual, res_ols.resid) # Given the estimate of scale as `sum(v_t^2 / f_t) / (T - d)` (see # Harvey, 1989, 4.2.5 on p. 183), then llf_recursive is equivalent to the # full OLS loglikelihood (i.e. without the scale concentrated out). desired = mod_ols.loglike(res_ols.params, scale=res_ols.scale) assert_allclose(res.llf_recursive, desired) # Alternatively, we can constrcut the concentrated OLS loglikelihood # by computing the scale term with `nobs` in the denominator rather than # `nobs - d`. scale_alternative = np.sum( (res.standardized_forecasts_error[0, 1:] * res.filter_results.obs_cov[0, 0]**0.5)**2) / mod.nobs llf_alternative = np.log( norm.pdf(res.resid_recursive, loc=0, scale=scale_alternative**0.5)).sum() assert_allclose(llf_alternative, res_ols.llf) # Prediction actual = res.forecast(10, design=np.ones((1, 1, 10))) assert_allclose(actual, res_ols.predict(np.ones((10, 1)))) # Sums of squares, R^2 assert_allclose(res.ess, res_ols.ess) assert_allclose(res.ssr, res_ols.ssr) assert_allclose(res.centered_tss, res_ols.centered_tss) assert_allclose(res.uncentered_tss, res_ols.uncentered_tss) assert_allclose(res.rsquared, res_ols.rsquared) # Mean squares assert_allclose(res.mse_model, res_ols.mse_model) assert_allclose(res.mse_resid, res_ols.mse_resid) assert_allclose(res.mse_total, res_ols.mse_total) # Hypothesis tests actual = res.t_test('m1 = 0') desired = res_ols.t_test('m1 = 0') assert_allclose(actual.statistic, desired.statistic) assert_allclose(actual.pvalue, desired.pvalue, atol=1e-15) actual = res.f_test('m1 = 0') desired = res_ols.f_test('m1 = 0') assert_allclose(actual.statistic, desired.statistic) assert_allclose(actual.pvalue, desired.pvalue, atol=1e-15) # Information criteria # Note: the llf and llf_obs given in the results are based on the Kalman # filter and so the ic given in results will not be identical to the # OLS versions. Additionally, llf_recursive is comparable to the # non-concentrated llf, and not the concentrated llf that is by default # used in OLS. Compute new ic based on llf_alternative to compare. actual_aic = aic(llf_alternative, res.nobs_effective, res.df_model) assert_allclose(actual_aic, res_ols.aic) actual_bic = bic(llf_alternative, res.nobs_effective, res.df_model) assert_allclose(actual_bic, res_ols.bic)
def test_ols(): # More comprehensive tests against OLS estimates mod = RecursiveLS(endog, dta['m1']) res = mod.fit() mod_ols = OLS(endog, dta['m1']) res_ols = mod_ols.fit() # Regression coefficients, standard errors, and estimated scale assert_allclose(res.params, res_ols.params) assert_allclose(res.bse, res_ols.bse) # Note: scale here is computed according to Harvey, 1989, 4.2.5, and is # the called the ML estimator and sometimes (e.g. later in section 5) # denoted \tilde \sigma_*^2 assert_allclose(res.filter_results.obs_cov[0, 0], res_ols.scale) # OLS residuals are equivalent to smoothed forecast errors # (the latter are defined as e_t|T by Harvey, 1989, 5.4.5) # (this follows since the smoothed state simply contains the # full-information estimates of the regression coefficients) actual = (mod.endog[:, 0] - np.sum(mod['design', 0, :, :] * res.smoothed_state, axis=0)) assert_allclose(actual, res_ols.resid) # Given the estimate of scale as `sum(v_t^2 / f_t) / (T - d)` (see # Harvey, 1989, 4.2.5 on p. 183), then llf_recursive is equivalent to the # full OLS loglikelihood (i.e. without the scale concentrated out). desired = mod_ols.loglike(res_ols.params, scale=res_ols.scale) assert_allclose(res.llf_recursive, desired) # Alternatively, we can constrcut the concentrated OLS loglikelihood # by computing the scale term with `nobs` in the denominator rather than # `nobs - d`. scale_alternative = np.sum(( res.standardized_forecasts_error[0, 1:] * res.filter_results.obs_cov[0, 0]**0.5)**2) / mod.nobs llf_alternative = np.log(norm.pdf(res.resid_recursive, loc=0, scale=scale_alternative**0.5)).sum() assert_allclose(llf_alternative, res_ols.llf) # Prediction actual = res.forecast(10, design=np.ones((1, 1, 10))) assert_allclose(actual, res_ols.predict(np.ones((10, 1)))) # Sums of squares, R^2 assert_allclose(res.ess, res_ols.ess) assert_allclose(res.ssr, res_ols.ssr) assert_allclose(res.centered_tss, res_ols.centered_tss) assert_allclose(res.uncentered_tss, res_ols.uncentered_tss) assert_allclose(res.rsquared, res_ols.rsquared) # Mean squares assert_allclose(res.mse_model, res_ols.mse_model) assert_allclose(res.mse_resid, res_ols.mse_resid) assert_allclose(res.mse_total, res_ols.mse_total) # Hypothesis tests actual = res.t_test('m1 = 0') desired = res_ols.t_test('m1 = 0') assert_allclose(actual.statistic, desired.statistic) assert_allclose(actual.pvalue, desired.pvalue, atol=1e-15) actual = res.f_test('m1 = 0') desired = res_ols.f_test('m1 = 0') assert_allclose(actual.statistic, desired.statistic) assert_allclose(actual.pvalue, desired.pvalue, atol=1e-15) # Information criteria # Note: the llf and llf_obs given in the results are based on the Kalman # filter and so the ic given in results will not be identical to the # OLS versions. Additionally, llf_recursive is comparable to the # non-concentrated llf, and not the concentrated llf that is by default # used in OLS. Compute new ic based on llf_alternative to compare. actual_aic = aic(llf_alternative, res.nobs_effective, res.df_model) assert_allclose(actual_aic, res_ols.aic) actual_bic = bic(llf_alternative, res.nobs_effective, res.df_model) assert_allclose(actual_bic, res_ols.bic)