"""Tests of GLSAR and diagnostics against Gretl Created on Thu Feb 02 21:15:47 2012 Author: Josef Perktold License: BSD-3 """ import os import numpy as np from numpy.testing import (assert_almost_equal, assert_equal, assert_allclose, assert_array_less) from statsmodels.regression.linear_model import OLS, GLSAR from statsmodels.tools.tools import add_constant from statsmodels.datasets import macrodata import statsmodels.stats.sandwich_covariance as sw import statsmodels.stats.diagnostic as smsdia import statsmodels.stats.outliers_influence as oi def compare_ftest(contrast_res, other, decimal=(5,4)): assert_almost_equal(contrast_res.fvalue, other[0], decimal=decimal[0]) assert_almost_equal(contrast_res.pvalue, other[1], decimal=decimal[1]) assert_equal(contrast_res.df_num, other[2]) assert_equal(contrast_res.df_denom, other[3]) assert_equal("f", other[4]) class TestGLSARGretl: def test_all(self): d = macrodata.load_pandas().data #import datasetswsm.greene as g #d = g.load('5-1') #growth rates gs_l_realinv = 400 * np.diff(np.log(d['realinv'].values)) gs_l_realgdp = 400 * np.diff(np.log(d['realgdp'].values)) #simple diff, not growthrate, I want heteroscedasticity later for testing endogd = np.diff(d['realinv']) exogd = add_constant(np.c_[np.diff(d['realgdp'].values), d['realint'][:-1].values]) endogg = gs_l_realinv exogg = add_constant(np.c_[gs_l_realgdp, d['realint'][:-1].values]) res_ols = OLS(endogg, exogg).fit() #print res_ols.params mod_g1 = GLSAR(endogg, exogg, rho=-0.108136) res_g1 = mod_g1.fit() #print res_g1.params mod_g2 = GLSAR(endogg, exogg, rho=-0.108136) #-0.1335859) from R res_g2 = mod_g2.iterative_fit(maxiter=5) #print res_g2.params rho = -0.108136 # coefficient std. error t-ratio p-value 95% CONFIDENCE INTERVAL partable = np.array([ [-9.50990, 0.990456, -9.602, 3.65e-018, -11.4631, -7.55670], # *** [ 4.37040, 0.208146, 21.00, 2.93e-052, 3.95993, 4.78086], # *** [-0.579253, 0.268009, -2.161, 0.0319, -1.10777, -0.0507346]]) # ** #Statistics based on the rho-differenced data: result_gretl_g1 = dict( endog_mean = ("Mean dependent var", 3.113973), endog_std = ("S.D. dependent var", 18.67447), ssr = ("Sum squared resid", 22530.90), mse_resid_sqrt = ("S.E. of regression", 10.66735), rsquared = ("R-squared", 0.676973), rsquared_adj = ("Adjusted R-squared", 0.673710), fvalue = ("F(2, 198)", 221.0475), f_pvalue = ("P-value(F)", 3.56e-51), resid_acf1 = ("rho", -0.003481), dw = ("Durbin-Watson", 1.993858)) #fstatistic, p-value, df1, df2 reset_2_3 = [5.219019, 0.00619, 2, 197, "f"] reset_2 = [7.268492, 0.00762, 1, 198, "f"] reset_3 = [5.248951, 0.023, 1, 198, "f"] #LM-statistic, p-value, df arch_4 = [7.30776, 0.120491, 4, "chi2"] #multicollinearity vif = [1.002, 1.002] cond_1norm = 6862.0664 determinant = 1.0296049e+009 reciprocal_condition_number = 0.013819244 #Chi-square(2): test-statistic, pvalue, df normality = [20.2792, 3.94837e-005, 2] #tests res = res_g1 #with rho from Gretl #basic assert_almost_equal(res.params, partable[:,0], 4) assert_almost_equal(res.bse, partable[:,1], 6) assert_almost_equal(res.tvalues, partable[:,2], 2) assert_almost_equal(res.ssr, result_gretl_g1['ssr'][1], decimal=2) #assert_almost_equal(res.llf, result_gretl_g1['llf'][1], decimal=7) #not in gretl #assert_almost_equal(res.rsquared, result_gretl_g1['rsquared'][1], decimal=7) #FAIL #assert_almost_equal(res.rsquared_adj, result_gretl_g1['rsquared_adj'][1], decimal=7) #FAIL assert_almost_equal(np.sqrt(res.mse_resid), result_gretl_g1['mse_resid_sqrt'][1], decimal=5) assert_almost_equal(res.fvalue, result_gretl_g1['fvalue'][1], decimal=4) assert_allclose(res.f_pvalue, result_gretl_g1['f_pvalue'][1], rtol=1e-2) #assert_almost_equal(res.durbin_watson, result_gretl_g1['dw'][1], decimal=7) #TODO #arch #sm_arch = smsdia.acorr_lm(res.wresid**2, maxlag=4, autolag=None) sm_arch = smsdia.het_arch(res.wresid, nlags=4) assert_almost_equal(sm_arch[0], arch_4[0], decimal=4) assert_almost_equal(sm_arch[1], arch_4[1], decimal=6) #tests res = res_g2 #with estimated rho #estimated lag coefficient assert_almost_equal(res.model.rho, rho, decimal=3) #basic assert_almost_equal(res.params, partable[:,0], 4) assert_almost_equal(res.bse, partable[:,1], 3) assert_almost_equal(res.tvalues, partable[:,2], 2) assert_almost_equal(res.ssr, result_gretl_g1['ssr'][1], decimal=2) #assert_almost_equal(res.llf, result_gretl_g1['llf'][1], decimal=7) #not in gretl #assert_almost_equal(res.rsquared, result_gretl_g1['rsquared'][1], decimal=7) #FAIL #assert_almost_equal(res.rsquared_adj, result_gretl_g1['rsquared_adj'][1], decimal=7) #FAIL assert_almost_equal(np.sqrt(res.mse_resid), result_gretl_g1['mse_resid_sqrt'][1], decimal=5) assert_almost_equal(res.fvalue, result_gretl_g1['fvalue'][1], decimal=0) assert_almost_equal(res.f_pvalue, result_gretl_g1['f_pvalue'][1], decimal=6) #assert_almost_equal(res.durbin_watson, result_gretl_g1['dw'][1], decimal=7) #TODO c = oi.reset_ramsey(res, degree=2) compare_ftest(c, reset_2, decimal=(2,4)) c = oi.reset_ramsey(res, degree=3) compare_ftest(c, reset_2_3, decimal=(2,4)) #arch #sm_arch = smsdia.acorr_lm(res.wresid**2, maxlag=4, autolag=None) sm_arch = smsdia.het_arch(res.wresid, nlags=4) assert_almost_equal(sm_arch[0], arch_4[0], decimal=1) assert_almost_equal(sm_arch[1], arch_4[1], decimal=2) ''' Performing iterative calculation of rho... ITER RHO ESS 1 -0.10734 22530.9 2 -0.10814 22530.9 Model 4: Cochrane-Orcutt, using observations 1959:3-2009:3 (T = 201) Dependent variable: ds_l_realinv rho = -0.108136 coefficient std. error t-ratio p-value ------------------------------------------------------------- const -9.50990 0.990456 -9.602 3.65e-018 *** ds_l_realgdp 4.37040 0.208146 21.00 2.93e-052 *** realint_1 -0.579253 0.268009 -2.161 0.0319 ** Statistics based on the rho-differenced data: Mean dependent var 3.113973 S.D. dependent var 18.67447 Sum squared resid 22530.90 S.E. of regression 10.66735 R-squared 0.676973 Adjusted R-squared 0.673710 F(2, 198) 221.0475 P-value(F) 3.56e-51 rho -0.003481 Durbin-Watson 1.993858 ''' ''' RESET test for specification (squares and cubes) Test statistic: F = 5.219019, with p-value = P(F(2,197) > 5.21902) = 0.00619 RESET test for specification (squares only) Test statistic: F = 7.268492, with p-value = P(F(1,198) > 7.26849) = 0.00762 RESET test for specification (cubes only) Test statistic: F = 5.248951, with p-value = P(F(1,198) > 5.24895) = 0.023: ''' ''' Test for ARCH of order 4 coefficient std. error t-ratio p-value -------------------------------------------------------- alpha(0) 97.0386 20.3234 4.775 3.56e-06 *** alpha(1) 0.176114 0.0714698 2.464 0.0146 ** alpha(2) -0.0488339 0.0724981 -0.6736 0.5014 alpha(3) -0.0705413 0.0737058 -0.9571 0.3397 alpha(4) 0.0384531 0.0725763 0.5298 0.5968 Null hypothesis: no ARCH effect is present Test statistic: LM = 7.30776 with p-value = P(Chi-square(4) > 7.30776) = 0.120491: ''' ''' Variance Inflation Factors Minimum possible value = 1.0 Values > 10.0 may indicate a collinearity problem ds_l_realgdp 1.002 realint_1 1.002 VIF(j) = 1/(1 - R(j)^2), where R(j) is the multiple correlation coefficient between variable j and the other independent variables Properties of matrix X'X: 1-norm = 6862.0664 Determinant = 1.0296049e+009 Reciprocal condition number = 0.013819244 ''' ''' Test for ARCH of order 4 - Null hypothesis: no ARCH effect is present Test statistic: LM = 7.30776 with p-value = P(Chi-square(4) > 7.30776) = 0.120491 Test of common factor restriction - Null hypothesis: restriction is acceptable Test statistic: F(2, 195) = 0.426391 with p-value = P(F(2, 195) > 0.426391) = 0.653468 Test for normality of residual - Null hypothesis: error is normally distributed Test statistic: Chi-square(2) = 20.2792 with p-value = 3.94837e-005: ''' #no idea what this is ''' Augmented regression for common factor test OLS, using observations 1959:3-2009:3 (T = 201) Dependent variable: ds_l_realinv coefficient std. error t-ratio p-value --------------------------------------------------------------- const -10.9481 1.35807 -8.062 7.44e-014 *** ds_l_realgdp 4.28893 0.229459 18.69 2.40e-045 *** realint_1 -0.662644 0.334872 -1.979 0.0492 ** ds_l_realinv_1 -0.108892 0.0715042 -1.523 0.1294 ds_l_realgdp_1 0.660443 0.390372 1.692 0.0923 * realint_2 0.0769695 0.341527 0.2254 0.8219 Sum of squared residuals = 22432.8 Test of common factor restriction Test statistic: F(2, 195) = 0.426391, with p-value = 0.653468 ''' ################ with OLS, HAC errors #Model 5: OLS, using observations 1959:2-2009:3 (T = 202) #Dependent variable: ds_l_realinv #HAC standard errors, bandwidth 4 (Bartlett kernel) #coefficient std. error t-ratio p-value 95% CONFIDENCE INTERVAL #for confidence interval t(199, 0.025) = 1.972 partable = np.array([ [-9.48167, 1.17709, -8.055, 7.17e-014, -11.8029, -7.16049], # *** [4.37422, 0.328787, 13.30, 2.62e-029, 3.72587, 5.02258], #*** [-0.613997, 0.293619, -2.091, 0.0378, -1.19300, -0.0349939]]) # ** result_gretl_g1 = dict( endog_mean = ("Mean dependent var", 3.257395), endog_std = ("S.D. dependent var", 18.73915), ssr = ("Sum squared resid", 22799.68), mse_resid_sqrt = ("S.E. of regression", 10.70380), rsquared = ("R-squared", 0.676978), rsquared_adj = ("Adjusted R-squared", 0.673731), fvalue = ("F(2, 199)", 90.79971), f_pvalue = ("P-value(F)", 9.53e-29), llf = ("Log-likelihood", -763.9752), aic = ("Akaike criterion", 1533.950), bic = ("Schwarz criterion", 1543.875), hqic = ("Hannan-Quinn", 1537.966), resid_acf1 = ("rho", -0.107341), dw = ("Durbin-Watson", 2.213805)) linear_logs = [1.68351, 0.430953, 2, "chi2"] #for logs: dropping 70 nan or incomplete observations, T=133 #(res_ols.model.exog <=0).any(1).sum() = 69 ?not 70 linear_squares = [7.52477, 0.0232283, 2, "chi2"] #Autocorrelation, Breusch-Godfrey test for autocorrelation up to order 4 lm_acorr4 = [1.17928, 0.321197, 4, 195, "F"] lm2_acorr4 = [4.771043, 0.312, 4, "chi2"] acorr_ljungbox4 = [5.23587, 0.264, 4, "chi2"] #break cusum_Harvey_Collier = [0.494432, 0.621549, 198, "t"] #stats.t.sf(0.494432, 198)*2 #see cusum results in files break_qlr = [3.01985, 0.1, 3, 196, "maxF"] #TODO check this, max at 2001:4 break_chow = [13.1897, 0.00424384, 3, "chi2"] # break at 1984:1 arch_4 = [3.43473, 0.487871, 4, "chi2"] normality = [23.962, 0.00001, 2, "chi2"] het_white = [33.503723, 0.000003, 5, "chi2"] het_breusch_pagan = [1.302014, 0.521520, 2, "chi2"] #TODO: not available het_breusch_pagan_konker = [0.709924, 0.701200, 2, "chi2"] reset_2_3 = [5.219019, 0.00619, 2, 197, "f"] reset_2 = [7.268492, 0.00762, 1, 198, "f"] reset_3 = [5.248951, 0.023, 1, 198, "f"] #not available cond_1norm = 5984.0525 determinant = 7.1087467e+008 reciprocal_condition_number = 0.013826504 vif = [1.001, 1.001] names = 'date residual leverage influence DFFITS'.split() cur_dir = os.path.abspath(os.path.dirname(__file__)) fpath = os.path.join(cur_dir, 'results/leverage_influence_ols_nostars.txt') lev = np.genfromtxt(fpath, skip_header=3, skip_footer=1, converters={0:lambda s: s}) #either numpy 1.6 or python 3.2 changed behavior if np.isnan(lev[-1]['f1']): lev = np.genfromtxt(fpath, skip_header=3, skip_footer=2, converters={0:lambda s: s}) lev.dtype.names = names res = res_ols #for easier copying cov_hac = sw.cov_hac_simple(res, nlags=4, use_correction=False) bse_hac = sw.se_cov(cov_hac) assert_almost_equal(res.params, partable[:,0], 5) assert_almost_equal(bse_hac, partable[:,1], 5) #TODO assert_almost_equal(res.ssr, result_gretl_g1['ssr'][1], decimal=2) assert_almost_equal(res.llf, result_gretl_g1['llf'][1], decimal=4) #not in gretl assert_almost_equal(res.rsquared, result_gretl_g1['rsquared'][1], decimal=6) #FAIL assert_almost_equal(res.rsquared_adj, result_gretl_g1['rsquared_adj'][1], decimal=6) #FAIL assert_almost_equal(np.sqrt(res.mse_resid), result_gretl_g1['mse_resid_sqrt'][1], decimal=5) #f-value is based on cov_hac I guess #res2 = res.get_robustcov_results(cov_type='HC1') # TODO: fvalue differs from Gretl, trying any of the HCx #assert_almost_equal(res2.fvalue, result_gretl_g1['fvalue'][1], decimal=0) #FAIL #assert_approx_equal(res.f_pvalue, result_gretl_g1['f_pvalue'][1], significant=1) #FAIL #assert_almost_equal(res.durbin_watson, result_gretl_g1['dw'][1], decimal=7) #TODO c = oi.reset_ramsey(res, degree=2) compare_ftest(c, reset_2, decimal=(6,5)) c = oi.reset_ramsey(res, degree=3) compare_ftest(c, reset_2_3, decimal=(6,5)) linear_sq = smsdia.linear_lm(res.resid, res.model.exog) assert_almost_equal(linear_sq[0], linear_squares[0], decimal=6) assert_almost_equal(linear_sq[1], linear_squares[1], decimal=7) hbpk = smsdia.het_breuschpagan(res.resid, res.model.exog) assert_almost_equal(hbpk[0], het_breusch_pagan_konker[0], decimal=6) assert_almost_equal(hbpk[1], het_breusch_pagan_konker[1], decimal=6) hw = smsdia.het_white(res.resid, res.model.exog) assert_almost_equal(hw[:2], het_white[:2], 6) #arch #sm_arch = smsdia.acorr_lm(res.resid**2, maxlag=4, autolag=None) sm_arch = smsdia.het_arch(res.resid, nlags=4) assert_almost_equal(sm_arch[0], arch_4[0], decimal=5) assert_almost_equal(sm_arch[1], arch_4[1], decimal=6) vif2 = [oi.variance_inflation_factor(res.model.exog, k) for k in [1,2]] infl = oi.OLSInfluence(res_ols) #print np.max(np.abs(lev['DFFITS'] - infl.dffits[0])) #print np.max(np.abs(lev['leverage'] - infl.hat_matrix_diag)) #print np.max(np.abs(lev['influence'] - infl.influence)) #just added this based on Gretl #just rough test, low decimal in Gretl output, assert_almost_equal(lev['residual'], res.resid, decimal=3) assert_almost_equal(lev['DFFITS'], infl.dffits[0], decimal=3) assert_almost_equal(lev['leverage'], infl.hat_matrix_diag, decimal=3) assert_almost_equal(lev['influence'], infl.influence, decimal=4) def test_GLSARlag(): #test that results for lag>1 is close to lag=1, and smaller ssr from statsmodels.datasets import macrodata d2 = macrodata.load_pandas().data g_gdp = 400*np.diff(np.log(d2['realgdp'].values)) g_inv = 400*np.diff(np.log(d2['realinv'].values)) exogg = add_constant(np.c_[g_gdp, d2['realint'][:-1].values], prepend=False) mod1 = GLSAR(g_inv, exogg, 1) res1 = mod1.iterative_fit(5) mod4 = GLSAR(g_inv, exogg, 4) res4 = mod4.iterative_fit(10) assert_array_less(np.abs(res1.params / res4.params - 1), 0.03) assert_array_less(res4.ssr, res1.ssr) assert_array_less(np.abs(res4.bse / res1.bse) - 1, 0.015) assert_array_less(np.abs((res4.fittedvalues / res1.fittedvalues - 1).mean()), 0.015) assert_equal(len(mod4.rho), 4) if __name__ == '__main__': t = TestGLSARGretl() t.test_all() ''' Model 5: OLS, using observations 1959:2-2009:3 (T = 202) Dependent variable: ds_l_realinv HAC standard errors, bandwidth 4 (Bartlett kernel) coefficient std. error t-ratio p-value ------------------------------------------------------------- const -9.48167 1.17709 -8.055 7.17e-014 *** ds_l_realgdp 4.37422 0.328787 13.30 2.62e-029 *** realint_1 -0.613997 0.293619 -2.091 0.0378 ** Mean dependent var 3.257395 S.D. dependent var 18.73915 Sum squared resid 22799.68 S.E. of regression 10.70380 R-squared 0.676978 Adjusted R-squared 0.673731 F(2, 199) 90.79971 P-value(F) 9.53e-29 Log-likelihood -763.9752 Akaike criterion 1533.950 Schwarz criterion 1543.875 Hannan-Quinn 1537.966 rho -0.107341 Durbin-Watson 2.213805 QLR test for structural break - Null hypothesis: no structural break Test statistic: max F(3, 196) = 3.01985 at observation 2001:4 (10 percent critical value = 4.09) Non-linearity test (logs) - Null hypothesis: relationship is linear Test statistic: LM = 1.68351 with p-value = P(Chi-square(2) > 1.68351) = 0.430953 Non-linearity test (squares) - Null hypothesis: relationship is linear Test statistic: LM = 7.52477 with p-value = P(Chi-square(2) > 7.52477) = 0.0232283 LM test for autocorrelation up to order 4 - Null hypothesis: no autocorrelation Test statistic: LMF = 1.17928 with p-value = P(F(4,195) > 1.17928) = 0.321197 CUSUM test for parameter stability - Null hypothesis: no change in parameters Test statistic: Harvey-Collier t(198) = 0.494432 with p-value = P(t(198) > 0.494432) = 0.621549 Chow test for structural break at observation 1984:1 - Null hypothesis: no structural break Asymptotic test statistic: Chi-square(3) = 13.1897 with p-value = 0.00424384 Test for ARCH of order 4 - Null hypothesis: no ARCH effect is present Test statistic: LM = 3.43473 with p-value = P(Chi-square(4) > 3.43473) = 0.487871: #ANOVA Analysis of Variance: Sum of squares df Mean square Regression 47782.7 2 23891.3 Residual 22799.7 199 114.571 Total 70582.3 201 351.156 R^2 = 47782.7 / 70582.3 = 0.676978 F(2, 199) = 23891.3 / 114.571 = 208.528 [p-value 1.47e-049] #LM-test autocorrelation Breusch-Godfrey test for autocorrelation up to order 4 OLS, using observations 1959:2-2009:3 (T = 202) Dependent variable: uhat coefficient std. error t-ratio p-value ------------------------------------------------------------ const 0.0640964 1.06719 0.06006 0.9522 ds_l_realgdp -0.0456010 0.217377 -0.2098 0.8341 realint_1 0.0511769 0.293136 0.1746 0.8616 uhat_1 -0.104707 0.0719948 -1.454 0.1475 uhat_2 -0.00898483 0.0742817 -0.1210 0.9039 uhat_3 0.0837332 0.0735015 1.139 0.2560 uhat_4 -0.0636242 0.0737363 -0.8629 0.3893 Unadjusted R-squared = 0.023619 Test statistic: LMF = 1.179281, with p-value = P(F(4,195) > 1.17928) = 0.321 Alternative statistic: TR^2 = 4.771043, with p-value = P(Chi-square(4) > 4.77104) = 0.312 Ljung-Box Q' = 5.23587, with p-value = P(Chi-square(4) > 5.23587) = 0.264: RESET test for specification (squares and cubes) Test statistic: F = 5.219019, with p-value = P(F(2,197) > 5.21902) = 0.00619 RESET test for specification (squares only) Test statistic: F = 7.268492, with p-value = P(F(1,198) > 7.26849) = 0.00762 RESET test for specification (cubes only) Test statistic: F = 5.248951, with p-value = P(F(1,198) > 5.24895) = 0.023 #heteroscedasticity White White's test for heteroskedasticity OLS, using observations 1959:2-2009:3 (T = 202) Dependent variable: uhat^2 coefficient std. error t-ratio p-value ------------------------------------------------------------- const 104.920 21.5848 4.861 2.39e-06 *** ds_l_realgdp -29.7040 6.24983 -4.753 3.88e-06 *** realint_1 -6.93102 6.95607 -0.9964 0.3203 sq_ds_l_realg 4.12054 0.684920 6.016 8.62e-09 *** X2_X3 2.89685 1.38571 2.091 0.0379 ** sq_realint_1 0.662135 1.10919 0.5970 0.5512 Unadjusted R-squared = 0.165860 Test statistic: TR^2 = 33.503723, with p-value = P(Chi-square(5) > 33.503723) = 0.000003: #heteroscedasticity Breusch-Pagan (original) Breusch-Pagan test for heteroskedasticity OLS, using observations 1959:2-2009:3 (T = 202) Dependent variable: scaled uhat^2 coefficient std. error t-ratio p-value ------------------------------------------------------------- const 1.09468 0.192281 5.693 4.43e-08 *** ds_l_realgdp -0.0323119 0.0386353 -0.8363 0.4040 realint_1 0.00410778 0.0512274 0.08019 0.9362 Explained sum of squares = 2.60403 Test statistic: LM = 1.302014, with p-value = P(Chi-square(2) > 1.302014) = 0.521520 #heteroscedasticity Breusch-Pagan Koenker Breusch-Pagan test for heteroskedasticity OLS, using observations 1959:2-2009:3 (T = 202) Dependent variable: scaled uhat^2 (Koenker robust variant) coefficient std. error t-ratio p-value ------------------------------------------------------------ const 10.6870 21.7027 0.4924 0.6230 ds_l_realgdp -3.64704 4.36075 -0.8363 0.4040 realint_1 0.463643 5.78202 0.08019 0.9362 Explained sum of squares = 33174.2 Test statistic: LM = 0.709924, with p-value = P(Chi-square(2) > 0.709924) = 0.701200 ########## forecast #forecast mean y For 95% confidence intervals, t(199, 0.025) = 1.972 Obs ds_l_realinv prediction std. error 95% interval 2008:3 -7.134492 -17.177905 2.946312 -22.987904 - -11.367905 2008:4 -27.665860 -36.294434 3.036851 -42.282972 - -30.305896 2009:1 -70.239280 -44.018178 4.007017 -51.919841 - -36.116516 2009:2 -27.024588 -12.284842 1.427414 -15.099640 - -9.470044 2009:3 8.078897 4.483669 1.315876 1.888819 - 7.078520 Forecast evaluation statistics Mean Error -3.7387 Mean Squared Error 218.61 Root Mean Squared Error 14.785 Mean Absolute Error 12.646 Mean Percentage Error -7.1173 Mean Absolute Percentage Error -43.867 Theil's U 0.4365 Bias proportion, UM 0.06394 Regression proportion, UR 0.13557 Disturbance proportion, UD 0.80049 #forecast actual y For 95% confidence intervals, t(199, 0.025) = 1.972 Obs ds_l_realinv prediction std. error 95% interval 2008:3 -7.134492 -17.177905 11.101892 -39.070353 - 4.714544 2008:4 -27.665860 -36.294434 11.126262 -58.234939 - -14.353928 2009:1 -70.239280 -44.018178 11.429236 -66.556135 - -21.480222 2009:2 -27.024588 -12.284842 10.798554 -33.579120 - 9.009436 2009:3 8.078897 4.483669 10.784377 -16.782652 - 25.749991 Forecast evaluation statistics Mean Error -3.7387 Mean Squared Error 218.61 Root Mean Squared Error 14.785 Mean Absolute Error 12.646 Mean Percentage Error -7.1173 Mean Absolute Percentage Error -43.867 Theil's U 0.4365 Bias proportion, UM 0.06394 Regression proportion, UR 0.13557 Disturbance proportion, UD 0.80049 '''