M/Ph TdZddlmZddlmZddlZddlZddl m Z ddl m Z m Z ddlmZmZddlmZmZmZmZdd lmZmZmZmZmZmZdd lmZgd Zd Z d Z!dZ"GddZ#d7dZ$d7dZ% d8dZ& d9dZ'eddd:ddddddZ(eddd;dZ)eddd:dZ*d ej+d!e,d"dfd#Z-dd+Z1d?d,Z2 d@d.Z3dAd/Z4d0Z5ed1d dBd4Z6d5Z7dCd6Z8dS)Da Various Statistical Tests Author: josef-pktd License: BSD-3 Notes ----- Almost fully verified against R or Gretl, not all options are the same. In many cases of Lagrange multiplier tests both the LM test and the F test is returned. In some but not all cases, R has the option to choose the test statistic. Some alternative test statistic results have not been verified. TODO * refactor to store intermediate results missing: * pvalues for breaks_hansen * additional options, compare with R, check where ddof is appropriate * new tests: - breaks_ap, more recent breaks tests - specification tests against nonparametric alternatives )deprecate_kwarg)IterableN)stats)OLSRegressionResultsWrapper)anderson_statistic normal_ad)kstest_exponential kstest_fit kstest_normal lilliefors) array_like bool_like dict_like float_likeint_like string_like)lagmat)r r r r r compare_cox compare_jacorr_breusch_godfreyacorr_ljungboxacorr_lmhet_archhet_breuschpaganhet_goldfeldquandt het_white spec_white linear_lmlinear_rainbowlinear_harvey_collierrz_The exog in results_x and in results_z are nested. {test} requires that models are non-nested. c|jd|jdkrdStj||dd}|||zz }tjtj||f|jdkS)a Check if a larger exog nests a smaller exog Parameters ---------- small : ndarray exog from smaller model large : ndarray exog from larger model Returns ------- bool True if small is nested by large FNrcondr)shapenplinalglstsq matrix_rankc_)smalllargecoeferrs \/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/statsmodels/stats/diagnostic.py_check_nested_exogr1@sx" {1~ A&&u 9??5%t? 4 4Q 7D %$, C 9 ucz!2 3 3u{1~ EEct|tstdt|tstdtj|jj|jjstd|jj}|jj}d}|j d|j dkr|pt||}n|pt||}|S)Nz2results_x must come from a linear regression modelz2results_z must come from a linear regression modelz/endogenous variables in models are not the sameFr#) isinstancer TypeErrorr'allclosemodelendog ValueErrorexogr&r1) results_x results_zxznesteds r0_check_nested_resultsr@Xs i!9 : :NLMMM i!9 : :NLMMM ;y,io.C D DLJKKKAA FwqzQWQZ3-a333-a33 Mr2ceZdZdZdS) ResultsStorec8t|d|jjS)N_str)getattr __class____name__)selfs r0__str__zResultsStore.__str__lstVT^%<===r2N)rG __module__ __qualname__rIr2r0rBrBks#>>>>>r2rBFct||r(ttd|jj}|jj}|jjjd}|j|z }|j|z }|j }t|| } | j } t| | } | j } |tj| j| |z z} |dz tj|tj| z z}|tj| j| z| dzz }|tj|z }dt$jtj|z}|rNt-}| |_| |_||_||_||_||_t$j|_|||fS||fS)ad Compute the Cox test for non-nested models Parameters ---------- results_x : Result instance result instance of first model results_z : Result instance result instance of second model store : bool, default False If true, then the intermediate results are returned. Returns ------- tstat : float t statistic for the test that including the fitted values of the first model in the second model has no effect. pvalue : float two-sided pvalue for the t statistic res_store : ResultsStore, optional Intermediate results. Returned if store is True. Notes ----- Tests of non-nested hypothesis might not provide unambiguous answers. The test should be performed in both directions and it is possible that both or neither test rejects. see [1]_ for more information. Formulas from [1]_, section 8.3.4 translated to code Matches results for Example 8.3 in Greene References ---------- .. [1] Greene, W. H. Econometric Analysis. New Jersey. Prentice Hall; 5th edition. (2002). zCox comparisontestr@)r@r9 NESTED_ERRORformatr7r:r8r&ssr fittedvaluesrfitresidr'dotTlogsqrtrnormsfabsrBres_dxres_xzxc01v01qpvaluedist)r;r<storer=r>nobssigma2_xsigma2_zyhat_xr_err_zxr`err_xzx sigma2_zxrarbrcpvalress r0rrpsLY 22E,,2B,CCDDDAA ? &q )D}t#H}t#H  #F ^^   ! !F \F&!nn  ""GmG26&(F33d::I )rvh''"&*;*;; rjrttstatrnros r0rrsLY 22C,,.,AABBBAA  #F BOVQK00 1 1 5 5 7 7F N1 E >! D  nn 76?++  dC $;r2 nonrobustc6t||r(ttd|jj}|jj}|jj}d}||||||}||||||} tj|| gddggdS)a Davidson-MacKinnon encompassing test for comparing non-nested models Parameters ---------- results_x : Result instance result instance of first model results_z : Result instance result instance of second model cov_type : str, default "nonrobust Covariance type. The default is "nonrobust` which uses the classic OLS covariance estimator. Specify one of "HC0", "HC1", "HC2", "HC3" to use White's covariance estimator. All covariance types supported by ``OLS.fit`` are accepted. cov_kwargs : dict, default None Dictionary of covariance options passed to ``OLS.fit``. See OLS.fit for more details. Returns ------- DataFrame A DataFrame with two rows and four columns. The row labeled x contains results for the null that the model contained in results_x is equivalent to the encompassing model. The results in the row labeled z correspond to the test that the model contained in results_z are equivalent to the encompassing model. The columns are the test statistic, its p-value, and the numerator and denominator degrees of freedom. The test statistic has an F distribution. The numerator degree of freedom is the number of variables in the encompassing model that are not in the x or z model. The denominator degree of freedom is the number of observations minus the number of variables in the nesting model. Notes ----- The null is that the fit produced using x is the same as the fit produced using both x and z. When testing whether x is encompassed, the model estimated is .. math:: Y = X\beta + Z_1\gamma + \epsilon where :math:`Z_1` are the columns of :math:`Z` that are not spanned by :math:`X`. The null is :math:`H_0:\gamma=0`. When testing whether z is encompassed, the roles of :math:`X` and :math:`Z` are reversed. Implementation of Davidson and MacKinnon (1993)'s encompassing test. Performs two Wald tests where models x and z are compared to a model that nests the two. The Wald tests are performed by using an OLS regression. zTesting encompassingrNcP||tj||ddzz }tj|\}}}tjtjj} |ddt|jz| z} tj || k} ||dd| fz} tj || g} | jd}| jd}t||  ||}tj ||f}tj||dd| df<||dd}|j|j}}t%|jt%|j}}||||fS) Nr$rT)axiskeepdimsr#)cov_typecov_kwdsuse_fscalar)r'r(r)svdfinfodoubleepsmaxr&r^hstackrrVzeroseye wald_test statisticrdintdf_numdf_denom)r8abcov_estrr/usvrtolnon_zeroaugaug_regk_akror_matrixrOstatrdrrs r0 _test_nestedz*compare_encompassing.._test_nested-st!biooa$o77:::)--$$1ahry!!%eede++c#)nn)rrdrr)indexcolumns) r@r9rRrSr7r8r:pd DataFrame) r;r<r cov_kwargsrxr=r>rx_nestedz_nesteds r0compare_encompassingrslY 22K,,2H,IIJJJAAA...&|Aq!Xz::H|Aq!Xz::H <8,"Cj H H H J J JJr2Tc dddlm}t|d}t|dd}t|dd}||d krt d |dkrt d |jd}|r|d z } ||| d } |sH||dzzt j| d | d zdz|t jd | d zz z z} n%|t j| d | d zdzz} d} t j | t j |z} t j |  t j |z}|| kr.| t jd |t j |zz } n| dt jd |zz } t j | }td |}t|d}t jd |d z}n|4t jd t|dzd|zd zt}np|1t jd t|dzdd zt}n=t!|t"s(t|d}t jd |d z}t|dd}| } ||| d } | d | d zdz|t jd | d zz z }||dzzt j||d z z}||z }t j|t j}|dk}t(j||||||<|st/j||d|S|t j| d | d zdz|d z z}t j|t j}t(j||||||<t/j||||d|S)a Ljung-Box test of autocorrelation in residuals. Parameters ---------- x : array_like The data series. The data is demeaned before the test statistic is computed. lags : {int, array_like}, default None If lags is an integer then this is taken to be the largest lag that is included, the test result is reported for all smaller lag length. If lags is a list or array, then all lags are included up to the largest lag in the list, however only the tests for the lags in the list are reported. If lags is None, then the default maxlag is min(10, nobs // 5). The default number of lags changes if period is set. boxpierce : bool, default False If true, then additional to the results of the Ljung-Box test also the Box-Pierce test results are returned. model_df : int, default 0 Number of degrees of freedom consumed by the model. In an ARMA model, this value is usually p+q where p is the AR order and q is the MA order. This value is subtracted from the degrees-of-freedom used in the test so that the adjusted dof for the statistics are lags - model_df. If lags - model_df <= 0, then NaN is returned. period : int, default None The period of a Seasonal time series. Used to compute the max lag for seasonal data which uses min(2*period, nobs // 5) if set. If None, then the default rule is used to set the number of lags. When set, must be >= 2. auto_lag : bool, default False Flag indicating whether to automatically determine the optimal lag length based on threshold of maximum correlation value. Returns ------- DataFrame Frame with columns: * lb_stat - The Ljung-Box test statistic. * lb_pvalue - The p-value based on chi-square distribution. The p-value is computed as 1 - chi2.cdf(lb_stat, dof) where dof is lag - model_df. If lag - model_df <= 0, then NaN is returned for the pvalue. * bp_stat - The Box-Pierce test statistic. * bp_pvalue - The p-value based for Box-Pierce test on chi-square distribution. The p-value is computed as 1 - chi2.cdf(bp_stat, dof) where dof is lag - model_df. If lag - model_df <= 0, then NaN is returned for the pvalue. See Also -------- statsmodels.regression.linear_model.OLS.fit Regression model fitting. statsmodels.regression.linear_model.RegressionResults Results from linear regression models. statsmodels.stats.stattools.q_stat Ljung-Box test statistic computed from estimated autocorrelations. Notes ----- Ljung-Box and Box-Pierce statistic differ in their scaling of the autocorrelation function. Ljung-Box test is has better finite-sample properties. References ---------- .. [*] Green, W. "Econometric Analysis," 5th ed., Pearson, 2003. .. [*] J. Carlos Escanciano, Ignacio N. Lobato "An automatic Portmanteau test for serial correlation"., Volume 151, 2009. Examples -------- >>> import statsmodels.api as sm >>> data = sm.datasets.sunspots.load_pandas().data >>> res = sm.tsa.ARMA(data["SUNACTIVITY"], (1,1)).fit(disp=-1) >>> sm.stats.acorr_ljungbox(res.resid, lags=[10], return_df=True) lb_stat lb_pvalue 10 214.106992 1.827374e-40 r)acfr=periodToptionalmodel_dfFNr#zperiod must be >= 2zmodel_df must be >= 0)nlagsfftrQg333333@lagsdtype r)lb_stat lb_pvalue)r)rrbp_stat bp_pvalue)statsmodels.tsa.stattoolsrrrr9r&r'cumsumaranger[rZr^rargmaxminrr4r full_likenanrchi2r]rr)r=r boxpiercerr return_dfauto_lagrrgmaxlagsacfq_sacfrc thresholdthreshold_metricsacf2 qljungboxadj_lagsrnloc qboxpiercepvalbps r0rrGsPj.-----1cA fh 6 6 6F*u===H fkk.///!||0111 71:D#&s1F... ?dQh'iQvz\ 2a 7#'")Avz*B*B#B!DEEEFFBId1VaZ<&8A&=>>>F GAt ,-- 6$<<++-- =  ) )ryD11BF4LL@AFFq29Q#5#556Fy  1d||f%%yD1H%%  yC 1v:66:#FFF yC 2..2#>>> h ' '&f%%yD1H%% dF% 0 0 0D XXZZF 3qE * * *D 6A: ! #tbi6A:.F.F'F GEq!BIe$4$4TAX$>>IhH < 26 * *D Q,C inhsm<= 2. ddof : int, default 0 The number of degrees of freedom consumed by the model used to produce resid. The default value is 0. cov_type : str, default "nonrobust" Covariance type. The default is "nonrobust` which uses the classic OLS covariance estimator. Specify one of "HC0", "HC1", "HC2", "HC3" to use White's covariance estimator. All covariance types supported by ``OLS.fit`` are accepted. cov_kwargs : dict, default None Dictionary of covariance options passed to ``OLS.fit``. See OLS.fit for more details. Returns ------- lm : float Lagrange multiplier test statistic. lmpval : float The p-value for Lagrange multiplier test. fval : float The f statistic of the F test, alternative version of the same test based on F test for the parameter restriction. fpval : float The pvalue of the F test. res_store : ResultsStore, optional Intermediate results. Only returned if store=True. See Also -------- het_arch Conditional heteroskedasticity testing. acorr_breusch_godfrey Breusch-Godfrey test for serial correlation. acorr_ljung_box Ljung-Box test for serial correlation. Notes ----- The test statistic is computed as (nobs - ddof) * r2 where r2 is the R-squared from a regression on the residual on nlags lags of the residual. rWr#ndimrNrrrrQrbothtrimrrrzFTr)rrrr&rrr'r+onesrBrrVfloatfvaluef_pvaluersquaredrrr]rrrrrrdresolsusedlag)rWrrfrrrrrgrxdallxshort res_storerrfvalfpvallmlmpvalr test_stats r0rrs'@ ugA . . .E8Z00H!)zJ:|44J ;q>D emTQYF ++ R## 5D>6 7 7 7E ;q>D E"'4)$$e+ ,E D566]FIG qqq,7Q;,/ 0 0 4 4h@J5LLF   D &/ " "E;TkV_ ,r7++9bh|44bfWooFGG$$XU4$HH 9& ' 'y'(( '! # 64 1164&&r2c.t|dz|||S)aL Engle's Test for Autoregressive Conditional Heteroscedasticity (ARCH). Parameters ---------- resid : ndarray residuals from an estimation, or time series nlags : int, default None Highest lag to use. store : bool, default False If true then the intermediate results are also returned ddof : int, default 0 If the residuals are from a regression, or ARMA estimation, then there are recommendations to correct the degrees of freedom by the number of parameters that have been estimated, for example ddof=p+q for an ARMA(p,q). Returns ------- lm : float Lagrange multiplier test statistic lmpval : float p-value for Lagrange multiplier test fval : float fstatistic for F test, alternative version of the same test based on F test for the parameter restriction fpval : float pvalue for F test res_store : ResultsStore, optional Intermediate results. Returned if store is True. Notes ----- verified against R:FinTS::ArchTest rQ)rrfr)r)rWrrfrs r0rrKs J EQJe5t D D DDr2resultsroc&tj|j}|jdkrt d|jj}|jd}|td|dz}tj tj ||f}t|dddf|d}|jd}tj tj|df|f}|| d}||}ntj||f}|jd} t!||} | tj|| | |z } | j} | j} t-tj| } t-tj| } || jz}t0j||}|r#t7}| |_||_||| | |fS||| | fS) a Breusch-Godfrey Lagrange Multiplier tests for residual autocorrelation. Parameters ---------- res : RegressionResults Estimation results for which the residuals are tested for serial correlation. nlags : int, optional Number of lags to include in the auxiliary regression. (nlags is highest lag). store : bool, default False If store is true, then an additional class instance that contains intermediate results is returned. Returns ------- lm : float Lagrange multiplier test statistic. lmpval : float The p-value for Lagrange multiplier test. fval : float The value of the f statistic for F test, alternative version of the same test based on F test for the parameter restriction. fpval : float The pvalue for F test. res_store : ResultsStore A class instance that holds intermediate results. Only returned if store=True. Notes ----- BG adds lags of residual to exog in the design matrix for the auxiliary regression with residuals as endog. See [1]_, section 12.7.1. References ---------- .. [1] Greene, W. H. Econometric Analysis. New Jersey. Prentice Hall; 5th edition. (2002). r#zFModel resid must be a 1d array. Cannot be used on multivariate models.rNrrrr)r'asarrayrWsqueezerr9r7r:r&r concatenaterrr+rrqrrVf_testrrrdrrrrr]rBrr)rorrfr=exog_oldrgrrr:k_varsrftrrrrrs r0rrssV 39%%''Av{{122 2y~H 71:D }B "" +,,A 1QQQW:u6 2 2 2E ;q>D E"'4)$$e+ ,E uvvYF%011 Z]F    " " $ $F rveVVe^<< = =B 9D IE D!! " "D "*U## $ $E  B Z]]2u % %F ' NN ! ! 64 1164&&r2r= test_namereturnc|d}tj||dz dk|dkzr|jddkrt |ddS)z Check validity of the exogenous regressors in a heteroskedasticity test Parameters ---------- x : ndarray The exogenous regressor array test_name : str The test name for the exception rr~r#rQzI test requires exog to have at least two columns where one is a constant.N)rr'anyrr&r9)r=rx_maxs r0_check_het_testrs EEqEMME FUQUUU]]*q0UaZ@ A A 71:>> 3 3 3    >r2ct|dd}t|dt|dddz}|s|tj|z }|j\}}t ||}|j}|j} |r ||j zn |j dz } | tj | |dz || fS)u Breusch-Pagan Lagrange Multiplier test for heteroscedasticity The tests the hypothesis that the residual variance does not depend on the variables in x in the form .. :math: \sigma_i = \sigma * f(\alpha_0 + \alpha z_i) Homoscedasticity implies that :math:`\alpha=0`. Parameters ---------- resid : array_like For the Breusch-Pagan test, this should be the residual of a regression. If an array is given in exog, then the residuals are calculated by the an OLS regression or resid on exog. In this case resid should contain the dependent variable. Exog can be the same as x. exog_het : array_like This contains variables suspected of being related to heteroscedasticity in resid. robust : bool, default True Flag indicating whether to use the Koenker version of the test (default) which assumes independent and identically distributed error terms, or the original Breusch-Pagan version which assumes residuals are normally distributed. Returns ------- lm : float lagrange multiplier statistic lm_pvalue : float p-value of lagrange multiplier test fvalue : float f-statistic of the hypothesis that the error variance does not depend on x f_pvalue : float p-value for the f-statistic Notes ----- Assumes x contains constant (for counting dof and calculation of R^2). In the general description of LM test, Greene mentions that this test exaggerates the significance of results in small or moderately large samples. In this case the F-statistic is preferable. **Verification** Chisquare test statistic is exactly (<1e-13) the same result as bptest in R-stats with defaults (studentize=True). **Implementation** This is calculated using the generic formula for LM test using $R^2$ (Greene, section 17.6) and not with the explicit formula (Greene, section 11.4.3), unless `robust` is set to False. The degrees of freedom for the p-value assume x is full rank. References ---------- .. [1] Greene, W. H. Econometric Analysis. New Jersey. Prentice Hall; 5th edition. (2002). .. [2] Breusch, T. S.; Pagan, A. R. (1979). "A Simple Test for Heteroskedasticity and Random Coefficient Variation". Econometrica. 47 (5): 1287–1294. .. [3] Koenker, R. (1981). "A note on studentizing a test for heteroskedasticity". Journal of Econometrics 17 (1): 107–112. exog_hetrQrzThe Breusch-PaganrWr#)rrr'meanr&rrVrrressrrr]) rWrrobustr=rxrgnvarsrrrrs r0rrsH 8Za000AA*+++5'***a/A   N'KD% AYY]]__F =D OE#) =  vzA~B uz}}R++T5 88r2c`t|dd}t|dd|jddf}t|d|j\}}tj|\}}|d d |f|d d |fz}|j\}}|||dz zd z |zksJt |dz|} | j} | j} || j z} | j tj |dz ksJtj| | j } | | | | fS) a White's Lagrange Multiplier Test for Heteroscedasticity. Parameters ---------- resid : array_like The residuals. The squared residuals are used as the endogenous variable. exog : array_like The explanatory variables for the variance. Squares and interaction terms are automatically included in the auxiliary regression. Returns ------- lm : float The lagrange multiplier statistic. lm_pvalue :float The p-value of lagrange multiplier test. fvalue : float The f-statistic of the hypothesis that the error variance does not depend on x. This is an alternative test variant not the original LM test. f_pvalue : float The p-value for the f-statistic. Notes ----- Assumes x contains constant (for counting dof). question: does f-statistic make sense? constant ? References ---------- Greene section 11.4.1 5th edition p. 222. Test statistic reproduces Greene 5th, example 11.3. r:rQrrWrr#)rr&zWhite's heteroskedasticityNrP)rr&rr' triu_indicesrrVrrrdf_modelr(r*rrr])rWr:r=rxrgnvars0i0i1rrrrrrs r0rr.sBJ 4a(((A5'!'!*aAAAAA34447LD& _V $ $FB QQQU8a2h D*KD% Ffqj)B.7 7 7 7 7 a   " " $ $F =D OE  B ?bi33D99A= = = = = Z]]2v / /F vtU ""r2 increasingctj|}tj|}|j\}}||dz}nd|kr|dkrt||z}||} n'd|kr|dkr|t||zz} n||z} |2tj|dd|f} || }|| ddf}t |d||d|} t || d|| d} | j| jz } |dvr.tj | | j | j }d}n|dvr1tj d| z | j | j }d }n|d vrltj | | j | j }tj | | j | j }dt||z}d }ntd |rtt!}d |_| |_||_| j | j f|_| |_| |_||_||_d|| ||_| |||fS| ||fS)ak Goldfeld-Quandt homoskedasticity test. This test examines whether the residual variance is the same in 2 subsamples. Parameters ---------- y : array_like endogenous variable x : array_like exogenous variable, regressors idx : int, default None column index of variable according to which observations are sorted for the split split : {int, float}, default None If an integer, this is the index at which sample is split. If a float in 0= 2rz,power must be an integer or list of integersrz.power must contains distinct integers all >= 2r"Nr:rz+Model contains only constant or binary data)PCAnipals)ncomp standardizedemeanmethodcg|]}|zSrLrL).0prs r0 z linear_reset..Es";";";3!8";";";r2r)rr)'r4rr5boolr7 k_constantr:r&r9rrrrr'rarray ExceptionrlensetrrrUrrallstatsmodels.multivariate.pcar'tolistpopdata const_idxfactorsrrFr8rVrr)ropowerr$rrrr:binaryr'retainpcaaug_exog mod_classmodnrestrnparamsr_matrs @r0 linear_resetrHs| c3 4 4KIJJJ CI !!1cin&:1&=&B&B011 1I{$BDDDI:|dCCCJ eW % %E% O 199122 2 !UQYc222 MHU#...EE M M MKLL L M :??c#e**ooQ??!!@MNN N 9>DHj)**111d73 f  in4888+++a8H8H0HI## ::<< LJKK K!!!fW*o444444 > !Ysy|,,3355F JJs39>344 5 5 5aaai.Cc#QD,@,@cn--h@@@k!!!RaR% y$";";";";U";";";;<::L 8D IOE 9>D !"" " h + + .!(JQeLLLHH(C(( &$: Hy~/9>>@@) H H H!GHHH HD$,, $,,4:a=11DJ##H--Dz$t*--HhH~#)>Tav fe $ $ M&''''C'''' !IJJJ46*++FFfh//Fv((((q(((( !KLLL&/* 000000 3#Csus{T122BBy$ 3 3 31%Csus{T122BBB 3uT:mCCCj&&c Cy WSAH%, - - 4 4S 9 9F Xftd{* + + 2 23 7 7F A&&344 4 && ! !E utE{ + + / / 1 1Fl &q)G JE B %ZD7N +e 3fo EE 7::eTG^V_ = =D $;s)C++!D &/IAJ/.J/c |d}tj|}tj|||ddddff}|j\}}t ||}|tj|dz |dzdz |}||jz}tj ||dz } || |fS)ae Lagrange multiplier test for linearity against functional alternative # TODO: Remove the restriction limitations: Assumes currently that the first column is integer. Currently it does not check whether the transformed variables contain NaNs, for example log of negative number. Parameters ---------- resid : ndarray residuals of a regression exog : ndarray exogenous variables for which linearity is tested func : callable, default None If func is None, then squares are used. func needs to take an array of exog and return an array of transformed variables. Returns ------- lm : float Lagrange multiplier test statistic lm_pval : float p-value of Lagrange multiplier tes ftest : ContrastResult instance the results from the F test variant of this test Notes ----- Written to match Gretl's linearity test. The test runs an auxiliary regression of the residuals on the combined original and transformed regressors. The Null hypothesis is that the linear specification is correct. Nc,tj|dS)NrQ)r'r>)r=s r0funczlinear_lm..func s8Aq>> !r2r#rQ) r'rrqr&rrVrrrrrr]) rWr:rzexog_auxrgrlsftestrlm_pvals r0rrsF | " " " :d  Ddd4122;&7&7899H:LD& UH   ! ! # #B IIbfVaZ!a@@ A AE  BjmmB ++G w r2czt|dd}t|dd}|jddks+tjtj|ddkst dtj|jd\}}tj|d d |f|d d |fzdd}d }d }|||dzz}tj |d } tj | tj |k} |d d tj| df}||z} | tj| z } tj|j| } |tj|dz }|| d d d fz}|j|}| tj || }|jd}t&j||}|||fS)a White's Two-Moment Specification Test Parameters ---------- resid : array_like OLS residuals. exog : array_like OLS design matrix. Returns ------- stat : float The test statistic. pval : float A chi-square p-value for test statistic. dof : int The degrees of freedom. See Also -------- het_white White's test for heteroskedasticity. Notes ----- Implements the two-moment specification test described by White's Theorem 2 (1980, p. 823) which compares the standard OLS covariance estimator with White's heteroscedasticity-consistent estimator. The test statistic is shown to be chi-square distributed. Null hypothesis is homoscedastic and correctly specified. Assumes the OLS design matrix contains an intercept term and at least one variable. The intercept is removed to calculate the test statistic. Interaction terms (squares and crosses of OLS regressors) are added to the design matrix to calculate the test statistic. Degrees-of-freedom (full rank) = nvar + nvar * (nvar + 1) / 2 Linearly dependent columns are removed to avoid singular matrix error. References ---------- .. [*] White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity. Econometrica, 48: 817-838. r:rQrrWr#rrWzPWhite's specification test requires at least twocolumns where one is a constant.Ng+=gvIh%<=r)moder)rr&r'rptpr9rdeletevarr(qrr^diagonalr[whererrXrYsolverrr])rWr:r=erratolrtolrrmasksqesqmndevsrdevxrrdofrns r0rrsd 4a(((A5'***AwqzA~~RVBF1aLLC$788~<== =_QWQZ ( (FB 9Qqqq"uX!!!R%(!Q / /D D D  # #C T $$A 6!**,,  "'#,, .D 28TE??1%% &D a%CRWS\\!H tvx  A "'$Q''' 'DHQQQW D  4A 55A&& ' 'D *Q-C :==s # #D s?r2 olsresultsrWrJc t|tstd|jj}|jj}t |dddd|jdf}|||}||}|j\}}||}tj tj ||fz} tj tj |z} tj tj |z} tj tj |z} |d|} tj | | jdkrd }t||d|}tj tj| j| |tj|zz}tj| j|}tj||}|| |dz <tj||dz |}|| |dz <||dz |z | |dz <dtj||dz tj|||dz z| |dz <t%||D]}|||dzddf}||}tj||}tj|| |<||z }tj|| |<tj||j}dtj||z}|tj||j|z z }|||z|z z}|| |<tj|| |<| tj| z }||z }||dd }|tj|z }||dz d}|d krd }n!|d krd}n|dkrd}ntd|tj|zd|ztjd||z ztj|z ztjdgdggz} | | | |||| fS)u Calculate recursive ols with residuals and Cusum test statistic Parameters ---------- res : RegressionResults Results from estimation of a regression model. skip : int, default None The number of observations to use for initial OLS, if None then skip is set equal to the number of regressors (columns in exog). lamda : float, default 0.0 The weight for Ridge correction to initial (X'X)^{-1}. alpha : {0.90, 0.95, 0.99}, default 0.95 Confidence level of test, currently only two values supported, used for confidence interval in cusum graph. order_by : array_like, default None Integer array specifying the order of the residuals. If not provided, the order of the residuals is not changed. If provided, must have the same number of observations as the endogenous variable. Returns ------- rresid : ndarray The recursive ols residuals. rparams : ndarray The recursive ols parameter estimates. rypred : ndarray The recursive prediction of endogenous variable. rresid_standardized : ndarray The recursive residuals standardized so that N(0,sigma2) distributed, where sigma2 is the error variance. rresid_scaled : ndarray The recursive residuals normalize so that N(0,1) distributed. rcusum : ndarray The cumulative residuals for cusum test. rcusumci : ndarray The confidence interval for cusum test using a size of alpha. Notes ----- It produces same recursive residuals as other version. This version updates the inverse of the X'X matrix and does not require matrix inversion during updating. looks efficient but no timing Confidence interval in Greene and Brown, Durbin and Evans is the same as in Ploberger after a little bit of algebra. References ---------- jplv to check formulas, follows Harvey BigJudge 5.5.2b for formula for inverse(X'X) updating Greene section 7.5.2 Brown, R. L., J. Durbin, and J. M. Evans. “Techniques for Testing the Constancy of Regression Relationships over Time.” Journal of the Royal Statistical Society. Series B (Methodological) 37, no. 2 (1975): 149-192. z!res a regression results instancerMrTr#r)rrrr&Nz"The initial regressor matrix, x[:skip], issingular. You must use a value of skip large enough to ensure that the first OLS estimator is well-defined. )rg?g333333?rJgtV?gGz?g}?5^I?z#alpha can only be 0.9, 0.95 or 0.99rQgr )r4rr5r7r8r:rr&r'rrr(r*r9rerXrYrrangerrgr[rrrr3)!rorKlamdarLrMrxr=rgrrparamsrresidrypredrvarrawx0err_msgy0xtxixtybetayipredrxiyiresiditmpr rresid_scalednrrsigma2rresid_standardizedrcusumrrcusumcis! r0rNrNmsJz c3 4 4=;<<< A A(Jed  777H hK hK'KD% |frxu ...G Vbhtnn $F Vbhtnn $Ffrx~~%G 5D5B yR  28A;..!!! 5D5B 9==b))EBF5MM,AA B BD &r  C 6$  DGD1H VAdQhK & &FF4!8{V+F4!8BF1TAX;tQtax[0I0IJJJGD1H 4  $$ qQwz] qTD!!Jv&&q fJv&&q fT24   C bfS#%((2--sV|b(//111 Z^^ RWW---M +C455 ! % %1 % - -F'"'&//9  + 2 2 4 4F }}  $  $ >???BGCLL 1q529Qt +D+D#Drw H H $  3%#(()H GV%8- H r2c|jj}t|jd|jddf}|j\}}|dz}t j||dddfz||z f}|d}||dddddf|dddddfz dz}|dddddf|dddddfz d} t j t j t j || } t jgddtfd t fg } | | fS) a Test for model stability, breaks in parameters for ols, Hansen 1992 Parameters ---------- olsresults : RegressionResults Results from estimation of a regression model. Returns ------- teststat : float Hansen's test statistic. crit : ndarray The critical values at alpha=0.95 for different nvars. Notes ----- looks good in example, maybe not very powerful for small changes in parameters According to Greene, distribution of test statistics depends on nvar but not on nobs. Test statistic is verified against R:strucchange References ---------- Greene section 7.5.1, notation follows Greene rWrr#)r&rQN))rQg)\(?)gffffff?)g@)gGz@rgcritr)r7r:rrWr&r'r+rrsumtracerXr(rer3rr) rr=rWrgrresid2rscorerrhcrit95s r0 breaks_hansenrse< A z'Q H H HE'KD% aZF q5D>!FV[[]]$:; >>F f9r2ctj|}t|}|dz}|dkr |||z z |z}|tj|z }tj|}gd}tj |}|||fS)u Cusum test for parameter stability based on ols residuals. Parameters ---------- resid : ndarray An array of residuals from an OLS estimation. ddof : int The number of parameters in the OLS estimation, used as degrees of freedom correction for error variance. Returns ------- sup_b : float The test statistic, maximum of absolute value of scaled cumulative OLS residuals. pval : float Probability of observing the data under the null hypothesis of no structural change, based on asymptotic distribution which is a Brownian Bridge crit: list The tabulated critical values, for alpha = 1%, 5% and 10%. Notes ----- Tested against R:structchange. Not clear: Assumption 2 in Ploberger, Kramer assumes that exog x have asymptotically zero mean, x.mean(0) = [1, 0, 0, ..., 0] Is this really necessary? I do not see how it can affect the test statistic under the null. It does make a difference under the alternative. Also, the asymptotic distribution of test statistic depends on this. From examples it looks like there is little power for standard cusum if exog (other than constant) have mean zero. References ---------- Ploberger, Werner, and Walter Kramer. “The Cusum Test with OLS Residuals.” Econometrica 60, no. 2 (March 1992): 271-285. rQr))r#gGz?)rg(\?)rgQ?) r'rrgr5rrr[r^rr kstwobignr])rWrrg nobssigma2rsup_brrns r0breaks_cusumolsresidr+sT Ju   # # % %E u::D1*!!##J axx4$;/$6  ,,,A F1IIMMOOE - - -D ?  e $ $D $ r2)F)rzN)NFrNTF)NF)NFr)T)NNNrF)r!r"FrzN)NN)rQNFN)N)NrWrJN)r)9rstatsmodels.compat.pandasrcollections.abcrnumpyr'pandasrscipyr#statsmodels.regression.linear_modelrrstatsmodels.stats._adnormrr statsmodels.stats._lillieforsr r r r statsmodels.tools.validationrrrrrrstatsmodels.tsa.tsatoolsr__all__rRr1r@rBrrrrrrrr]r^rrrrrHr!r rrrNrrrLr2r0rs'0655555$$$$$$MMMMMMMMCCCCCCCC ,+++++ L L L FFF0&>>>>>>>> DDDDN6666r9D$(TJTJTJTJnFJ,1X$X$X$X$v7##e'tkde'e'e'e'$#e'P7##$E$E$E$#$ENE""O'O'O'#"O'd rz c d    ,O9O9O9O9d7#7#7#t9=7<q!q!q!q!h5!!9>26p:p:p:"!p:f(+(+(+(+V?Dkkkk\....bPPPfu%%<@$(MMM&%M`***Z999999r2