M/Ph:dZddlZddlmZmZmZddlmZm Z m Z m Z m Z m Z ddlmZddlmZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmcm Z!ddl"m#Z#GddeZ$GddeZ%GddeZ&e!j'e&e%dS)zC Dynamic factor model Author: Chad Fulton License: Simplified-BSD N)MLEModel MLEResultsMLEResultsWrapper) is_invertible prepare_exogconstrain_stationary_univariate!unconstrain_stationary_univariate!constrain_stationary_multivariate#unconstrain_stationary_multivariate)PCA)OLS)VAR)ARIMA)Bunch)_is_using_pandas)lagmat)cache_readonly)AppenderceZdZdZ dfd ZdZd Zd Zdd Zd Z d Z dZ dZ dZ dZddZedZedZedZedZdZdZfdZ ddZxZS) DynamicFactoruu Dynamic factor model Parameters ---------- endog : array_like The observed time-series process :math:`y` exog : array_like, optional Array of exogenous regressors for the observation equation, shaped nobs x k_exog. k_factors : int The number of unobserved factors. factor_order : int The order of the vector autoregression followed by the factors. error_cov_type : {'scalar', 'diagonal', 'unstructured'}, optional The structure of the covariance matrix of the observation error term, where "unstructured" puts no restrictions on the matrix, "diagonal" requires it to be any diagonal matrix (uncorrelated errors), and "scalar" requires it to be a scalar times the identity matrix. Default is "diagonal". error_order : int, optional The order of the vector autoregression followed by the observation error component. Default is None, corresponding to white noise errors. error_var : bool, optional Whether or not to model the errors jointly via a vector autoregression, rather than as individual autoregressions. Has no effect unless `error_order` is set. Default is False. enforce_stationarity : bool, optional Whether or not to transform the AR parameters to enforce stationarity in the autoregressive component of the model. Default is True. **kwargs Keyword arguments may be used to provide default values for state space matrices or for Kalman filtering options. See `Representation`, and `KalmanFilter` for more details. Attributes ---------- exog : array_like, optional Array of exogenous regressors for the observation equation, shaped nobs x k_exog. k_factors : int The number of unobserved factors. factor_order : int The order of the vector autoregression followed by the factors. error_cov_type : {'diagonal', 'unstructured'} The structure of the covariance matrix of the error term, where "unstructured" puts no restrictions on the matrix and "diagonal" requires it to be a diagonal matrix (uncorrelated errors). error_order : int The order of the vector autoregression followed by the observation error component. error_var : bool Whether or not to model the errors jointly via a vector autoregression, rather than as individual autoregressions. Has no effect unless `error_order` is set. enforce_stationarity : bool, optional Whether or not to transform the AR parameters to enforce stationarity in the autoregressive component of the model. Default is True. Notes ----- The dynamic factor model considered here is in the so-called static form, and is specified: .. math:: y_t & = \Lambda f_t + B x_t + u_t \\ f_t & = A_1 f_{t-1} + \dots + A_p f_{t-p} + \eta_t \\ u_t & = C_1 u_{t-1} + \dots + C_q u_{t-q} + \varepsilon_t where there are `k_endog` observed series and `k_factors` unobserved factors. Thus :math:`y_t` is a `k_endog` x 1 vector and :math:`f_t` is a `k_factors` x 1 vector. :math:`x_t` are optional exogenous vectors, shaped `k_exog` x 1. :math:`\eta_t` and :math:`\varepsilon_t` are white noise error terms. In order to identify the factors, :math:`Var(\eta_t) = I`. Denote :math:`Var(\varepsilon_t) \equiv \Sigma`. Options related to the unobserved factors: - `k_factors`: this is the dimension of the vector :math:`f_t`, above. To exclude factors completely, set `k_factors = 0`. - `factor_order`: this is the number of lags to include in the factor evolution equation, and corresponds to :math:`p`, above. To have static factors, set `factor_order = 0`. Options related to the observation error term :math:`u_t`: - `error_order`: the number of lags to include in the error evolution equation; corresponds to :math:`q`, above. To have white noise errors, set `error_order = 0` (this is the default). - `error_cov_type`: this controls the form of the covariance matrix :math:`\Sigma`. If it is "dscalar", then :math:`\Sigma = \sigma^2 I`. If it is "diagonal", then :math:`\Sigma = \text{diag}(\sigma_1^2, \dots, \sigma_n^2)`. If it is "unstructured", then :math:`\Sigma` is any valid variance / covariance matrix (i.e. symmetric and positive definite). - `error_var`: this controls whether or not the errors evolve jointly according to a VAR(q), or individually according to separate AR(q) processes. In terms of the formulation above, if `error_var = False`, then the matrices :math:C_i` are diagonal, otherwise they are general VAR matrices. References ---------- .. [*] Lütkepohl, Helmut. 2007. New Introduction to Multiple Time Series Analysis. Berlin: Springer. NrFdiagonalTc  |_|_|_|_|o|dk_|_t |\_}jdk_t|dstj |d}|j dkr |j dnd} tdjjz_j| z_j} j} jdkr| jz } | | z } d_| dkr d} d} d_| dkrt%dj| krt%d jd vrt%d | d d t)j|f|| | d| jdkr dj_i_t=j_ fd} d}| d|\_!}| d|\_"}| d|\_#}| d|\_$}| d|\_%}xj&gdtO| (zz c_&dS)NrC)orderrFTzEThe dynamic factors model is only valid for multivariate time series.zGNumber of factors must be less than the number of endogenous variables.)scalarr unstructuredz3Invalid error covariance matrix type specification.initialization stationary)exogk_statesk_posdefc^j|}tj|||z}||z }||fSN) parametersnps_)keyoffsetlength param_sliceselfs i/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/statsmodels/tsa/statespace/dynamic_factor.py_slicez&DynamicFactor.__init__.._slices9_S)F%v 67K f F& &factor_loadingsr! error_covfactor_transitionerror_transition) k_factors factor_order error_order error_varerror_cov_typeenforce_stationarity))r:r5r6r7r8r9rk_exogmle_regressionrr' asanyarrayndimshapemax _factor_order _error_order _unused_state ValueError setdefaultsuper__init__ssm_time_invariantr&_initialize_loadings_initialize_exog_initialize_error_cov_initialize_factor_transition_initialize_error_transitionsumvaluesk_params_params_loadings _params_exog_params_error_cov_params_factor_transition_params_error_transition _init_keyslistkeys)r-endogr5r6r!r7r8r9r:kwargsk_endogr"r#r/r* __class__s` r.rGzDynamicFactor.__init__so %9!#('"6{Q,+400d#kAo t,, 4M%s333E%*JNN%+a.. 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'sT3G2G-H-H'HIIJ I$"; I "HIII HZ.0117799 40 1"h..&//116688t-..$ 221;1C1L1L1N1Nt-..$66C!#!3!3J4F!G!GJJy,CCC!# (:(@(C!D!D"*3355::<.?s\   4>**   ![^4 4    r0cpg|]2}tjD]}dj|d|3S)zbeta..)rr; exog_namesrs r.rz-DynamicFactor.param_names..Gse   4;''   :DOA& 9 9Q 9 9    r0rsigma2rc&g|] }d|zS)z sigma2.%sr)rrrs r.rz-DynamicFactor.param_names..Qs2k!n,r0rcRg|]$}t|dzD]}d|dz|dzfz%S)rzcov.chol[%d,%d])r)rrrs r.rz-DynamicFactor.param_names..VsZqs"QUAEN2r0c g|]A}tjD]*}tjD]}d|dz|dz|dzfz+BS)z L%d.f%d.f%dr)rr6r5)rrrkr-s r.rz-DynamicFactor.param_names..]s   4,--  4>**    QqS!A#qsO +     r0c g|]G}tjD]0}tjD]}d|dz||fz1HSzL%d.e(%s).e(%s)r)rr7r\)rrrrrr-s r.rz-DynamicFactor.param_names..fst/00t|,, "QqS+a.+a.$IIr0clg|]0}tjD]}d|dz||fz1Sr)rr7)rrrrr-s r.rz-DynamicFactor.param_names..msct/00"QqS+a.+a.$IIr0)rrr\r9r5r8)r- param_namesrs` @r.rzDynamicFactor.param_names9s  &       4<((          4<((      ( * * H: %KK  J . . t|,, KK N 2 2 t|,, K     4>**     >  t|,, KK t|,, K r0cg}j|fdttdjDz }jdkr$|fdtjDz }jr|dgz }|S)Ncng|]1}tjD]}|dkrd|dzzn d|dz|fz2S)rzf%drzf%d.L%dr)rrrr-s r.rz-DynamicFactor.state_names..{sk,,,4>**,,#$q&&eq1uooyAE1:/E,,,,r0rrczg|]7}tjD] }|dkr d|zn d||fz!8S)rze(%s)z e(%s).L%d)rr\rs r.rz-DynamicFactor.state_names..sq...t|,, ..01Avv'KN**"k!na%88....r0dummy)rrr@r6r7rC)r-namesrs` @r. state_nameszDynamicFactor.state_namesus&  ,,,,3q$"34455,,, ,  a   .....t/00... .E    gY E r0ctj|d}|j}tj|j|}||j||j<||j||j<|jdvr||jdz||j<n |jdkr||j||j<|j r|j dkr~||j  |j |j}|jdd |j d |j fj}t#||\}}|||j <n||j ||j <|j r|jdkr|jr||j |j|j}|j }|j |jz} |jd|| || fj}t#||\}}|||j<n||j} t3|jD]5} | |jz}| dz|jz} t5| || | || <6| ||j<n||j||j<|S) a[ Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation Parameters ---------- unconstrained : array_like Array of unconstrained parameters used by the optimizer, to be transformed. Returns ------- constrained : array_like Array of constrained parameters which may be used in likelihood evaluation. Notes ----- Constrains the factor transition to be stationary and variances to be positive. rndminrrrrrrrnN)r'rrrfr?rRrSr9rTr:r6rUrr5rArHrealr rr7r8rVr\rBrrr ) r- unconstrainedr constrainedunconstrained_matricescovrvariancerbrc coefficientsrs r.transform_paramszDynamicFactor.transform_paramss,a888 #h}2%@@@ $/ 0 D)* $+ , D%&  "8 8 8d45q8 . / / N 2 2d45 . /  $ ?):Q)>)>d<=EEND$688 # (;$.HINC12H#NN + (%**,, 6 7 7d<= 6 7  $ >)9A)=)=~ J!$"?@HH d&799'nt|3h{E#IuSy@AF5.55/$h)..00D9:: "$"?@EEGGt|,,11A 00Eq5D$44C.M$U3Y//1/1Ls++=I D9::d;< 5 6r0ctj|d}|j}tj|j|}||j||j<||j||j<|jdvr||jdz||j<n |jdkr||j||j<|j r|j dkr~||j  |j |j}|jdd |j d |j fj}t#||\}}|||j <n||j ||j <|j r|jdkr|jr||j |j|j}|j }|j |jz} |jd|| || fj}t#||\}}|||j<n||j} t3|jD]5} | |jz}| dz|jz} t5| || | || <6| ||j<n||j||j<|S) a Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer. Parameters ---------- constrained : array_like Array of constrained parameters used in likelihood evaluation, to be transformed. Returns ------- unconstrained : array_like Array of unconstrained parameters used by the optimizer. rrrrrrrrnN)r'rrrfr?rRrSr9rTr:r6rUrr5rArHrr rr7r8rVr\rBrrr ) r-rrrconstrained_matricesrrrrbrcrrs r.untransform_paramsz DynamicFactor.untransform_paramss h{!444 !!2%@@@ - . d+, ) * d'(  "8 8 8D23S8 $0 1 1 N 2 2D23 $0 1  $ =):Q)>)>D:;CCND$688 !(;$.HINC3(#// + (%**,, $8 9 9D:; $8 9  $ <)9A)=)=~ L =>FF d&799%nt|3h{E#IuSy@AF7,c33/$h)..00d;<<   =>CCEEt|,,66A 00Eq5D$44C9(s355!s++@L d;<<D9: $7 8r0ct|tjt |jdd}dtj|j|D\}}}}}|j ro|j dkrd|j dks |j dkrN| |}t||dk}|r|std|j rk|jdkrb|js |jdkrT| |}t||dk}|r|stddSdSdSdSdS)Nrc3>K|]}|VdSr%)r)rarrs r. z9DynamicFactor._validate_can_fix_params..OsKFJFJ CJJLLFJFJFJFJFJFJr0rrzCannot fix individual factor transition parameters when `enforce_stationarity=True`. In this case, must either fix all factor transition parameters or none.zCannot fix individual error transition parameters when `enforce_stationarity=True`. In this case, must either fix all error transition parameters or none.)rF_validate_can_fix_paramsr'cumsumrXr&rP array_splitrr:r6r5 issupersetlen intersectionrDr7r8) r-rix_factor_transition_nameserror_transition_namesfix_allfix_anyr]s r.rz&DynamicFactor._validate_can_fix_paramsKs ((555 YtDO224455 6 6ss ;FJFJ$&N43CR$H$HFJFJFJBAq)+A  $ "):Q)>)>~!!T%6%:%:%001HII 001HIIJJQN"7"$!"""  $ ")9A)=)=~ "!1A!5!5%001GHH 001GHHIIAM"7"$!""" " ")=)= "" "6!5""r0c||||}||j|j|j|j|j<|jdkr\||j|j|jj }tj |j |j |j|j <|jdvr||j|j|j<nn|jdkrctj|j|jf|j}||j||j<tj ||j |j|j<||j|j|j|jz|j|j<|jr:||j|j|j|j|j<dS||j|j|j<dS)a2 Update the parameters of the model Updates the representation matrices to fill in the new parameter values. Parameters ---------- params : array_like Array of new parameters. transformed : bool, optional Whether or not `params` is already transformed. If set to False, `transform_params` is called. Default is True.. Returns ------- params : array_like Array of parameters. Notes ----- Let `n = k_endog`, `m = k_factors`, and `p = factor_order`. Then the `params` vector has length :math:`[n imes m] + [n] + [m^2 imes p]`. It is expanded in the following way: - The first :math:`n imes m` parameters fill out the factor loading matrix, starting from the [0,0] entry and then proceeding along rows. These parameters are not modified in `transform_params`. - The next :math:`n` parameters provide variances for the error_cov errors in the observation equation. They fill in the diagonal of the observation covariance matrix, and are constrained to be positive by `transofrm_params`. - The next :math:`m^2 imes p` parameters are used to create the `p` coefficient matrices for the vector autoregression describing the factor transition. They are transformed in `transform_params` to enforce stationarity of the VAR(p). They are placed so as to make the transition matrix a companion matrix for the VAR. In particular, we assume that the first :math:`m^2` parameters fill the first coefficient matrix (starting at [0,0] and filling along rows), the second :math:`m^2` parameters fill the second matrix, etc. ) transformedincludes_fixedrrrrN) handle_paramsrRrr\r5rHrar;rSrr'rr!rhr9rTrqrfrrwrUr6r{r8rVrBr)r-rr r  complex_step exog_paramserror_cov_lowers r.updatezDynamicFactor.updateisX##F 3A$CC 4( ) 1 1$, O O #$ ;?? !23;; dk+++, ')vdi'E'E'GDHT^ $  "8 8 8t-. HT( ) )  N 2 2 h dl'C-3\;;;Ot-. D5 6(9:: HT( ) 41 2 : : 1DN B D D ,- > 7t45==L$"355 HT/ 0 0 0 t45 HT/ 0 0 0r0)NrFrT)Fr%)TFF)__name__ __module__ __qualname____doc__rGrJrKrLrkrlrMrNr}r~rrpropertyrrrrrrrr __classcell__r]s@r.rrsnn`=A@J&*fJfJfJfJfJfJP A A A B B B666...."999"NNN.???<CCC888$&I&I&IPFFFFLLXL@@X@D99X9vX,aaaFZZZx"""""<?D!S7S7S7S7S7S7S7S7r0rceZdZdZd fd ZedZedZ d dZ e e j jd fd Z xZ S) ram Class to hold results from fitting an DynamicFactor model. Parameters ---------- model : DynamicFactor instance The fitted model instance Attributes ---------- specification : dictionary Dictionary including all attributes from the DynamicFactor model instance. coefficient_matrices_var : ndarray Array containing autoregressive lag polynomial coefficient matrices, ordered from lowest degree to highest. See Also -------- statsmodels.tsa.statespace.kalman_filter.FilterResults statsmodels.tsa.statespace.mlemodel.MLEResults Nc >tj||||fi|tj|_t di|jj|jj|jj |jj |jj |jj |jj |jjd|_d|_|jj dkr~tj|j|jj}|jj }|jj }|||z|j|||j|_d|_|jj dkrtj|j|jj}|jj} |jj } |jj r?|| | z| j| | | j|_dStj| | | zf} || |jj<| j| | | |_dSdS)N)r\r:r5r6r7r8r9r;rr)rFrGr'infdf_residrmodelr\r:r5r6r7r8r9r; specificationcoefficient_matrices_varrrrUrrcoefficient_matrices_errorrVrfr) r-rrfilter_resultscov_typer[ ar_paramsr5r6r\r7matr]s r.rGzDynamicFactorResults.__init__s KKFKKK "  z)$(J$C- J3 :1-"j7j'& &   &)-% : "Q & &TZ%IJKK  ,I:2L!!)l":IFFHgiL99!  )+/' : !A % %TZ%HIJJ j(G*0Kz# B%%g &;WEEG''7K88///h;)>?@@2;DJ./EMM+w@@/// & %r0c&d}|j}|jdkr|d}|j}|j}t|j|||j||||fdd|}|j|j|||_|j|j||||f|_ |S)a Estimates of unobserved factors Returns ------- out : Bunch Has the following attributes shown in Notes. Notes ----- The output is a bunch of the following format: - `filtered`: a time series array with the filtered estimate of the component - `filtered_cov`: a time series array with the filtered estimate of the variance/covariance of the component - `smoothed`: a time series array with the smoothed estimate of the component - `smoothed_cov`: a time series array with the smoothed estimate of the variance/covariance of the component - `offset`: an integer giving the offset in the state vector where this component begins Nr)filtered filtered_covsmoothed smoothed_covr*) rr5r!rfiltered_statefiltered_state_covsmoothed_stater(smoothed_state_covr))r-outspecr*rcress r.rzDynamicFactorResults.factors s4! >A  F.C%C+F3J7 3F3Js 4JKD C ".#26#:> &2+F3Js ,BC  r0cddlm}|j}tj|j|jf}|jdnd}t|jD]s}||j ||}t|jD]@}|j j |}t|| j|||f<At|S)aH Coefficients of determination (:math:`R^2`) from regressions of individual estimated factors on endogenous variables. Returns ------- coefficients_of_determination : ndarray A `k_endog` x `k_factors` array, where `coefficients_of_determination[i, j]` represents the :math:`R^2` value from a regression of factor `j` and a constant on endogenous variable `i`. Notes ----- Although it can be difficult to interpret the estimated factor loadings and factors, it is often helpful to use the coefficients of determination from univariate regressions to assess the importance of each factor in explaining the variation in each endogenous variable. In models with many variables and factors, this can sometimes lend interpretation to the factors (for example sometimes one factor will load primarily on real variables and another on nominal variables). See Also -------- plot_coefficients_of_determination r) add_constantNr&r()statsmodels.toolsr2rr'rfr\r5r,rrr!rZrrrsquared) r-r2r/rwhichrr!rrZs r.coefficients_of_determinationz2DynamicFactorResults.coefficients_of_determination5s: 322222!xt~ >?? "19 zt~&& E EA< U 3A 677D4<(( E E+1!4%(%5%5%9%9%;%;%D QT"" Er0cddlm}m}||||}|j}| |jdk}|j}d}t j|j} |jD]} | |j d|} | d| d|zd | | | } |r_| d} | j| | d z z| j|jjn0| d | jg|dz }|S) a5 Plot the coefficients of determination Parameters ---------- endog_labels : bool, optional Whether or not to label the endogenous variables along the x-axis of the plots. Default is to include labels if there are 5 or fewer endogenous variables. fig : Figure, optional If given, subplots are created in this figure instead of in a new figure. Note that the grid will be created in the provided figure using `fig.add_subplot()`. figsize : tuple, optional If a figure is created, this argument allows specifying a size. The tuple is (width, height). Notes ----- Produces a `k_factors` x 1 plot grid. The `i`th plot shows a bar plot of the coefficients of determination associated with factor `i`. The endogenous variables are arranged along the x-axis according to their position in the `endog` array. See Also -------- coefficients_of_determination r) _import_mplcreate_mpl_figNr)rrz Factor %iz$R^2$)titleylabelrzEndogenous variables)xlabel)statsmodels.graphics.utilsr8r9rr\r6r'aranger add_subplotr5set_ylimsetbar get_widthxaxis set_ticksset_ticklabelsrr)r- endog_labelsfigfigsizer8r9r/r6plot_idx locationscoeffsaxbarswidths r."plot_coefficients_of_determinationz7DynamicFactorResults.plot_coefficients_of_determination_st> KJJJJJJJ nS'**!  <1,L)-(J%Idl++ 35  FH==B KK    FFx/F A A A66)V,,D 'Q))++""9uqy#8999'' (>????4555""2&&& MHH r0皙?Tc ~ &ddlm&|j}g}|jdkre|jdkrd|j|jfz}n d|jz}|||jdkr|d|jzn|d|jz|jdkr*|jrdnd}|d ||jfzt ||| }|rtj t|j} d&fd } |jj} |jj} |jj} |jj}|jj}|jj}| |jj}g}| |jj}g}t+| D]}||| z|d z| z}||||| z|d z| z}||tj||g}d|jj|z}| |||}|j|| |jj}g}|dkrht+| D]X}||z}||||z}||d|d zz}| |||}|j|Yg}|jdkr| |jj}t+| D]}|jr||z}|d z|z} n||jz}|d z|jz} ||| }!||!d|jj|z}| ||!|}|j|| |jj}"| ||"dd}|j|g}#|||||"gfD]P}$tj|$}$t|$dkr|#|$Qtj|#}#tjt=t?|  t?|#}%t|%dkr)| ||%dd}|j||S)Nr)summary_paramsz#DynamicFactor(factors=%d, order=%d)zStaticFactor(factors=%d)z %d regressorszSUR(%d regressors)rARz %s(%d) errors)alpharb model_namedisplay_paramsTcF||j||j||j||j|||f}fdt j|jj| D}|d|d|S)Ncxg|]6}r0d|dddn|7S)rNr)joinsplit)rname strip_ends r.rzDDynamicFactorResults.summary..make_table..sS7@ICHHTZZ__SbS1222Tr0F)ynamexnamerVuse_tr;) rbsezvaluespvaluesconf_intr'rdatarr)r-rr;r^r0rrVrTs ` r. make_tablez0DynamicFactorResults.summary..make_tablesT[.|D)4<+=}}U++D13HTY233D9@@BB &~c[,1eMMMMr0rzResults for equation %szResults for factor equation f%dz Results for error equation e(%s)zError covariance matrixF)r^zOther parameters)T)!statsmodels.iolib.summaryrTrr5r6appendr;r7r8rFsummaryr'r?rrrr\rArBrRrSr concatenatertablesrUrVrTrflattenrXrB difference)(r-rVrbseparate_paramsr/rW model_type error_typerjindicesrgr\r;r5r6rArBloading_indices loading_masks exog_indices exog_masksr loading_mask exog_maskrr;tablefactor_indices factor_masks factor_mask error_masks error_indicesrc error_maskerror_cov_maskmasksm inverse_maskrTr]s( ` @r.rjzDynamicFactorResults.summaryso<<<<<<! >A   1$$C#~t/@AB 8$.H   j ) ) ){Q!!/DK"?@@@   2T[@ A A A  a  "&.:dJ   oT=M0NN O O O''//u.."   g -iDK 0 011G M M M M M M Mj(GZ&F ,I:2L J4M:2L&dj&ABOM"4:#:;LJ7^^ - - $A M1q5I2E$EF$$\222 )VQUf4D)DE !!),,,~|Y&?@@1DJ4J14MM" 4u55%%e,,,,%TZ%IJNLay))11A -E"0 8M1M"NK '' 444>1EE&Jt[%@@EN))%0000K!## ' (K L w11A~9 !L 0 1u 4 !D$4 4 1u(88!.uSy!9J&&z222@!Z3A67E&JtZ??EN))%0000%TZ%ABNJt^8EKKKE N ! !% ( ( (E#Z!N#35 $ $HQKK''))q66A::LLOOON5))E8DW)@)@U)L)L$M$MNNL<  1$$" 47I-2444%%e,,,r0r%)NNN)rRNT)rrrrrGrrrr6rQrrrjrrs@r.rrs,1B1B1B1B1B1Bf))X)V''^'R?C=A>>>>@Xj ())EEEEE*)EEEEEr0rcneZdZiZejejeZiZejej eZ dS)rN) rrr_attrswrap union_dictsr _wrap_attrs_methods _wrap_methodsrr0r.rr(sR F"$"#4#@#)++KH$D$%6%D%-//MMMr0r)(rnumpyr'mlemodelrrrtoolsrrr r r r statsmodels.multivariate.pcar #statsmodels.regression.linear_modelr#statsmodels.tsa.vector_ar.var_modelrstatsmodels.tsa.arima.modelrstatsmodels.tools.toolsrstatsmodels.tools.datarstatsmodels.tsa.tsatoolsrstatsmodels.tools.decoratorsrstatsmodels.base.wrapperbasewrapperrstatsmodels.compat.pandasrrrrpopulate_wrapperrr0r.rs========== -,,,,,333333333333------))))))333333++++++777777'''''''''......a7a7a7a7a7Ha7a7a7Hfffff:fffR /////"3///1*,,,,,r0