def test_het_breusch_pagan(self):
res = self.res
bptest = dict(statistic=0.709924388395087, pvalue=0.701199952134347, parameters=(2,), distr="f")
bp = smsdia.het_breuschpagan(res.resid, res.model.exog)
compare_t_est(bp, bptest, decimal=(12, 12))
resid = fsm_results.resid
qq1 = sm.qqplot(resid, line ='45', fit=True, dist=stats.t)
y = fsm_df["price"]
y_hat = fsm_results.predict()
fig2, ax2 = plt.subplots()
ax2.set(xlabel="Price",
ylabel="Residuals")
ax2.scatter(x=y_hat, y=y_hat-y, color="blue", alpha=0.2);
lm, lm_p_value, fvalue, f_p_value = het_breuschpagan(y-y_hat, fsm_df[["price"]])
print("Lagrange Multiplier p-value:", lm_p_value)
print("F-statistic p-value:", f_p_value)
#necessary??
preds = fsm_results.predict()
def plot_predictions(y_true, y_hat):
fig, axis = plt.subplots()
axis.scatter(y_true,y_hat,label='Model Output', alpha=.5, edgecolor='black')
y_equalsx = np.linspace(0,y_true.max())
def quiz4():
df = pd.read_csv(utils.PATH.COURSE_FILE(4, 'botswana.tsv', 'week3'),
sep='\t')
print(df.head())
print(df.info())
# Q1
religion = df['religion'].value_counts()
print('---- Q1:', religion)
# Q2
df_nonna = df.dropna()
print('---- Q2:', df_nonna.shape)
# Q3
df['nevermarr'] = df['agefm'].apply(lambda x: 0 if x >= 0 else 1)
df.drop(['evermarr'], axis=1, inplace=True)
df['agefm'].fillna(0, inplace=True)
df['heduc'] = df.apply(lambda row: row['heduc']
if row['nevermarr'] == 0 else -1,
axis=1)
heduc_na = df['heduc'].isnull().value_counts()
print('---- Q3:', heduc_na)
# Q4
df['idlnchld_noans'] = df['idlnchld'].apply(lambda x: 0 if x >= 0 else -1)
df['idlnchld'].fillna(-1, inplace=True)
df['heduc_noans'] = df['heduc'].apply(lambda x: 0 if x >= 0 else -1)
df['heduc'].fillna(-1, inplace=True)
df['usemeth_noans'] = df['usemeth'].apply(lambda x: 0 if x >= 0 else -1)
df['usemeth'].fillna(-1, inplace=True)
df.dropna(inplace=True)
print(df.shape)
print('---- Q4:', df.shape[0] * df.shape[1])
# Q5
model = smf.ols('ceb ~ age + educ + religion + idlnchld + knowmeth + usemeth + agefm + heduc + urban + electric + radio + tv + bicycle +'\
'nevermarr + idlnchld_noans + heduc_noans + usemeth_noans', data=df)
fitted = model.fit()
sum = fitted.summary()
print('---- Q5"\n', sum)
# Q6
print('---- Q6: p=%f' %
het_breuschpagan(fitted.resid, fitted.model.exog)[1])
# Q7
model1 = smf.ols('ceb ~ age + educ + religion + idlnchld + knowmeth + usemeth + agefm + heduc + urban + electric + radio + tv + bicycle +'\
'nevermarr + idlnchld_noans + heduc_noans + usemeth_noans', data=df)
fitted = model1.fit(cov_type='HC1')
sum = fitted.summary()
print('---- Q7\n:', sum)
# Q8
model2 = smf.ols('ceb ~ age + educ + idlnchld + knowmeth + usemeth + agefm + heduc + urban + electric + bicycle +'\
'nevermarr + idlnchld_noans + heduc_noans + usemeth_noans', data=df)
fitted = model2.fit()
print('p=%f' % het_breuschpagan(fitted.resid, fitted.model.exog)[1])
model2 = smf.ols('ceb ~ age + educ + idlnchld + knowmeth + usemeth + agefm + heduc + urban + electric + bicycle +'\
'nevermarr + idlnchld_noans + heduc_noans + usemeth_noans', data=df)
fitted = model2.fit(cov_type='HC1')
print("---- Q8: F=%f, p=%f, k1=%f" %
model1.fit().compare_f_test(model2.fit()))
# Q9
model3 = smf.ols('ceb ~ age + educ + idlnchld + knowmeth + agefm + heduc + urban + electric + bicycle +'\
'nevermarr + idlnchld_noans + heduc_noans', data=df)
fitted = model3.fit()
print("---- Q9: F=%f, p=%f, k1=%f" %
model2.fit().compare_f_test(model3.fit()))
# Q10
model = smf.ols('ceb ~ age + educ + idlnchld + knowmeth + agefm + heduc + urban + electric + bicycle +'\
'nevermarr + idlnchld_noans + heduc_noans', data=df)
fitted = model.fit(cov_type='HC1')
sum = fitted.summary()
print('---- Q10\n:', sum)
return
def breusch_pagan():
names = [
"Lagrange multiplier statistic", "p-value", "f-value", "f p-value"
]
test = het_breuschpagan(res.resid, res.model.exog)
print(dict(zip(names, test)))
import pandas as pd import os import statsmodels.api as sm from statsmodels.stats.diagnostic import het_breuschpagan base_path = os.path.dirname(__file__) base_path = os.path.join(base_path, 'Aula 9') df = pd.read_stata(os.path.join(base_path, 'HPRICE1-Kayo.dta')) print(df.head()) # df.corr() X = df[['lotsize', 'sqrft', 'bdrms']] y = df['price'] X2 = sm.add_constant(X) # model = sm.RLM(y, X2, M=sm.robust.norms.HuberT()) model = sm.OLS(y, X2) results = model.fit(cov_type='HC0') # Statsmodels gives R-like statistical output print(results.summary()) name = ['Lagrange multiplier statistic', 'p-value', 'f-value', 'f p-value'] bp = het_breuschpagan(results.resid, results.model.exog) bp df = pd.DataFrame(name, bp) print(df)
#define input
X2 = sm.add_constant(X)
# create OLS model
model = sm.OLS(Y, X2)
# fit data
est = model.fit()
# test for heteroscedasticity (want p values over 0.05)
_, pval, _, f_pval = diag.het_white(est.resid, est.model.exog, retres=False)
print(pval, f_pval)
print('_' * 100)
_, pval, _, f_pval = diag.het_breuschpagan(est.resid, est.model.exog)
print(pval, f_pval)
print('_' * 100)
# values greater than 0.05 there is no heterodasticity
# test for autocorrelation want p values over 0.05)
lag = min(10, (len(X) // 5))
print("number of lags is {}".format(lag))
ibvalue, pval = diag.acorr_ljungbox(est.resid, lags=lag)
print(min(pval))
print('_' * 100)
# check residuals are normal (visual, close to line)
sm.qqplot(est.resid, line='s')
pylab.show()
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)
diag.het_goldfeldquandt(modelMR2.resid, modelMR2.model.exog) diag.het_breuschpagan(modelMR4.resid, modelMR4.model.exog) diag.het_white(modelMR2.resid, modelMR2.model.exog, retres = False) diag.(modelMR2.resid, modelMR2.model.exog) diag.acorr_ljungbox(modelMR2.resid)
def breusch_pagan_test(dataset, regression_dict, results_dict):
for key in regression_dict.keys():
BP_statistic = smd.het_breuschpagan(results_dict[key].resid,
dataset[regression_dict[key][1:]])
print("{}\nBP:\t{:.2f}\nP-Val:\t{:.4f}".format(key, BP_statistic[0],
BP_statistic[1]))
# Fit Random Forest Regressor according to best hyperparameters
regressor = RandomForestRegressor(random_state=0, n_estimators=500, max_depth=3, max_features=4,
min_samples_leaf=1, bootstrap=True)
regressor.fit(X, y)
# Obtain R2 squared and adjusted R-squared
r2 = regressor.score(X,y)
RAdjusted = 1-(((1-r2)*len(dataset['demand_Maa'])/(len(dataset['demand_Maa'])-5-1)))
# Apply 10-fold cross validation
maescores = cross_val_score(regressor, X, y, scoring='neg_mean_absolute_error', cv=10, n_jobs=-1)
# Obtain AIC and BIC value
y_hat = regressor.predict(X)
residuals = y - y_hat
res = het_breuschpagan(residuals, X)
sse = sum(residuals**2)
AIC = 2*4 - 2*np.log(sse)
BIC = 54*np.log(sse/54) + 4*np.log(54)
# Print scores
#print("The Mean Absolute Error of the Random Forest Regressor is " + str(abs(np.mean(maescores))))
print("The R2 squared of the Random Forest Regressor is " + str(r2))
print("The adjusted R2 squared of the Random Forest Regressor is " + str(RAdjusted))
print("The AIC score of the Random Forest Regressor is " + str(AIC))
print("The BIC score of the Random Forest Regressor " + str(BIC))
print("The p-value from the Breuschpagan test is " + str(res))
# Plot feature importance
feature_import = pd.DataFrame(data=regressor.feature_importances_, index=['own_price','compe_price','inc_per_capita','pro_exp','Annual_Prod_MSeeds'], columns=['values'])
feature_import.sort_values(['values'], ascending=False, inplace=True)
variance_inflation_factor(X.values, i) for i in range(X.shape[1])
]
vif["features"] = X.columns
vif
# <h3>3. Homoscedasticity</h3>
#
# When residuals do not have constant variance (they exhibit heteroscedasticity), it is difficult to determine the true standard deviation of the forecast errors, usually resulting in confidence intervals that are too wide/narrow. For example, if the variance of the residuals is increasing over time, confidence intervals for out-of-sample predictions will be unrealistically narrow.
# In[49]:
from statsmodels.stats.diagnostic import het_breuschpagan
from statsmodels.stats.diagnostic import het_white
# breuschpagan test
bp_test = het_breuschpagan(res.resids, df[independent_vars].dropna())
labels = ['BP Statistic', 'BP-Test p-value', 'F-Statistic', 'F-Test p-value']
print(pd.Series(zip(labels, bp_test)))
# In[50]:
# visualization of heteroscedasticity
fitted = res.fitted_values
residuals = res.resids
plt.figure(figsize=(12, 8))
sns.regplot(x=fitted, y=residuals)
plt.xlabel('Fitted Values')
plt.xlim([4, 6])
plt.show()
# <center><h2>WEIGHTED REGRESSION</h2></center>
y_pred1 = result1.predict(X1)
err1 = y1 - y_pred1
plt.scatter(y1, y_pred1)
plt.plot(xx, xx, color='k')
plt.ylabel('Predicted')
plt.xlabel('Real')
##Need to find another model, when price is higher, it seems under-estimate
plt.hist(err1, bins=50)
probplot(err1, plot=plt)
##
diagnostic.het_breuschpagan(err1, X1) ##result ftest, chi-square
diagnostic.het_breuschpagan(err1, X1[['bedrooms', 'bathrooms']])
##to delete outliers
cond = (house['price'] < 1000000) & (house['price'] >= 20000)
X2 = house[cond][varlist]
y2 = house[cond]['price']
X2 = sm.add_constant(X2)
y2.plot.kde()
model2 = sm.OLS(y2, X2)
result2 = model2.fit()
result2.summary()
y_pred2 = result2.predict(X2)
#plt.title('Stadium Percentage - Reg(p-value) - Normal Q-Q')
plt.savefig(savepath+'QQPlot.jpeg')
plt.show()
_,p=shapiro(y_train_res2)
print(p)
if p>0.05:
x="The residuals seem to come from Gaussian process"
else:
x="The normality assumption may not hold"
pvalue=pd.DataFrame(['Stadium Percentage Regression (p-value)',p,x]).T
pvalue=pvalue.rename(columns={0:'Model',1:'p-value',2:'Interpretation'})
savedfile=pvalue.to_csv('C:/Users/bcheasty/OneDrive - Athlone Institute Of Technology/Research Project/Data Set Creation/Data/Model/pvalue7.csv',index=False)
#Breuschpagan test
bp_test = het_breuschpagan(y_train_res, X_train2)
labels = ['LM Statistic', 'LM-Test p-value', 'F-Statistic', 'F-Test p-value']
bp=(dict(zip(labels, bp_test)))
bp=pd.DataFrame(bp,index=[0])
bp['Model']='Stadium Percentage - Reg(p-value)'
bp.loc[(bp['F-Test p-value']<0.05),'Interpretation']='The Residuals seem to be Heteroskedastic'
bp.loc[(bp['F-Test p-value']>0.05),'Interpretation']='The Residuals seem to be Homoskedastic'
bp=bp[['Model','F-Test p-value','Interpretation']]
savedfile=bp.to_csv('C:/Users/bcheasty/OneDrive - Athlone Institute Of Technology/Research Project/Data Set Creation/Data/Model/bp7.csv',index=False)
#Scale Loction
plot_lm_3 = plt.figure()
plt.scatter(y_train_pred, y_train_res, alpha=0.5,
c='steelblue', marker='o', edgecolor='white',
label='Training data');
sns.regplot(y_train_pred, y_train_res,
# Feel free to adjust the chart size
plt.figure(figsize = (15,7.5))
plt.plot(result['2016-7':'2016-9'].index,result.loc['2016-7':'2016-9','simple regression'])
plt.plot(result['2016-7':'2016-9'].index,result.loc['2016-7':'2016-9','fama_french'])
plt.plot(result['2016-7':'2016-9'].index,result.loc['2016-7':'2016-9','sample'])
plt.legend()
plt.show()
# In[22]:
plt.figure()
fama_model,resid.plot.density()
plt.show()
# In[23]:
print ('Residual mean:', np.mean(fama_model.resid))
# In[ ]:
from statsmodels.stats import diagnostic as dia
het = dia.het_breuschpagan(fama_model.resid,fama_df[['MKT','SMB','HML','RMW','CMA']][1:])
print ('p-value: ', het[-1])
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, maxlag=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, maxlag=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, maxlag=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)
plt.plot(RESID[k])
plt.ylabel('Residual amplitude',fontsize = 13)
plt.xlabel('Week',fontsize = 13)
import statsmodels.api as sm
sm.graphics.tsa.plot_acf(RESID[k],lags=25)
np.array(abs(RESID[k])).sum()/(len(RESID[k]))
from statsmodels.stats.diagnostic import het_breuschpagan
from statsmodels.stats.diagnostic import het_white
df_resid = pd.DataFrame(RESID[k],columns = ['resid'])
X= np.concatenate((np.ones(30).reshape(-1,1),x[-30:,:]),axis = 1)
white = het_white(list(df_resid.resid), X )
breuschpagan = het_breuschpagan(list(df_resid.resid), X )
from statsmodels.stats.diagnostic import het_breuschpagan
from statsmodels.stats.diagnostic import het_white
import pandas as pd
import statsmodels.api as sm
from statsmodels.formula.api import ols
statecrime_df = sm.datasets.statecrime.load_pandas().data
f ='violent~hs_grad+poverty+single+urban'
statecrime_model = ols(formula=f, data=statecrime_df).fit()
white_test = het_white(np.array(statecrime_model.resid)[-30:], x[-30:,:] )
np.array(statecrime_model.model.exog)[-30:,:]
IndVariable, IndVariable2, IndVariable3, IndVariable4, IndVariable5
]], DataExcel['demand_Maa'])))
print("Average demands:" + str(np.mean(DataExcel['demand_Maa'])))
print("Percentage MSE: " +
str(np.mean(Scores) / np.mean(DataExcel['demand_Maa'])))
print("-----------------------------------------------")
print(SecondRegressor.feature_importances_ * 100)
print("-----------------------------------------------")
r2 = r2_score(y_true=y_actual, y_pred=y_pred)
print("The R Squared is: " + str(r2))
AIC, BIC = AICCalc(y_actual, y_pred, 5)
print("The AIC is: " + str(AIC))
print("The BIC is: " + str(BIC))
print("-----------------------------------------------")
Res, lm_pvalue, fvalue, pvalue = het_breuschpagan(
Residuals, DataExcel[[
IndVariable, IndVariable2, IndVariable3, IndVariable4, IndVariable5
]])
print("P Value of the Test: " + str(pvalue))
print("-----------------------------------------------")
# Computing the adjusted R squared
RAdjusted = 1 - (((1 - r2) * len(DataExcel['demand_Maa']) /
(len(DataExcel['demand_Maa']) - 5 - 1)))
print("The adjusted R squared is: " + str(RAdjusted))
def AICCalc(Y_Actual, Y_Predicted, N_Variables):
res = Y_Actual - Y_Predicted
SSE = sum(res**2)
AIC = 2 * N_Variables - 2 * math.log(SSE)
BIC = len(Y_Actual) * math.log(
SSE / len(Y_Actual)) + N_Variables * math.log(len(Y_Actual))
