Time series fitting: use of 'trend' in VAR models via statsmodels.tsa.vector_ar.var_model.VAR - statsmodels

Question

I'm using the statsmodels.tsa.vector_ar.var_model.VAR package to fit some bivariate time series. That is, time series of vectors with 2 components: (x1, y1)T, ..., (xN, yN)T.
However, I do not understand how to use the parameter trend in the VAR.fit() function. There are 4 possible values for that parameter ({“c”, “ct”, “ctt”, “n”}) and the explanation given in the doc is: “c” - add constant “ct” - constant and trend “ctt” - constant, linear and quadratic trend “n” - co constant, no trend Note that these are prepended to the columns of the dataset.
What I understand is that the trend parameter allows to perform some pre-processing steps on the time series, before fitting the VAR model. In particular, I expect that, when trend is:
n: the VAR model yt = A1 yt-1 + ... + Ap yt-p is fitted on the series as it is;
c: the mean of the series is removed before the fit. So, defining mu_x = np.mean(x) and mu_y = np.mean(y) the mean values of the univariate series, the VAR model is fitted on the new series: (xi - mu_x, yi - mu_y)i=1,...,N;
ct: the VAR model is fitted on the series where the linear trend has been removed: (xi - a - b * i, yi - c - d * i)i=1,...,N;
ctt: the VAR model is fitted on the series where the quadratic trend has been removed: (xi - m - n * i - p* i2, yi - q - r * i - s* i2)i=1,...,N.
However, this is not what results from some attempts. Here after my code with a few examples.
1) I define a bivariate time series ts0 and, from that, 2 other transformed time series:
ts1: computed from ts0 by subtracting the mean;
ts2: compuetd from ts1 by removing the linear trend.
### original bivariate time series
ts0 = np.array([[-2.27390781, 4.89021106],
[ 0.56894665, 1.57356924],
[-1.54000883, -1.97090661],
[ 0.60917182, 0.3684891 ],
[-2.518067 , 0.42002855],
[-0.4788302 , -0.63284219],
[ 1.8208968 , 2.27831329],
[-1.65226058, -2.6647208 ],
[ 0.72437619, 1.09676352],
[-3.190304 , -0.48445386],
[ 0.41290842, 1.01441648]])
N = np.shape(ts0)[0] # number of time steps
### pre-processing of the time series by hand:
# remove the mean values
ts1 = ts0 - np.mean(ts0,axis=0) # translated time series
# remove the linear trend
time = np.arange(N)
b,d = (np.cov(time,ts1[:,0],ddof=1)[0,1]/np.var(time,ddof=1),
np.cov(time,ts1[:,1],ddof=1)[0,1]/np.var(time,ddof=1)) # ang. coef
a,c = (np.mean(ts1[:,0]) - b*np.mean(time),
np.mean(ts1[:,1]) - d*np.mean(time)) # intercept
x_lin,y_lin = ([a+b*i for i in time], [c+d*i for i in time]) # linear predictions
ts2 = ts1 - np.transpose([x_lin, y_lin]) # translated and rotated time series
In the following, I'll forecast the vector N+1 for the three time series, by setting different values for the parameter trend.
2) I expect the forecast for ts2 to return the same result regardeless of trend = n, c, or ct. This is not true:
### fit the series ts2
lag = 1
var_2 = VAR(pd.DataFrame(data=ts2))
var_2n = var_2.fit(lag, trend='n') # 3 fits
var_2c = var_2.fit(lag, trend='c')
var_2ct = var_2.fit(lag, trend='ct')
pred_2n = var_2n.forecast(y=ts2[-lag:,:], steps=1)[0] # 3 predictions
pred_2c = var_2c.forecast(y=ts2[-lag:,:], steps=1)[0]
pred_2ct = var_2ct.forecast(y=ts2[-lag:,:], steps=1)[0]
### plot
fig, axs = plt.subplots(1,2,figsize=(12,3))
# coord x
ax = axs[0]
ax.plot(time, ts2[:,0])
ax.hlines(xmin=time[0], xmax=time[-1], y=np.mean(ts2[:,0]), color='g', label='linear fit')
ax.scatter(N, pred_2n[0], marker='x', c='salmon', label='VAR(ts2, trend=n)')
ax.scatter(N, pred_2c[0], marker='D', c='orange', label='VAR(ts2, trend=c)', alpha=0.5)
ax.scatter(N, pred_2ct[0], marker='.', c='k', label='VAR(ts2, trend=ct)', alpha=0.5)
ax.grid(True)
ax.set_xlabel('time')
ax.set_ylabel('ts2: coord x')
ax.legend(loc='lower left')
# coord y
ax = axs[1]
ax.plot(time, ts2[:,1])
ax.hlines(xmin=time[0], xmax=time[-1], y=np.mean(ts2[:,1]), color='g', label='linear fit')
ax.scatter(N, pred_2n[1], marker='x', c='salmon')
ax.scatter(N, pred_2c[1], marker='D', c='orange')
ax.scatter(N, pred_2ct[1], marker='.', c='k')
ax.grid(True)
ax.set_xlabel('time')
ax.set_ylabel('ts2: coord y')
fig.tight_layout()
enter image description here
3) I expect to get the same prediction when:
I fit the ts2 series with trend=n and I transform back the points on the ts1 series;
I fit the ts1 series with trend=c.
This is not true:
### compute predictions
pred_2n_backto_1 = np.array([a,c]) + np.array([b,d])*N + pred_2n # prediction VAR(ts2, trend=n) back to ts1
var_1 = VAR(pd.DataFrame(data=ts1)) # prediction VAR(ts1, trend=c)
var_1c = var_1.fit(lag, trend='c')
pred_1c = var_1c.forecast(y=ts1[-lag:,:], steps=1)[0]
### plot
fig, axs = plt.subplots(1,2,figsize=(12,3))
# coord x
ax = axs[0]
ax.plot(time, ts1[:,0])
ax.plot(time, x_lin, c='g', label='linear fit')
ax.scatter(N, pred_2n_backto_1[0], marker='x', c='salmon', label='VAR(ts2, trend=n) back to ts1')
ax.scatter(N, pred_1c[0], marker='o', c='b', alpha=0.5, label='VAR(ts1, trend=c)')
ax.grid(True)
ax.set_xlabel('time')
ax.set_ylabel('ts1: coord x')
ax.legend(loc='lower left')
# coord y
ax = axs[1]
ax.plot(time, ts1[:,1])
ax.plot(time, y_lin, c='g', label='linear fit')
ax.scatter(N, pred_2n_backto_1[1], marker='x', c='salmon', label='VAR(ts2, trend=n) back to ts1')
ax.scatter(N, pred_1c[1], marker='o', c='b', alpha=0.5, label='VAR(ts1, trend=c)')
ax.grid(True)
ax.set_xlabel('time')
ax.set_ylabel('ts1: coord y')
fig.tight_layout()
enter image description here
4) More general investigation:
I compute predictions for the 3 series (ts0, ts1, ts2), with trend={n, c, ct};
I transform back those values through the pre-processing steps, in order to have all the predictions on the ts0 series;
I compare the 9 predictions.
I find that only some predictions are equals, but I don't understand the logic behind. In particular:
VAR(ts2, trend='ct'), VAR(ts1, trend='ct') and VAR(ts0, trend='ct') return the same predictions;
VAR(ts1, trend='c') and VAR(ts0, trend='c') also return the same predictions.
# pre-processing parameters
intercepts = np.array([a,c])
ang_coefs = np.array([b,d])
mean_ts0 = np.mean(ts0, axis=0)
# predictions for ts2 back to ts0
pred_2n_backto_0 = (intercepts + ang_coefs*N + pred_2n) + mean_ts0
pred_2c_backto_0 = (intercepts + ang_coefs*N + pred_2c) + mean_ts0
pred_2ct_backto_0 = (intercepts + ang_coefs*N + pred_2ct) + mean_ts0
# predictions for ts1 back to ts0
var_1 = VAR(pd.DataFrame(data=ts1))
var_1n = var_1.fit(lag, trend='n')
var_1c = var_1.fit(lag, trend='c')
var_1ct = var_1.fit(lag, trend='ct')
pred_1n = var_1n.forecast(y=ts1[-lag:,:], steps=1)[0]
pred_1c = var_1c.forecast(y=ts1[-lag:,:], steps=1)[0]
pred_1ct = var_1ct.forecast(y=ts1[-lag:,:], steps=1)[0]
pred_1n_backto_0 = pred_1n + mean_ts0
pred_1c_backto_0 = pred_1c + mean_ts0
pred_1ct_backto_0 = pred_1ct + mean_ts0
# predictions for ts0
var_0 = VAR(pd.DataFrame(data=ts0))
var_0n = var_0.fit(lag, trend='n')
var_0c = var_0.fit(lag, trend='c')
var_0ct = var_0.fit(lag, trend='ct')
pred_0n = var_0n.forecast(y=ts0[-lag:,:], steps=1)[0]
pred_0c = var_0c.forecast(y=ts0[-lag:,:], steps=1)[0]
pred_0ct = var_0ct.forecast(y=ts0[-lag:,:], steps=1)[0]
# compare predictions
list_pred_x = [pred_2n_backto_0[0], pred_2c_backto_0[0], pred_2ct_backto_0[0],
pred_1n_backto_0[0], pred_1c_backto_0[0], pred_1ct_backto_0[0],
pred_0n[0], pred_0c[0], pred_0ct[0]]
list_pred_y = [pred_2n_backto_0[1], pred_2c_backto_0[1], pred_2ct_backto_0[1],
pred_1n_backto_0[1], pred_1c_backto_0[1], pred_1ct_backto_0[1],
pred_0n[1], pred_0c[1], pred_0ct[1]]
list_labels = ['pred_2n_backto_0', 'pred_2c_backto_0', 'pred_2ct_backto_0',
'pred_1n_backto_0', 'pred_1c_backto_0', 'pred_1ct_backto_0',
'pred_0n', 'pred_0c', 'pred_0ct']
mat_x = mat_y = np.zeros((9,9))
for i in range(9):
x_i = list_pred_x[i]
y_i = list_pred_y[i]
for j in range(9):
x_j = list_pred_x[j]
y_j = list_pred_y[j]
mat_x[i,j] = np.log(np.fabs(x_i - x_j))
mat_y[i,j] = np.log(np.fabs(y_i - y_j))
fig = plt.figure(figsize=(16,5))
plt.subplot(1,2,1)
plt.imshow(mat_x, cmap='plasma')
plt.colorbar()
plt.title('predictions for coord x:' + '\n' + r'$val_{ij} = log(|pred_i - pred_j|)$')
plt.xticks(np.arange(9), list_labels, rotation=90)
plt.yticks(np.arange(9), list_labels)
plt.subplot(1,2,2)
plt.imshow(mat_y, cmap='plasma')
plt.colorbar()
plt.title('predictions for coord y:' + '\n' + r'$val_{ij} = log(|pred_i - pred_j|)$')
plt.xticks(np.arange(9), list_labels, rotation=90)
plt.yticks(np.arange(9), list_labels)
plt.show()
enter image description here
Could someone explain to me the logic of the trend parameter in the statsmodels.tsa.vector_ar.var_model.VAR package ?
Thank you in advance :)