Issues in example 14.3.1.3 page 289 about quantq HOT 1 CLOSED

The unbiased estimator of a VAR(p) process is YZ'(ZZ')^-1 as described in equation (14.22) on p. 288 where Z is a Kp+1 x 1 matrix defined as Zt = (1,y_t, ..., y_{t-p+1})'.

The code of Zt is given in p. 293. It looks like you are using the variable Zt of the vector autogressive model with exogenous variable VARX(p,q) defined as Zt = (Y_0,...,Y_{T-1}; X_0,..., X_{T-1}; .... x_1,...,x_T)' (see eq. 14.29 on p 297) instead of Zt defined in eq. 14.22 on p.288.

I will create two name spaces: one for VAR and one for VARX such that Zt for VAR is not overrriden by Zt for VARX.

To reproduce the results on p. 291, please use the code of the book.

.quantQ.ts.vec:{[a]
// a -- matrix
i:(count raze a);
j:1;
v:(flip a)[0;];
while[j<i;
v:v,/(flip a)[j;];
j:j+1];
:v;
};

.quantQ.ts.zt:{[y;t;p]
// y -- historical data (list of floating numbers)
// t -- time point (positive integer)
// p -- lags (positive integer) corresponding to the order of the VAR(p) process
:1f,.quantQ.ts.vec[y[;(t-1)-til p]];
};

.quantQ.ts.Z:{[y;p]
// y -- historical data (list of floating numbers)
// p -- lags (positive integer) corresponding to the order of the VAR(p) process
:flip {[y;i;p]
.quantQ.ts.zt[y;i;p]}[y;;p] each p + til (count y[0])-p;
};

.quantQ.ts.Y:{[y;p]
// y -- historical data (list of floating numbers)
// p -- lags (positive integer) corresponding to the order of the VAR(p) process
:y[;p+til (count y[0;])-p];
};

.quantQ.ts.varpest:{[y;p]
// y -- historical data (list of floating numbers) used to estimate the VAR(p) model.
// p -- order of the VAR(p) model
:.quantQ.ts.Y[y;p] mmu (flip .quantQ.ts.Z[y;p]) mmu
inv[.quantQ.ts.Z[y;p] mmu flip .quantQ.ts.Z[y;p]];
};

.quantQ.ts.covarianceResidual:{[y;p]
// y -- historical data (list of floating numbers)
// p -- order of the VAR(p) process
T:(count y[0])-p;
K:count y;
coef:1f%((T-Kp)-1f);
:coef(.quantQ.ts.Y[y;p] mmu flip .quantQ.ts.Y[y;p]) -
(.quantQ.ts.Y[y;p] mmu flip .quantQ.ts.Z[y;p]) mmu
inv[.quantQ.ts.Z[y;p] mmu flip .quantQ.ts.Z[y;p]] mmu
(.quantQ.ts.Z[y;p] mmu flip .quantQ.ts.Y[y;p]);
};

.quantQ.ts.eye:{[k]
// k -- rank of matrix
:`float$(til k)=/:til k;
};

.quantQ.ts.beta:{[y;p]
// y -- historical data (list of floating numbers)
// p -- order of the VAR(p) process
:(1f%(count y[0])-p) * (.quantQ.ts.Y[y;p] mmu
(.quantQ.ts.eye[(count y[0])-p] - ((flip .quantQ.ts.Z[y;p]) mmu
inv[.quantQ.ts.Z[y;p] mmu flip .quantQ.ts.Z[y;p]] mmu
.quantQ.ts.Z[y;p]))) mmu flip .quantQ.ts.Y[y;p];
};

.quantQ.ts.varOrder:{[y;m]
// y -- historical data (list of floating numbers
// m -- maximum order of the VAR process
T:count y[0];
K:count y;
i:m;
k:v1:v2:();
while[i>0;
lr:T*(log .quantQ.mat.det[.quantQ.ts.beta[y;i-1]])-(log .quantQ.mat.det[.quantQ.ts.beta[y;i]]);
k:k,(m-i);
v1:v1,lr;
v2:v2,.quantQ.ts.pValueChi2[lr;K*K];
i:i-1];
:([p:k]lr:v1;pvalue:v2);
};

.quantQ.ts.pValueChi2:{[chi;nu]
// chi -- instance of distribution (floating)
// nu -- degrees of freedom of the Chi2 distribution
u:chi%2f;
v:nu%2f;
p:(u-v)+1f;
pi:3.141592654f;
term1:exp[v-u]%(psqrt[2pi]);
term2:(u % v) xexp v;
term3:1f-((v-1f)%((pp)+2fu));
term4:(12f*(v xexp 1.5f))%((12fv)+1);
:term1term2term3term4;
};

from quantq.