Niccolò's Blog: VAR estimation and cumulative response to shocks for the Hasbrouck model with different specifications

The aim of this project is the estimation of a VAR process molded on the Hasbrouck model. In particular, the model considered has the following reduced form (RF) specification:

and the following structural form (SF) representation:

where y_t= [ r_t x_t]’, and r_t represents returns from a stock and obtained as the log difference of the mid-quotes in two subsequent periods, i.e.

and x_t = {1,-1}, depending on whether a transaction is initiated by the buyer or the seller, respectively (i.e. whether the realized price is above or below the mid-quote).

The data set includes trades and quotes from the ITCH feed for Amgen (AMGN) and from TAQ for John Deere (DE). The files containing this information are provided below:

Amgen
John Deere

The results discussed below are obtained by running the following Matlab code:

Note that the estimated coefficients refer to the matrix of coefficients generated by the product

for the lags of the two variables r_t and x_t in the SF representation of the VAR model, whereas they refer to the entry A₁₂ in the matrix of coefficients A for the contemporaneous effect of x_t on r_t again in the SF representation.

(a) Estimation of the Hasbrouck VAR:

Stock: AMGN

Equation: r_t

R²

1.4050E-01

Variable	lag	coefficient	t-statistic
constant		1.4872E-06	2.3930E+00
r	1	-1.0112E-01	-1.4629E+01
	2	3.4773E-03	-1.1039E+00
	3	1.9697E-02	1.2987E+00
	4	3.9301E-03	-6.4438E-01
	5	3.9358E-02	4.1271E+00
x	0	-1.3247E-08	-2.7536E+01
	1	3.9012E-08	3.0929E+01
	2	2.6534E-08	6.7267E+00
	3	1.5311E-08	2.8532E+00
	4	3.7657E-09	-9.7301E-01
	5	2.7018E-09	2.0719E-01

Equation: x_t

R²

5.5220E-01

Variable	lag	coefficient	t-statistic
constant		-1.9380E-02	-3.3940E+00
r	1	-1.9325E+02	-2.8418E+00
	2	-1.3483E+02	-1.9670E+00
	3	-1.3676E+02	-1.9956E+00
	4	-3.1328E+01	-4.5879E-01
	5	7.7282E+01	1.2155E+00
x	1	6.1244E-01	7.0156E+01
	2	7.6485E-02	7.2777E+00
	3	5.9862E-02	5.6831E+00
	4	2.8602E-02	2.7174E+00
	5	3.7386E-02	4.1068E+00

Stock: DE

Equation: r_t

R²

2.5751E-02

Variable	lag	coefficient	t-statistic
constant		3.1689E-06	1.0511E+01
r	1	-4.6338E-03	-7.8262E-01
	2	8.6860E-03	1.4647E+00
	3	8.7373E-03	1.4731E+00
	4	2.2391E-02	3.7789E+00
	5	5.5227E-03	9.3799E-01
x	0	4.8916E-06	8.9877E+00
	1	3.5896E-06	5.7774E+00
	2	1.4471E-06	2.3093E+00
	3	1.0292E-07	1.6421E-01
	4	-4.5744E-08	-7.3548E-02
	5	-1.4585E-06	-2.6497E+00

Equation: x_t

R²

6.9150E-01

Variable	lag	coefficient	t-statistic
constant		-2.4893E-02	-7.7720E+00
r	1	-2.2958E+03	-3.7301E+01
	2	-7.0211E+02	-1.1156E+01
	3	-2.8913E+02	-4.5852E+00
	4	-1.7222E+02	-2.7334E+00
	5	8.8098E+00	1.4070E-01
x	1	5.5359E-01	9.5800E+01
	2	1.5162E-01	2.2951E+01
	3	9.3421E-02	1.4062E+01
	4	4.4952E-02	6.8013E+00
	5	7.3822E-02	1.2645E+01

(b) Re-estimation of the VAR using signed volumes (xV_t):

Stock: AMGN

Equation: r_t

R²

3.9045E-02

Variable	lag	coefficient	t-statistic
constant		1.4872E-06	1.9603E+00
r	1	-1.0112E-01	-1.1902E+01
	2	3.4773E-03	4.0721E-01
	3	1.9697E-02	2.3098E+00
	4	3.9301E-03	4.6273E-01
	5	3.9358E-02	4.6970E+00
xV	0	-1.3247E-08	-4.4139E+00
	1	3.9012E-08	1.2884E+01
	2	2.6534E-08	8.6614E+00
	3	1.5311E-08	4.9825E+00
	4	3.7657E-09	1.2271E+00
	5	2.7018E-09	8.8587E-01

Equation: x_t

R²

1.2657E-01

Variable	lag	coefficient	t-statistic
constant		-1.1615E+01	-5.4069E+00
r	1	-3.4625E+04	-1.4378E+00
	2	-1.9473E+04	-8.0455E-01
	3	-3.2947E+04	-1.3631E+00
	4	3.9506E+04	1.6412E+00
	5	3.8296E+04	1.6125E+00
xV	1	1.4283E-01	1.6812E+01
	2	1.4354E-01	1.6696E+01
	3	1.2109E-01	1.4000E+01
	4	9.4511E-02	1.0912E+01
	5	6.2098E-02	7.1966E+00

Stock: DE

Equation: r_t

R²

3.8311E-03

Variable	lag	coefficient	t-statistic
constant		7.9210E-07	2.7816E+00
r	1	5.0775E-03	8.7730E-01
	2	1.0051E-02	1.7380E+00
	3	7.5085E-03	1.2985E+00
	4	2.1588E-02	3.7348E+00
	5	5.4057E-03	9.3608E-01
xV	0	1.2569E-09	3.2170E+00
	1	1.8315E-09	4.6824E+00
	2	1.5372E-09	3.9278E+00
	3	1.1583E-09	2.9587E+00
	4	1.2367E-09	3.1589E+00
	5	9.0419E-10	2.3111E+00

Equation: x_t

R²

9.2510E-03

Variable	lag	coefficient	t-statistic
constant		-3.6456E+01	-8.6528E+00
r	1	-2.9853E+04	-3.4819E-01
	2	-3.7331E+04	-4.3574E-01
	3	-4.2116E+04	-4.9167E-01
	4	2.3718E+04	2.7699E-01
	5	1.4167E+05	1.6560E+00
xV	1	5.6051E-02	9.6884E+00
	2	3.6705E-02	6.3351E+00
	3	3.2688E-02	5.6393E+00
	4	3.1312E-02	5.4016E+00
	5	3.4526E-02	5.9605E+00

(c ) Re-estimation of the VAR one last time using inside depth (id_t):

Stock: AMGN

Equation: r_t

R²

5.3874E-02

Variable	lag	coefficient	t-statistic
constant		-3.7280E-07	-4.8430E-01
r	1	-9.8169E-02	-1.1035E+01
	2	1.4969E-02	1.6748E+00
	3	3.1517E-02	3.5299E+00
	4	1.1959E-02	1.3455E+00
	5	3.8592E-02	4.6378E+00
xV	0	-1.0346E-08	-3.4599E+00
	1	4.2370E-08	1.4023E+01
	2	2.7035E-08	8.7679E+00
	3	1.3838E-08	4.4729E+00
	4	1.9472E-09	6.3012E-01
	5	9.0100E-10	2.9333E-01
id	1	5.4164E-07	6.1803E+00
	2	2.7731E-07	2.4696E+00
	3	-2.0101E-09	-1.7868E-02
	4	-1.0524E-07	-9.3551E-01
	5	-2.6622E-07	-3.0039E+00

Equation: x_t

R²

1.3345E-01

Variable	lag	coefficient	t-statistic
constant		-8.1477E+00	-3.7218E+00
r	1	-1.2791E+04	-5.0533E-01
	2	-3.0477E+04	-1.1985E+00
	3	-7.7233E+04	-3.0411E+00
	4	6.5406E+04	2.5868E+00
	5	3.8298E+04	1.6177E+00
xV	1	1.3959E-01	1.6393E+01
	2	1.3816E-01	1.5891E+01
	3	1.1763E-01	1.3450E+01
	4	1.0258E-01	1.1725E+01
	5	5.7986E-02	6.6453E+00
id	1	-7.0185E-02	-2.8145E-01
	2	-9.6914E-01	-3.0343E+00
	3	-1.0054E+00	-3.1421E+00
	4	2.0109E+00	6.2915E+00
	5	-8.2450E-01	-3.2708E+00

Equation: id_t

R²

7.9614E-01

Variable	lag	coefficient	t-statistic
constant		4.0629E-01	5.2063E+00
r	1	3.0637E+02	3.3953E-01
	2	-3.6986E+02	-4.0802E-01
	3	-1.6861E+03	-1.8624E+00
	4	-3.4692E+03	-3.8490E+00
	5	-1.6520E+03	-1.9575E+00
xV	1	3.1328E-03	1.0321E+01
	2	1.3425E-04	4.3314E-01
	3	-4.0323E-04	-1.2933E+00
	4	-9.0015E-05	-2.8861E-01
	5	3.4966E-04	1.1241E+00
id	1	7.7963E-01	8.7703E+01
	2	5.8936E-02	5.1763E+00
	3	-5.0136E-03	-4.3956E-01
	4	-4.2211E-03	-3.7048E-01
	5	9.6309E-02	1.0718E+01

Stock: DE

Equation: r_t

R²

4.0524E-03

Variable	lag	coefficient	t-statistic
constant		8.1449E-07	2.8587E+00
r	1	4.9502E-03	8.5525E-01
	2	9.9163E-03	1.7146E+00
	3	7.4213E-03	1.2834E+00
	4	2.1481E-02	3.7161E+00
	5	5.3622E-03	9.2855E-01
xV	0	1.3139E-09	3.3366E+00
	1	1.9066E-09	4.8369E+00
	2	1.6040E-09	4.0666E+00
	3	1.2605E-09	3.1951E+00
	4	1.2906E-09	3.2913E+00
	5	9.5463E-10	2.4362E+00
id	1	2.6491E-08	7.2184E-01
	2	-1.0909E-08	-2.1662E-01
	3	1.0118E-08	2.0092E-01
	4	-3.3625E-08	-6.6768E-01
	5	4.4848E-08	1.2182E+00

Equation: x_t

R²

2.4651E-02

Variable	lag	coefficient	t-statistic
constant		-3.7961E+01	-9.0763E+00
r	1	-3.9029E+03	-4.5871E-02
	2	-2.5239E+04	-2.9686E-01
	3	-3.2797E+04	-3.8583E-01
	4	3.3468E+04	3.9386E-01
	5	1.3998E+05	1.6490E+00
xV	1	4.9589E-02	8.5686E+00
	2	3.1439E-02	5.4250E+00
	3	2.8598E-02	4.9330E+00
	4	2.6752E-02	4.6428E+00
	5	2.8875E-02	5.0151E+00
id	1	7.4783E+00	1.3907E+01
	2	-8.9044E+00	-1.2057E+01
	3	-6.0254E-01	-8.1396E-01
	4	-1.4352E+00	-1.9387E+00
	5	5.5825E-01	1.0316E+00

Equation: id_t

R²

8.5838E-01

Variable	lag	coefficient	t-statistic
constant		-3.2536E-02	-7.2264E-01
r	1	8.0694E+01	8.8102E-02
	2	9.2587E+01	1.0117E-01
	3	-5.9768E+02	-6.5317E-01
	4	4.2327E+02	4.6272E-01
	5	-1.8022E+01	-1.9723E-02
xV	1	-1.9007E-04	-3.0509E+00
	2	-1.4832E-04	-2.3776E+00
	3	-1.3555E-04	-2.1721E+00
	4	3.5928E-06	5.7921E-02
	5	-6.4841E-05	-1.0461E+00
id	1	9.3718E-01	1.6190E+02
	2	1.0236E-02	1.2875E+00
	3	7.9848E-03	1.0020E+00
	4	-2.6881E-02	-3.3733E+00
	5	-1.0150E-02	-1.7423E+00