minimac: Minimum Multi-horizon Autocovariances

The Multi-Horizon Test for Market Frictions

This repository provides software packages in Matlab, Python and R for implementing the statistical test developed in Multi-Horizon Test for Market Frictions, published in the Journal of Econometrics.

What the test can do

• Check whether intraday prices are relatively free of microstructure noise.
• Diagnose whether the underlying returns are uncorrelated.
• Detect persistent returns driven by large drifts or drift bursts.
• Outperforms modern variance ratio and portmanteau tests.

The key statistic is defined as

$$ H(Y;\mathbf{K}_\ell)^n_t := \min_{k_j \in \mathbf{K}_\ell} H(Y;k_j)^n_t, $$

where $\mathbf{K}_\ell = {1,2,\ldots,2^\ell},\ \ell = 0,1,2,3$ and the standardized single-horizon testing statistic

$$ H(Y;k)_t^n:=\frac{F(Y;k)_t^n}{\sqrt{\Phi_kQ(Y)_t^n}} $$ is derived from $$ F(Y;k)_t^n:=\frac{1}{2k}\sum_{i=k+1}^{n_t-k}f(Y;k)_i^n,\quad f(Y;k)_i^n:=(Y_i^n-Y_{i-k}^n)(Y_{i+k}^n-Y_i^n),\quad \Phi_k:=\frac16\left(k+\frac1{2k}\right), \quad Q(Y)_t^n:=\sum_{i=2}^{n_t-1}(f(Y;1)_i^n)^2. $$

Getting Started

There are 3 folders in the directory:

• In the ./MATLAB folder, a function script minimac.m that depends on cvs.mat table and a test script test.m are provided.
• Similarly, each of ./R and ./Python contains a corresponding function script that depends on cvs.csv table and a test script/ipynb.

Usage and Examples

• The callable function stat=minimac(y,T) employs uniform syntax across mentioned languages, where y is log price and T is sample period of your dataset (T=1/252 represents data covers 1 day's trading hours).

• summary_minimac(stat) returns formal output for stat=minimac(y,T) testing result (also callable in all mentioned languages). Note that the function is in independent script for MATLAB, while for R and Python, it is integrated in corresponding function script of minimac.

• Before calling minimac(y,T), put function script (e.g., minimac.m, summary_minimac.m) and critical value table (e.g., cvs.mat) into your current directory.

• MATLAB:

  % Import data
  data = readtable('../SPY_20141215_1s.csv'); % import your intraday data here
  y = log(data.PRICE);
  
  % Testing for Market Friction
  stat = minimac(y,1/252);
  summary_minimac(stat)

• R:

  # Load function of minimac
  source("minimac.R")
  
  # Import data
  data <- read.csv("../SPY_20141215_1s.csv") # import your data here
  y <- log(data$PRICE)
  
  # Testing for Market Friction
  stat <- minimac(y,1/252)
  summary_minimac(stat)
  

• Python:

  ## Import modules
  from minimac import minimac, summary_minimac
  import pandas as pd
  import numpy as np
  
  ## Load data and run minimac
  data = pd.read_csv('../SPY_20141215_1s.csv') # input your data here
  y = np.log(data['PRICE'])
  stat = minimac(y,1/252)
  summary_minimac(stat)

• All languages will return identical output

  ================ minimac testing result ================
  >> Testing Statistics:
          K_0      K_1      K_2      K_3
  H -0.683775 -1.61216 -3.90376 -3.90376
  >> Rejection Status:
                 K_0 K_1 K_2 K_3
  1%, one-sided    0   0   1   1
  1%, two-sided    0   0   1   1
  5%, one-sided    0   0   1   1
  5%, two-sided    0   0   1   1
  10%, one-sided   0   1   1   1
  10%, two-sided   0   0   1   1