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FOSSEE_Signal_Processing_Toolbox >> FOSSEE_Signal_Processing_Toolbox > arch_test

# arch_test

Perform a Lagrange Multiplier (LM) test for conditional heteroscedasticity.

### Syntax

    [pval, lm] = arch_test(y, x, p)

### Parameters

y:

  Array-like. Dependent variable of the regression model.

x:

  Array-like. Independent variables of the regression model. If x is a scalar integer k, it represents the order of autoregression.

p:

  Integer. Number of lagged squared residuals to include in the heteroscedasticity model.

pval:

  Float. p-value of the LM test.

lm:

  Float. Lagrange Multiplier test statistic.

### Description

This function performs a Lagrange Multiplier (LM) test for conditional heteroscedasticity in a linear regression model: y = x * b + e The null hypothesis is no conditional heteroscedasticity, while the alternative is CH(p): y(t) = b(1) * x(t,1) + … + b(k) * x(t,k) + e(t), given y up to t-1 and x up to t, e(t) is N(0, h(t)) with h(t) = v + a(1) * e(t-1)^2 + … + a(p) * e(t-p)^2, and the null is a(1) == … == a(p) == 0. If the second argument is a scalar integer, k, perform the same test in a linear autoregression model of order k, i.e., with [1, y(t-1), …, y(t-k)] as the t-th row of x. Under the null, LM approximately has a chisquare distribution with p degrees of freedom and pval is the p-value (1 minus the CDF of this distribution at LM) of the test. If no output argument is given, the p-value is displayed.

### Examples

    // Perform an ARCH test for conditional heteroscedasticity in financial data

    // Generate synthetic financial returns data
    t = 1:500; // Time points
    y = 0.05 * t + 0.2 * sin(2 * %pi * t / 50) + 0.1 * rand(1, length(t)); // Simulated returns with trend and noise

    // Define lagged independent variable (autoregressive model of order 1)
    x = [ones(length(y)-1, 1), y(1:$-1)'] // Lagged values of y

    // Perform the ARCH test with 2 lags
    [pval, lm] = arch_test(y(2:$)', x, 2)

### See also

- ols — Ordinary least squares estimation.
- autoreg_matrix — Generate a regressor matrix for autoregressions from a time series.

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