Chow Test - Regression Stability Test

Returns the p-value of the regression stability test (i.e. whether the coefficients in two linear regressions on different data sets are equal).

 

Syntax

ChowTest(Y1, X1, Y2, X2, Mask, Intercept, Return_type)

Y1 is the response or the dependent variable data array of the first data set (one dimensional array of cells (e.g. rows or columns)).

X1 is the independent variables data matrix of the first data set, such that each column represents one variable.

Y2 is the response or the dependent variable data array of the second data set (one dimensional array of cells (e.g. rows or columns)).

X2 is the independent variables data matrix of the second data set, such that each column represents one variable.

Mask is the boolean array to select a subset of the explanatory variables in the model. If missing, all variables in X are included.

Intercept is the regression constant or the intercept value (e.g. zero). If missing, an intercept is not fixed and will be computed from the data set.

Return_type is a switch to select the return output (1 = p-value (default), 2 = test statistics, 3 = standardized residuals).

Method Description
1 p-value
2 test statistics
 

Remarks

  1. The data sets may include missing values.
  2. The model errors ($\varepsilon$) are assumed to be independent and identically distributed from a normal distribution with unknown variance.
  3. Each column in the explanatory (predictor) matrix corresponds to a separate variable.
  4. Each row in the explanatory matrix and corresponding dependent vector correspond to one observation.
  5. Observations (i.e. row) with missing values in X or Y are removed.
  6. Number of observation of each data set must be larger than the number of explanatory variables.
  7. In principle, the Chow test constructs the following regression models:
    • Model 1 (Data set 1):
      $y_t = \alpha_1 + \beta_{1,1}\times X_1 + \beta_{2,1}\times X_2 + \cdots + \epsilon$
    • Model 2 (Data set 2):
      $y_t = \alpha_2 + \beta_{1,2}\times X_1 + \beta_{2,2}\times X_2+ \cdots + \epsilon$
    • Model 3 (Data sets 1 + 2):
      $y_t = \alpha + \beta_1\times X_1 + \beta_2 \times X_2 + \cdots + \epsilon$
  8. The Chow test hypothesis:

    $$ H_{o}= \left\{\begin{matrix} \alpha_1 = \alpha_2 = \alpha \\ \beta_{1,1} = \beta_{1,2} = \beta_1 \\ \beta_{2,1} = \beta_{2,2} = \beta_2 \end{matrix}\right. $$
    $H_{1}: \exists \alpha_i \neq \alpha, \exists \beta_{i,j} \neq \beta_i$

    Where:
    • $H_{o}$ is the null hypothesis.
    • $H_{1}$ is the alternate hypothesis.
    • $\beta_{i,j}$ is the i-th coefficient in the j-th regression model (j=1,2,3).
  9. The Chow statistics are defined as follows:

    $$ \frac{(\textrm{SSE}_C -(\textrm{SSE}_1+\textrm{SSE}_2))/(k)}{(\textrm{SSE}_1+\textrm{SSE}_2)/(N_1+N_2-2k)}. $$

    Where:
    • $\textrm{SSE}$ is the sum of the squared residuals.
    • $K$ is the number of explanatory variables.
    • $N_1$ is the number of non-missing observations in the first data set.
    • $N_2$ is the number of non-missing observations in the second data set.
  10. The Chow test statistics follow an F-distribution with $k$, and $N_1+N_2-2\times K$ degrees of freedom.
  11. The ChowTest function is available starting with version 1.60 APACHE.

Files Examples

References

  • Chow, Gregory C. (1960). "Tests of Equality Between Sets of Coefficients in Two Linear Regressions". Econometrica 28 (3): 591–605.
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