statespace_diagnostics

Goodness of fit (DK §7.4) and diagnostic checking (DK §2.12, §7.5).

Residuals

subroutine standardized_residuals(fres, e)
  real(dp), intent(out) :: e(:, :)                   ! (p, n)
subroutine auxiliary_residuals(rep, sres, eps_std, eta_std)
  real(dp), intent(out) :: eps_std(:, :), eta_std(:, :)   ! (p, n), (r, n)
subroutine auxiliary_residuals_vector(rep, sres, eps_std, eta_std)
subroutine de_jong_penzer(rep, fres, sres, r_stat, e_stat)
  real(dp), intent(out) :: r_stat(:, :), e_stat(:, :)     ! (m, n), (p, n)
subroutine least_squares_residuals(rep, first, last, vplus, info)
  real(dp), intent(out) :: vplus(:, :)               ! (p, n)
integer function diagnostic_start(rep, fres)

standardized_residuals

\(e_t = L_t^{-1} v_t\) over the observed elements, with \(F_{t,oo} = L_t L_t'\) (Cholesky, as in statsmodels); NaN for missing elements and in the diffuse periods (statsmodels reports 0).

auxiliary_residuals

The smoothed disturbances standardized element by element by \(Var(\hat\varepsilon_t) = H_t - Var(\varepsilon_t | Y)\) and \(Var(\hat\eta_t) = Q_t - Var(\eta_t | Y)\), for detecting outliers and structural breaks (DK §2.12.2, §7.5). NaN where the variance is zero.

auxiliary_residuals_vector

The same standardized as vectors, \(B_t \hat\varepsilon_t\) with \(B_t' B_t = Var(\hat\varepsilon_t)^{-1}\) (DK §7.5).

de_jong_penzer

The shock statistics of de Jong and Penzer (1998) (DK §7.5): \(r_{i,t} / \sqrt{N_{ii,t}}\) for the state equation and \(e_{i,t} / \sqrt{D_{ii,t}}\) for the observation equation, with \(e_t = F_t^{-1} v_t - K_t' r_t\) and \(D_t = F_t^{-1} + K_t' N_t K_t\) (DK §4.5.3). NaN where undefined.

least_squares_residuals

DK §6.2.4: with regression coefficients in states first..last (diffuse, constant), the residuals at their full-sample estimate \(\hat\beta\), computed as the innovations of the model with those states fixed at \(\hat\beta\). The innovations of the original model are the recursive residuals.

diagnostic_start

First period of the residuals to use in tests: after the burn-in and the diffuse periods.

Tests

On a residual series, NaNs skipped, as in statsmodels (DK §2.12.1):

subroutine ljung_box(x, stat, pvalue, model_df)
  real(dp), intent(in) :: x(:)
  real(dp), intent(out) :: stat(:), pvalue(:)     ! lags 1..size(stat)
  integer, intent(in), optional :: model_df
subroutine jarque_bera(x, jb, pvalue, skew, kurtosis)
subroutine breakvar_test(x, stat, pvalue, h)
  integer, intent(in), optional :: h              ! default n / 3

ljung_box

\(Q(k) = n(n+2) \sum_{j=1}^k \rho_j^2 / (n - j)\) against \(\chi^2_{k - model\_df}\). Pairs with a NaN are skipped.

jarque_bera

Normality from skewness and kurtosis, against \(\chi^2_2\).

breakvar_test

Heteroskedasticity: H(h), the sum of the last h squared residuals over the sum of the first h, against F(h, h); two-sided p-value.

Goodness of fit

real(dp) function r2_diffuse(rep, fres, i) result(r2)
subroutine prediction_error_variance(rep, F, info)
  real(dp), intent(out) :: F(:, :)

r2_diffuse

\(R^2_D = 1 - SSE / \sum_t (\Delta y_t - \overline{\Delta y})^2\), against a random walk with drift (Harvey 1989), for series i.

prediction_error_variance

The steady-state \(\bar F\) of a time-invariant model (DK §4.3.4, §7.4; see steady_state). DK’s printed values in §8.2 differ by the factor (n - d)/n; see Road casualties and the seat belt law (DK §8.2).