Residual Diagnostics for VAR Systems

Why Diagnose a Fitted VAR?

A fitted vector autoregression can produce plausible coefficients and forecasts while still missing important features of the data. Its residuals may retain serial dependence, depart substantially from multivariate normality, or show changing variance. The estimated dynamics may also fail the model’s stability condition. Diagnostics help identify these problems and indicate which conclusions from the fitted system need caution.

No single diagnostic establishes that a VAR is correctly specified. Each addresses a particular feature, and each relies on assumptions about the fitted model and the sample. A non-significant result means the test did not find sufficiently strong evidence against its null; it does not demonstrate that the null is true.

Model and Diagnostic Targets

Consider a \(k\)-variable VAR(\(p\)) with deterministic terms collected in \(d_t\):

Here \(y_t\) and \(u_t\) are \(k\)-vectors, \(A_1,\ldots,A_p\) are coefficient matrices, and \(u_t\) is the innovation vector. After estimation, \(\hat u_t\) denotes the fitted residual vector and \(\hat\Sigma\) its estimated covariance matrix. The basic reference model for the diagnostics below has zero-mean innovations, finite positive-definite covariance, and no serial correlation. Constant conditional covariance is an additional assumption tested by variance diagnostics; multivariate normality is an additional distributional assumption, not a requirement for every VAR estimator or inference procedure.

The residuals are estimates of the unobserved innovations, not the innovations themselves. Their estimated nature matters: reference distributions for many diagnostics are approximate, especially in small samples or when the system has many parameters relative to the number of observations.

Testing Residual Serial Correlation

Residual autocorrelation suggests that the VAR’s conditional mean may not have captured all predictable movement. A multivariate portmanteau test aggregates residual autocovariances over a chosen range of lags. Define

For a chosen maximum lag \(h\), the portmanteau statistic is

The null is that residual autocorrelations through lag \(h\) are jointly zero. For a fitted VAR(\(p\)), a commonly used large-sample reference is chi-square with \(k^2(h-p)\) degrees of freedom, when \(h>p\). This approximation is sensitive to sample size and lag choices; some software reports a small-sample-adjusted version. A complementary option is a residual LM test, which adds lagged residuals to the system’s regressions and tests whether their coefficients are jointly zero.

The degrees-of-freedom adjustment reflects that the fitted VAR has already used data to estimate its dynamics. Under the null, the residual autocovariance entries behave approximately like jointly centered sampling errors. The statistic sums their standardized squared magnitudes across lags; estimating the VAR parameters reduces the effective number of unrestricted autocorrelation directions, motivating the \(k^2(h-p)\) reference in the usual asymptotic approximation.

Checking Multivariate Normality

Normality diagnostics are relevant when an analysis relies on Gaussian innovations, for example to interpret Gaussian likelihood-based results or construct distributional forecasts. One standard family is Mardia’s multivariate skewness and kurtosis measures. Let \(z_t=\hat\Sigma^{-1/2}\hat u_t\) be whitened residuals, with their sample mean removed when calculating the measures. Define

Under multivariate normality, the skewness statistic \(T b_{1,k}/6\) is asymptotically chi-square with \(k(k+1)(k+2)/6\) degrees of freedom. The kurtosis statistic can be standardized as

which is approximately standard normal under the null. Large absolute values indicate unusual kurtosis. These approximations require care with fitted residuals and finite samples. A rejection indicates evidence against multivariate normality, not which alternative distribution generated the innovations.

Screening for Changing Variance

A stable residual mean does not imply stable residual variance. Begin with plots of residuals over time, squared residuals, and residual cross-products. Look for clusters of large residuals, changing scale, or periods in which variables move together more strongly. Such patterns are useful warnings, but visual inspection alone does not quantify uncertainty.

A simple formal screen for a particular equation regresses its squared residuals on variables suspected of explaining scale changes. For example, an auxiliary regression on an intercept and a pre-specified time-period indicator gives the Breusch–Pagan-style statistic \(T R^2\), where \(R^2\) is from that auxiliary regression. Under the null of constant conditional variance and the relevant regularity conditions, it is approximately chi-square with degrees of freedom equal to the number of non-intercept regressors. This screen is equation-specific unless a suitable joint system test is used. It also does not replace a method designed for a particular conditional variance structure.

Checking Dynamic Stability

The fitted VAR’s companion matrix summarizes its dynamic propagation. The estimated system is stable when every eigenvalue of that matrix has modulus less than 1. Equivalently, the roots of the VAR characteristic equation lie outside the unit circle. This criterion concerns the fitted dynamic coefficients. It is distinct from checking whether those coefficients remain constant over the sample, which may require recursive estimates, rolling estimates, or a formal parameter-constancy test.

Reading Diagnostics Together

A useful diagnostic workflow separates the questions rather than treating one p-value as a verdict:

These checks are complementary. A VAR can have uncorrelated residuals but non-normal innovations, or stable roots but changing coefficients. Conversely, a test may fail to reject because the sample is too small to detect a relevant departure. Diagnostic results should inform model revision and uncertainty, not be used as a mechanical pass/fail certificate.

Common Mistake