Why Test Nested Time-Series Models?
A more flexible time-series model can usually fit the observed data at least as well as a restricted version of itself. For example, an AR(2) model can reproduce an AR(1) model when its second autoregressive coefficient is zero. The improvement in fit alone does not tell us whether the additional parameter provides enough evidence to justify the more flexible specification.
A likelihood-ratio test (LRT) measures that improvement while accounting for the number of restrictions being tested. It is a formal test for nested models, not a universal method for deciding between any two candidate forecasts. The method is aligned with the SOA Exam SRM (Statistics for Risk Modeling) syllabus.
Test Statistic and Decision
Let \(L_R\) be the maximized likelihood under the restricted model and \(L_U\) the maximized likelihood under the unrestricted model. Both likelihoods must be evaluated using the same observed series, the same observations, and a comparable likelihood convention. Since the unrestricted model includes the restricted model, \(L_U\) cannot be smaller than \(L_R\), apart from numerical fitting error.
Here \(\ell_R=\log L_R\) and \(\ell_U=\log L_U\) are the maximized log-likelihoods. Under the null hypothesis and standard regularity conditions, \(G^2\) has an approximate chi-square distribution with \(r\) degrees of freedom. At significance level \(\alpha\), reject the restrictions when \(G^2\) exceeds the upper-tail critical value \(\chi^2_{r,1-\alpha}\), or equivalently when the p-value is less than \(\alpha\).
For time-series models, likelihoods may be computed exactly or conditionally. A conditional likelihood, for example, may condition on initial observations. What matters for this comparison is that both nested fits use a consistent convention and the same effective sample. Comparing likelihoods computed on different observations or with incompatible conditioning does not produce the usual LRT.
Why the Chi-Square Reference Distribution Appears
The chi-square result is an asymptotic result: it describes the approximate null distribution as the series length grows. Its justification comes from the local behavior of the log-likelihood near the true parameter value. The following outline explains why the number of restrictions determines the degrees of freedom.
Under regularity conditions, the log-likelihood is locally approximately quadratic. Its curvature is described by the information matrix, and the parameter estimates are asymptotically normal.
The unrestricted fit can move in all parameter directions. The restricted fit is confined to the subset satisfying the null restrictions. Their log-likelihood difference measures the loss from enforcing those restrictions.
After the likelihood's curvature is used to standardize the parameter differences, the loss is asymptotically a sum of \(r\) squared standard normal quantities. That sum has a \(\chi^2_r\) distribution.
Thus, for a single restriction, the asymptotic statistic is approximately \(\chi^2_1\); for two independent restrictions, it is approximately \(\chi^2_2\). The degrees of freedom are not the total number of parameters in the larger model. They are the number of independent restrictions imposed by the null.
Regularity Conditions and Important Exceptions
The usual chi-square calibration requires more than simply having two fitted models. In a time-series setting, the result is typically used when the models are correctly specified, the likelihood is well behaved, and the dependence in the process permits the usual large-sample likelihood approximation. In practical terms, check the following before interpreting a p-value:
- Genuine nesting: the restricted model must be obtained from the unrestricted model through the stated parameter constraints.
- Same data and likelihood basis: use the same response observations, and consistent treatment of initial values, differencing, and likelihood calculations.
- Identifiability: distinct parameter values should correspond to distinct distributions, at least locally around the null parameter.
- Interior null value: the null parameter should not lie on a boundary of the allowed parameter space.
- Regular likelihood: the likelihood should be sufficiently smooth, with nonsingular information and a finite number of restrictions.
- Adequate time-series conditions: the fitted process should satisfy conditions supporting the large-sample approximation, such as appropriate stationarity and dependence assumptions for the model being used.
The standard chi-square result can fail when a null value lies on a boundary, as can happen in some variance-component tests, or when parameters are not identifiable under the null. Unit-root restrictions also require special care: their test statistics do not generally follow the ordinary chi-square reference distribution. ARMA parameter cancellation can create identification problems, and a fit at or near a stationarity or invertibility boundary can undermine the usual approximation. In such cases, do not apply the ordinary chi-square test automatically; a problem-specific reference distribution or another inferential method may be needed.
Worked Example: Testing an Additional AR Coefficient
A hypothetical monthly payment index is modeled using Gaussian innovations. The two fitted models use the same 120 observations and the same conditional likelihood, with the first observations treated consistently. The restricted AR(1) model has maximized log-likelihood \(\ell_R=-86.3\). The unrestricted stationary AR(2) model has \(\ell_U=-84.1\). Test whether the lag-2 coefficient is needed at the 5% level, assuming the regularity conditions hold.
Method. The null restriction is that the AR(2) coefficient at lag 2 equals zero. This is one independent restriction, so the reference distribution is \(\chi^2_1\).
Calculate the statistic. The log-likelihood improvement is \(\ell_U-\ell_R=-84.1-(-86.3)=2.2\). Therefore, \(G^2=2(2.2)=4.4\).
Find the p-value and decide. For a \(\chi^2_1\) variable, the upper-tail probability at 4.4 is approximately 0.0359. Since \(0.0359<0.05\), reject the null restriction at the 5% level. There is evidence, under the stated model and conditions, that the added lag-2 term improves the fit beyond what would typically be expected under the restricted model.
Check. The unrestricted log-likelihood, \(-84.1\), is greater than the restricted value, \(-86.3\), so the statistic is positive as required. The common mistake is to use \(2(\ell_R-\ell_U)\), which gives \(-4.4\); the LRT statistic is \(2(\ell_U-\ell_R)\), and cannot be negative when the fits are correctly nested and maximized.
Worked Example: Jointly Testing Two Higher-Order Terms
A hypothetical weekly claim-count series is fitted with Gaussian errors after a fixed transformation. The restricted AR(2) model sets the lag-3 and lag-4 coefficients to zero. The unrestricted stationary AR(4) model estimates both terms. Both use the same observations and exact likelihood. The maximized log-likelihoods are \(\ell_R=-152.8\) and \(\ell_U=-149.2\). Test the two restrictions jointly at the 5% level, assuming the null is interior and the regularity conditions hold.
Method. The null is \(H_0:\phi_3=\phi_4=0\), where \(\phi_j\) denotes the AR coefficient at lag \(j\). There are two independent restrictions, so \(r=2\).
Calculate the statistic. The log-likelihood difference is \(-149.2-(-152.8)=3.6\). Thus \(G^2=2(3.6)=7.2\).
Calculate the p-value. For \(\chi^2_2\), the upper-tail probability at 7.2 is \(e^{-7.2/2}=e^{-3.6}\), approximately 0.0273. Since \(0.0273<0.05\), reject the joint null. The data provide evidence that at least one of the two higher-order coefficients contributes, under the fitted model and stated assumptions.
Check. The 5% critical value for \(\chi^2_2\) is approximately 5.991, and \(7.2>5.991\), agreeing with the p-value decision. This is a joint test: it does not establish that both coefficients are individually nonzero. A common mistake is to use two degrees of freedom for the number of parameters in the AR(4) model; the correct degrees of freedom are two because the null imposes two restrictions.
Nonnested Candidates Are Not an LRT
Two candidate models may be fitted to the same series without either being a restricted version of the other. For example, an ARMA(1,1) model and an AR(2) model are generally nonnested: an ARMA model has a moving-average component, while an AR(2) specification does not generally reproduce it by setting one or more AR(2) parameters to zero. The fact that one candidate has a higher maximized likelihood does not create a valid chi-square test.
Worked Example: Do Not Test Between Nonnested Models
A hypothetical quarterly exposure index is modeled with Gaussian innovations. An ARMA(1,1) candidate has maximized log-likelihood \(-101.6\), and an AR(2) candidate has maximized log-likelihood \(-100.9\). Both use the same 96 observations and the same likelihood convention. An analyst proposes an LRT with one degree of freedom because the AR(2) fit has one more parameter. Determine whether that test is valid.
Check nesting first. The stated models are not generally nested. The ARMA(1,1) specification is not obtained from the AR(2) model by imposing a single regular restriction, or vice versa. Therefore, the proposed one-degree-of-freedom chi-square reference distribution is not justified.
Interpret the likelihoods carefully. The difference is \(-100.9-(-101.6)=0.7\), so the AR(2) model has the higher maximized log-likelihood. Doubling this difference gives \(1.4\), but \(1.4\) is not a calibrated LRT statistic here. It has no p-value from the usual nested-model chi-square result.
Conclusion. The raw likelihood comparison says only that the AR(2) fit achieved a slightly higher likelihood on this common sample. It does not establish statistically significant superiority. Use a model-comparison approach appropriate to the question—such as a suitable information criterion or time-ordered forecast evaluation—rather than inventing an LRT. The trap is treating a difference in parameter counts as proof of nesting; nesting is about the relationship between the model specifications, not just their sizes.
Practical Interpretation
An LRT answers a targeted question: does the improvement in maximized likelihood support rejecting a specified set of restrictions? It does not prove that the unrestricted model is the best forecasting model, that its extra parameters will forecast well, or that the null model is true when the test is not significant. In particular, a non-rejection can reflect limited information in the available series.
Before calculating a test, write down the null restrictions and verify that the alternative contains the null as a special case. Then check that the likelihoods are comparable, count the independent restrictions, calculate \(2(\ell_U-\ell_R)\), and assess the result using the correct reference distribution. Finally, state the conclusion in context and qualify it by the assumptions. This sequence prevents two common errors: assigning degrees of freedom by counting all parameters, and turning a raw fit comparison into a formal test when the candidates are nonnested.
Check Your Understanding
For each problem, check both the model relationship and the numerical calculation before making a conclusion.
- A Gaussian AR(1) model is nested in an AR(2) model by setting the lag-2 coefficient to zero. On the same 80 observations, the maximized log-likelihoods are \(-75.0\) and \(-73.2\), respectively. Find the LRT statistic, degrees of freedom, and 5% decision. (Answer: \(G^2=3.6\), \(1\) degree of freedom, p-value approximately \(0.0578\); do not reject at 5%.)
- An AR(3) null model is compared with a nested AR(5) alternative that adds two unrestricted coefficients. The maximized log-likelihoods are \(-210.4\) and \(-206.9\). Assuming regularity, find the statistic and p-value, and decide at 5%. (Answer: \(G^2=7.0\), \(2\) degrees of freedom, p-value approximately \(0.0302\); reject the two restrictions jointly.)
- A nested model comparison tests one coefficient restriction. The restricted and unrestricted log-likelihoods are \(-44.8\) and \(-42.9\). State the statistic and whether the restriction is rejected at 5%, assuming the ordinary chi-square approximation applies. (Answer: \(G^2=3.8\), \(1\) degree of freedom, p-value approximately \(0.0513\); do not reject at 5%.)
- An ARMA(1,1) candidate and an AR(2) candidate are fitted to the same series. The AR(2) likelihood is larger, and it has one more estimated coefficient. Is an ordinary one-degree-of-freedom LRT justified solely by these facts? (Answer: No; the specifications are generally nonnested, and parameter-count difference alone does not establish nesting.)
- A nested Gaussian time-series comparison imposes two restrictions, but under the null a variance parameter is constrained to equal zero, a boundary of its allowed range. Should the usual \(\chi^2_2\) calibration automatically be used? (Answer: No; the boundary condition violates a standard regularity condition, so the ordinary chi-square reference distribution may not apply.)