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A Novel Approach to Predictive Accuracy Testing in Nested Environments

We introduce a new approach for comparing the predictive accuracy of two nested models that bypasses the difficulties caused by the degeneracy of the asymptotic variance of forecast error loss differentials used in the construction of commonly used predictive comparison statistics. Our approach continues to rely on the out of sample MSE loss differentials between the two competing models, leads to nuisance parameter free Gaussian asymptotics and is shown to remain valid under flexible assumptions that can accommodate heteroskedasticity and the presence of mixed predictors (e.g. stationary and local to unit root). A local power analysis also establishes its ability to detect departures from the null in both stationary and persistent settings. Simulations calibrated to common economic and financial applications indicate that our methods have strong power with good size control across commonly encountered sample sizes.


Introduction
This paper is concerned with comparing the forecasting performance of two nested models through tests that rely on out of sample mean squared error (MSE) loss differentials. Our proposed approach bypasses the widely documented complications caused by the degenerate asymptotic variances of these differentials in nested environments while also leading to nuisance parameter free standard normal asymptotics. Our approach remains valid under both stationary and persistent predictors thus also greatly expanding its practical relevance in Economics and Finance.
Since the early work of Diebold and Mariano (1995), West (1996), West and McCracken (1998) and others a vast body of theoretical research has been concerned with developing new methods for comparing the out of sample predictive ability of competing models. Such tests typically compare the out of sample forecast errors generated from two models under a variety of loss functions and forecasting schemes (e.g. recursive, rolling or fixed updating of model parameter estimates) with the aim of testing the null hypothesis of equal predictive accuracy. Most of the test statistics introduced in this literature are based on estimated out of sample MSE loss differentials associated with the two competing forecast error series and have been shown to be asymptotically normally distributed provided that the models being compared are non-nested and a set of standard regularity conditions hold.
The fundamental difficulties that arise as one moves from a non-nested to a nested environment have also generated a vast and growing literature aiming to operationalise and adapt the above approach to nested models. In a nested modelling context a key complication comes from the fact that under the null of interest the population errors of the two models are identical thus leading to sample MSE loss differentials that are identically zero in the limit with null asymptotic variances. These in turn result in test statistics that are not well defined asymptotically and in the failure of normal approximations for popular test statistics such as the Diebold-Mariano-West statistic (henceforth referred to as DMW).
Alternative normalisations applied to the MSE loss differentials in nested contexts have subsequently been shown to lead to test statistics with well defined but no longer Gaussian limiting distributions expressed as functionals of stochastic integrals in Brownian Motions McCracken (2001, 2005), McCracken (2007), Hansen and Timmermann (2015)). With the exception of restrictive frameworks that rule out heteroskedasticity or allow only a single additional predictor in the nesting model these distributions typically depend on a variety of model specific parameters that cannot be eliminated via standard HAC type corrections, requiring simulation based approaches for their implementation (see (West, 2006, pp. 126-127), and (Clark and McCracken, 2013, pp. 10-15)). The asymptotics of these test statistics are further influenced by how the in-sample observations are allowed to grow relative to the out of sample observations and the particular choice of the forecasting scheme used to generate forecasts. In an important recent contribution to this literature Hansen and Timmermann (2015) greatly simplified the non-standard formulation of some of these limiting distributions by showing that they can be expressed as differences of independent chi-square random variables with analytically tractable densities under homoskedastic environments.
Rather than relying on these non-standard and non-Gaussian distributions this same literature has also proposed to bypass the difficulties of nested model comparisons by continuing to use normal approximations for adjusted versions of DMW type statistics. In Clark and West (2007) for instance the authors introduced an adjustment to the spread of the out of sample MSEs of the two competing models and argued that although asymptotic normality cannot be established per se such an approach results in reasonably accurate inferences with acceptable size distortions. The adjustment essentially corrects for the fact that under the null hypothesis of equal predictive accuracy the MSE of the larger model is contaminated with estimation noise.
In this paper we introduce an alternative formulation of the out of sample MSE loss differential between two models that is not subject to the variance degeneracy problem of existing procedures. This subsequently allows us to construct novel test statistics for testing the null hypothesis of equal out of sample population MSEs which are shown to have simple nuisance parameter free normal distributions. The main idea underlying our proposed approach is based on the observation that MSE comparisons across two competing models need not be performed within the same out of sample span of available forecast error observations. These can be performed over partially overlapping segments instead, leading to test statistics that accumulate MSE spreads over all possible such segments. This new setting can trivially accommodate desirable features such as conditional heteroskedasticity and persistent predictors and is also shown to lead to both consistent and locally powerful test statistics. As we discuss further below, our approach can also be adapted to broader contexts where the nestedness of models is an important consideration for inferences such as model selection testing.
Besides conventional forecasting objectives, nested models are commonly encountered environments when it comes to testing economic hypotheses and validating theories. Notable examples include forecast accuracy comparisons against random walk models in the exchange rate literature spurred by the early work of Meese and Rogoff (1983) and more recently reconsidered in Rossi (2005), Molodotsova and Papell (2009) amongst others, equity premium predictability issues as recently investigated in Ferson, Nallareddy and Biqin (2013), Avdis and Wachter (2017) and numerous others. This paper's key aim is to propose a way of addressing and resolving a long standing issue that has generated a vast agenda on the formal comparison of such models via their out-of-sample predictive accuracy. The important auxiliary debate on the advantages or disadvantages of using out-of-sample versus in-sample approaches is not part of our focus. It is also important to emphasise that our interest here is on testing population level predictive ability when forecasts are generated recursively as opposed to finite sample based predictive ability as considered for instance in Giacomini and White (2006). This latter approach is able to avoid the complications induced by the nestedness of models being compared by proceeding via a rolling fixed window based forecasting scheme so that the issue of competing models becoming identical in the limit can be bypassed.
Throughout this paper we also followed the common practice of referring to statistics based on MSE differentials obtained from competing estimated models as Diebold-Mariano type statistics. It is important to acknowledge however that the specific testing approach initially developed by the authors was not concerned with model specific considerations or specification testing motives as its underlying theory was developed for given sequences of forecast errors assumed to satisfy certain regularity conditions (see Diebold (2015)).
Nevertheless, the forecasting literature of the past decade has generally amalgamated the notion of forecast evaluation with the evaluation of models on the basis of their forecasting abilities.
The paper is organised as follows. Section 2 introduces the nested forecasting environment and establishes the limiting null distributions of two novel test statistics. Section 3 concentrates on their asymptotic power properties, establishing their consistency and ability to detect local departures from the null. Section 4 introduces a simple adjustment to the same statistics shown to further enhance their power properties without affecting their null distributions. Section 5 provides a comprehensive finite sample evaluation of our methods based on two DGPs calibrated to commonly encountered applications. Section 6 provides an overview of our key results and discusses extensions. All proofs are relegated to the appendix.

Models and Test Statistics: Theory
We consider the following predictive regressions where the x it 's are the (p i × 1) vectors of predictors, δ 1 and β i the (p 1 × 1) and (p i × 1) parameter vectors and v t and u t the random disturbance terms. We let x t = (x ′ 1t , x ′ 2t ) ′ and β = (β ′ 1 , β ′ 2 ) ′ and set p = p 1 +p 2 . Here model (1) is nested within the larger model in (2) and under β 2 = 0 we have δ 1 ≡ β 1 and v t+1 ≡ u t+1 . The formulation of the above two nested models is standard and parallels closely the most commonly encountered setting considered in the predictive accuracy testing literature as for instance in Hansen and Timmermann (2015).
One step ahead forecasts of y t+1 from (1) and (2) are generated recursively asŷ 1,t+1|t = x ′ 1tδ 1t andŷ 2,t+1|t = x ′ tβ t for t = k 0 , . . . , T −1 whereδ 1t = ( t j=1 x 1j−1 x ′ 1j−1 ) −1 t j=1 x ′ 1j−1 y j , β t = ( t j=1 x j−1 x ′ j−1 ) −1 t j=1 x ′ j−1 y j and the resulting pseudo out of sample forecast errors are then obtained asê 1,t+1 = y t+1 − x ′ 1tδ 1t andê 2,t+1 = y t+1 − x ′ tβ t . Here k 0 is the sample location used to initiate the first recursive forecasts that lead to the first out of sample forecast errorsê 1,k 0 +1 andê 2,k 0 +1 and subsequently resulting in (T − k 0 ) out of sample forecast error observations. Throughout this paper we take k 0 to be a given fraction of the sample size, setting k 0 = [T π 0 ]. Following the early work of Diebold and Mariano (1995), West (1996) and others, a common approach for comparing the predictive accuracy of the two models under MSE loss involves testing the null hypothesis using a test statistic based on suitably normalised versions of the sample average MSE loss (4) Within a non-nested setting and a strictly stationary and ergodic environment Diebold and Mariano (1995) and West (1996) established a standard normal limit theory for this class of test statistics (e.g. √ T − k 0 D T /σ D T ) leading to their systematic use in applied work and a voluminous literature on their refinements. Within a nested context where u t+1 ≡ v t+1 however it is straightforward to also show that dating the limiting standard normal approximation and the use of these test statistics for inference purposes. This degeneracy problem is not solely confined to the Diebold-Mariano type statistics but universally affects all existing methods that compare forecast errors (or models) in nested settings with recursively generated forecasts.
These observations have led to a vast body of research on out of sample predictive accuracy testing in nested models due to their importance in empirical applications in areas such as asset pricing and the modelling of expected returns in particular. For their validity, inferences in nested contexts such as (1) and (2) must rely on the observation that under H 0 it is (T − k 0 )D T rather than √ T − k 0 D T that has a non-degenerate limit which could be used for developing suitable inferences (see McCracken (2001, 2005), Hansen and Timmermann (2015)). This however is also problematic due to the non-standard and non-pivotal nature of the resulting asymptotic distributions.
These take the form of functionals of stochastic integrals in Brownian Motions and with the exception of some special cases contain nuisance parameters that are difficult to remove via standard HAC type normalisations. Even under special instances such as conditional homoskedasticity these distributions continue to depend on the number of extra predictors included in the nesting models and the fraction of the sample used to build the first recursive forecasts. Equally importantly these results have been obtained under stationarity and ergodicity assumptions ruling out the important and frequently encountered case of predictors having roots near unity in their autoregressive representations.
Instead of evaluating the two sequences of squared forecast errors {ê 2 1,t+1 } and {ê 2 2,t+1 } over the entire and same interval [k 0 + 1, T ] as it is done in the formulation of all commonly used test statistics based on D T we here propose to compare the two out of sample MSEs over partially overlapping segments of the [k 0 + 1, T ] interval instead. For this purpose we introduce the following generalised MSE spread where ℓ 1 and ℓ 2 control the range over which the two squared forecast error sequences are evaluated. Note that setting ℓ 1 = ℓ 2 = T −k 0 in (5) reduces it to D T which can be viewed as a special case of D T (ℓ 1 , ℓ 2 ). In line with the analysis based on (4) we take and ℓ 2 = [(T − k 0 )λ 2 ] with λ 1 and λ 2 referring to the fraction of the (T − k 0 ) squared forecast errors associated with models (1) and (2) respectively.
From a theoretical standpoint, proceeding with the use of D T (ℓ 1 , ℓ 2 ) instead of D T has no bearing on the null hypothesis being tested in the sense that when u t+1 ≡ v t+1 the population counterpart of D T (ℓ 1 , ℓ 2 ) also equals zero. A key feature of (5) that distinguishes it from D T however is that the variance of its suitably normalised version will no longer be degenerate provided that ℓ 1 = ℓ 2 (equivalently λ 1 = λ 2 ). This normalised version of (5) which forms the basis of our proposed test statistics is given by Note that (6) is simply the normalised difference in the means of the two sample MSEs evaluated over the two relevant segments of the effective sample size. A key point to observe here is that the variance of (6) no longer collapses to zero in the limit provided that λ 1 = λ 2 for λ i ∈ (0, 1]. To illustrate and motivate this point heuristically let us replace bothê 2 1t+1 andê 2 2t+1 in (6) suggesting that a test statistic based on D T (ℓ 1 , ℓ 2 ) will not have a degenerate distribution as it was the case under the use of D T within our nested context.
The quantity in (6) forms the building block of our proposed test statistics for testing the null hypothesis in (3) against one sided right tail based alternatives as it is the norm in this literature. We consider two types of test statistics that operationalise (6). Our choice is guided by the simplicity of the ensuing asymptotics and their intuitive interpretation while recognising that alternative constructions/normalisations of D T (ℓ 1 , ℓ 2 ) may also be considered.
Our second test statistic is based on averaging (6) across the ℓ j 's. This can be implemented by letting ℓ 1 ∈ [1, T − k 0 ] for a given ℓ 0 2 (e.g. ℓ 0 2 = T − k 0 ) so that the MSE of the smaller model accumulates progressively as ℓ 1 increases. More generally, this averaging can be performed over any desired range of ℓ 1 ∈ [1, T − k 0 ]. To allow this level of generality we introduce the fractional parameter τ 0 ∈ [0, 1) and write where the choice of τ 0 determines the user-chosen range of ℓ 1 over which the average of (8) and (9) allow for a broad range of theoretically feasible magnitudes for λ 0 1 and λ 0 2 one naturally expects that choosing both to lie in the vicinity of unity would capture the greatest amount of information from both models. As we show further below such a choice does indeed lead to remarkably powerful tests with excellent size control. Given a sequence of forecast errors available to the investigator the practical implementation of either (8) or (9) is also as straightforward as calculating standard DMW type test statistics.
To establish the limiting properties of our test statistics under the null hypothesis in (3) we introduce a set of high level assumptions ensuring a flexible environment that encompasses the vast majority of settings considered in the literature while also allowing for a richer temporal structure. As we wish to highlight the generality and usefulness of our methods based on the use of (5)-(6) we abstain from primitive conditions that may unnecessarily suggest a restrictive scope for their use.
We note that condition A(i) holds for both j = 1 and j = 2 highlighting the fact that we operate within a nested environment with (1) being the true model. This requirement is trivially satisfied under a very broad range of settings such as the environment considered in Hansen and Timmermann (2015) or more generally all the settings used to obtain the large sample properties of DMW type statistics in nested models. An important feature to also highlight here is the fact that A(i) does not restrict the persistence properties of the predictors which could be highly persistent in the sense of following local to unit root processes for instance. This greatly expands and enriches the environment in which predictive accuracy inferences have commonly been introduced. The robustness of A(i) to the persistence properties of the predictors is an important and useful feature of the squared forecast errors as opposed to their level for which a result such as A(i) would not hold. For more primitive conditions illustrating specialised environments under which A(i) holds see Deng and Perron (2008a) and Hansen and Timmermann (2015) for strictly stationary and ergodic/mixing settings and Berenguer-Rico and Nielsen (2019) for environments where A(i) is shown to hold under both stationary and unit-root or near unit-root regressors.
Assumption A(ii) requires the centered squared errors driving (1)-(2) to satisfy a functional central limit theorem with σ 2 referring to their long run variance. The absolute summability of the autocovariances of η t ensures that σ 2 the limit of exists. Examples of processes which satisfy Assumption A(ii) include a broad range of conditionally heteroskedastic ARCH/GARCH processes under suitable existence of moments restrictions. For a detailed set of primitive assumptions ensuring that the stated FCLT holds see (Giraitis, Kokoszka and Leipus, 2000, Theorem 5.1), (Giraitis, Kokoszka and Leipus, 2001, Example 2.2 and Theorem 2.1), Berkes, Hörmann and Horvath (2008), Linder (2009) and references therein.
Assumption A(iii) requires that the long run variance of η t be estimated consistently.
Such an estimator could be trivially constructed using least squares residuals from either the null or alternative models. Under conditional homoskedasticity an obvious candidate would beσ 2 hom = T −1 t=k 0η 2 t+1 /(T − k 0 ) while under dependent errors (e.g. if the u ′ t s follow a GARCH type process) a Newey-West type formulation as in Deng and Perron (2008b) would be suitable and ensure that A(iii) holds.

PROPOSITION 2. Under Assumptions A(i)-(iii), the null hypothesis in
The variance components of the distributional outcomes in (10) and (12) are of course known to the investigator so that both Z 0 T (λ 0 1 , λ 0 2 ) and Z T (τ 0 ; λ 0 2 ) can be trivially standardised as and to proceed with standard normal inferences for testing H 0 .
The results in (10)-(15) highlight the simplicity and practicality of our proposed inferences while at the same time offering a solution to an important problem that has not been satisfactorily resolved in this literature. The test statistics in (16) and (17) allow us to generalise the widely used DMW style forecast accuracy testing approach to a broad class of empirically relevant models including specifications with highly persistent predictors with or without conditional heteroskedasticity.
We naturally expect the quality of inferences (e.g. power, size vs power trade-offs) to be influenced by the specific choices of (λ 0 1 , λ 0 2 ) in (16) and (τ 0 , λ 0 2 ) in (17). Although the above null asymptotics hold under a very broad range of parameterisations for those user-inputs a formal analysis of their local asymptotic power allows us to provide precise and tight guidelines ensuring excellent power properties with good size control.

Asymptotic Power and Test Parameterisations
We here deviate from Assumption A(i) in order to evaluate the large sample behaviour of S 0 T (λ 0 1 , λ 0 2 ) and S T (τ 0 ; λ 0 2 ) when the DGP is given by (2). Assumption A(i) continues to hold for j = 2 but no longer for j = 1 since model (1) is misspecified due to the omitted x 2,t predictors. Asê 2 1,t+1 will now be contaminated by those omitted predictors we expect the stochastic properties of the latter (e.g. the variance of the x 2,t 's and their interactions with the x 1,t 's) to influence the power properties of both test statistics. Unlike their null distributions we thus also expect the test statistics to diverge at different rates depending on whether the predictors are stationary or highly persistent. Our analysis of the consistency and local power properties of S 0 T (λ 0 1 , λ 0 2 ) and S T (τ 0 ; λ 0 2 ) is guided by these two distinct scenarios which we consider separately.

Consistency and Local Power under Stationarity
We initially concentrate on the case where the predictors driving both (1) and (2) are stationary and ergodic. Specifically, we operate under the following set of high level assumptions that mirror closely the most common environments considered in the predictive accuracy testing literature.
ASSUMPTION B1: The above assumptions mirror closely the environment of Hansen and Timmermann (2015) and can be viewed as more primitive conditions ensuring that Assumption A(i) holds. B1(i) requires that the predictors satisfy a uniform law of large numbers and rules out trending or local to unit root predictors while B1(ii) ensures that {x t u t+1 } satisfies a multivariate functional central limit theorem. Our main result regarding the asymptotic power properties of the two tests within such a stationary environment is now summarised in Proposition 3 below.
PROPOSITION 3. (i) Suppose model (2) holds with β 2 = 0 and fixed, then under assumption B1 and as T → ∞ we have S 0 The above results highlight the consistency of both test statistics as well as their abil-ity to detect local departures from the null hypothesis under stationary settings. To gain further insights into the specific role played by key factors influencing power it is useful to also present the explicit asymptotic local power functions of the two tests for a given size α ∈ (0, 1). These will in turn be used to provide explicit guidance on selecting suitable parameterisations of our two test statistics (i.e. (λ 0 1 , λ 0 2 ) in (16) and (τ 0 ; λ 0 2 ) in (17)).
We note that both test statistics diverge at the same rate of √ T and power is monotonic in the sense that both ψ 0 and ψ are non-decreasing as ||γ|| gets large. For a given significance level, the larger the two non-centrality parameters are the greater the associated probabilities of rejecting the null hypothesis.
The expressions in (18)-(19) are particularly useful for highlighting the factors that influence power by shifting the center of the null asymptotic standard normal distributions away from zero. Viewing the asymptotic local power functions Φ(ψ 0 − q α ) and Φ(ψ − q α ) in Corollary 1 as providing approximations to the correct decision frequencies of the two test statistics under a sufficiently large T and specific alternatives, we note that for a given size α both test statistics are expected to exhibit a stronger ability to detect departures from the null when the variances of the omitted predictors are large and their correlation with the included predictors small. This feature is particularly important since it hints at the fact that the presence of nearly integrated predictors may help enhance power, a scenario we formally consider further below.
To highlight these points with greater clarity it is useful to focus on the simplified case of two centered predictors x t = (x 1,t , x 2.t ) so that (18)-(19) simplify as with ρ 12 = Corr[x 1,t , x 2,t ]. All other things being equal, power is expected to deteriorate under a noisy omitted predictor that has low variance (low E[x 2 2,t ]) and/or that is highly correlated with the included predictor (e.g. |ρ 12 | ≈ 1). Interestingly, this also suggests that an ideal setting in terms of power implications is one where omitted predictors are highly persistent while included predictors are stationary so that ρ 12 ≈ 0 with E[x 2 2,t ] large. Another important factor affecting power is the variance of the u 2 t 's which impacts the magnitudes of ψ 0 and ψ via σ ≡ V [u 2 t+1 ]. Within an NID errors setting for instance we u so that all other things being equal, an environment with high kurtosis will have a detrimental impact on the power properties of both test statistics.
Power enhancing choices for (λ 0 1 , λ 0 2 ) and (τ 0 ; λ 0 2 ) The non-centrality parameters in (18)-(19) are also useful for assessing the impact of (λ 0 1 , λ 0 2 ) and (τ 0 , λ 0 2 ) on both the absolute and relative local powers of the two tests and for providing useful guidance on suitable choices for those user-inputs. From Corollary 1, will be preferable, in terms of its local power, to a test of the same size based , holding all other parameters entering (11) it follows that those two parameters should be set near their boundary of 1 and in close vicinity of one another (e.g.
Given these preferred parameterisations of the two test statistics it is also useful to evaluate whether either of the two statistics is expected to dominate the other in the sense of ψ 0 being greater or smaller than ψ over particular regions of the pairs (λ 0 1 , λ 0 2 ) and (τ 0 , λ 0 2 = 0.5τ 0 + 0.5), holding all other parameters constant. Given the standard normal asymptotics of both test statistics a useful metric for comparing their local powers is Pitman's Asymptotic Relative Efficiency (ARE) which here takes particularly simple forms, following directly from Corollary 1. To avoid confusion between the λ 0 2 parameters used in S 0 T (λ 0 1 , λ 0 2 ) and S T (τ 0 , λ 0 2 ) we write the two competing statistics as S 0 T (λ 0 1 , λ 0 2 ) and S T (τ 0 ; λ 0 2 ′ ) with λ 0 2 ′ = 0.5τ 0 + 0.5 as in (22). Their ARE is now given by and more specifically From (24) we can observe a clear trade-off between τ 0 and the magnitudes of λ 0 1 and λ 0 2 used in S 0 T (λ 0 1 , λ 0 2 ). If we focus on λ 0 1 = 1 it follows from (24) that ARE ≥ 1 for which is a monotonically increasing function of τ 0 and highlights the fact that the average based statistic will dominate S 0 T (λ 0 1 = 1, λ 0 2 ) in terms of its local power (i.e. ARE < 1) unless impractically large magnitudes of λ 0 2 are used in its implementation. If the average based statistic is implemented with τ 0 = 0.8 for instance, its power properties will dominate If it is implemented with τ 0 = 0.9 the average based statistic will again dominate S 0 T (λ 0 1 = 1, λ 0 2 ) unless λ 0 2 > 0.9908. These values suggest that the average based statistic with τ 0 set in the vicinity of unity (e.g. S(τ 0 = 0.8; λ 0 ′ 2 = 0.9)) will dominate S 0 T (λ 0 1 = 1, λ 0 2 ) in terms of its local power unless impractically large magnitudes of λ 0 2 are used in S 0 T (λ 0 1 = 1, λ 0 2 ).

Consistency and Local Power under Persistence
We now consider an environment where the p predictors x t entering (1)-(2) are modelled as local to unit root processes specified as . . , p and ǫ t some stationary and ergodic random disturbance process. The new set of assumptions under which we establish our results are now summarised in Assumption B2 below where J C (s) = (J 1C (s), J 2C (s)) ′ denotes a p-dimensional Ornstein-Uhlenbeck process whose two components J 1C (s) and J 2C (s) are associated with the dynamics of x 1,t and x 2,t respectively.

PROPOSITION 4. (i) Suppose model
A key message that is conveyed by Proposition 4 when contrasted with Proposition 3 is the important impact of persistence on the power properties the test statistics. The presence of persistent predictors leads to a faster divergence rate for both statistics as reflected in the faster convergence rate towards zero of β 2 that can be accommodated. With highly persistent predictors, both test statistics diverge at the same T 3/2 rate compared with a rate of T 1/2 when predictors were stationary.
A more explicit formulation of the departure from the null distribution in this local to unit-root context can also be highlighted through the following formulations of the limiting distributions of the two test statistics under the local alternative of interest.
COROLLARY 2. Suppose model (2) holds with β 2 = γ/T 3/4 for γ = 0. Under assumption B2 and as T → ∞ we have It is here interesting to compare (29)-(30) with the non-centrality parameters (18)- (19) obtained in the stationary context. The two pairs are essentially analogous in the sense that the constant population moments of predictors (i.e. the Q i,j 's) are now replaced by stochastic integrals in Ornstein-Uhlenbeck processes (i.e. J i,C ). The homogeneity throughout the sample of the limit moment matrix in Assumption B1(i) is of course no longer valid in the context of the stochastic integrals in (29)-(30). The above results also imply that the role played by the pairs (λ 0 1 , λ 0 2 ) and (τ 0 , λ 0 2 ) in this local to unit-root context will mirror our earlier analysis based on a stationary setting, supporting the same practical implementation of both test statistics in terms of their parameterisations i.e. the power enhancing choices for (λ 0 1 , λ 0 2 ) and (τ 0 , λ 0 2 ) discussed above continue to hold in the current context.

Adjusting for Estimation Noise and Further Power Enhancements
Here we explore a particular adjustment that can be applied to our two test statistics with the purpose of boosting their asymptotic local power properties without affecting their limiting null distributions. Our proposed adjustment adapts the well known Clark and West adjustment to DMW type statistics introduced in Clark and West (2007) to the particular context of our two test statistics in (16)-(17). Interestingly, in the specific framework of our inferences based on S T (λ 0 1 , λ 0 2 ) and S T (τ 0 ; λ 0 2 ) this adjustment mirrors the idea of introducing a power enhancement component to conventional test statistics developed in Fan, Liao and Yao (2015) where the authors proposed to augment Wald type statistics with a component that vanishes asymptotically under the null while diverging under alternatives of interest.
The original motivation behind Clark and West's adjustment relied on the intuition that under the null hypothesis estimation noise contaminates theê 2,t+1 's due to the estimation of parameters that are zero in the population. This in turn translates into an inflated MSE 2 resulting in test statistics that are severely undersized. Clark and West proposed to correct for such distortions by suitably adjusting the magnitudes of the forecast errors estimated from the larger model.
To lay down the context and with no loss of generality it is useful to operate within a simplified version of (1)-(2), setting δ 1 = 0 and β 1 = 0 so thatê 1,t+1 = u t+1 and We can now write Z T (ℓ 1 , ℓ 2 ) in (6) as Although it is implicit in our Assumption A(i) that the last two terms in the right hand side of (31) vanish asymptotically under the null hypothesis, in finite samples the rightmost quadratic form is likely to pull down the spread in MSEs causing their null distribution to be Clark and West (2007)'s proposal was to reformulate sample MSE spreads with an adjusted version ofê 2 2,t+1 , say e 2 2,t+1 , given by This is also the approach we adapt to our own context by replacingê 2 2,t+1 in (16)-(17) with e 2 2,t+1 as defined above. As we show further below and in the particular context of our inferences based on S T (λ 0 1 , λ 0 2 ) and S T (τ 0 ; λ 0 2 ) this adjustment does not affect the limiting null distributions of these two test statistics but leads to an increase in the associated non-centrality parameters under the local alternatives of interest.
The expression in (31) is also useful for highlighting what distinguishes our framework that operates under ℓ 1 = ℓ 2 with a standard approach that sets ℓ 1 = ℓ 2 = T − k 0 as for instance in all Diebold-Mariano type statistics. Under ℓ 1 = ℓ 2 we note that the first two terms in the right hand side of (31) cancel out so that the asymptotic behaviour of the expression is determined by the two rightmost quadratic forms whose non-normalised versions have been shown to be O p (1) with non-standard limits (see McCracken (2001, 2005)). Allowing ℓ 1 = ℓ 2 essentially forces the asymptotics of the MSE spreads to be driven solely by the first two components in the right hand side of (31).
Letting S 0 T,adj (λ 0 1 , λ 0 2 ) and S T,adj (τ 0 ; λ 2 ) denote the adjusted versions of our two test statistics it immediately follows from (32) that we have and The expressions in (33) and (34) highlight the fact that the adjustment to the MSE of the larger model results in test statistics that are augmented versions of S T (λ 0 1 , λ 0 2 ) and S T (τ 0 ; λ 0 2 ). As we establish formally below the presence of the additional terms h 0 T (λ 0 1 , λ 0 2 ) and h T (τ 0 ; λ 0 2 ) leaves the limiting null distributions unchanged as both quantities vanish asymptotically. Under the alternative both h 0 T (λ 0 1 , λ 0 2 ) and h T (τ 0 ; λ 0 2 ) diverge to infinity at the same rate as S T (λ 0 1 , λ 0 2 ) and S T (τ 0 ; λ 0 2 ) implying that S T,adj (λ 0 1 , λ 0 2 ) and S T,adj (τ 0 ; λ 0 2 ) will also share the consistency and local power characteristics of their unadjusted counterparts in the sense of diverging to infinity as ||γ|| → ∞. More importantly however, the presence of h 0 T (λ 0 1 , λ 0 2 ) and h T (τ 0 ; λ 0 2 ) does result in different (strictly larger) non-centrality parameters that make these adjusted statistics have more favourable power properties. These features are formalised in Proposition 5 and Corollary 3 below.
Proposition 5 essentially implies that all our results regarding the null limiting distributions of S T (λ 0 1 , λ 0 2 ) and S T (τ 0 ; λ 0 2 ) and their general power properties (consistency and detectability of local departure from the null) continue to hold for their adjusted counterparts while Corollary 3 documents important differences in their specific non-centrality terms.
Indeed, the results in Corollary 3 are particularly interesting and useful for the practical assessment of the power properties of the adjusted versus unadjusted statistics. We have an environment whereby the limiting distributions of the two types of test statistics are the same under the null hypothesis while their non-centrality parameters differ under the alternative, pointing to a more favourable behaviour for the adjusted statistics when it comes to detecting local departures from the null.
Letting ψ 0 adj and ψ adj denote the non-centrality parameters associated with the adjusted statistics, Corollary 3(i) establishes that in a stationary context we have ψ 0 adj = 2ψ 0 and ψ adj = 2ψ so that ψ 0 adj /ψ 0 = 2 and ψ adj /ψ = 2. In the case of persistent predictors the comparison between ξ 0 and ξ 0 adj and between ξ and ξ adj also indicates that the adjusted quantities will stochastically dominate their non-adjusted counterparts in the sense that for some given critical value q and this is again expected to translate into more favourable power outcomes for the adjusted statistics under persistent predictors as well.

Empirical Size and Power
In this section we investigate the size and power properties of S 0 T (λ 0 1 = 1, λ 0 2 ) and S T (τ 0 ; λ 0 2 ) together with their adjusted versions across two DGPs calibrated to commonly encountered applications and sample sizes in Macroeconomics and Finance. The experiments are designed to emphasises the role of the pairs (λ 0 1 , λ 0 2 ) and (τ 0 ; λ 0 2 ) on inferences with the choice of their magnitudes guided by the analysis surrounding our results in Corollaries 1 and 2. The implementation of the first statistic is restricted to S 0

DGP1
A specification that mimics a frequently encountered setting in the asset pricing literature is one where the null model is the martingale difference sequence, y t+1 = u t+1 , and the larger model the single predictor based predictive regression, y t+1 = βx t + u t+1 in line with commonly encountered magnitudes from the equity premium predictability literature we set σ 2 u = 3, σ 2 v = 0.01, ρ uv = −0.8 and experiment with φ 1 ∈ {0.75, 0.95, 0.98}. The conditionally homoskedastic setting takes (u t , v t ) ∼ NID(0, Σ) while conditional heteroskedasticity is modelled via an ARCH(1) specification, writing . This latter choice naturally influences the magnitudes of σ 2 u and ρ uv chosen above and we parameterise {α 0 , α 1 } in a way that maintains the same magnitude for σ 2 u as in the conditionally homoskedastic case i.e. α 0 /(1 − α 1 ) = σ 2 u . For this purpose we set (α 0 , α 1 ) = (1.8, 0.4) throughout. Size experiments set β = 0 while for the power properties of the tests we fix the sample size at T = 500 and evaluate correct decision frequencies as β moves away from the null with β ∈ {0, −1.5, −1.75, −2.0, −2.25, −2.5, −3, −3.5}. For β = γ/T 1/4 this is equivalent to |γ| increasing with γ ∈ {0, −7.1, −8.3, −9.5, −10.6, −11.8, −14.2, −16.6}. Lastly, all of the above experiments are conducted using two alternative estimators for σ. The first one denotedσ 2 hom is suitable under conditional homoskedasticity while the second one denoted σ 2 nw is its robustified version à la Newey-West. Both estimators are based on the residuals from the model estimated under the alternative. Table 1 presents size estimates for S 0 T (λ 0 1 = 1, λ 0 2 ) and S 0 T,adj (λ 0 1 = 1, λ 0 2 ) across a broad range of parameterisations in a conditionally homoskedastic setting. For both test statistics we note good to excellent matches of the nominal size of 10% across all choices of λ 0 2 for T ≥ 500. The adjusted statistic S 0 T,adj (λ 0 1 = 1, λ 0 2 ) in particular has empirical sizes that almost perfectly match 10% for all magnitudes of λ 0 2 when T = 1000. Under φ 1 = 0.75 for instance, S 0 T,adj (λ 0 1 = 1, λ 0 2 = 0.8) has resulted in an empirical size of 10.5% for T = 1000 and 10.6% for T = 500. The corresponding figures for φ 1 = 0.95 were 10.1% and 10.5% respectively, thus also highlighting the robustness of the test statistics to the degree of persistence of the predictors as expected from our results in Propositions 1-2. Similar outcomes also characterise S 0 T,adj (λ 0 1 = 1, λ 0 2 = 0.9) suggesting that these test statistics maintain good size control for T ≥ 500 even when λ 0 2 is as large as 0.90 and λ 0 1 set equal to one.

Table 1
Continuing with the track record of S 0 T,adj (λ 0 1 = 1, λ 0 2 ), Table 1 also suggests a mild deterioration of its empirical size when λ 0 2 is very near its boundary of 1 (e.g. for λ 0 2 = 0.95) with a mild tendency to over-reject the null an extra 1% of the times under T = 1000 and about 2% of the times under T = 500. Within this same boundary region the unadjusted statistic S 0 T (λ 0 1 = 1, λ 0 2 ) is characterised by a mild degree of undersizeness instead, with its empirical sizes clustered in the vicinity of 8.5%-9.5%.
Overall the outcomes in Table 1 have highlighted remarkably stable size properties for both test statistics across the different magnitudes of λ 0 2 including when it is set at 0.9. This is also reassuring given our earlier theoretical power analysis which pointed at choices that satisfy λ 0 1 ≈ λ 0 2 with both set in the vicinity of 1 in the practical implementation of S 0 T (λ 0 1 , λ 0 2 ) and S 0 T,adj (λ 0 1 , λ 0 2 ). Before proceeding further it is also useful to briefly rationalise the size behaviour of these two statistics when λ 0 2 is chosen to lie almost at its boundary as when we set λ 0 2 = 0.95. In such instances we noted the mild undersizeness of S 0 T (λ 0 1 = 1, λ 0 2 ) and mild oversizeness of S 0 T,adj (λ 0 1 = 1, λ 0 2 ) when operating with small to moderately sized samples. A magnitude of λ 0 2 that is close to 1 essentially translates into more 'MSE content' from the larger model and hence a greater exposure to estimation noise when the null model holds true. This results in the unadjusted S 0 T (λ 0 1 = 1, λ 0 2 ) test statistic's distribution being pushed leftward with fewer than expected rejections of the null. On the other hand the adjusted statistic which aims to correct for estimation error via (32) sees its correction factor's contribution increase as λ 0 2 → 1, a correction factor that is overly inflated in small samples. For comparison purposes the last column of Table 1 also includes the corresponding size estimates for the DM and CW statistics. These conform with the consensus view that the DM statistic is severely undersized under such nested settings while the CW statistics' empirical sizes are clustered around 5% for a nominal size of 10%, in line with the simulation results in Clark and West (2007).
We next focus on the same DGP when its errors are driven by an ARCH(1) process. Table 2 presents size outcomes when inferences ignore the presence of conditional heteroskedasticity while Table 3 presents the same outcomes with Newey-West adjusted long run variance normalisers. The point of considering both is to provide a gauge of the effectiveness of the Newey-West correction in the current context. As expected, the presence of omitted conditional heteroskedasticity leads to important size distortions (Table 2) regardless of sample size (e.g. empirical sizes in the vicinity of 17%). Table 3 highlights the effectiveness of the Newey-West corrections in bringing these inflated size estimates close to their nominal counterparts albeit with some remaining degree of oversizeness under small to moderate sample sizes. Under φ 1 = 0.75 and T = 1000 for instance we note a decline in the empirical size of S 0 T,adj (λ 0 1 = 1, λ 0 2 = 0.90) from 16.5% to 11.8% thanks to the use of robust standard errors.

Tables 4-6
Tables 5-6 focus on the effects of conditional heteroskedasticity on the outcomes of Table 4 and essentially mirror our earlier analysis demonstrating the effectiveness of the Newey-West adjusted statistics for correcting the size distortions induced by the presence of omitted ARCH effects.

Tables 7, 10
This is in complete agreement with our earlier theoretical power analysis (see (22)) where we showed that holding all else constant the power of the test statistic is at its peak under λ 0 2 = 0.5τ 0 + 0.5. Thus the empirical sizes peaking for λ 0 2 in the vicinity of 0.5(0.5) + 0.5 = 0.75 and 0.5(0.8) + 0.5 = 0.90 highlight the size vs power trade-offs that will characterise this average based test statistic. Regarding the unadjusted statistics S T (τ 0 = 0.5; λ 0 2 ) and S T (τ 0 = 0.8; λ 0 2 ) we note a tendency for them to be undersized (e.g. empirical sizes in the vicinity of 8% under τ 0 = 0.5 and 6% under τ 0 = 0.8) and this undersizeness deteriorating as λ 0 2 → 1 and φ 1 near 1. This behaviour conforms with the intuition that estimation noise caused by the estimation of parameters that are zero in the population pushes the test statistic too much to the left, a feature that was the key motivation behind Clark and West's adjustment to the DM statistic. Note for instance that these distortions are substantially dampened when the test statistic is implemented with smaller magnitudes of λ 0 2 for which it shows good to excellent size control. Tables 8-9 and 11-12 replicate the analysis under conditional heteroskedasticity (omitted vs Newey-West corrected) for τ 0 = 0.5 and τ 0 = 0.8 respectively.
Tables 8-9, 11-12 We note that omitting the presence of conditional heteroskedasticity leads to substantial spurious over-rejections of the null while the Newey-West corrected versions of the two statistics are again able to efficiently correct these distortions by bringing the empirical sizes close to their nominal counterparts.

Empirical Power
Table 13 presents empirical power estimates for S 0 T (λ 0 1 = 1, λ 0 2 ) and S 0 T,adj (λ 0 1 = 1, λ 0 2 ) across λ 0 2 ∈ {0.80, 0.95, 0.90, 0.95}. The sample size is fixed at T = 500 and power is evaluated as the DGP moves further away from the null hypothesis. The choices of λ 0 2 are dictated by our theoretical results in Corrollaries 1-2 which pointed to magnitudes satisfying λ 0 1 ≈ λ 0 2 ≈ 1. For both test statistics we note the tendency of their empirical power to converge to 1 as |γ| is allowed to increase. We can also clearly observe the particularly favourable impact that the degree of persistence of predictors has on power. As expected from our findings in Propositions 3-4 and their corollaries, power improves as λ 0 2 → 1 and as φ 1 → 1.

Tables 13, 15 and 17
Given the good size control displayed by S 0 T,adj (λ 0 1 = 1, λ 0 2 ) Table 13 clearly highlights the benefits of basing inferences on this adjusted version of our first test statistic calibrated to λ 0 1 = 1 and λ 0 2 ≈ 0.9, also noting that its performance improves considerably as φ 1 → 1. For β 2 = −2 for instance it displays power in the vicinity of 70% under φ 1 = 0.75 and 100% under φ 1 = 0.95 or φ 1 = 0.98. It is here important to relate our simulation outcomes with the theoretical results of Corollary 3 where we established an ARE of 2 for the adjusted statistic S 0 T,adj (λ 0 1 , λ 0 2 ) relative to its unadjusted counterpart. This theoretical power enhancement is clearly supported by the empirical power estimates of Table 13. Compare for instance the empirical power of 50.9% for the unadjusted statistic under φ 1 = 0.75 and β = −2.25 with 80.3% for its adjusted version.
At this stage it is also important to recall that our theoretical analysis based on the asymptotic relative efficiency of S 0 T,adj (λ 0 1 , λ 0 2 ) versus S T,adj (τ 0 ; λ 0 2 ) clearly pointed to the potentially superior power performance of the average based statistic, for larger magnitudes of τ 0 in particular. This is clearly corroborated by the comparison between the power outcomes in Table 13 and Tables 15 (τ 0 = 0.5) and 17 (τ 0 = 0.8). Focusing on the "optimal" choice of λ 0 2 = 0.5τ 0 + 0.5 = 0.90 when τ 0 = 0.8 we note from Table 17 that S T,adj (τ 0 = 0.8; λ 0 2 = 0.9) clearly dominates all configurations of S 0 T,adj (λ 0 1 , λ 0 2 ) in terms of its power properties, typically resulting in relative power gains in excess of 10 percentage points.
Before proceeding further it is also useful to examine the power behaviour of the DM and CW statistics in comparison to S T (τ 0 = 0.8; λ 0 2 = 0.90) (Table 17). Despite being severely undersized and theoretically unsuitable in the present nested context we note that the DM statistic does show a reasonable ability to detect departures from the null. However we can also observe that it is uniformly dominated by S T,adj (τ 0 = 0.8; λ 0 2 = 0.90) which under φ 1 = 0.75 for instance exceeds its power by about ten percentage points. Comparing the power performance of S T,adj (τ 0 = 0.8; λ 0 2 = 0.90) with that of the CW statistic we note that these two test statistics display very similar power outcomes across most scenarios.

Empirical Size
Tables 19-22 present empirical size estimates corresponding to the null DGP under β 1 = β 2 = β 3 = 0 for S 0 T (λ 0 1 = 1, λ 0 2 ), S T, (τ 0 = 0.00; λ 0 2 ), S T (τ 0 = 0.50; λ 0 2 ) and S T (τ 0 = 0.80; λ 0 2 ) respectively together with their adjusted versions. As DGP2 contains a larger number of predictors than DGP1 we expect the impact of estimation error on the MSE of the second/larger model to be more pronounced under the null. This is indeed corroborated by the size estimates in Table 19 where we note the undersizeness of the unadjusted S 0 T (λ 0 1 , λ 0 2 ) statistic which is biased downward and thus results in too few rejections of the null (e.g. 6.4% under T = 1000 versus a nominal size of 10%). Furthermore, its undersizeness tends to deteriorate for larger magnitudes of λ 0 2 as this translates into an increased influence of the larger model's MSE.

Tables 19-22
The adjusted version of the test statistic S 0 T,adj (λ 0 1 , λ 0 2 ) on the other hand is very effective in adjusting for estimation noise (e.g. the earlier empirical size of 6.4% is now pushed up to 10.4%), while also showing a tendency to "over-adjust" in small to moderately size samples, in particular for larger magnitudes of λ 0 2 . Table 20 presents the corresponding size estimates for the average based statistic S T,adj (τ 0 = 0.0; λ 0 2 ) Here we note a substantially improved match of nominal sizes which are clustered in the vicinity of 10% for the adjusted version of the test statistic. Under T = 500 for instance S T,adj (τ 0 = 0.0; λ 0 2 = 1) is characterised by an empirical size of 11.2% compared with 8.4% for its unadjusted version S T (τ 0 = 0.0; λ 0 2 = 1). Note that with τ 0 = 0 the averaging is performed across all feasible magnitudes of ℓ 1 , including very small ones which correspond to the inclusion of test statistics with only a small proportion of model 1's MSE. Comparing Table 20 with Table 22 which repeats the experiment for τ 0 = 0.80 we note greater oversizeness for S T (τ 0 = 0.80; λ 0 2 = 1) in small to moderately sized samples but stabilising in the vicinity of 11.4% under T = 1000 for S T,adj (τ 0 = 0.80; λ 0 2 = 1).

Empirical Power
For this DGP our power experiments focus on documenting the rejection frequencies of the null hypothesis for a fixed alternative as the sample size is allowed to increase. Tables 23-26 for S 0 T (λ 0 1 , λ 0 2 ), S T (τ 0 = 0.0; λ 0 2 ), S T (τ 0 = 0.5; λ 0 2 ) and S T (τ 0 = 0.8; λ 0 2 ) and their adjusted versions respectively. Under either T = 500 or T = 1000 all test statistics have powers at or near 100%.
The main findings from our simulation experiments can be summarised as follows.
(i) The adjusted versions of the two test statistics S 0 T,adj (λ 0 1 , λ 0 2 ) and S T,adj (τ 0 ; λ 0 2 ) displayed good to excellent size control across most parameterisations of their respective inputs (λ 0 1 , λ 0 2 ) and (τ 0 , λ 0 2 ). (ii) Both test statistics are consistent and have non-trivial local asymptotic power while their finite sample power properties are strongly influenced by the respective magnitudes of those same inputs. The guidelines provided by our theoretical power analysis do however lead to highly favourable power outcomes with implementations such as S 0 T,adj (λ 0 1 = 1, λ 0 2 ≈ 0.9) and S T,adj (τ 0 ≈ 0.8; λ 0 2 ≈ 0.9) standing out in terms of their size/power trade-offs, especially for moderately sized samples such as T ≥ 500. (iii) The proposed methods are valid irrespective of the degree of persistence of the predictors as also corroborated by our finite sample simulations. (iv) Our Monte-Carlo analysis did show that Clark and West's CW statistic which although not grounded on formal standard normal asymptotics also performed particularly well in terms of its power, despite relatively important size distortions. Strictly speaking the CW statistic has been introduced for handling nested models estimated via a rolling as opposed to a recursive approach since from a theoretical standpoint it continues to suffer from the variance degeneracy problem characterising DM type constructions.

Conclusions
The main motivation of this paper has been to provide a way of bypassing the variance degeneracy problem that arises in the context of out-of-sample nested model comparisons.
We did so by developing two new test statistics shown to have nuisance parameter free standard normal asymptotics and good power properties including in the close vicinity of the null hypothesis. Our proposed inferences can trivially accommodate conditional heteroskedasticity and are also shown to be robust to the presence of highly persistent predictors.
Although our proposed test statistics require two user inputs each, both have been shown to display good to excellent size control across a very broad range of such parameterisations in a multitude of empirically relevant settings. Although these user-inputs do have considerable influence on the finite sample power properties of both test statistics their choices can be accurately guided by examining the power functions associated with each test statistic, as demonstrated by our simulations and their theoretical backing.
It is important to recognise that our proposed test statistics do involve the discarding of some information albeit very limited and with its amount under the control of the user.
As a result some power loss is of course unavoidable but the absence of any alternative approach that uses more information while achieving the same purpose in an environment that can accommodate both stationary and persistent predictors as well as conditional heteroskedasticity makes such power losses only notional. Our simulation results have indeed shown that very little needs to be discarded for our methods to work well and to provide reliable inferences. In this sense they are not subject to the disadvantages of sample splitting based techniques used for instance in the goodness of fit literature (e.g. half-sample methods).
We next focus on the local asymptotic behaviour of the two test statistics. Using (37) in conjunction with Lemma A1, under the local alternative with β 2 = γ/T 1/4 we can write It now follows directly from (46) in Lemma A1, Assumption B1(iii) and Slutsky's theorem that as required. The result for Z T (τ 0 ; λ 0 2 ) follows identical arguments and is therefore omitted.
1. Under Assumption B2 and as T → ∞ we have for λ ∈ [0, 1] and the statement in (70) follows directly using (68) in (74) and appealing to the continuous D → N (2ψ 0 , 1) and similarly for S 0 T,adj (λ 0 1 , λ 0 2 ) D → N (2ψ, 1) which establishes the fact that Proposition 3 continues to hold for the two adjusted statistics in addition to part (i) of Corollary 3. The result for the case of persistent predictors follows identical lines, making use of Lemma A3 and Proposition 3 which in turn establishes part (ii) of Corollary 3 and that Proposition 4 also holds for the two adjusted statistics.