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Corrected Forecast Combinations
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Consider the motivating example from bates1969combination, in which the passenger miles flown in 1953 were predicted by two models, exponential smoothing (ES) and Box-Jenkins' ARIMA model (BJ), as well as by their equally weighted (EW) combination, reproduced in Table (ref). The final line in the table reports the mean squared forecasting error (MSFE). The final column presents our contribution, explained below.
\sisetup{ table-number-alignment = center, table-format = 3.1, detect-all, input-symbols = {−}, table-align-text-post = false }
This motivating example, showing that the average of two forecasts is better than the individual forecasts, started the area of forecast combination, with a multitude of papers appearing every year; see WANG20231518 for the latest over 50-year review. One simple fact, however, remained overlooked over all these years: the errors of the combined forecast exhibit strong autocorrelation of around 54%. This information can be used in different ways, but the simplest option is to correct the combined forecast with the error from the previous period, i.e., if we denote the actual observation $y_t$ at time $t$, the combined forecast $f^{\text{EW}}_{t | t-1}$ based on the information available at time $t-1$, and its error $e^{\text{EW}}_{t | t-1}=y_{t}-f^{\text{EW}}_{t | t-1}$ observable at time $t$, then the combined forecast for the next period $t+1$ can be corrected using $$ f^{\text{CEW}}_{t+1 | t} = f^{\text{EW}}_{t+1 | t} + 0.5\, e^{\text{EW}}_{t | t-1} $$ (where we use the correction of 0.5 for simplicity) and its error is $$ e^{\text{CEW}}_{t+1 | t} = y_{t+1} - f^{\text{CEW}}_{t+1 | t} = e^{\text{EW}}_{t+1 |t} - 0.5\, e^{\text{EW}}_{t | t-1}, $$ which is reported in the final column in Table (ref). What is surprising is that correcting for first-order autocorrelation yields even greater improvement than the original combination. Specifically, the change in MSFE from the best individual forecast to the combination (from 188 to 150) is a 20% reduction, while the change in MSFE from the combination to the corrected combination (from 150 to 103) is a 31% reduction.
Figure (ref) depicts this improvement. The back plane shows the performance of two original forecasts, ES and BJ, and the equally weighted combination, EW. The quadratic curve in this plane passing through all three points has been studied in claeskens2016forecast and holds only when a fixed weight (such as 0.5) is used. Once we correct EW, a new dimension emerges: the plane corresponding to changes in the correction factor. It also contains a new curve that starts at EW and illustrates the improvements achieved by using a fixed correction factor, such as 0.5, which yields point CEW on that curve. The improvement in MSFE from the correction, i.e., from EW to CEW, is visually larger than the improvement from the combination, i.e., from ES or BJ to EW, in line with the numerical results in Table (ref).
This motivating example shows that, although the vast forecast combination literature operates in the back plane and has made substantial progress over the years, see WANG20231518, a new dimension focusing on the properties of the combined forecast can be highly beneficial.
Furthermore, the optimal forecasting weights introduced by bates1969combination exhibit poor empirical performance, leading to the forecast combination puzzle of SW2004: “repeated finding that simple combination forecasts outperform sophisticated adaptive combination methods in empirical applications”. The optimal weight problem can be estimated using ordinary least squares (OLS) in a linear regression, as noted by granger1984improved. However, as many textbooks show, autocorrelated errors in regression models make OLS estimators inefficient, potentially leading to undesirable consequences. Generalized least squares (GLS) estimation is more efficient than OLS in the presence of autocorrelated errors, a point recently reiterated by POOJARI2025100109. By focusing on the properties of the combined optimal forecast, we should be able to shed more light on the forecast combination puzzle.
The paper is organized as follows. Section (ref) formalizes our idea with the correction of the mean ((ref)) and the optimal forecast ((ref)) via the two-step procedure used in the motivating example, before calling on the conditional framework of COW2024 in Subsection (ref) that allows simultaneous estimation of the optimal weights and the correction factor. We also provide a generalization of our approach via GLS in Subsection (ref). The empirical illustration using the U.S. Survey of Professional Forecasters is presented in Section (ref). Finally, Section (ref) concludes.
Assume that we wish to forecast $y_{T+h}\in\mathbb{R}$ using the vector of $h$-step-ahead forecasts $\vf_{T+h | T}=(f_{1,T+h | T},f_{2,T+h | T},\dots,f_{n,T+h | T})' \in \mathbb{R}^n$. Following GiacominiWhite:2006, we assume that $y_t$ may follow a complex process marked by measurement issues, structural changes, and nonstationarity induced by distribution changes. Following aiolfi2006persistence, we map all forecasts to the real number line and limit our analysis to linear combinations with weights $ \vw=(w_1,w_2,\dots,w_n)' \in \mathbb{R}^n$ to produce the combined forecast $f_{c,T+h | T}=\vw'\vf_{T+h | T}$. We denote the vector of forecasting errors as $\ve_{T+h | T}=y_{T+h}\viota - \vf_{T+h | T}, $ where $\viota$ is a vector of ones, and the error of the combined forecast as $e_{c,T+h | T}=y_{T+h}-f_{c,T+h | T}=\vw'\ve_{T+h | T}.$
We first formalize the correction to the mean forecast used in the motivating example. We then embed it in the conditional framework of COW2024 and conclude with the generalization.
Let us consider the equal weights $\vw^{\text{EW}} = \viota/n$ (where $\viota$ is a $n\times 1$ vector of ones), a generalization of the motivating example with two forecasts in the introduction. The combined forecast $f_{T+h | T}^{\text{EW}}=\viota'\vf_{T+h | T}/n$ and the corresponding forecasting error $e^{\text{EW}}_{T+h | T} = y_{t+h} - f_{T+h | T}^{\text{EW}}$ can be corrected using a correction factor $b_T^{\text{EW}}$ available at time $T$ to produce the corrected combined forecast $f_{T+h | T}^{\text{CEW}} = f_{T+h | T}^{\text{EW}} + b_{T}^{\text{EW}}$. In our motivating example, $b_{T}^{\text{EW}} = 0.5 \, e^{\text{EW}}_{T | T-1}$ with $h=1$, a simple function of the previous forecasting error. However, a general correction $b_{T}^{\text{EW}}(I_T,\vgamma) $ that involves information $I_T$, including additional variables, and parameters $\vgamma$ is also possible and presented in Section (ref). Here, we show a simple possibility of estimating one parameter.
When dealing with the simple correction $b_{T}^{\text{EW}} = \gamma \, e^{\text{EW}}_{T | T-1}$, it seems that the correction factor of 0.5 is a reliable starting point in many applications, but with sufficient data available, one can use historical data to find the optimal correction. In each period $t$, one needs to solve the optimization
and use the historically optimal (in the period $[t_0,t]$) factor $\widehat{\gamma}_{t}$ to correct forecast $f^{\text{CEW}}_{t+h | t} = f^{\text{EW}}_{t+h | t} +\widehat{\gamma}_{t}\, e^{\text{EW}}_{t | t-h}$ at the future moment $t+h$.
The optimal correction, in terms of the mean squared error (MSE), is achieved by $b_{T}^{\text{EW}} = \E(e_{T+h | T}^{\text{EW}} | I_T)$ as given by the following theorem.
In applications, one needs to specify how to model $b_{T}^{\text{EW}} = \E(e_{T+h | T}^{\text{EW}} | I_T)$ and estimate it, which will affect the MSE. Therefore, keeping the model simple and minimizing additional estimation, as in our motivating example, should be a favorable correction strategy.
We now turn our attention to the optimal weights. Assuming that the loss function $L(\cdot)$ depends only on $e_{c,T+h | T}$, the classical unconditionally optimal weights solve the problem
Under mean squared error loss, $L(e)=e^2$, only the first two conditional moments influence the optimal weights, and the optimization problem can be solved explicitly. We also assume that the forecasts are unbiased, $\E(\ve_{T+h | T})=\vzeros$; thus, we solve the optimization problem ((ref)) subject to the restriction that the weights sum up to one, $\vw'\viota=1$, which produces the optimal weights of bates1969combination
where $\Sigma = \E[\ve_{T+h | T}\ve'_{T+h | T}]$, and the optimal BG forecast $f_{T+h | T}^{\text{BG}}=(\vw^{\text{BG}})'\vf_{T+h | T}$. The existence of the first two moments of the errors, $\ve_{T+h | T}$, is sufficient for the results derived in this section to be valid under our general data assumptions.
If a suitable correction $b_T^{\text{BG}}$ is available, then the conditional and unconditional inequalities for the reduction in error will hold for the corrected BG forecast $f_{T+h | T}^{\text{CBG}} = f_{T+h | T}^{\text{BG}} + b_T^{\text{BG}} = (\vw^{\text{BG}})'\vf_{T+h | T} + b_T^{\text{BG}}$, as stated in the following theorem.
In fact, any other combination (not necessarily optimal) or even the original individual forecasts can be corrected in a similar manner to improve performance. All previous corrections were two-step procedures, in which the original forecasts were combined in the first step and corrected in the second step.
The correction term, such as $b_{T}^{\text{EW}}$ or $b_{T}^{\text{BG}}$, will generally depend on a vector of unknown parameters $\vgamma\in\mathbb{R}^k$ that need to be estimated or selected. From now on, we explicitly denote this dependence as $b_{T}^{\text{EW}}(\vgamma)$ or $b_{T}^{\text{BG}}(\vgamma)$. In our motivating example in the introduction, this is a scalar first-order autoregressive parameter, and its value is simply set to 0.5. However, if $\vgamma$ is estimated, then the two-step procedure is the simplest option, i.e., given the weights, $\vw^{\text{EW}}$, $\vw^{\text{BG}}$ (or any other chosen weights) and their corresponding combinations $f_{T+h | T}^{\text{EW}}$, $f_{T+h | T}^{\text{BG}}$, we can use the corresponding combined forecasting errors to estimate $\vgamma$ and correct the combined forecast $f_{T+h | T}^{\text{CZZ}} = f_{T+h | T}^{\text{ZZ}} + b_{T}^{\text{ZZ}} (\widehat{\vgamma}^{\text{ZZ}} )$, where ZZ stands for EW, BG or any other combination that one decides to use. This procedure may be effective in many situations due to its simplicity.
One expects benefits from jointly estimating $\vw$ and $\vgamma$, at least in theory, so we now turn our attention to a one-step procedure in which weights and corrections are determined simultaneously. Given $I_T$, the conditionally optimal combination weights of COW2024 solve the problem
For the conditional expectation in the optimization ((ref)) to be well defined, one must specify how the conditional information is used. While COW2024 used $I_T$, the information set available at time $T$, to extract the conditional bias of individual models producing the conditionally optimal weights, we will focus on the combined error.
We assume that the combined forecast error, $e_{c,T+h | T}$, can be decomposed into two parts, $e_{c,T+h | T} = b_{T} +\xi_{T+h}$, where $b_{T}=\E(e_{c,T+h | T}|I_T)$ and $\E(\xi_{T+h}|I_T)=0$. A simple example where this decomposition arises naturally is when forecasting errors exhibit autocorrelation, as in our motivating example in the introduction. In this case, the information set $I_T$ consists of previous forecasting errors, i.e., $I_T = \{e_{c,T | T-1}\}$, and the conditional bias follows a first-order autoregressive model $e_{c,T+1 | T} = \gamma e_{c,T | T-1} + \xi_{T+1}$ (for $h=1$), with $b_{T}=\gamma e_{c,T | T-1}$ and the remainder $\xi_{T+1}=e_{c,T+1 | T}-\gamma e_{c,T | T-1}$.
We now approach the optimal weight problem by noting that, as granger1984improved observed, the unconditional optimal weights can be estimated using ordinary least squares (OLS) in the linear regression
subject to the restriction $w_1 + \dots + w_n =1$. The implicit assumption is that $e_{c,t+h | t}$ are independent and identically distributed (iid). If the errors are correlated, i.e., the iid assumption does not hold, then the OLS estimator, $\widehat{\vw}^{\text{OLS}}$ (and its corresponding forecast $f_{T+h | T}^{\text{OLS}}$), remains unbiased and consistent but is not necessarily efficient. We can therefore estimate the corrected forecast combination using
which is equivalent to
POOJARI2025100109 refer to this transformation as the Hildreth–Lu method and also present other alternatives that may be more attractive. We will focus on the simple version and the corresponding estimators $(\widehat{\vw}^{\text{GLS}},\widehat{\gamma}^{\text{GLS}})$, which can be used to combine and correct at the same time
When considering the forecasting errors, $e^{\text{OLS}}_{T+h | T} = y_T - f_{T+h | T}^{\text{OLS}}$ and $e_{T+h | T}^{\text{GLS}} = y_T - f_{T+h | T}^{\text{GLS}}$, one naturally expects the GLS to perform better. This result is formalized by the following theorem.
We now extend our approach to allow general time-series dependence in the forecasting error term. Let $\vy = (y_{1+h},\dots,y_{T})'$, $\mathbf{F} = (\vf_{1+h|1},\dots,\vf_{T|T-h})'$, $\vf_{t+h|t} = (f_{1,t+h|t},\dots,f_{n,t+h|t})'$, and $\ve_{c} = (e_{c,1+h|1},\dots,e_{c,T|T-h})'$. In matrix notation, we write the model ((ref)) as $\vy = \mathbf{F}\vw + \ve_{c}$. The time-series dependence in the forecasting errors can be modeled with a general covariance matrix $\E[\ve_{c}\ve_{c}']= \Omega(\vgamma)$. Suppose that $\ve_{c}$ follows a stationary ARMA$(p,q)$ process and that $\Omega(\vgamma)$ is known. Then we solve the following optimization problem
subject to the restriction $\vw'\viota=1$, which produces the optimal weights
where $\widetilde{\Sigma}=\widetilde{\ve}' \,\widetilde{\ve} $, $\widetilde{\ve} = \Omega^{-1/2}(\vgamma) \ve$, and $\ve = (\vy\viota'-\mathbf{F})$.
In practice, $\Omega(\vgamma)$ is unknown, but the efficient generalized least squares $(\widehat{\vw}^{\text{GLS}},\widehat{\vgamma}^{\text{GLS}})$ can be estimated simultaneously using the maximum likelihood estimation (MLE) method proposed by harvey1979maximum or a two-step procedure proposed by zinde1991estimation. Let $R(\vw)=(\vy - \mathbf{F}\vw)'\Omega^{-1}(\vgamma) (\vy - \mathbf{F}\vw)$ and $R_{n}(\vw)=(\vy - \mathbf{F}\vw)'\Omega^{-1}(\widehat{\vgamma})(\vy - \mathbf{F}\vw)$ denote the theoretical and empirical risks, respectively. They differ only in the use of the true parameter $\gamma$ versus its estimator $\widehat{\vgamma}$. The following theorem shows that the theoretical minimum risk $R(\vw^\text{opt})$ and the empirical GLS risk $R_{n}(\widehat{\vw}^\text{GLS})$ are approximately equal as long as the covariance matrix estimate $\Omega^{-1}(\widehat{\vgamma})$ is close to the true value $\Omega(\vgamma)$.
While our theory allows for general correction, parsimonious models often deliver good forecasts in practice. Therefore, the practical advice is to use simple corrections, as in our motivating example in the introduction, before adding complexity and estimation variability that could harm forecasting performance.
The analysis uses individual-level forecasts from the U.S. Survey of Professional Forecasters (SPF), conducted quarterly by the Federal Reserve Bank of Philadelphia. The SPF collects forecasts from a panel of professional economists for a range of key U.S. macroeconomic variables across multiple horizons. We focus on one-step-ahead forecasts ($h=1$) and four major macroeconomic indicators: the civilian unemployment rate (UNEMP), real gross domestic product (RGDP), industrial production growth (INDPROD), and consumer price inflation (CPI). The data were accessed in October 2025\footnote{\url{https://www.philadelphiafed.org/surveys-and-data/data-files}}. Figure (ref) presents a graphical overview of the data.
We first investigate the performance of the corrected mean forecast in our data. Because we have several indicators, we introduce a subscript to distinguish among them, making the correction specific to each indicator
where $i\in$\{UNEMP, RGDP, INDPROD, CPI\}. The forecasts and correction elements remain the same as before, but with the additional subscript $i$. To assess the effects of events such as the global financial crisis (GFC) and the COVID pandemic, as well as performance in historical and more recent periods, we consider the periods specified in Table (ref).
Table (ref) shows the relative root mean squared forecasting errors (RMSFE) across different periods and various correction factors, with the different panes corresponding to UNEMP, RGDP, INDPROD, and CPI. The results are relative to the original mean forecast for each row. Let us consider the UNEMP results in the first pane. The COVID period is characterized by large outliers that are propagated by the correction ((ref)), as shown in Figure (ref). Therefore, when the period 2020-2022 is included in the evaluation, it is better to use the original mean forecast, i.e., all values in the rows corresponding to 1969Q1-2025Q2 and 2000Q1-2025Q2 are greater than one, and the optimal correction parameter is $\gamma_i=0$ (no correction). However, once this relatively short period is excluded from the evaluation, the optimal correction is in the neighbourhood of 0.5 (used in the motivating example in the introduction) for 1969Q1-2025Q2$^*$, 2000Q1-2025Q2$^*$, and all other periods. This treatment of the COVID period aligns with the `set to missing' approach of Hyndman03042025 and is sufficient for our purposes; however, other approaches can also be used.
It appears that a correction factor of 0.5 is a reliable starting point for our application; however, with sufficient historical data, one can use historical data to determine the optimal correction factor via the simple optimization procedure ((ref)) introduced earlier, with the additional stabilizing restriction $-1<\widehat{\gamma}_{i,t}<1$. The final column in Table (ref) shows this possibility. As in the evaluation, the COVID period is detrimental to correction, so the values corresponding to COVID are excluded from the sum in ((ref)). The correction with the historically optimal correction factor is beneficial in all periods (once COVID is excluded). Moreover, the estimated corrected factor $\widehat{\gamma}_{i,t}$ delivers the performance close to the optimal fixed correction factor. In fact, the estimated correction factor is very stable for most of the period, ranging between 0.4 and 0.54, as can be seen in Figure (ref).
The results for RGDP are more pronounced. Even with the COVID period included, we observe around a 10% improvement from the correction in the 1969Q1--2025Q2 period, and this improvement increases once COVID is excluded from the evaluation in 1969Q1--2025Q2$^*$. In all periods considered, the correction is beneficial, i.e., the minimum relative RMSFE is substantially less than 1 in all rows when a fixed correction factor is used. Again, a correction factor of around 0.5 delivers good performance in all cases. When using the historically optimal correction factor, as given in the last column, performance drops only marginally (with the largest deterioration in the distant 1969Q1--1999Q4 period), and the corrected mean forecast always outperforms the mean. The results for INDPROD further strengthen support for the fixed correction factor of 0.5, with only a marginal drop in performance when using the historically optimal correction factor across all periods.
CPI is challenging to forecast, see COW2024, and beating simple benchmarks, such as a simple average or random walk variations, by more than a few percent is notoriously difficult. Keeping this in mind, we can examine the CPI results that diverge from other indicators. The correction yields only minor improvements over the mean forecasts, and the best results are achieved with a correction factor close to 0.3. The important exception is the most recent period after COVID, 2022Q1--2025Q2, where the fixed factor 0.5 yields a 30% improvement. Again, estimating the historically optimal correction factor affects performance only marginally, except for the recent 2022Q1--2025Q2 period, where the drop is more substantial, yet the final results are 16% better than the original mean forecast.
We now turn to the correction of the optimal forecast. The main difficulty here is that the SPF is highly unbalanced, with many forecasters skipping periods, new forecasters joining the panel, and old ones dropping out, making the computation of the correlation matrix $\Sigma$ for the optimal weights ((ref)) challenging. For this reason, we limit our demonstration to UNEMP for the period 2000Q1--2019Q4, before COVID. We further select 6 forecasters with at least 70 observations (out of 80). Even with this restriction, there are only a few time periods in which all of them submit their forecasts. We therefore need to implement a simple imputation using their previously submitted forecasts for the missing periods. Other forms of imputation that prevent data leakage (i.e., using future information to impute the past) are also possible, see ahn2022comparison, but since a relatively small number of observations are missing, the final results should not be sensitive. The precision of the individual forecasts is reported in the top section of Table (ref).
The bottom part of Table (ref) reports the precision of the combination methods. The simple mean of the forecasts outperforms each individual forecast (except one), consistent with the vast empirical literature; see, for example, WANG20231518. The errors of the mean forecast exhibit high autocorrelation, as shown in Figure (ref). Applying a correction factor $\gamma_i=0.5$ yields close to a 50% improvement, which increases by another 5% when using the optimal fixed correction factor $\gamma_i = 0.65$. Estimating the historically optimal correction factor slightly deteriorates forecasting performance but still provides a sizable improvement of close to 45% over the original mean forecast. As expected, the correction removes the autocorrelation in the forecasting errors, as shown in Figure (ref).
When running the restricted regression ((ref)) of granger1984improved over the full period 2000Q1--2019Q4, the optimal (in-sample) weights improve upon equal weights by construction. However, a more realistic approach is to use only the available observations $[t_0,t]$ to estimate the optimal (out-of-sample) weights and then use them to forecast $t+1$. The performance of this approach corresponds to the optimal combination in Table (ref). The optimal combination is slightly worse than the mean forecast, which is consistent with the vast empirical literature on the forecast combination puzzle, see claeskens2016forecast. The optimal forecast can be corrected using the default fixed factor 0.5, the optimal fixed factor 0.7, or a historically optimal factor $-1<\widehat{\gamma}_{i,t}<1$. All options outperform the mean forecast, which partially addresses the forecast combination puzzle, i.e., the corrected optimal forecast now performs better than the mean forecast, as expected by theory but often violated in practice. The improvements are dramatic, with the default $\gamma_i=0.5$ correction performing more than 40% better than the mean. The optimal fixed correction factor $\gamma_i=0.7$ further improves performance by another 3.5%. Estimating the historical correction factor diminishes the improvement, yielding results similar to the default correction with $\gamma_i=0.5$.
Finally, the last row of Table (ref) reports the case in which the estimation of the optimal weights and the correction factor is done simultaneously via GLS, as given by equation ((ref)) in Section (ref). This one-step procedure yields better results than the two-step forecasts analyzed in the previous paragraph because it is more efficient. Its performance is almost catching up to the corrected mean forecast using the default correction factor of 0.5. Our practical recommendation is to use the corrected mean forecast until further research reveals insights into the modified forecasting puzzle: the corrected optimal forecast is better than the mean forecast, yet it is still outperformed by the corrected mean forecast. However, the GLS forecast is now a formidable contender and might outperform the corrected mean forecast in some applications.
Figure (ref) investigates the autocorrelation of the regression forecasting errors. The OLS regression, which is equivalent to the optimal weights of bates1969combination, produces forecasting errors that exhibit significant autocorrelation at multiple lags, as shown in Figure (ref). Correcting the OLS forecast removes this autocorrelation, as shown in Figure (ref). The same result is obtained by estimating the weight and correction simultaneously via GLS, as shown in Figure (ref).
This paper shows that an important and largely overlooked source of forecast improvement lies not in refining combination weights, but in correcting combined forecasts when their errors are serially dependent. Building on the classic bates1969combination example, we demonstrate that combined forecast errors often exhibit substantial autocorrelation and that exploiting this predictability can deliver gains that exceed those achieved by forecast combination itself. We formalize this idea within the conditional-risk framework of COW2024 and link forecast correction to efficient estimation of combination weights under time-series dependence using GLS. Empirical evidence from the U.S. Survey of Professional Forecasters across multiple variables and subsamples strongly supports the effectiveness of this approach.
A central empirical finding is the robustness of a simple, easily implementable correction: adding a fraction of the previous combined forecast error to the next-period combined forecast. Across most variables and subsamples—excluding periods dominated by extreme structural disruptions such as COVID—a correction factor near 0.5 consistently yields sizable improvements in forecast accuracy. The gains from this parsimonious adjustment are comparable to those from estimating historically optimal correction factors, while avoiding additional estimation uncertainty. As a result, the corrected mean forecast is a practical and robust default choice for applied forecasting.
The correction perspective also offers new insight into the forecast combination puzzle. Although optimal-weight combinations estimated via OLS often perform poorly out of sample, correcting these forecasts substantially improves their accuracy and frequently restores their theoretical advantage over simple averages. Jointly estimating weights and the correction factor via GLS further improves efficiency and eliminates residual autocorrelation in forecast errors, although in our application, the corrected mean forecast remains highly competitive.
These findings point to a promising and largely unexplored direction for future research. We encourage researchers to give explicit attention to the time-series properties of combined forecast errors and to treat forecast correction as a complementary dimension to forecast combination. Rather than replacing existing techniques, correction can be naturally integrated into the multitude of established combination methods, as well as newer approaches, including machine-learning-based forecast ensembles. Incorporating simple correction mechanisms into both classical and modern forecasting frameworks has the potential to deliver substantial accuracy gains at minimal additional cost, and we view this as a fertile area for further theoretical and empirical work.
The code for the motivating example and the empirical investigation is available at \url{https://github.com/a-vasnev/GLS}.