Equilibrium climate sensitivity controls uncertainty in regional climate change over the 21st century

aDepartment of Atmospheric Sciences, University of Illinois Urbana-Champaign, Urbana, IL; bDepartment of Geology, University of Illinois Urbana-Champaign, Urbana, IL; cformerly at Joint Institute for the Study of the Atmosphere and Ocean, University of Washington, Seattle, WA; dDepartment of Atmospheric Sciences, University of Washington, Seattle, WA; eSchool of Oceanography, University of Washington, Seattle, WA; fDepartment of Earth and Space Sciences, University of Washington, Seattle, WA

global warming and transient regional warming uncertainty arises due to the large scales over which the atmosphere mixes energetic perturbations. The dominant source of variance in both ECS and 21st-century warming are low-latitude feedbacks, whereas TCR is more sensitive to uncertainties in CO 2 radiative forcing and ocean heat uptake. The results imply that throughout the tropics and midlatitudes, and especially over land, most of the uncertainty in local temperature projections is due to model differences in tropical cloud feedbacks.
climate sensitivity | local climate change | temperature | moisture transport W e do not live in the global mean. Both impacts and perceptions of climate change depend on the patterns of regional climatic change, and how these patterns interact with population density (1). Similarly, policies designed to limit the effects of climate change are often decided by considering targets for regional warming (2). While the overall pattern of regional warming has long been known (5,6), little progress has been made in reducing the significant uncertainty in the magnitude of regional climate change (4). Uncertainties in the magnitude of regional warming expected by the end of 21 st century come in roughly equal amounts from uncertainties in emission pathways, and uncertainties in the physical response of the system (3,7). In order to better forecast, mitigate, and adapt to the impacts of climate change, it is crucial to understand and reduce uncertainties in the regional temperature response to greenhouse gas emissions. Projections of regional 21 st century temperature change have been made with ensembles of numerical climate models (8). The ensemble-mean pattern of warming is characterized by amplified warming over land and high latitudes (Fig. 1A). This pattern of both land and polar amplification arises through the combined effects of less-negative high-latitude radiative feedbacks (9,10) and atmospheric energy-transport tendencies, the latter of which act to transport moist-static energy downgradient both zonally (11) and meridionally (12). The ensemble spread in regional warming exhibits a similar pattern (Fig. 1B). The ensemble spread is largest where the ensemblemean warming is largest, with the notable exception of the Southern Ocean and the sub-polar North Atlantic Ocean .
Here we show how these uncertainties in regional 21 st century warming relate to better-quantified uncertainties in metrics of global temperature response, namely equilibrium climate sensitivity (ECS) and the transient climate response (TCR). We then illustrate how this pattern of uncertainty is driven by the same physical mechanisms that give rise to the ensemble-mean warming pattern -radiative feedbacks and atmospheric and oceanic heat transport. Finally, we explain why 21 st century warming is better predicted by ECS than by TCR (13).

Patterns of Regional Uncertainty
We focus on regional temperature changes in an ensemble of 25 General Circulation Models (GCMs) from phase five of the Coupled Model Intercomparison Project, (CMIP5 (14), Table  S1). These model were the basis for climate-change projections in the fifth assessment report of the Intergovernmental Panel on Climate Change (IPCC AR5 (8)). We define 21 st -century

Significance Statement
Societal impacts of anthropogenic global warming are determined by local climatic changes in temperature and precipitation. Uncertainties in the physical climate response to carbon emissions accounts for approximately as much of the spread in predictions of future warming as the uncertainty in socioeconomic scenarios. We characterize the uncertainty patterns for end-of-century temperature response, and show that they are tightly related to uncertainties in Earth's long-term climate sensitivity. Our results show that constraining equilibrium climate sensitivity, such as from observational or palaeoclimate records, could lead to improved predictions of near-term regional temperature changes in the tropics and mid-latitudes, especially over land. In terms of processes, the largest improvement in regional temperature projections will be gained by reducing uncertainty in subtropical cloud feedbacks.

author contributions
The authors declare no conflict of interest 1 To whom correspondence should be addressed. E-mail: cristi@illinois.edu temperature change as the difference between averages over years 1980-2000 in historical simulations and 2080-2100 in simulations using representative concentration pathways (RCPs, (15)). To maximize the signal-to-noise ratio of forced response to natural variability, we primarily use the high-emissions RCP8.5 scenario leading to 8.5 W/m 2 of radiative forcing by the end of the century (Fig. 1), although the main results hold across scenarios (Figs. S1-S3).

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The spread in multi-model projections can be analyzed using empirical orthogonal functions (EOFs) * , which are a way to partition the ensemble spread in temperature changes into a basis of fixed patterns (the EOFs) and their associated principal components that describes how much the spread of a given model projects onto a given EOF. An analysis of the inter-model spread of 21 st -century temperature changes reveals a leading EOF of model-spread that is very similar * Often known as Principal Component Analysis (PCA) in other fields of science.
to the ensemble-mean pattern (Fig. 1C,D). Across scenarios, only the leading EOF is well separated and contains 59% of the total variance in the RCP-8.5 ensemble spread (Fig. 1E).
Across models, the magnitude of the dominant EOF pattern is highly coherent with a models' global-mean temperature change and ECS (Fig. 1F). For RCP 8.5, these squared correlations are r 2 = 98% and r 2 = 90%, respectively. The first EOF explains 60% of global variance, however it explains much higher amounts of variance over large parts of the world, with values of r 2 in the 70% to 90% range over most of the tropics, mid-latitudes, and land (Fig. 1G). Of note, the coherence with TCR is substantially less (r 2 = 60%), consistent with recent results suggesting ECS is a better predictor of 21 st century temperature changes than TCR (13). Seasonal anomalies show similar results in terms of uncertainty being dominated by a single EOF, with a pattern similar to the ensemble mean, and a high correlation with ECS (Figs. S4-S7). The relative spread after accounting for the leading EOF is negligible over much of the Earth (Fig. 1H). Over most of the continents, the spread is in the range of 10-15% of the ensemblemean response, with slightly higher values on the order of 30% over equatorial Africa and parts of South America. The regions where the leading EOF does not account for a significant part of the inter-model spread are regions of significant ocean heat uptake. The correlation of the dominant's EOF projection with a model's ECS and the similarity between its pattern and the ensemble mean implies that a majority of the model spread in future projections can be captured by scaling of the ensemblemean pattern by the spread in ECS. The fact that within a single model the pattern of warming scales with global-mean temperature has long been used to infill projections between the standard scenarios (16)(17)(18)(19). These results suggest that the same pattern scaling applies across models. Furthermore, the high correlation between the leading EOF and ECS implies that reducing uncertainty in ECS would significantly reduce uncertainty in regional temperature projections.

What physical mechanism drive uncertainty?
We can simplify the problem by adopting a zonal-mean view. Indeed, the correlation structure of ECS with zonal-mean temperature changes ( Fig. 2A) is similar to that with local temperature. Across RCP scenarios the squared correlation co-efficient r 2 between ECS and zonal-mean 21 st century warming is high over low-and mid-latitudes and decreases substantially for high-latitudes, especially over the Southern Ocean. The correlation is lower for RCP4.5 and RCP2.6, likely because the forced response is smaller in these scenarios, and natural variability contributes more relative uncertainty.
The ensemble spread in temperature changes can be traced to ensemble spread in top-of-atmosphere (TOA) radiative feedbacks, ocean heat uptake, and radiative forcing (20). We take a more detailed look at this partition for 21 st century warming in RCP8.5, by analyzing the squared correlation of zonal-mean temperature with radiative feedbacks and ocean heat uptake (see methods). Spread in local radiative forcing is not available or readily computable for CMIP5 historical and RCP simulations, but is expected to explain a small fraction of the variance (Fig. S8). The primary driver of uncertainty in low-and mid-latitude (∼ 45 • S−60 • N) 21 st century warming is the large uncertainty in tropical and subtropical feedbacks (∼ 20 • S−30 • N) (Fig. 2B). In contrast, uncertainty over the Southern Ocean is driven primarily by local radiative feedbacks and ocean heat uptake (Fig. 2B,C).
The strong correlation of regional warming with ECS arises because the same low-latitude feedbacks that control uncertainty in regional temperature changes over a large part of the Earth also control uncertainty in ECS (21). The strong D R A F T correlation of Southern Ocean temperature with both TCR and ocean heat uptake suggests that TCR may also be more correlated to the spread in heat uptake. This would explain why TCR is more-weakly correlated with low-and mid-latitude temperature changes where the spread is controlled almost exclusively by radiative feedbacks.
The dominance of a single pattern of uncertainty, and thus the high correlation of local temperatures to global metrics of temperature change, is due to the large (meridional and zonal) mixing scales in the atmosphere. The relative contribution of low-latitude radiative feedback uncertainty from a given latitude, φ λ , is approximately the same for temperature spread at any point in the low-and mid-latitudes (as evidenced by the strong self-similarity of r 2 as a function of latitude φT in Fig 2B).

A model for mixing scales
In order to understand what sets this partition of uncertainty between different physical processes at different latitudes, we need to understand how the atmosphere responds to perturbations in local feedbacks, ocean heat uptake, and radiative forcing. Zonal-mean energy budget anomalies can be decomposed into TOA radiative forcing R f , surface ocean heat uptake Q, horizontal atmospheric heat-flux divergence ∇ · H, and a radiative response at the TOA that is assumed to be linearly proportional to temperature perturbations RT = λ · T : where φ denotes latitude, and ∇ φ denotes the meridional gradient in spherical coordinates. For the remainder of the section, we will drop the φ notation, with the understanding that terms represent zonal means. In order to model the meridional mixing scales, we need a dynamical rule for atmospheric heat transport, and we approximate atmospheric heat transport as governed by downgradient diffusion of near-surface moist static energy (MSE) h, following several recent studies (12,20,(22)(23)(24)(25), with h = cpT + Lν q, where cp is the specific heat at constant pressure, Lν is the latent heat of vaporization, q is change in specific humidity, and D is a constant diffusion coefficient. This simple moist energy balance model (MEBM) has been shown to reproduce the CMIP5 ensemble spread in both temperature (20) and meridional heat-transport anomalies (12). Crucially, the MEBM allows us to directly attribute the spread in the temperature response to the spread in the different forcing terms in Eqn. 1. We find the MEBM is able to reproduce the correlation structure and mixing scales of zonal-mean temperature with ECS, TCR (Fig. 2D), radiative feedbacks (Fig. 2E), and ocean heat uptake (Fig. 2F).
Response patterns. Using the dynamical rule for heat transport allows us to compute the response function to a local energetic perturbation, in a way that cannot be done with a statistical decomposition of the energy budget in GCMs. Let T be the ensemble-mean response to an ensemble-mean abrupt CO2 forcing, R f , with ensemble-mean radiative feedbacks, λ , and heat uptake Q . An individual model's response can be described as a perturbation δT from the ensemble mean, which can be tied to that model's perturbations from the ensemble mean in radiative forcing, δR f , heat uptake, δQ , and feedbacks, δλ: where second-order perturbations have been neglected, and each term is understood to represent zonal-mean values at a latitude φ. The response function ∂T (φi)/∂F (φj) for a perturbation in any of the forcing terms, F , on the right-hand side of Eqn. 3 is depicted in Fig. 3A for perturbations imposed at a set of 5 representative latitudes (Methods). A perturbation concentrated on the equator leads to a relatively uniform response across all latitudes. However, a perturbation concentrated in polar regions leads to a response that is strongly polar amplified with relatively little influence on the tropics. The response function of the MEBM is supported by the response of full GCMs to narrow-band perturbations in either ocean heat uptake (24) or CO2 forcing (10).
To understand this response structure we need to consider how the system closes its energy budget. If a local perturbation δF is applied (right-hand side of Eqn. 3), the system responds by radiatively damping to space and by meridionaly exporting MSE (left-hand side terms in Eqn. 3). The meridional export of MSE, h, is accomplished through a sensible heat term cpT , and a latent heat term Lν q. Since q scales exponentially with temperature following Clausius-Clapeyron, the relative importance of latent to sensible heat will also increase dramatically with temperature.
In high latitudes, the lower climatological temperature requires that the lion's share of the energy export has to be accomplished via transport of sensible heat. Consequently, for a constant diffusion coefficient of MSE, the effective coefficient for diffusion of temperature is four times lower at the poles compared to the equator (22). Thus, the efficiency of energy export (meridional energy exported per degree of local temperature change) -is weaker in the high latitudes. Radiative

D R A F T
feedbacks are also less negative in the high latitudes. The lower efficiency of both TOA radiative damping and meridional heat export results in a large and zonally confined high-latitude temperature response being required to balance high-latitude forcing (12) (Fig. 3A).
In contrast, the larger climatological temperatures in the low-latitudes results in more efficient energy export through meridional fluxes of latent heat. Thus, a perturbation δF in low-latitudes does not have to be balanced by a large local increase in temperature. Instead the energy is exported through meridional moisture transport, leading to a smaller but near global increase in temperature and radiative damping.
The decrease of both radiative damping and meridional heat transport efficiency with increasing latitude can help explain the patterns of both the ensemble-mean response and the ensemble spread. The large high-latitude response required to balance local forcing leads to polar amplificationa basic feature of the climate response to CO2 forcing. Land amplification, the other ubiquitous feature of the response to radiative forcing, can also be explained by the tendency of atmospheric heat transport to minimize gradients in zonal near-surface MSE (11): since the relative humidity is lower over land than over oceans, equal changes in MSE requires more warming over land than over ocean, because a greater fraction of the change in MSE has to be accomplished through changes in sensible heat content.
Patterns of uncertainty. The ensemble spread in zonal-mean temperature response T at a given latitude φ can be decomposed into contributions from the ensemble spread in feedbacks, heat uptake, and radiative forcing at every latitude φj, where we have made the assumption that perturbations in each of the three forcing terms on the right-hand side are independent of each other, and between latitudes. The variance of the CMIP5 ensemble spread in radiative forcing, heat uptake, and radiative feedbacks are shown in Fig. 3B. The highest source of inter-model spread comes from the spread in low-latitude feedbacks and Southern Ocean heat uptake, with some additional variance in high-latitude feedbacks, and heat uptake in the northern mid-latitudes. This combination of the structure of the response function (Fig. 3A) and process uncertainty (Fig. 3B) explains the partition of uncertainty observed in Fig. 2.
High-latitude warming is most sensitive to local feedbacks, forcing, and heat-uptake (10,25), and thus uncertainty in high-latitude warming is dominated by uncertainty in these local terms (20). Consequently, high-latitude warming in poorly predicted predicted by ECS, which is dominated by low-latitude processes. Outside the high latitudes, the response is more uniform, and warming uncertainty is dominated by the the large uncertainty in low-latitude feedbacks.
The large mixing scale of the tropical response also explains why there is a single pattern that dominates the spread in the model response to a given forcing scenario, and why this pattern is highly correlated to ECS. There are multiple independent sources of uncertainty in the radiative feedbacks, i.e., multiple degrees of freedom in the δλj vector. However, most of the uncertainty in radiative feedbacks is concentrated in the , and 2100 (black circle) relative to pre-industrial. The angle between two vectors gives a qualitative indication of the correlation between the ensemble spreads they represent. If the two sources of uncertainty, σ λ and σ R f +Q were independent, than the relation between correlation and angle would be exact, and uncertainty in 21st century warming (black square) could be represented as the vector difference between uncertainties in warming relative to pre-industrial in years 2100 and 2000 (green square).

D R A F T
low latitudes, and the response function to low-latitude perturbations is nearly uniform on global scales (Fig. 3). Thus, the cumulative response at every latitude will be a nearly identical weighted average of the response to each low-latitude perturbation, leading to the self-similarity exhibited in Fig. 2B,E, and a high correlation between local and global temperature. Since ECS is by definition an average of the temperature response, it will be strongly correlated with local temperature change.

ECS, TCR, and the time-evolution of uncertainty
We have so far discussed the large skill of ECS in predicting both global and regional transient temperature changes over the 21 st century. However, as recently noted in ref. (13), the larger skill of ECS relative to TCR is a somewhat surprising result, given that TCR was specifically designed to summarize the transient response.
To understand how it comes about, we consider how the partition of variance evolves in time. As previously noted, we do not have a way to readily diagnose regional uncertainty in radiative forcing over historical and RCP simulations for all models. However, given the large correlation between regional and global uncertainty, we can draw insight by understanding how the variance, σ 2 T , of the ensemble spread in global temperatures evolves in time. We do this by estimating the contributions to the ensemble spread in global temperature, ECS, and TCR from the spread in each model's net equilibrium radiative feedback, σ 2 λ , and the joint contribution from the spread in global-mean radiative forcing and ocean-heat uptake, σ 2 R f +Q , estimated using a multiple linear regression (Fig. 4, Methods). Note that this decomposition is only an approximation, since the spread in feedbacks is not independent from the spread in radiative forcing and ocean-heat uptake, and thus the individual variances σ 2 λ and σ 2 R f +Q will not exactly sum up to σ 2 T . We find that the relative contribution of the uncertainty in radiative feedbacks to the uncertainty in global temperatures increases with time-scale. This can be understood from the structure of Eqn. 4, which shows that impact of uncertainty in radiative feedbacks on ensemble spread σ 2 T is proportional to the ensemble-mean warming T . Following an increase in radiative forcing, temperature will slowly increase, while heat uptake is a maximum immediately after the imposed forcing, declining to zero at equilibrium. As long as the growth rate of temperature is larger than the growth rate of radiative forcing, the relative importance of radiative feedbacks will increase in time. This increased relative importance of radiative-feedback uncertainty is consistent with the predictions of ref. (26) based on a simple energy balance model with separate fast and slow time scales (27).
Consistent with ref. (28), Fig. 4C shows that uncertainty in 20 th century warming is dominated by uncertainty in forcing and heat uptake, whereas uncertainty in warming over the 21 st century is dominated by uncertainty in radiative feedbacks. For ECS, uncertainty in radiative feedbacks contribute roughly ∼ 5 − 6 times more variance than uncertainty in radiative forcing. TCR is defined as the temperature at the time of doubling (year 69) in a simulation with a 1% per year increase in CO2, when radiative feedbacks contribute roughly the same amount of variance as radiative forcing and ocean heat uptake.
As the relative contribution of radiative feedback uncertainty increases with time, so will the correlation of warming uncertainty with ECS (Fig. 4D). For anomalies relative to a 1880-1950 baseline, the relative contributions to uncertainty will resemble the contributions to ECS more than that to TCR starting in the second part of the 21 st century. This is qualitatively illustrated in Fig. 4D as the angle relative to ECS becomes smaller than the angle relative to TCR sometime between years 2050 and 2100. By rebasing the anomalies relative to a 1980-2000 baseline, we are removing the spread in temperatures predictions for the baseline 1980-2000 period. Since this removes more spread due to radiative forcing and heat uptake than due to radiative feedbacks, the uncertainty contributions to rebased temperatures will shift more towards more relative contributions from radiative feedbacks. This explains why 21 st century warming trends are even more correlated with ECS than warming considered relative to pre-industrial baselines (13). This is qualitatively illustrated in Fig. 4D by the fact that temperatures relative to year 2000 tend to form a sharper angle with ECS than temperatures relative to pre-industrial conditions.

Conclusions
Uncertainty in both annual-mean and seasonal regional warming has a relatively simple structure, with a low number of degrees of freedom. Over most of the mid-and low-latitudes, and especially over land, regional uncertainty is dominated by a single pattern of uncertainty. We note that no dominant pattern of uncertainty is evident in precipitation (not shown), and that further work should also asses uncertainty in other metrics of climate change such as changes in temperature variability, extremes, and threshold-crossing events.
This single dominant pattern of regional warming arises because of the large mixing scales of atmospheric heat-transport. While the single pattern of uncertainty in temperature comes from mixing uncertainty in radiative feedbacks across a wide range of latitudes, by far the largest source of uncertainty are the equatorial and sub-tropical radiative feedbacks.
It is possible that there are sources of uncertainty that GCMs do not properly account for, which could lead to additional uncertainty in local temperatures. For example, uncertainties in land-use changes that are poorly represented in models can alter local temperature by altering the landsurface Bowen ratio (29,30), without necessarily violating the assumption of weak MSE gradients. However, the fact that a simple, zonally-symmetric MEBM is able to capture the dominant structure and magnitude of regional temperature uncertainty in full-complexity GCM adds theoretical support to the notion that regional uncertainty is strongly dominated by large-scale energetic constraints.
Our analysis suggests that there are limits to how much we could improve regional climate projections by improving representations of regional processes. This is because even if an improved representation of regional processes were to be achieved within GCMs, the large uncertainty in regional projections coming from global-scale processes would remain.
Going forward, the greatest gain in reducing uncertainties in regional temperature predictions around the world can be obtained through improved representations of subtropical clouds processes, or through improved global constraints on ECS derived from either improved observations or from paleoclimate proxies.

Materials and Methods
GCM data. Models used reflect availability of data from Earth System Grid Federation (https://esgf-node.llnl.gov/search/cmip5/) and Center for Environmental Data Analysis (ceda.ac.uk) as of March 10, 2020. The models used for the MEBM analysis reflect the models calibrated in ref. (20). The models used for the analysis in fig. 4 reflects the models for which adjusted radiative forcing was computed along RCP8.5 in ref. (31). 25 available GCMs were used for the RCP 8.5 and RCP 4.5 ensembles and 14 available GCMs were used for the RCP6.0 and RCP2.6 ensembles. MEBM parameters are calibrated to 19 GCMs. The complete list of models used for each analysis is presented in table S1. Unless otherwise noted in Table S1, values of ECS, TCR, and the radiative forcing, F 2× , associated with a doubling of atmospheric CO 2 , are taken from ref. (32). Partition of Regional Uncertainty. The partition of uncertainty in global temperature for purposes of Fig. 2A-C is computed using temperature anomalies from an ensemble of 19 RCP8.5 simulations for which zonal-mean radiative feedbacks have been previously diagnosed in ref. (20). We compute point-wise squared cross-correlation r 2 of ensemble spread in zonal-mean 21 st century warming in RCP8.5 simulations with ensemble spread in model ECS, zonal-mean 21 st surface heat uptake, and the zonal-mean radiative feedbacks from ref. (20).

MEBM.
Radiative forcing and feedbacks cannot be diagnosed from available RCP8.5 and 1pctCO2 simulations. Hence we use MEBM parameters (radiative feedbacks, λ, radiative forcing per CO 2 doubling, R f ) and ocean heat uptake patterns calibrated to an abrupt quadrupling simulation ref. (20). To maximize the ability of the MEBM to replicate the CMIP5 ensemble, we set the diffusion coefficient to D 0 =1.35 W/m 2 /K per (12). A latitude grid of 101 points is used with a constant ∆x = ∆(sin φ) in order to create an equal-area grid.
The ECS for the MEBM is computed by integrating the MEBM with heat fluxes set to zero. TCR is defined as the global temperature response at the time of doubling in an experiment where CO 2 is increasing at 1% per year. To compute the MEBM TCR we set the effective radiative forcing to half of the value diagnosed from the abrupt quadrupling experiments. We also keep the same patterns of heat uptake, but rescale them by the ratio of global-mean heat uptake in the ensemble mean 1pctCO2 experiment at doubling relative to heat uptake in 2090 in an RCP8.5 simulation, using 20-year windows.
The MEBM response function, ∂T i /∂F j is computed by perturbing the radiative forcing term R f using Gaussian perturbations, δR f (φ i ) ∝ exp[−(x i − x j )/w 2 ], centered at 40 equidistant latitudes, φ j =asin(x 0 ), and normalized to unit net radiative forcing of R f (φ i ) = 1W/m 2 . The response function for perturbations at other locations is interpolated from the 40 gaussian bump integrations. The response function presented in Fig. 3A is computed for each calibration of the MEBM to a GCM, then averaged across the ensemble.
Partition of Global Uncertainty. The partition of uncertainty in global temperature for purposes of Fig. 4 is computed using temperature anomalies from an ensemble of 18 GCMs. For RCP scenarios, the spread in temperature projections δT (t) is regressed simultaneously against the equilibrium radiative feedback, radiative forcing, and heat uptake, δT (t) = β λ (t) · δλ + β Rf (t) · δR f (t) + β Q (t) · δQ(t) + error. [5] The regression is done every year, using a 20 year averaging window. The global equilibrium radiative feedbacks, λ, are computed as λ = F 2× /ECS. For ECS and TCR, the spread in radiative forcing comes from the spread in F 2× . For the historical+RCP8.5 ensembles, the spread in adjusted radiative forcing, R f , is taken from ref. (31). For all simulations, net heat uptake is computed as the net imbalance at the top of the atmosphere. The variance due to radiative feedbacks, σ 2 λ (t) is computed as the standard deviation ofβ λ (t) · δλ, while the variance due to radiative forcing σ 2 λ (t) and heat uptake is computed as the the standard deviation of β Rf (t) · δR f (t) +β Q (t) · δQ(t).β j (t) denotes the least-squares estimate of the regression coefficients.